Simulations and Technology Development for
Time-Domain Astronomy: TESS and InGaAs Detectors
ARCHNES
by
MASSACHUSETTS INSTITUTE
OF rECHNOLOLGY
Peter William Sullivan
B.S., Cornell University (2009)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
JUN 3 0 2015
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Master of Science in Physics
at the
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February 2015
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January 16, 2015
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Simulations and Technology Development for Time-Domain
Astronomy: TESS and InGaAs Detectors
by
Peter William Sullivan
Submitted to the Department of Physics
on January 16, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Physics
Abstract
Optical time-domain astronomy is in the midst of significant developments due to new
instruments, data processing techniques, and space missions. The Kepler mission has identified thousands of exoplanet systems with the transit technqiue, allowing the first statistical
analyses of the exoplanet population. The upcoming TransitingExoplanet Survey Satellite
(TESS) mission will complement Kepler by identifying the brightest transiting exoplanet
systems on the sky. These planets will offer the most insight into their radius, mass, and
composition. At approximately the same time that TESS launches, the Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) will reach full operation, allowing
the detection of gravitational waves from the mergers of compact objects out to hundreds
of Mpc. Finding the electromagnetic counterparts to gravitational-wave sources, which
may appear as kilonovae events, will give far greater insight into these mergers. While the
Large Synoptic Survey Telescope will be a capable facility for following up triggers from
gravitational-wave observatories at A < 1pm, a wide-field infrared telescope operating at
A > lpm would complement its capabilities.
In this Thesis, I first describe a simulation of the TESS mission. The simulation predicts
TESS should find hundreds of small transiting planets; those with the brightest host stars
will be favorable for radial-velocity determinations of their masses and spectrophotometric
measurements of their atmospheres. However, approximately half of the transit-like signals that TESS detects will turn out to be astrophysical false-positives arising from stellar
eclipsing binaries. I will discuss how planets can be distinguished from eclipsing binaries
using the TESS data and a ground-based follow-up campaign.
Secondly, I describe the development of InGaAs detectors for near-IR imaging and photometry. These detectors are a less-expensive alternative to the HgCdTe detectors in current
use, enabling wide-area IR surveys and IR observations with small telescopes to take place.
Using camera hardware that I designed and constructed, I have tested InGaAs detectors as
they have matured into an appropriate technology for near-IR time-domain astronomy. I
will discuss their possible use for following up the TESS detections and gravitational-wave
triggers.
3
During my time at MIT, I have also pursued other research topics that are beyond the
scope of this Thesis. They include observations with the FIRE spectrograph at Magellan,
where I measured the NIR continuum sky brightness (Sullivan & Simcoe, 2012) and obtained the spectra of several high-redshift QSOs. Neutral hydrogen gas was detected in one
QSO spectrum at a redshift of 7 with no corresponding metal lines, indicating that the gas
could either be the site of Population-HI star formation, or it is the intergalactic medium
prior to reionization (Simcoe et al., 2012). I also obtained FIRE spectra of GJ 3470, which
hosts a transiting Neptune-sized planet, to improve estimates of the stellar and planetary
properties (Demory et al., 2013), and of the Phoenix Galaxy Cluster, which is the most
luminous site of star formation known (McDonald et al., 2012). In addition, I explored the
technique of using narrow-band images to detect Lyman-a absorbers and emitters, which
allows us to correlate their spatial densities at a redshift of 3.
Thesis Supervisor: Robert A. Simcoe
Title: Associate Professor, Department of Physics
4
Acknowledgments
I would like to thank the following members of the TESS Science Team for their input
into Chapter 1: Jacob Bean, Tabetha Boyajian, Eric Gaidos, Daniel Huber, Jon Jenkins,
Geoffrey Marcy, Roberto Sanchis-Ojeda, and Keivan Stassun. We also acknowledge Anthony Smith, Kristin Clark, Michael Chrisp, Barry Burke, and Vyshnavi Suntharalingam at
MIT-Lincoln Laboratory for their input on the TESS hardware. We thank Leo Girardi for
his help with TRILEGAL and Gibor Basri for sharing the stellar variability data with us.
I've enjoyed working on TESS with Joel Villasenor, Ed Morgan, Roland Vanderspek, and
George Ricker, and I thank the program for funding a significant part of my time at MIT.
In preparing Chapter 2, I would like to thank the staff of MIT's Wallace Observatory,
Tim Brothers and Michael Person. The staff of the MIT CCD Laboratory assisted with our
development of InGaAs detectors: Richard Foster and Steve Kissel consulted on the design
and testing, and Fred Miller and Tom Hoyle helped with the fabrication. Andrew Hood
and Falgun Patel at FLIR provided useful technical support for their product. Hardware for
the InGaAs development project was purchased with the MIT-Kavli Institute Development
fund. I would also like to thank Edward Hugo Darlington for his mentorship in detectors
when I worked at JHU/APL.
This research makes use of data products from the Two Micron All Sky Survey, which is
a joint project of the University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics and Space
Administration and the National Science Foundation.
On a more personal note, I thank my family of Matt, Chuck, and Donna Sullivan, my
grandmother Sheila Noyes, the Sullivans out in Medfield, my cousin Liz, and my friends
Jesse, Carla, Matthew, Mike, Jessamyn, Kat, Andy, Greg, and Aaron for providing stress
relief and encouragement during my time here. I thank Laura Lopez for her love and
support. This thesis is dedicated to my brave aunt, Deborah Molano.
Rob Simcoe and Josh Winn taught me many lessons through the courses they instructed
(8.902 and 8.901), the papers we wrote, observing runs we took together, and the personal
conversations we enjoyed along the way. I am grateful for their mentorship.
5
6
Contents
The Transiting Exoplanet Survey Satellite:
15
1.1
Introduction . . . . . . . . . . . . .
15
1.2
Brief Overview of TESS
17
.
Sky Coverage . . . . . . .
18
1.2.2
Spectral Response
19
1.2.3
Simplified sensitivity of TESS
.
1.2.1
20
23
1.3.1
Model Queries
24
1.3.2
Properties of low-mass stars
26
1.3.3
Stellar Multiplicity . . . .
29
1.3.4
Luminosity Function . . .
32
1.3.5
Stellar Variability . . . . .
34
Eclipsing Systems . . . . . . . . .
37
1.4.1
Planets
37
1.4.2
Eclipsing Binaries
.
Star Catalog . . . . . . . . . . . .
.
.
. . . .
.
.
.
. . . . . .
.
1.4
. . . . . .
. . . . . . . . . .
.
1.3
.
Simulations of planet detections and astrophysical false positives
41
1.5
Best Stars for Transit Detection . .
43
1.6
Instrument Model . . . . . . . . .
45
1.6.1
Pixel response function . .
46
1.6.2
Synthetic images . . . . .
47
1.6.3
Determination of optimal aper ture
50
1.6.4
Noise Model
51
.
.
.
.
.
. . . .
. . . . . . .
.
1
7
1.7
1.8
1.9
1.6.5
Duration of observations . . . . . . . . . . . . . . . . . . . . . . .
1.6.6
Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.6.7
Selection of target stars . . . . . . . . . . . . . . . . . . . . . . . .
1.6.8
Full-frame images
Survey Yield
55
58
. . . . . . . . . . . . . . . . . . . . . . . . . . 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.7.1
Transiting Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.7.2
False positives
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Distinguishing False Positives from Planets
. . . . . . . . . . . . . . . . . 67
1.8.1
Ellipsoidal Variations . . . . . . . . . . . . . . . . . . . . . . . . . 68
1.8.2
Secondary Eclipse Detection . . . . . . . . . . . . . . . . . . . . . 70
1.8.3
Ingress and Egress Detection . . . . . . . . . . . . . . . . . . . . . 71
1.8.4
Centroid Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
1.8.5
Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
1.8.6
Statistical Discrimination . . . . . . . . . . . . . . . . . . . . . . . 77
Prospects for Follow-Up Observations . . . . . . . . . . . . . . . . . . . . 80
1.9.1
Radial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1.9.2
Atmospheric Characterization
. . . . . . . . . . . . . . . . . . . . 83
1.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2
InGaAs Detectors for Near-IR Imaging and Photometry
2.1
87
Motivations for InGaAs Instruments . . . . . . . . . . . . . . . . . . . . . 89
2.1.1
IR Transit Photometry
2.1.2
Wide-Field IR Transient Searches . . . . . . . . . . . . . . . . . . 90
. . . . . . . . . . . . . . . . . . . . . . . . 89
2.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.3
APS640C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4
2.3.1
Design
2.3.2
Detector Characterization
2.3.3
Photometric Testing in the Laboratory . . . . . . . . . . . . . . . . 98
2.3.4
Testing on the Sky
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
. . . . . . . . . . . . . . . . . . . . . . 96
. . . . . . . . . . . . . . . . . . . . . . . . . . 103
AP1121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8
2.4.1
Design Changes
. . . . . . . . . . . . . . . . . . . . . . . . . . .105
2.4.2
Linearity
2.4.3
Gain and Read Noise . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.4.4
Dark Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
2.4.5
Persistence
2.4.6
Laboratory Photometry . . . . . . . . . . . . . . . . . . . . . . . .110
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
2.5
Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
2.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A Tables
115
9
10
List of Figures
1-1
Polar projection of the TESS sky coverage . . . . . . . . . . . . . . . . . .
18
1-2
The TESS spectral response . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1-3
Limiting magnitudes for planet detection as a function of stellar radii . . . . 22
1-4
Radius-magnitude relation for simulated stars compared to empirical observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1-5
Luminosity function compared to observations . . . . . . . . . . . . . . . . 35
1-6
Star counts as function of apparent magnitude and galactic coordinates . . . 36
1-7
Input distributions of stellar variability . . . . . . . . . . . . . . . . . . . . 38
1-8
Input planet occurrence rates in the period-radius plane . . . . . . . . . . .
1-9
The distribution in relative period differences for multi-planet systems . . . 40
39
1-10 Surface density of eclipsing binaries compared to Kepler . . . . . . . . . . 43
1-11 Expected number of transiting planets on the sky
1-12 Sample images of the pixel response function
1-13 The TESS pixel response function
. . . . . . . . . . . . . . 44
. . . . . . . . . . . . . . . . 48
. . . . . . . . . . . . . . . . . . . . . . 49
1-14 Noise model for TESS photometry . . . . . . . . . . . . . . . . . . . . . . 54
1-15 Determination of the SNR threshold
. . . . . . . . . . . . . . . . . . . . .
1-16 Selection of the target stars shown on the H-R diagram
57
. . . . . . . . . . .
59
1-17 Distribution of target stars on the IC-Teff plane . . . . . . . . . . . . . . . .
60
1-18 Mean numbers of planets and eclipsing binaries detected in the simulation .
62
. . . . . . . . . . . . . . . . .
63
1-19 Sky maps of the simulated TESS detections
1-20 The output distribution of planets on the radius-period plane
. . . . . . . .
64
1-21 Small planets detected in or near the habitable zone . . . . . . . . . . . . .
65
1-22 Completeness of the TESS survey
66
. . . . . . . . . . . . . . . . . . . . . .
11
1-23 Detecting the ellipsoidal variations of eclipsing binary false posi tives . . . .
69
. . . . . . . 72
1-25 Detecting the ingress/egress of eclipsing binary false positives
. . . . . . . 74
.
1-24 Detecting the secondary eclipses of false positives . . . . . .
1-26 Detecting the centroid shifts of eclipsing binary false positives . . . . . . . 76
1-27 Contrast of blended eclipsing binary systems
. . . . . . . 78
1-28 Follow-up photometry of the TESS candidates . . . . . . . .
. . . . . . . 79
1-29 Detection ratios on the period-depth plane . . . . . . . . . .
. . . . . . . 81
1-30 Detection ratios as a function of galactic latitude . . . . . . .
.... ........
1-31 Mass measurement of the TESS planets
. . . . . . . . . . .
. . . . . . . 83
1-32 Transit spectroscopy of the TESS planets . . . . . . . . . . .
. . . . . . . 85
.
.
.
.
.
.
. . . . . . . .
81
2-1
InGaAs QE and atmospheric transmission . . . . . . . . . . . . . . . . . . 89
2-2
A comparison of source follower and CTIA architectures . . . . . . . . . . 92
2-3
The assembled camera shown in the laboratory. . . . . . . . . . . . . . . . 94
2-4
Effective read noise of the APS640C . . . . . . . . . . . . . . . . . . . . . 97
2-5
Dark current of the APS640C detector . . . . . . . . . . . . . . . . . . . . 98
2-6
Laboratory photometry with simulated stars and their cross-correlation . . .
2-7
Precision of laboratory photometry with the APS604C
2-8
Results from testing the APS640C on the sky . . . . . . . . . . . . . . . . 104
2-9
Nonlinearity of the AP1121 . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2-10 Read noise of the AP.121
99
. . . . . . . . . . . 102
. . . . . . . . . . . . . . . . . . . . . . . . . . 107
2-11 Dark current of the AP1121 . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2-12 Persistence of the API121
. . . . . . . . . . . . . . . . . . . . . . . . . . 110
2-13 Stability of the API 121 with photometry in the laboratory
12
. . 112
List of Tables
. . . . . . .
. . . . . . 20
1.2
TRILEGAL input settings
. . . . . . . . . . . . . . . . . . .
. . . . . . 25
1.3
Binary properties as function of the mass of the primary . . . .
. . . . . . 32
1.4
J-band luminosity function . . . . . . . . . . . . . . . . . . .
. . . . . . 33
1.5
Methods of distinguishing false positives from transiting planets
. . . . . . 76
2.1
Noise budgets for the APS640C testing
. . . . . . . . . . . .
. . . . . . 102
2.2
Observational modes for InGaAs cameras . . . . . . . . . . .
. . . . . . 113
A. 1
Catalog of simulated TESS detections. . . . . . . . . . . . . .
. . . . . . 116
.
Fluxes in the TESS bandpass and Ic - T colors
.
.
.
.
.
.
1.1
13
14
Chapter 1
The Transiting Exoplanet Survey
Satellite:
Simulations of planet detections and
astrophysical false positives
Sullivan, P.W.; Winn, Joshua N.; et al.
to be submitted to the Astrophysical Journal
1.1
Introduction
Transiting exoplanets offer opportunities to explore the compositions, atmospheres, and
orbital dynamics of planets beyond the solar system. The Transiting Exoplanet Survey
Satellite (TESS) is a NASA-sponsored Explorer mission that will monitor several hundred
thousand Sun-like and smaller stars for transiting planets (Ricker et al., 2014). The brightest dwarf stars in the sky are the highest priority for TESS since they facilitate follow-up
measurements of the planet masses and atmospheres. After launch (currently scheduled for
late 2017), TESS will spend two years observing nearly the entire sky using four wide-field
cameras.
15
Previous wide-field transit surveys, such as HAT (Bakos et al., 2004), TrES (Alonso
et al., 2004), XO (McCullough et al., 2005), WASP (Pollacco et al., 2006), and KELT
(Pepper et al., 2007), have been conducted with ground-based telescopes. These surveys
have been very successful in finding giant planets that orbit bright host stars, but they
have struggled to find planets smaller than Neptune because of the obstacles to achieving
fine photometric precision beneath the Earth's atmosphere. In contrast, the space missions
CoRoT (Auvergne et al., 2009) and Kepler (Borucki et al., 2010) achieved outstanding
photometric precision, but targeted relatively faint stars within restricted regions of the sky.
This has made it difficult to measure the masses or study the atmospheres of the small
planets discovered by CoRoT and Kepler, except for the brightest systems in each sample.
TESS aims to combine the merits of wide-field surveys with the fine photometric precision and long intervals of uninterrupted observation that are possible in a space mission.
Compared to Kepler, TESS will examine stars that are generally brighter by ~3 magnitudes
over a solid angle that is larger by a factor of 400. However, in order to complete the survey
within the primary mission duration of two years, TESS will not monitor stars for nearly as
long as Kepler did; it will mainly be sensitive to planets with periods <20 days.
This Chapter presents simulations of the population of transiting planets that TESS
will detect and the population of eclipsing binary stars that produce photometric signals
resembling those of transiting planets. These simulations were originally developed to
inform the design of the mission. They are also being used to plan the campaign of groundbased observations required to distinguish planets from eclipsing binaries as well as followup measurements of planetary masses and atmospheres. In the future, these simulations
could inform potential proposals for an extended mission.
We have organized this Chapter as follows: Section 1.2 provides an overview of TESS
and the types of stars that will be searched for transiting planets. Sections 1.3-1.5 present
our model for the relevant stellar and planetary populations. Section 1.3 describes the
properties and luminosity function of the stars in our simulation. Section 1.4 describes
the assignment of transiting planets and eclipsing binary companions to these stars. Section 1.5 combines these results to forecast the properties of the brightest transiting planet
systems on the sky regardless of how they might be detected. This information is relevant
16
to planning a transit survey and sets expectations for the transiting planets most favorable
for characterization. Sections 1.6-1.8 then describe the detection of the simulated planets
specifically with TESS. Section 1.6 details our model for the photometric performance of
the TESS cameras. Sub-section 1.6.7 describes how stars will be prioritized for observation
by the spacecraft. Section 1.7 presents the simulated detections of planets and their properties. Section 1.7 also shows the detections of astrophysical false-positives, and Section 1.8
investigates the possibilities for distinguishing them from planets using TESS data and supplementary data from ground-based telescopes. Finally, Section 1.9 discusses the prospects
for following up the TESS detections to study the planets' masses and atmospheres.
1.2
Brief Overview of TESS
TESS employs four refractive cameras, each with a field of view of 240 x 24' imaged by an
array of four 2kx2k charge-coupled devices (CCD). This gives a pixel scale of 21."1. The
four camera fields are stacked vertically to create a combined field that is 24' wide and 96'
tall, captured by 64M pixels. Each camera has an entrance pupil diameter of 105 mm and
an effective collecting area of 69 cm 2 after accounting for transmissive losses in the lenses
and their coatings. (The quantum efficiency of the CCDs is considered separately.)
Each camera will acquire a new image every 2 seconds. The readout noise, which has
a design goal of 10 e- pix- RMS, is incurred with every 2 sec image. This places the read
noise at or below the zodiacal photon-counting noise, which ranges from 10-16 e- pix- 1
RMS for a 2 sec integration time (see Section 1.6.4).
Due to limitations in data storage and telemetry, it will not be possible to transmit
all the 2 sec images back to Earth. Instead, TESS will stack these images to create two
basic data products with a longer effective exposure time. First, the subset of pixels that
surround pre-selected "target stars" will be stacked at a 2 min cadence. Second, the fullframe images ("FFIs") will be stacked at a 30 min cadence. The selection of the target stars
will prioritize the detection of small planets; this described further in Section 1.6.7. The
FFIs will allow a wider range of stars to be searched for transits, and they also enable other
scientific investigations that require time-domain photometry of bright sources.
17
Ecliptic Polar Projection [deg]
13
90
45
0
1
-90
-45
0
45
90
Figure 1-1: Polar projection illustrating how each ecliptic hemisphere is observed with
13 pointings. Each pointing has a duration of 27.4 days, or two spacecraft orbits, and
four TESS cameras have a combined field-of-view of 24x96'. The number of pointings
that TESS provides to a given star is primarily a function of the star's ecliptic latitude.
Coverage near the ecliptic (dashed line) is sacrificed in favor of coverage near the ecliptic
poles, which receive nearly continuous coverage for 355 days.
1.2.1
Sky Coverage
TESS will observe from a 13.7-day elliptical orbit around the Earth. Over two years, it
will observe the sky using 13 pointings in each ecliptic hemisphere; two spacecraft orbits
(27.4 days) are devoted to each pointing. Because the cameras are fixed to the spacecraft,
the spacecraft must re-orient for every pointing. The pointings are spaced equally in ecliptic
longitude, and they are positioned such that the top camera is centered on the ecliptic pole
and the bottom camera reaches down to an ecliptic latitude of 60. Figure 1-1 shows the sky
coverage resulting from this arrangement.
Over ten thousand "target stars" can be observed with each pointing, so TESS will
observe over 2 x 10 5 target stars during its primary mission.
18
V
Jc:
R:
TESS
I 0.80
Q1 0.60.4
0.2-
500
600
800
700
900
1100
1000
Wavelength [nm]
Figure 1-2: The TESS spectral response, which is the product of the CCD quantum efficiency and the longpass filter curve. Shown for comparison are the filter curves for the
familiar Johnson-Cousins V, R, and Ic filters as well as the SDSS z filter. Each curve
is normalized to its peak. The vertical dotted lines indicate the wavelengths at which the
point-spread function is evaluated for our optical model (see Section 1.6.2).
1.2.2
Spectral Response
The spectral response of the TESS cameras is limited at its red end by the quantum efficiency of the CCDs. TESS employs the MIT Lincoln Laboratory CCID-80 detector, a
back-illuminated CCD with a depletion depth of 100 pm. This relatively deep depletion
allows for sensitivity to wavelengths slightly longer than 1000 nm.
At its blue end, the spectral response is limited by a longpass filter with a cut-on wavelength of 600 nm. Figure 1-2 shows the the complete spectral response, defined as the
product of the quantum efficiency and filter transmission curves.
It is convenient to define a TESS magnitude T normalized such that Vega has T = 0.
We calculate the T = 0 photon flux by multiplying the template AOV spectrum provided by
Pickles (1998) by the TESS spectral response curve and then integrating over wavelength.
We assume Vega has a flux density of FA
=
3.44 x i0 9 erg s
cm
2
A
1
at A
=
5556
A
2
1
(Hayes, 1985). We find that T = 0 corresponds to a flux of 4.03 x 10-6 erg s- cm- , and
a photon flux of 1.51 x 106 ph s-1 cm-2.
19
1
Table 1.1: Fluxes in the TESS bandpass and Ic - T colors.
Spectral Type
[K]
IC = 0 photon flux
[106 ph s-1 cm- 2 ]
[mmag]
M9V
M5V
M4V
M3V
M1V
K5V
K3V
KLV
G2V
F5V
FOV
AOV
2450
3000
3200
3400
3700
4100
4500
5000
5777
6500
7200
9700
2.38
1.43
1.40
1.38
1.39
1.41
1.43
1.45
1.45
1.48
1.48
1.56
306
-191
-202
-201
-174
-132
-101
-80.0
-69.5
-40.0
-34.1
35.0
Teff
Ic - T
The mapping between Tff and spectral type is based on data compiled by
E. Mamajekl. The photon flux at T = 0 is 1.514 x 101 ph s-- cm-2
By repeating the calculation for different template spectra from the Pickles (1998) library, we obtain the photon fluxes for stars of other spectral types. These are shown in
Table 1.1. To facilitate comparisons with the standard Johnson-Cousins Ic band (which is
nearly centered within the T-band), Table 1.1 also provides synthetic Ic - T colors. We
note that the IC - T color for an AOV star is +0.035, which is equal to the apparent Ic
magnitude defined for Vega.
1.2.3
Simplified sensitivity of TESS
The most important stellar characteristics that affect planet detectability are apparent magnitude and stellar radius. Here, we provide a simple calculation for the limiting apparent
magnitude, as a function of stellar radius, that permits TESS to detect planets smaller than
Neptune (Rp < 4 Re). This gives an overview of TESS's planet detection capabilities
and establishes the necessary depth of our more detailed simulations of the population of
nearby stars.
We assume the noise in the photometric observations to be the quadrature sum of read
noise and the photon-counting noise from the target star and the zodiacal background (see
20
Section 1.6.4 for the more comprehensive noise model). We require a signal-to-noise ratio
of 7.3 for detection (see Section 1.6.6 for the rationale). We assume that the total integration
time during transits is 6 hours, which may represent two or more transits of shorter duration.
Using these assumptions, Figure 1-3 shows the limiting apparent magnitude as a function
of stellar radius at which transiting planets of various sizes can be detected.
To gauge the necessary depth of the detailed simulations, we consider the detection of
small planets around two types of stars represented in Figure 1-3, a Sun-like star and an
M dwarf with Tff
3200 K. These two choices span the range of spectral types that TESS
will prioritize; stars just larger than the Sun give transit depths that are too shallow, and
dwarf stars just cooler than 3200 K are too faint in the TESS bandpass.
For the Sun-like star, a 4 Re planet produces a transit depth of 0.13%. The limiting
magnitude for transits to be detectable is T = 13.9. This also corresponds to K,
=
13.2
and a maximum distance of 1.4 kpc assuming no extinction.
For the M dwarf with Tff =3200 K, we assume R,
=
0.155 RD, based on the Dart-
mouth Stellar Evolution Database (Dotter et al., 2008) for solar metallicity and an age of
1 Gyr. We consider a planet of size 2.8 Re rather than 4 Re since Dressing & Charbonneau
(2013) found that M dwarfs very rarely have close-in planets larger than 2.8 Re. For the
2.8 R@ planet, the transit depth is 2.7% and the limiting apparent magnitude for detection
is T
=
17.3. This corresponds to K, = 15.0 and a maximum distance of 190 pc assuming
no extinction.
A similar calculation can be carried out for eclipsing binary stars. Some TESS target
stars will turn out to be eclipsing binaries, and others will be blended with faint binaries
in the background. The maximum eclipse depth for an eclipsing binary is approximately
50%, which occurs when two identical stars undergo a total eclipse. Assuming the period
is 1 day, and that TESS observes the system for 27.4 days, the limiting apparent magnitude
for detection of the eclipse signals is T < 21, corresponding to many kiloparsecs.
To summarize, TESS is sensitive to small planets around Sun-like stars within
,,
1 kpc.
For M dwarfs, the search distance is AS 200 pc. Eclipsing binaries can be detected across
the Milky Way. These considerations set the required depth of our simulations of the stellar
population, which must also take into account the structure of the galaxy and extinction.
21
1012S14
16-
-
12
10
2
.8 R
16
-----
1818 -
. -,-4.0 RE
0.1
1
0.5
1.4
Radius [R®]
Figure 1-3: The limiting magnitude for planet detection as a function of stellar radius for
three planetary radii. Here, detection is defined as achieving a signal-to-noise ratio greater
than 7.3 based on 6 hours of integration time during transits. The noise model includes
read noise and photon-counting noise from the target star and a typical level of zodiacal
light. While the TESS bandpass is similar to the Ic band, the sensitivity curve is flatter in
K, magnitudes. M dwarfs rarely have short-period planets larger than 2.8 Re, so the 4 RD
line becomes dashed over their radii.
22
1.3
Star Catalog
Due to the wide dynamic range of apparent magnitudes that we need to consider, and sensitive dependence of transit detections on stellar radii, we use a synthetic stellar population
rather than a real catalog. The basis for our stellar population is TRILEGAL, an abbrevia-
tion for the TRIdimensional modeL of thE GALaxy (Girardi et al., 2005). TRILEGAL is a
Monte Carlo population synthesis code that models the Milky Way with four components:
a thin disk, a thick disk, a halo, and a bulge. Each of these components contains stars with
the same initial mass function but with a different spatial distribution, star formation rate,
and age-metallicity relation. For stars with masses 0.2-7 Me, TRILEGAL uses the Padova
evolutionary tracks (Girardi et al., 2000) to determine the stellar radius, surface gravity, and
luminosity as a function of age. For stars less massive than 0.2 M®, TRILEGAL uses a
brown dwarf model (Chabrier et al., 2000). Apparent magnitudes in various photometric
bands are computed using a spectral library drawing upon several theoretical and empirical
sources. A disk extinction model is used to redden the apparent magnitudes depending on
the location of the star. TRILEGAL does not include the Magellanic Clouds, nor does it
model any star clusters in the Milky Way.
The star counts predicted by the TRILEGAL model were originally calibrated against
the Deep Multicolor Survey (DMS) and ESO Imaging Survey (EIS) of the South Galactic
Pole. The model was also found to be consistent with the EIS coverage of the Chandra
Deep Field South (Groenewegen et al., 2002). More recently, TRILEGAL was updated
and re-calibrated against the shallower 2MASS and Hipparcos catalogs while maintaining
agreement with the DMS and EIS catalogs (Girardi et al., 2005).
Given a specified line of sight and solid angle, TRILEGAL returns a magnitude-limited
catalog of simulated stars, including properties such as mass, age, metallicity, surface gravity, distance, and extinction. Apparent magnitudes are reported in the Sloan griz, 2MASS
JHK,, and Kepler bandpasses; at our request, L. Girardi kindly added the TESS bandpass
to TRILEGAL. When necessary, we translate between the Sloan and Johnson-Cousins filters using the transformations for Population I stars provided by Jordi et al. (2006).
We find it necessary to adjust the properties of the population of low-mass stars (M <
23
0.78 M 0 ) to bring them into satisfactory agreement with more recent determinations of the
absolute radii and luminosity function of these stars. These modifications are described in
Sections 1.3.2 and 1.3.4. In addition, we employ our own model for stellar multiplicity that
is described in Section 1.3.3.
1.3.1
Model Queries
The TRILEGAL simulation is accessed through a web-based interface. 2 We use the default
input parameters for the simulation (Table 1.2); the postfacto adjustments that we make
to dwarf properties, binarity, and the disk luminosity function are discussed below. The
runtime of a TRILEGAL query is limited to 10 minutes, so we build an all-sky catalog by
performing repeated queries over regions with small solid angles.
We divide the sky into 3072 equal-area tiles using the HEALPix scheme (G6rski et al.,
2005). Each tile subtends a solid angle of 13.4 deg2 . Within the 164 tiles closest to the
galactic disk and bulge, the stellar surface density is too large for the necessary TRILEGAL
computations to complete within the runtime limit. We omit these tiles from consideration
since the high background and incidence of false-positive eclipsing binaries will make these
areas difficult to search for transiting planets. This leaves 2908 tiles covering 95% of the
sky.
For each of the 2908 sightlines through the centers of tiles, we make three queries to
TRILEGAL:
1. The "bright catalog" with K, < 15 and a solid angle of 6.7 deg. This is intended
to include any star that could be searched for transiting planets; the magnitude limit
of K, < 15 is based on the considerations in Section 1.2.3.
Using the K, band
to set the limiting magnitude is a convenient way to allow the catalog to have a
fainter T mag limit for M stars than for FGK stars. The full solid angle of 13.4
deg2 cannot be simulated due to the 10-minute maximum runtime of the simulation.
Instead, we simulate a 6.7 deg 2 field and simply duplicate each star in the catalog.
Once duplicated, we assign coordinates to each star randomly from a probability
2
http://stev.oapd.inaf.it/cgi-bin/trilegal
24
Table 1.2: TRILEGAL input settings.
Parameter
Galactic radius of Sun
Value
8.70 kpc
Galactic height of Sun 24.2 pc
IMF (log-normal, Chabrier 2001)
Characteristic mass
Dispersion
0.1 MD
0.627 Mo
Thin Disk
Scale height (sech 2 )
94.69 pc
Scale radius (exponential) 2.913 kpc
Surface density at Sun 55.4 MD pc-2
Thick Disk
Scale height (sech 2) 800 pc
Scale radius (exponential) 2.394 kpc
Density at Sun 10-3A1 pc-3
Halo (R114 Oblate Spheriod)
Major axis
2.699 kpc
Oblateness 0.583
Density at Sun 10- 4 MA pc-3
Bulge (Triaxial, Vanhollebeke et al. 2009)
Scale length
2.5 kpc
Truncation length 95 pc
Bar: y/x aspect ratio 0.68
Bar-Sun angle 150
z/x ratio 0.31
Central Density 406 M pc~ 3
Disk Extinction
Scale height (exponential) 110 pc
Scale radius (exponential) 100 kpc
Extinction at Sun (dAv/dR) 0.15 mag kpc
Av(z = oo) 0.0378 mag
Randomization (RMS) 10%
25
1
distribution that is spatially uniform across the entire tile. Across all of the tiles, this
catalog contains 2.11 x 10' stars.
2. The "intermediate catalog" with T < 21 and a solid angle of 0.134 deg 2 . This
is intended to include stars for which TESS would be able to detect a deep eclipse
of a binary star. We use this catalog to assign blended background binaries to the
target stars in the bright catalog and also to evaluate background fluxes. This deeper
query is limited to a smaller solid angle (1/100th of the area of the tile) to limit
computational time. The simulation then re-samples from these stars 100 times when
assigning background stars to the target stars. We also restrict this catalog to K, > 15
in the simulation to avoid double-counting stars from the bright catalog. Across all
tiles, this catalog contains 1.19 x 109 stars.
3. The "faint catalog" with 21 < T < 27 and a solid angle of 0.0134 deg 2 . This is used
only to calculate background fluxes due to unresolved background stars. The limiting
magnitude is not critical because the surface brightness due to unresolved stars is
dominated by stars at the brighter end rather than the fainter end of the population of
unresolved stars. Stars from this catalog are re-sampled 1000 times. Across all tiles,
this catalog contains 7.39 x 10 9 stars.
1.3.2
Properties of low-mass stars
Low-mass dwarf stars are of particular importance for TESS because they are abundant
in the solar neighborhood and their small sizes facilitate the detection of small transiting
planets. Although the TRILEGAL model is designed to provide simulated stellar populations with realistic distributions in spatial coordinates, mass, age, and metallicity, we
noticed that the radii of low-mass stars for a given luminosity or Tff in the TRILEGAL
output were smaller than have been measured in recent observations or calculated in recent
theoretical models.
Figure 1-4 illustrates the discrepancy. It compares the radius-magnitude relation employed by TRILEGAL with that of the more recent Dartmouth models (Dotter et al., 2008)
as well as empirical data based on optical interferometry of field stars and analysis of
26
Q 0.5-
EB Radii
0.2
0 Interferometric Radii
-
0.1
Padova 1 Gyr, [Fe/H]=0
Dartmouth 1 Gyr, [Fe/H]=0
2
4
8
6
Absolute Ic
10
12
Figure 1-4: Radius-magnitude relation for simulated stars compared to empirical observations. The Padova models (red curve) are employed by default within the TRILEGAL simulation. These models seem to underestimate the radii of low-mass stars; the
Dartmouth models (green curve) give better agreement. For stars of mass 0.14-0.78 MID
(dashed boundaries), we overwrite the TRILEGAL-supplied properties with Dartmouthbased properties for a star of the given mass, age, and metallicity. The interferometric
measurements plotted here are from Boyajian et al. (2012), and the eclipsing-binary measurements come from a variety of sources (see text). The scatter in radius for Ic < 5 arises
from stellar evolution.
27
eclipsing binary stars. The interferometric radius measurements are from Boyajian et al.
(2012). The measurements based on eclipsing binaries are from the compilation of Andersen (1991) that has since been maintained by J. Southworth 3 . We also make use of the
systems in Winn et al. (2011 b) that were considered to determine properties of Kepler-16.
The published data specify Tff rather than absolute Ic magnitude; in preparing Figure 1-4, we converted Tff into absolute 1 c using the temperature-magnitude data compiled
by E. Mamajek 4 and Pecaut & Mamajek (2013).
Figure 1-4 shows that the Dartmouth stellar-evolutionary models give better agreement
with measured radii, especially those from interferometry. Therefore, to bring the key
properties of the simulated stars into better agreement with the data, we replaced the TRILEGAL output for the apparent magnitudes and radii of low-mass stars (0.15-0.78 M)
with the properties calculated with the Dartmouth models. To make these replacements,
we use a trilateral interpolation in mass, age, and metallicity to determine the absolute
magnitudes, Tff, and radii from the grid of Dartmouth models. For simplicity, we assume
the helium abundance is solar for all stars. Furthermore, motivated by Fuhrmann (1998),
we only select the grid points that adhere to the following one-to-one relation between
[a/Fe] and [Fe/H]:
[Fe/H > 0
-
[Fe/H] = -0.05
[Fe/H] < -0.1
[a/Fe] = 0.0
=-
<
[a/Fe]
(1.1)
+0.2
(1.2)
[a/Fe] = +0.4
(1.3)
In calculating the apparent magnitudes of the stars with properties overwritten from the
Dartmouth models, we preserve the distance modulus from TRILEGAL and apply reddening corrections using the same extinction model that TRILEGAL uses. TRILEGAL reports
the extinction Al for each star, and for bands other than V, we use the AA/Av ratios from
Cardelli et al. (1989).
3
4
http://www.astro.keele.ac.uk/jkt/debcat/
http://www.pas.rochester.edu/-emamajek/EEMdwarfUBVIJHKcolorsTeff.txt
28
1.3.3
Stellar Multiplicity
Binary companions to the TESS target stars have three important impacts on the detection
of transiting planets. First, whenever a "target star" is really a binary, there are potentially
two stars that can be searched for transiting planets. The effective size of the search sample
is thereby increased. However, this works against the second effect: if there is a transit
around one star, the constant light from the unresolved companion dilutes the observed
transit depth, making it more difficult to detect the transit. Even if the transit is detectable,
the radius of the planet may be underestimated from the diluted depth. The third effect is
that a planet around one member of a close binary has a limited range of periods within
which its orbit would be dynamically stable.
Furthermore, eclipsing binaries that are blended with target stars, or that are bound
to the target star in hierarchical triple or quadruple systems, can produce eclipses that resemble planetary transits. Because eclipsing binaries produce larger signals than planetary
transits, the population of eclipsing binaries needs to be simulated down to fainter apparent
magnitudes than the target stars.
To capture these effects in our simulations, we need a realistic description of stellar
multiplicity. We are guided by the review of Duchene & Kraus (2013). The multiplicity
fraction (MF) is defined as the fraction of systems that have more than one star; it is the
sum of the binary fraction (BF), triple fraction (TF), quadruple fraction (QF), and so on.
However, our simulations just consider systems with up to 4 stars.
The MF has been observed to increase with the mass of the primary, which is reflected
in our simulation. In our TRILEGAL queries, every star is originally a binary, and we
decide randomly whether to keep the secondary based on the primary mass and the MF
values in Table 1.3. Next, we turn a fraction of the remaining binaries into triple and
quadruple systems according to the desired TF and QF. The MF, TF, and QF are adopted
as follows:
1. For primary stars of mass 0.1-0.6 M, we adopt the MF of 26% from Delfosse et al.
(2004). For systems with n = 3 or 4 components, the fraction of higher-order systems is taken to be 3.92-
from Duchene & Kraus (2013).
29
2. For stars of mass 0.8-1.4 Al, we draw on the results of Raghavan et al. (2010).
Primary masses of 0.8-1.0 MD have a MF of 41%, while primary masses of 1.01.4 M have a MF of 50%. The fraction of higher-order systems is 3.82-" for both
ranges (Duchene & Kraus, 2013).
3. For stars of mass 0.6-0.8 AlD, we adopt an intermediate MF of 34%. The fraction of
higher-order systems is
3 .7 2-n
4. For primaries more massive than 1.4 M, we use the results for A stars from Kouwenhoven et al. (2007), giving a MF of 75%. We assume that the fraction of higher-order
systems is 3 . 7 2Next, we consider the properties of the binary systems. TRILEGAL originally creates binaries with a uniform distribution in the mass ratio between the secondary and the
primary, q, between 0.1 and 1. However, a more realistic distribution in q is
dN
dq
dq
q ,
(1.4)
where the power-law index y is allowed to vary with the primary mass, as specified in
Table 1.3. When we select the binary systems to obtain the desired MF, we choose the
systems to re-create this distribution in q over the range 0.1< q <1.0.
The period P is not specified by TRILEGAL, so we assign it from a log-normal distribution. Duchene & Kraus (2013) parametrizes the distribution in terms of the mean semimajor axis (d) and the standard deviation in log P; both parameters vary with the primary
mass as shown in Table 1.3. We convert from d to P with Kepler's third law.
The orbital inclination i is drawn randomly from a uniform distribution in cos i. The
orbital eccentricity e is drawn randomly from a uniform distribution, between zero and a
maximum value
emax =
1
-
tan- 1 (2 [log P - 1.5])
+
1
(1.5)
where P is specified in days, to provide a good fit to the range of eccentricities shown in
Figure 14 of Raghavan et al. (2010). The argument of pericenter W is drawn randomly from
a uniform distribution between 0' and 3600.
30
For the systems that are designated as triples, we assign the properties using the approach originally suggested by Eggleton (2009). Although there is no physical reason why
this method should work well, it has been found to reproduce the multiplicity properties
of a sample of Hipparcos stars (Eggleton & Tokovinin, 2008). First, we create a binary
according to the prescriptions described above with a period Po. Then, we split the primary
or secondary star (chosen randomly) into a new pair of stars. The new pair of stars orbit
their barycenter with a higher-order period PHOP according to
PHOP = 0.2 x 10,2u
(1.6)
PO
where u is uniformly distributed between 0 and 1. This way, PHOP is < 1/5 the orbital
period of the original binary system, a rudimentary method for enforcing dynamical stability. The mass of a star is conserved when it is split, so the barycenter of the original binary
remains the same, and the orbital period of the companion star about this barycenter is
unchanged. The original prescription given by Eggleton (2009) assigns P from a distribution peaking in log(Po) at 105 days and allows the new period to vary over 5 decades. Our
assumed distribution for log(Po) peaks at shorter values (for stars ,<1 MD), so in our implementation, the higher-order orbital period can only vary over 2 decades to avoid generating
unphysically short periods.
While the total mass of a new pair of stars is equal to that of the origial star, the mass
ratio needs to be assigned. The parent distribution of this assignment is taken from the
sample of triples presented in Figure 16 of Raghavan et al. (2010). We model this distribution by setting q = 1.0 for 23% of the pairs and drawing q from a normal distribution with
(A, a2 ) = (0.5, 0.04) for the other 77% of the pairs. Finally, for each star in a higher-order
pair, we calculate the absolute and apparent magnitudes, radius, and Tf from the new stellar mass in combination with the age and metallicity inherited from the original star. We do
so using the same interpolation onto the Dartmouth model grid described in Section 1.3.2.
For the systems that are turned into quadruples, we create a binary and then split both
stars using the procedure described above. This results in two higher-order pairs that orbit
.
one another with the original binary period P0
31
Table 1.3: Binary properties as function of the mass of the primary.
Mass [M®]
<0.1
0.1-0.6
0.6-0.8
0.8-1.0
1.0-1.4
>1.4
1.3.4
MF
0.22
0.26
0.34
0.41
0.50
0.75
a [AU]
4.5
5.3
20
45
45
350
o-(log P)
0.5
1.3
2.0
2.3
2.3
3.0
4.0
0.4
0.35
0.3
0.3
-0.5
TF
n/a
0.067
0.089
0.11
0.14
0.20
QF
n/a
0.017
0.023
0.030
0.037
0.055
Luminosity Function
After modifying the TRILEGAL simulation to improve upon the properties of low-mass
stars and assign multiple-star systems, we ensure that the luminosity function (LF) is in
agreement with observations. For this purpose, we rely on two independent J-band LFs
reported in the literature. The first LF is from Cruz et al. (2007). It is based on volumelimited samples: a 20 pc sample for M
> 11 and an 8 pc sample for MA
< 11 (Reid et al.,
2003). Both samples use 2MASS photometry and are limited to J < 16. The second LF,
from Bochanski et al. (2010), is based on data from the Sloan Digital Sky Survey for stars
with 16 < r < 22. The resulting LF is reported for the range 5 < Mj < 10. Where the
Cruz et al. (2007) and Bochanski et al. (2010) LFs overlap, we use the mean of the two
LFs reported for single and primary stars (the brightest member of a multiple system). This
results in the "empirical LF' to which the TRILEGAL LF is adjusted.
Next, we compute the LF of our TRILEGAL-based catalog by selecting all of the single
and primary disk stars with distances within 30 pc. Then, we bin the stars according to
Mj and compare the result to the empirical LF. For each Mj bin, we find the ratio of
the TRILEGAL LF to the empirical LF. This ratio ranges from 0.5 to 11 across all of the
magnitude bins.
We then return to each HEALPix tile individually, and we bin the stars by Ali. Using
the ratio computed above for each Mj bin, we select stars at random for duplication or
deletion to bring the simulated LF into agreement with the empirical LF. This process
results in a net reduction of ~30% in the total number of stars in the catalog and a shift in
the LF peak towards brighter absolute magnitudes.
32
Table 1.4: J-band luminosity function in 10- 3 stars pc-3
3.25
3.75
4.25
4.75
5.25
5.75
6.25
6.75
7.25
7.75
8.25
8.75
9.25
9.75
10.25
10.75
11.25
11.75
12.25
12.75
13.25
13.75
14.25
14.75
15.25
Primaries and Singles
Systems
Individual Stars
0.85
1.44
2.74
3.85
1.55
1.79
3.01
3.37
7.74
7.15
7.62
4.84
5.25
3.56
1.95
2.16
1.75
1.11
0.73
0.55
0.45
0.02
0.00
0.00
0.00
0.94
1.74
2.87
3.38
1.54
1.91
3.12
4.04
7.90
7.10
7.03
4.89
4.75
3.49
2.11
2.10
1.56
1.07
0.76
0.52
0.36
0.02
0.00
0.00
0.00
1.08
1.72
3.10
4.55
2.19
2.27
3.57
4.15
8.82
8.57
9.29
6.64
6.50
4.72
2.68
2.67
2.21
1.52
1.08
0.84
0.69
0.06
0.02
0.00
0.02
33
The left panel of Figure 1-5 shows the LF of the TRILEGAL simulation before and
after this adjustment. The final LF is also quantified in Table 1.4. Each column of the table
considers stellar multiplicity in a different fashion: "Singles and Primaries" counts single
stars and the brightest member of a multiple system; "Systems" counts the combined flux
of all stars in a system, regardless of whether it is single or multiple; and "Individual Stars"
counts the primary and secondary members separately.
As a sanity check, we make some further comparisons between our simulated LF and
other published luminosity functions. Figure 1-5 shows a comparison to the 10 pc RECONS sample (Henry et al., 2006), the Hipparcos catalog (Perryman et al. 1997 and van
Leeuwen 2007), and the Ic-band LF of Zheng et al. (2004). The agreement with the Hipparcos sample is good up until V
8, where the Hipparcos sample becomes incomplete.
The RECONS LF has a lower and blunter peak, and the Zheng et al. (2004) LF has a
sharper and taller peak than the simulated LF, but are otherwise in reasonable agreement.
As another sanity check, we examine star counts as a function of limiting apparent
magnitude in Figure 1-6. We compare the number of stars per unit magnitude per square
&
degree in the simulated stellar population against star counts from the classic Bahcall
Soneira (1981) star-count model in the IC band as well as actual star counts from the
2MASS point source catalog (Cohen et al., 2003) in the J band. In all cases, multiple
systems are counted as a single "star" with a magnitude equal to the total system magnitude.
The agreement seems satisfactory; we note that the comparison with 2MASS becomes less
reliable at faint magnitudes because of photometric uncertainties as well as extra-galactic
objects in the 2MASS catalog.
1.3.5
Stellar Variability
Intrinsic stellar variability is a potentially significant source of photometric noise for the
brightest stars that TESS observes. To each star in the simulation, we assign a level of intrinsic photometric variability depending on the spectral type. Our assignments are based
on the variability of Kepler stars reported by Basri et al. (2013). For each star, they calculated the median differential variability (MDV) on a 3-hour timescale by binning the
34
25
ho2 0
- - - Uncorrected Sim.
-
Cruz (2007)
-
Bochanski (2010)
-
Corrected Sim.
-
Hipparcos
2
Zheng (2004)
RECONS
-
.2(
-Simulation
5
C1
1.
0
0
-1
5
5
0
5
Absolute J
10
0
5
10
Absolute Ic
15
0
5
10
15
Absolute V
20
Figure 1-5: The luminosity function of the simulated stellar population compared with
various published determinations. Left.-Comparison with the J-band LFs of Cruz et al.
(2007) and Bochanski et al. (2010) before and after we correct the LF of the simulation. The
stellar multiplicity and dwarf properties have already been adjusted in the "Uncorrected"
LF. Center-Comparisonwith the Ic-band LF of Zheng et al. (2004) and the Hipparcos
sample (Perryman et al. 1997 and van Leeuwen 2007). Right.-Comparison with Hipparcos and the 10 pc RECONS sample (Henry et al., 2006). For the J- and V-band LFs,
we count the single, primary, and secondary stars separately, since binaries are generally
resolved in the surveys with which we are comparing. For the Ic band, we count the system magnitude of binary systems since we assume they are unresolved in the Zheng et al.
(2004) survey. The range of absolute magnitudes from the Hipparcos catalog are dominated
by single and primary stars, so this distinction is less important.
35
b=
b
90*
=
30', 1 =
b
180'
= 304,1 = 90'
103
103
101
101
bO
10
C2
10
IC
IC
IC
bO
10
33
bO
10
CO2
10
10
1 5-
10
15
5
10
15
5
10
1510
Figure 1-6: Star counts as function of apparent magnitude and galactic coordinates. In
the IC band (top row), we compare the star counts in our simulated catalog (black) to those
from Bahcall & Soneira (1981) (blue). In the J band (bottom row), we compare our catalog
(black) to the 2MASS point source catalog (red).
light curve into 3-hour segments and then calculating the median of the absolute differences between adjacent bins. Since each transit is a flux decrement between one segment
of a lightcurve relative to the much longer timeseries, rather than two adjacent segments of
equal length, the noise statistic relevant to transit detection is ~ V"2 smaller than the MDV.
G. Basri kindly provided their sample from Figures 7-10, which is is divided into four
subsamples according to stellar Tff. We select 100 stars in each subsample with Kp < 11.5
to minimize the contributions of instrumental noise from Kepler. Since red giants exhibiting pulsations can contaminate the subsample with Tff < 4500 K, particularly at brighter
apparent magnitudes, we select stars with 12.5 < Kp < 13.1 for these temperatures.
Figure 1-7 shows the resulting distributions of variability. Each star in our simulated
population is randomly assigned a variability from one of the 100 stars in the appropriate Tef subsample. The variability of the Teff < 4500 K subsample is roughly 5 times
greater than that of solar-type stars. However, M dwarfs are the faintest stars that TESS will
observe, so instrumental noise and background will dominate their photometric error.
36
Since the photometric variations associated with stellar variability exhibit strong correlations on short timescales, we assume that the level of noise due to intrinsic variability is
independent of transit duration: we do not adjust it according to t-1/2 as would be the case
for white noise. However, we do assume that stellar variations are independent from one
transit to the next, so the noise contribution from stellar variability scales with the number
of transits as N-11 2 . In summary, the standard deviation in the relative flux due to stellar
variability, after phase-folding all of the transits together, is taken to be
orv
1.4
=
MDV(3 hr)N
1/2
N127
(1.7)
Eclipsing Systems
We next assign planets to the simulated stars, and we identify the transiting planets as
well as the eclipsing binaries. We then calculate the properties of the transits and eclipses
relevant to their detection and follow-up.
1.4.1
Planets
The planet assignments are based on several recent studies of Kepler data. For FGK stars,
we adopt the planet occurrence rates from Fressin et al. (2013). For Tff < 4000 K, we
adopt the occurrence rates from Dressing & Charbonneau (2013). We note that Dressing
& Charbonneau (2013) did not report any planets larger than 2.8 RID. Also, unlike Fressin
et al. (2013), they did not correct for astrophysical false positives. We have made no attempt
to adjust the published occurrence rates for false positives.
In both cases, the published results are provided as a matrix of occurrence rates over
bins in planetary radius and period. Because the bins are relatively coarse, we allow the
radius and period of a given planet to vary randomly within the limits of each bin. Periods
are assigned from a uniform distribution in log P. (We omit planets for which the selected
period would place the orbital distance within 2 R,, on the grounds that tidal forces would
destroy any such planets.) For planets assigned from the Fressin et al. (2013) rates, the
37
Teff> 6000K
0.30.20.1
0
5000K < Teff < 6000K
0.30.20.1
-
w4
0
Q
4500K < Teff < 5000K
0.30.2-
0.1
"I
0
Teff < 4500K
0.30.2-
0.1
0'
10
100
1000
c-v [ppm]
Figure 1-7: The input distributions of the intrinsic stellar variability o-v per transit in parts
per million (ppm). Each star in our catalog is assigned a variability statistic from these
distributions according to its effective temperature. We calculate ov from the 3-hour MDV
statistic in Basri et al. (2013) using Equation 1.7.
38
22
22
*
22-*
7
22
6
61
6-6
4-
4
4-
2.
2
2
2
1.25
O.B.
1.25
1.25
1.25
4-
0.8
,
V
0.8
*.8
2
5.9
17
50
145
418
0.50
0.8
8
2
17
50
145
418
Period [days]
dN/dlog(R)
Period [days]
5.9
0
1
2
dN/dlog(R)
0.8
0.6
P
S0.4-
0.5
%.
2
.
0.
.Period
7
5
[days]
4
1
0.8
154
2
5.9rd [d7 5]
145
418
Figure 1-8: Probability density function for planet occurrence in the period-radius plane.
Left.-For stars with Tff > 4000 K, we use the planet occurrence rates reported by Fressin
et al. (2013). Right.-For stars with Tff < 4000 K, we use the planet occurrence rates
reported by Dressing & Charbonneau (2013).
radii of the smallest planets are chosen uniformily on (0.8, 1.25) Re. For the smallest
Dressing & Charbonneau (2013) planets, the radius is chosen uniformily on (0.5, 1.4) Re.
For larger planets (from both sets of occurrence rates), the radius is chosen according to
the distribution
dN x R-
7
P
dR-p
(1.8)
These intra-bin distributions were chosen ad hoc to provide a relatively smooth function in
the radius-period plane. The final distributions are illustrated in Figure 1-8.
We allow our simulation to assign more than one planet to a given star with independent
probability. The only exceptions are that we require the periods of adjacent planetary orbits
to have ratios of at least 1.2, and planets around a star with a binary companion cannot have
orbital periods that are within a factor of 5 of the binary orbital period. The result is that
53% of the transiting systems around FGK stars and 55% of those around M stars are
multiple-planet systems. Figure 1-9 shows the resulting distribution of period ratios. The
orbits of multi-planet systems are assumed to be perfectly coplanar, both for simplicity and
from the evidence for low mutual inclinations in compact multi-planet systems (Fabrycky
39
-x
104
0
0.5
100
10
1
(Pmt - Pi.)/P.
0.1
Figure 1-9: The distribution in the relative period difference for multi-planet systems. In
systems with more than two planets, the minimum period difference is counted. The distribution peaks near 1, which corresponds to a period ratio of 2:1. All systems with at least
one transiting member and an apparent magnitude of IC < 12 are counted.
et al., 2014; Figueira et al., 2012). As a sanity check, we compare the proportion of planets
in multi-transiting systems in our simulated stellar population to the proportion of multitransiting Kepler candidates. In our simulation, 26.2% of planets around FGK stars and
33.6% of planets around M stars reside in multi-transiting systems. Out of the 4,178 Kepler
objects of interest, 41% are in multi-transiting systems.
Since TESS is most sensitive to close-in planets, we assume that all planetary orbits are
circular. The orbital inclinations i are assigned randomly from a uniform distribution in
cos i. We identify the transiting systems as those with
b
Ibi < 1, where
a cos i
R*
is the transit impact parameter. We then calculate the properties of the planets and their
transits and occultations. The transit duration t is given by Eqns. (18) and (19) of Winn
(2011) in terms of the mean stellar density p*:
/1/3
=13
hr (
365 days
-1/3
/
40
*
P0
13V1
- b2.
(1.10)
The depth of the transit 61 is given by (Rp/R,)2 . The depth of the occultation (secondary
eclipse) is found by estimating the effective temperature of the planet (T,) and then computing the photon flux FP within the TESS bandpass from a blackbody of radius RP. The
photon flux from the planet is then divided by the combined photon flux from the planet
and the star:
62
P
(111)
+ IF*
The equilibrium planetary temperature Tp is determined by assuming radiative equilibrium
with an albedo of zero and isotropic radiation (from a recirculating atmosphere), giving
Tp =Teff
(1.12)
2a
We also keep track of the relative insolation of the planet S/Se, defined as
S
So
1.4.2
2
( T 4
a = -2(R)
1 AU
Re
--
5777 K
(1.13)
Eclipsing Binaries
We identify the eclipsing binaries by computing the impact parameters b 1 and b2 of the
primary and secondary eclipses, respectively:
acosi (1-
b1,2 = a o
R1 ,2
1
e2
2(1.14)
esinw/
(see Eqns. 7-8 of Winn 2011). Non-grazing primary eclipses are identified with the criterion
b1R1 < R1 - R2,
(1.15)
while grazing primary eclipses have larger impact parameters:
R, - R 2
<
b i R1 < R1 + R2 .
41
(1.16)
The eclipse depth of non-grazing primary eclipses is given by
R2=,
R1
r,
(1.17)
+ IP2
where F1 and F 2 are the photon fluxes from each star. In the event that R 2 > R 1, the area
ratio is set equal to unity; in that case, the primary undergoes a total eclipse. We neglect
limb-darkening in these calculations for simplicity. Secondary eclipses are identified and
quantified in a similar manner. We discard eclipsing binaries when the assigned parameters
imply a < R1 or a < R 2 . We also exclude systems where a is less than the Roche limit aR
for either star, assuming they are tidally locked:
aRl,2
= R 2 ,1
3 M1,2
1/3
(M2,1)1/
(1.18)
For grazing eclipses, the area ratio (R2 /R 1 ) 2 is replaced with the overlap area of two
uniform disks with the appropriate separation of their centers, given by Eqns. (2.14-5) of
Kopal (1979). The durations and timing of eclipses are calculated from Eqns. (14-16) of
Winn (2011).
For primaries with 1 c < 12, our simulated stellar population has 97461 eclipsing binaries over the 95% of the sky that is covered by the simulation. Another 21441 systems
contain eclipsing pairs in a hierarchical system. As another sanity check, we compare the
simulated density of eclipsing systems on the sky to the catalog of eclipsing binaries in the
Kepler field. We use Version 2 of the compilation 5 from Prsa et al. (2011) and Slawson
et al. (2011) to plot the density of eclipsing binaries as a function of apparent system magnitude in Figure 1-10. Within the range of 0.5 < P < 50 days, this catalog contains 1.85
EBs deg-
2
with Kp < 12. A 203 deg 2 subsample of our TRILEGAL catalog, taken from
15 HEALPix tiles and centered on galactic coordinates 1 = 76' and b = 13.4' for similarity
to the Kepler field, contains 1.04 EBs deg- 2 with Kp < 12.
42
-Kepler
100
Simulation
-
S10-17
10-2
8
10
9
11
12
Kp
Figure 1-10: Surface density of eclipsing binaries as a function of limiting magnitude in the
Kepler bandpass. The blue curve represents actual observations by Slawson et al. (2011).
The red curve is from our simulated stellar population in the vicinity of the Kepler field.
All eclipsing systems with 0.5 < P < 50 days are shown.
Best Stars for Transit Detection
1.5
to
Now that planets have been assigned to all of the stars with K, < 15, it is interesting
might be
explore the population of nearby transiting planets independently from how they
detected by TESS or other surveys. This helps to set expectations for the brightest systems
that can reasonably be expected to exist with any desired set of characteristics.
First, we identify the brightest stars with transiting planets. Figure 1-11 shows the
cumulative number of transiting planets as a function of the limiting apparent magnitude
a
of the host star. This is equal to the total number of planets that would be detected in
of
95% complete magnitude-limited survey (since our HEALPix tiles cover this fraction
the sky). We include the stars with 2000 < Tff < 6500 K and R* < 1.3R® that host
of
planets with periods <20 days. To reduce the statistical error, we combine the outcomes
5 trials.
The brightest star with a transiting planet of size 0.8-2 Re has an apparent magnitude
IC = 4.8. The tenth brightest such star has Ic = 6.6. For transiting planets of size 2-4 RD,
5
http://keplerebs.villanova.edu/v
2
43
C10
Rp>4
--
1
-2<R<4
C4
10
2 ~- Rp <2
0.5< S <2
-
10
X
100
55 Cnc. e x
D 209458b
HD 189733b
010
4
8
6
10
IC
Figure 1-11: Expected number of transiting planets that exist, regardless of detectability,
over the 95% of the sky covered by the simulation. The cumulative number of transiting
planets is plotted as a function of the limiting apparent Ic magnitude of the host star. The
mean of five realizations is shown. We count all planets having orbital periods between 0.520 days and host stars with effective temperatures 2000-6500 K and radii 0.08-1.3 RG. The
planet populations are categorized by radius ranges as shown in the figure. Also marked
are the apparent magnitudes of a few well-known systems with very bright host stars; their
locations relative to the simulated cumulative distributions suggest that these systems are
among the very brightest that exist on the sky.
44
the brightest host star has Ic = 5.3 and tenth brightest has 1 c = 7.6. One must look deeper
in order to find potentially habitable planets with periods shorter than 20 days; if we require
0.8 < Rp/Re < 2 and 0.5 < S/S® < 2, the brightest host star has 1 c = 8.7 and the tenth
brightest has Ic = 11.8.
In reality, the brightest host stars could be brighter or fainter than the expected magnitudes. In Figure 1-11 we also show the brightest known transiting systems for some of the
categories. Their agreement with the simulated cumulative distributions suggest that some
of the very brightest transiting systems have already been discovered.
1.6
Instrument Model
Now that the simulated population of transiting planets and eclipsing binaries has been
generated, the next step is to calculate the signal-to-noise ratio (SNR) of the transits and
eclipses when they are observed by TESS. The signal is the fractional loss of light during
a transit or an eclipse (6), and the noise (o) is calculated over the duration of each event.
The noise is the quadrature sum of all the foreseeable instrumental and astrophysical components.
Evaluation of the SNR is partly based on the parameters of the cameras already described in Section 1.2. We also need to describe how well the TESS cameras can concentrate the light from a star into a small number of pixels. The same description will be
used to evaluate the contribution of light from neighboring stars that is also collected in the
photometric aperture.
Our approach is to create small synthetic images of each transiting or eclipsing star, as
described below. These images are then used to determine the optimal photometric aperture
and the SNR of the photometric variations.
The synthetic images are also used to study the problem of background eclipsing binaries. Transit-like events that are apparent in the total signal measured from the photometric
aperture could be due to the eclipse of any star within the aperture. With only the photometric signal, there is no way to determine which star is eclipsing If the timeseries of the x
and y coordinates of the flux-weighted center of light (the "centroid" is also examined, then
45
in some cases, one can determine which star is undergoing eclipse. The synthetic images
allow us to calculate the centroid during and outside of transits and eclipses. We will later
show, in Section 1.8.4, how background eclipsing binaries tend to produce a larger and
more measurable centroid shift during eclipse than transiting planets. This can be used to
distinguish the two.
1.6.1
Pixel response function
The synthetic images are constructed from the pixel responsefunction (PRF), which describes the fraction of light from a star that is collected by a given pixel. It is calculated by
numerically integrating the point-spreadfunction (PSF) over the boundaries of pixels. The
photometric aperture for a star is the collection of pixels over which the electron counts are
summed to create the photometric signal; they are selected to maximize the photometric
SNR of the target star. (We assume in this work that the sum over the photometric aperture
is unweighted.)
The TESS lens uses seven elements with two aspheres to deliver a tight PSF over a large
focal plane and over a wide bandpass. Due to off-axis and chromatic aberrations, the TESS
PSF must be described as a function of field angle and wavelength. We calculate the PSF
at four field angles from the center (0) to the corner (170) of the field of view. Chromatic
aberrations arise both from the refractive elements of the TESS camera and from the deepdepletion CCDs absorbing redder photons deeper in the silicon. We calculate the PSF for
nine wavelengths, evenly spaced by 50 nm, between 625 and 1025 nm. These wavelengths
are shown with dashed lines in Figure 1-2. These wavelengths also correspond to a set
of bandpass filters that will be used in the laboratory to measure the performance of each
flight TESS camera.
The TESS lens has been modeled with the Zemax ray-tracing software. We use the
Zemax model to trace 250,000 simulated rays through the camera optics for each field
angle and wavelength. The model is set to the predicted operating temperature of -75
0
C.
Rays are propagated through the optics and then into the silicon of the CCD. A probabilistic
model is used to determine the depth of travel in the silicon before the photons are converted
46
to electrons. Finally, the diffusion of the electrons within the remaining depth of silicon is
modeled to arrive at the PSF.
Pointing errors from the spacecraft will effectively enlarge the PSF because the 2 sec
exposures are summed into 2 min stacks without compensating for these errors. The spacecraft manufacturer (Orbital Sciences) has provided a simulated time series of spacecraft
pointing errors from a model of the spacecraft attitude control system. Using two minutes
of this time series, we offset the PSF according to the pointing error and then stack the
resulting time series of PSFs. The root-mean-squared (rms) amplitude of the pointing error
is ~1", which is small in comparison to the pixel size and the full width half-maximum
of the PSF. Thus, the impact of pointing errors on short timescales turns out to be minor.
Long-term drifts in the pointing of the cameras will also introduce photometric errors, but
this effect is budgeted in the systematic error described in Section 1.6.4.
The full width half-maximum of the PSF is usually smaller than the 15 jpm size of a
single pixel. The PSF is therefore undersampled, so it must be recalculated for a given
offset and orientation between the PSF and the pixel boundaries. We numerically integrate
the PSF over a grid of 16 x 16 pixels to arrive at the PRF. We do so over a 10 x 10 grid of
sub-pixel centroid offsets and two different azimuthal orientations (00 and 450) with respect
to the pixel boundaries. For the corner PSF (at a field angle of 170), only the 45' azimuth
angle is considered.
We can also view the PRF in terms of the cumulative fraction of light collected by a
given number of pixels. In Figure 1-13, we average over all of the centroid offsets and
both azimuthal angles. For clarity, only three of the field angles and three values of Tff
are shown. There is little change in the PRF across the range of Tff, but the PRF degrades
significantly at the corners of the field.
1.6.2
Synthetic images
For each target star with eclipses or transits, we create a synthetic image in the following
manner. First, we determine the appropriate PRF based on the star's color and location in
the camera field. We calculate the field angle from its ecliptic coordinates and the direction
47
in which the relevant TESS camera is pointed. We randomly assign an offset between
the star and the nearest pixel center, and we randomly assign an azimuthal orientation of
either 0' or 45'. We then look up the nine wavelength-dependent PRFs for the appropriate
field angle, centroid offset, and azimuthal angle. The nine PRFs are summed with weights
according to the stellar effective temperature.
The weight of a given PRF is proportional to the stellar photon flux integrated over
the wavelengths that the PRF represents. Outside of the main simulation, we considered a
Vega-normalized stellar template spectrum of each spectral type from the Pickles (1998) library. We multiplied each template spectrum by the spectral response function of the TESS
camera, and we integrated the photon flux for each of the nine PRF bandpasses. Next, we
fitted a polynomial function to the relationship between the stellar effective temperatures
and the photon flux in each bandpass. During the simulation, the polynomial functions
are used to quickly calculate the appropriate PRF weights as a function of stellar effective
temperature.
Figure 1-12: Sample images of the pixel-response function (PRF). Left.--A target star. The
PRFs computed for 9 wavelengths have been stacked to form a single image. The weight
of each PRF in the sum depends on the the stellar effective temperature. Right.-Fainter
stars in the vicinity of the target star. We sum the flux from neighboring stars, with PRFs
weighted according to the Teff of each star, in the same fashion as the target stars.
Once the PRFs are summed, the result is a synthetic 16 x 16 -pixel image of each target
star. We only consider the central 8 x 8 pixels when determining the optimal photometric
aperture; the left panel of Figure 1-12 shows an example.
After synthesizing the image of each eclipsing or transiting target star, a separate 16 x
16 image is synthesized of all the relevant neighboring stars and companion stars. The
48
1.0-
0.8
-
0.6
0.4
-Center
(00)
Edge (120)
Comer (170)
0.2-
0
1
10
Pixels in aperture
100
Figure 1-13: The TESS pixel response function (PRF), after sorting and summing to show
the cumulative fraction of light collected for a given number of pixels in the photometric
aperture. We show this fraction for three field angles and three values of stellar effective
temperature. The dotted line is for Tff = 3000 K, the solid line is for 5000 K, and the
dashed line is for 7000 K. These temperatures span most of the range for which planets
smaller than 4 Re are detectable with TESS.
neighboring stars are drawn from all three star catalogs described in Section 1.3. The stars
are assumed to be uniformly distributed across each HEALPix tile, allowing us to randomly
generate the distances between the target star and the neighboring stars. Stars from the
target catalog are added to the synthesized image if they are within a radius of 6 pixels from
the target star. Stars from the intermediate catalog are added if they are within 4 pixels, and
stars from the faint catalog are added if they are within 2 pixels. The synthesized images
are created in the same manner as described above: by weighting, shifting, and summing
the PRFs associated with each star. The right panel of Figure 1-12 shows an example.
Synthetic images are also created for the eclipsing binary systems drawn from the intermediate catalog, but a slightly different approach is taken. For each eclipsing binary, we
search for any target stars within 6 pixels. If any are found, the brightest is added to the list
of target stars with apparent transits or eclipses. Separate synthetic images are created for
the target star, the eclipsing binary, and the non-eclipsing neighboring stars. Hierarchical
49
binaries are treated in a similar fashion; the non-eclipsing component is treated as the target star, and a separate synthetic image is created for the eclipsing pair so that its apparent
depth can be diluted. While this approach may appear to strongly depend upon the somewhat arbitrary magnitude limits adopted for the different catalogs, this is not really the case.
Both the eclipsing binaries from the target catalog and the background eclipsing binaries
from the intermediate catalog end up being diluted by neighboring stars drawn from all of
the catalogs.
1.6.3
Determination of optimal aperture
For each target star that is associated with an eclipse or transit (whether it is due to the
target star itself or a blended eclipsing binary), we select the pixels that provide the optimal
photometric aperture from the central 8 x 8 pixels of its synthetic image. Starting with
the three brightest pixels in the PRF, we add pixels in order decreasing brightness one at
a time. At each step, we sum the flux of the pixels from the synthetic image of target star
and from the synthetic image of the neighboring stars. We also consider the read noise
and zodiacal noise, which are discussed in Section 1.6.4. As the number of pixels in the
photometric aperture increases, more photons are collected from the target star, but more
noise is accumulated from the readout, sky background, and neighboring stars. The optimal
photometric aperture maximizes the SNR of the target star even if the eclipse is produced
by a blended binary. We assume that neighboring stars are identified a priori, but not their
eclipses.
Once the optimal aperture is determined, we calculate the dilution parameter D, which
is the factor by which the true eclipse or transit depth is reduced by blending with other
stars in the photometric aperture. Specifically, the dilution parameter is defined as the ratio
of the total flux in the aperture from the neighboring stars
(FN)
and target star
(IT)
to the
flux from the target star:
D =
EN ~+ PT
.
(1.19)
For blended binaries and hierarchical systems, the denominator is replaced with the flux
50
from the binary FB, and the target star becomes a source of dilution:
D =
+(1.20)
.
With this definition, D = 1 signifies an isolated system, and in general, D > 1. This
parameter is later reported for all detected eclipses under the "Dil." column of Table A. 1.
1.6.4
Noise Model
The photometric noise model includes the photon-counting noise from all of the stars in
the photometric aperture, photon-counting noise from zodiacal light, stellar variability, and
instrumental noise. Stellar variability and background stars are randomly assigned from
distributions, while the other noise terms are more deterministic in nature. Figure 1-14
shows the relative photometric noise as a function of apparent magnitude and also breaks
down the contributions from the deterministic sources of noise. Each subsection below
describes the noise terms in more detail.
Zodiacal Light
Although TESS avoids the telluric sky background by observing from space, it its still
affected by the zodiacal light (ZL) and its associated photon-counting noise. Our model
of the zodiacal flux is based on the spectrum measured by the Space Telescope Imaging
Spectrograph on the Hubble Space Telescope.6 We multiply this ZL spectrum by the TESS
spectral response function and integrate over wavelength. This gives the photon flux of
2.56 x 10-3 10-0.4(V-22.8) ph s-
cm- 2 arcsec -2,
(1.21)
where V is the V-band surface brightness of the ZL in mag arcsec-2. For TESS, the pixel
scale is 21."1 and the effective collecting area is 69 cm 2 . To model the spatial dependence
of V, we fit the tabulated values of V as a function of helio-ecliptic coordinates 7 with a
6
7
http://www.stsci.edu/hst/stis/performance/background/skybg.html
http://www.stsci.edu/hst/stis/documents/handbooks/currentIHB/cO6_exptime6.html#68957
51
function
V = Vma - AV b - 0 0 2
(1.22)
(900
where b is the ecliptic latitude and Vma and AV are free parameters. Because TESS will
generally be pointed in the anti-solar direction (near helio-ecliptic longitude 1 ~ 1800),
and because V depends more strongly on latitude than longitude in that region, we only
fitted to the data with 1 > 120' and weighted the points in proportion to (1
-
900)2. The
least-squares best-fit has Vm, = 23.345 mag and AV = 1.148 mag.
Based on these results, we find that the zodiacal light collected in a 2 sec image ranges
from 95-270 e- pix-
1
depending on ecliptic latitude. The photon-counting noise associated
with this signal varies from 10-16 e- pix- 1 RMS, as mentioned in Section 1.2.
Instrumental Noise
The read noise of the CCDs is assumed to be 10 e- pix-
1
RMS in each 2 sec exposure,
which is near or below the level of photon-counting noise from the ZL. Both the read noise
and ZL noise grow in proportion to the square root of the number of pixels used in the
photometric aperture.
Our noise model for TESS cameras also includes a systematic error term of 60 ppm hr1 / 2
This is an engineering requirement on the design rather than an estimate of a particular
known source of error. We assume that the systematic error is uncorrelated and scales
with the total observing time as t- 1/ 2 . (It is unlikely that the systematic error grows above
60 ppm for timescales shorter than one hour, but such timescales are not relevant to transits
and eclipses.)
It is thought that the systematic error of the TESS cameras will primarily stem from
pointing errors that couple to the photometry through non-uniformity in the pixel response.
These pointing errors come from the attitude control system, velocity aberration, thermal
effects, and mechanical flexure. In addition, long-term drifts in the camera electronics can
contribute to the systematic error.
The data reduction pipeline will use the same co-trending techniques that were used by
the Kepler mission to mitigate these effects, but the exact level of residual error that TESS
52
will be able to achieve is unknown at this time.
Cosmic Rays
Typical back-illuminated CCDs have depletion depths of 10-50 pm. In contrast, the TESS
CCDs have a 100 ttm depletion depth. This is desirable to enhance the quantum efficiency
at long wavelengths, but it also makes the detectors more susceptible to cosmic rays (CRs)
since the pixel volume is larger and the maximum amount of charge collected per event can
be larger.
To assess the effect of cosmic rays, we consider a typical cosmic ray flux of 5 events s
and minimally-ionizing events that deposit 100 e-
tm
1
1
cm-2
within silicon. Each pixel has an
optical exposure time of 2 sec. The accumulated images also spend an average of 1 sec in
the frame-store region of the CCD, where they are still vulnerable to cosmic rays. Given
these parameters, for each 2 min stack of values from one pixel, there is a 10% chance
of experiencing a cosmic ray event with an energy deposition above the combined read
and zodiacal noise of 110 e-. The distribution in the energy deposition values has a peak
near 1500 e-, which is comparable to the photon-counting noise of bright stars observed
with 2 min cadence. Electrons from cosmic rays will therefore add significantly to the
photometric noise, but will not be easily detected in the 2 min or 30 min data products.
Cosmic rays are far more conspicuous in the 2 sec images. Therefore, it is probably
best to remove the contaminated pixel values before they are combined into the 2 min and
30 min stacks. The Data Handling Unit on TESS will apply a digital filter that rejects outlier
values during the stacking process either periodically or adaptively. A possible side-effect
of this filter, depending on the algorithm used, is a reduction in the signal-to-noise ratio to
the degree that uncontaminated data is also rejected in the absence of cosmic rays.
The exact algorithm that will be used to mitigate cosmic-ray noise is still being studied.
For the present simulations we have budgeted for a 3% loss in the SNR. In the simulation
code, we simply raise the detection threshold (described in Section 1.6.6) by 3% to compensate for the reduced SNR, and we assume that there are no other residual effects from
cosmic rays.
53
10
Star noise
- ----
1
Zodiacal noise
Read noise
-
Sys. noise
10
10
a> 101
----
----------
-
30
a)
0
U:)
10
0
4
6
14
8
10
12
Apparent Magnitude [Ic]
16
Figure 1-14: Noise model for TESS photometry. Top.-Expected standard deviation of
measurements of relative flux, as a function of apparent magnitude, based on 1 hour of
data. For the brightest stars, the precision is limited by the systematic noise floor of 60
ppm. For the faintest stars, the precision is limited by noise from the zodiacal light (shown
here for an ecliptic latitude of 300). Over the range Ic ~ 8-13, the photon-counting noise
from the star is the dominant source of uncertainty. Bottom.-The number of pixels in the
optimal photometric aperture, chosen to maximize the SNR. The scatter in the simulated
noise performance and number of pixels is due to the random assignment of contaminating
stars and centroid offsets in the PRF.
54
1.6.5
Duration of observations
The SNR of transits or eclipses will depend critically on how long the star is observed.
Figure 1-1 is a sky map showing the number of times that TESS will point at a given location
as a function of ecliptic coordinates. As noted above, the simulations assign coordinates
to each star through a uniform random distribution across the HEALPix tile to which it
belongs. The star's ecliptic coordinates are then converted to x and y pixel coordinates for
each TESS pointing. We tally the number of pointings for which the target falls within the
field-of-view of a TESS camera. The total amount of observing time is calculated as the
total duration of all consecutive pointings.
The duty cycle of observations must also be considered. At each orbital perigee, TESS
interrupts observations in order to transmit data to Earth and perform other housekeeping
operations. This takes approximately 0.6 days. We model this interruption in the simulation, so each 13.6-day spacecraft orbit actually results in 13.0 days of data.
The presence of the Earth or Moon in the field-of-view of any camera will also prohibit
observations. We do not model this effect since predicting their presence depends upon the
specific launch date of TESS. However, our simulations do show that if observations are
interrupted near TESS's orbital apogee in addition to its perigee, then the planet yields are
approximately proportional to the duty cycle of observations.
1.6.6
Detection
The model for the detection process is highly simplified: we adopt a threshold for the
signal-to-noise ratio, and we declare a signal to be detected if the total SNR exceeds
the threshold. In other words, the detection probability is modeled as a step function of
the computed SNR. (The matched-filter technqiues of the TESS pipeline probably have a
smoother profile, such as a standard error function [Jenkins et al. 1996]). For transiting
planets, all of the observed transits contribute to the total SNR. For eclipsing binaries, we
allow both the primary and secondary eclipses to contribute to the total SNR.
The choice of an appropriate SNR threshold was discussed in detail by Jenkins et al.
(2002) in the context of the Kepler mission. Their criterion was that the threshold should be
55
sufficiently high to prevent more than one "detection" from being a purely statistical fluke
after analyzing all of the data from the entire mission. We adopt the same criterion here.
Since the number of astrophysical false positives is at least several hundred (as discussed
below), this criterion allows statistical false positives to be essentially ignored.
To determine the appropriate threshold, we use a separate Monte Carlo simulation of
the transit search. We produce 2 x 10' lightcurves containing uncorrelated, Gaussian noise
and analyze them for transits in a similar manner as will be done with real data. Then,
we find the SNR threshold that results in approximately one statistical false positive. Each
lightcurve consists of 38,880 points, representing two 27.4-day TESS pointings with 2minute sampling. We chose a timeseries length of two pointings rather than one to account
for the stars observed with overlapping pointings.
To search for transits, we scan through a grid of trial periods, times of transit, and transit
durations. At each grid point, we identify the data points belonging to the candidate transit
intervals. The SNR is computed as the mean of the in-transit data values divided by the
uncertainty in the mean.
The grid of transit durations t starts with 28 min (14 samples) and each successive grid
point is longer by 4 min (2 samples). The grid of periods P is the range of periods that are
compatible with the transit duration. The periods are calculated by inverting Eqn. (1.10):
P
=
(365 days)
(
)-
78 min
pD
(1 - b
We allow P to vary over a sufficient range to include plausible stellar densities p/p
(1.23)
from
0.5 to 100. The fractional step size in the period AP/P is then 3AT/T, which has a
minimum value of 0.43 for the shortest periods. We consider orbital periods ranging from
1.7 hr (which is below the period corresponding to Roche limit) to 27.4 days (half of the
nominal observing interval). The transit phase is stepped from zero to the orbital period in
increments of one-half the transit duration.
Figure 1-15 shows how the number of false-positive detections scales with the detection
threshold. We find that a SNR of 7.1 produces approximately one statistical false positive
within the library of 2 x 10 5 light curves. By coincidence, this is equal to the SNR threshold
56
.. 106
U2
4-
10
0
------------------
o10
--------
Cd
5.5
-
50
7.5
7
6
6.5
Detection Threshold [a]
-
-
8
-
5
40
30
20
E-30
10
5
5.5
6
6.5
7
7.5
Significance of detections [o]
8
Figure 1-15: Determination of the SNR threshold. Top.-The statistical false-positive rate
for the TESS mission as a function of the detection threshold. We do not want more than
one statistical false positive to occur (red dashed line), which dictates a threshold of 7.1.
Bottom.-The SNR distribution of transits near the threshold from the full TESS simulation
(presented in Section 1.7.1). The small slope of this distribution near 7.1 suggests that the
planet yield is not extremely sensitive to the detection process or threshold.
57
of 7.1 that was calculated for Kepler mission by Jenkins et al. (2002). TESS searches twice
as many stars as the 10 5 considered in the Kepler study, and over a larger dynamic range in
period; Kepler searches for planets with longer periods using longer intervals of data.
To account for the expected reduction in SNR due to the cosmic-ray rejection algorithm
(see Section 1.6.4), we adopt a slightly higher threshold of 7.3 in this work. In addition,
we only consider a transit or eclipse to be detected if two or more events are observed.
We also record the single events that exceed the SNR threshold, but we do not count them
as "detections" in the tallies and the discussion that follows. The planets detected with a
single transit generally have longer periods than the multiple-transit detections. However,
they will require additional ground-based follow-up observations to determine the orbital
period and discriminate against astrophysical or statistical false positives.
1.6.7
Selection of target stars
From the 2.11 x 10 7 stars in the K, < 15 catalog, we must select the 2 x 105 target stars for
which pixel data will be saved and transmitted with 2 min time sampling. In our simulation,
the target stars are not chosen from a magnitude cut; rather, they are individually chosen
according to the prospects for detecting the transits of small planets.
In the simulation, we have complete knowledge of the properties of each star, which
makes it straightforward to determine whether a fiducial transiting planet with a given radius and period could be detected with TESS. We adopt an orbital period of 10 days; for
each 27.4-day pointing that TESS spends observing a star, we assume that 2 transits are observed. The stellar radius and mass are used to calculate the transit duration with a 10-day
period, thereby determining the total exposure time during transits. Then, we use the simplified noise model from Section 1.2.3 that considers the read noise and photon-counting
noise of the star and zodiacal light. We then check to see if the fiducial transiting planet
would be detectable with a signal-to-noise ratio exceeding of 7.3.
The number of stars meeting this detection criterion depends strongly on the radius of
the fiducial planet. Starting from small values, we increase the radius until the number
of stars for which the planet would be detectable is 2 x 10 5 . This is achieved for R, =
58
-6
-4
41P J
n~*
V111,
-2
6
8
10
All K, < 6 Stars
" TESS Target Stars
"
30000
6000
12000
Effective Temperature [K]
3000
5
Figure 1-16: Selection of the 2 x 10 target stars on the H-R diagram. For clarity, only
the simulated stars with apparent K, < 6 are shown, and a random selection of 1% of the
target stars are shown. Nearly all main-sequence dwarfs smaller than the Sun are selected
as target stars; only a small fraction of larger stars are selected.
2.3 RD. Through this procedure, the target star catalog is approximately complete for
planets smaller than 2.3 Re 10-day orbits. There is a higher density of target stars assigned
near the ecliptic poles due to the longer duration of TESS observations in those regions.
In selecting the target stars, we do not assume prior knowledge of whether a star is part
of a multiple-star system. If it is, we assume that all components of the system fall within
a single photometric aperture, and they are all observed at the 2 min cadence.
Figure 1-16 illustrates the selection of the target stars in a Hertzsprung-Russell diagram.
For clarity, we show a magnitude-limited subsample (K, < 6) of our "bright" catalog as
5
well as a randomly-selected subsample of the 2 x 10 target stars. Nearly all main-sequence
stars with Tff < 6000 K are selected as target stars. Stars that are larger than sun are only
included if they have a sufficiently bright apparent magnitude. White dwarfs could also be
interesting targets for TESS, but we do not include objects with Tff >15000K in our target
catalog since the occurrence rates of planets around white dwarfs is unknown.
Figure 1-17 shows the target stars plotted against observable quantities in the apparent
59
5-
4-D
1.5x
cd
1
bD
3000
7000
5000
Effective Temperature [K]
Figure 1-17: The distribution of the TESS target stars on the Ic - Tff plane (top) and in
Teff only (bottom). For clarity, only 10% of the target stars are plotted in the top panel.
IC - Teff plane. The distribution of the effective temperatures of the target stars is bi-modal,
with a sharp peak near 3400 K and a broader peak near 5500 K.
In reality, it will not be quite so straightforward to select the target stars for TESS. While
proper-motion surveys (e.g., Lepine & Shara 2005) can readily distinguish red giants from
dwarf stars, it is much more difficult to distinguish dwarfs from subgiant stars (Stassun
et al., 2014). Ultimately, the selection of the TESS target stars may rely on parallaxes from
the ongoing Gaiamission (Perryman et al., 2001). Errors in selecting the target stars might
be mitigated by simply observing a larger number of stars at 2 min cadence. There is also
the possibility of detecting transits in the full-frame images, which is described below.
1.6.8
Full-frame images
TESS will record and downlink a continuous sequence of full-frame images (FFIs) with an
effective integration time of 30 min or less. Transiting planets can still be detected with
60
30 min sampling, but the longer time window of the FFIs reduces the sensitivity to events
with a short duration. Our simulation estimates the yield of transiting planets from the FFIs
in the following fashion.
First, we identify all the transiting or eclipsing stars that are not among the pre-selected
2 x 105 target stars. We assign to each system a random phase between the beginning of a
30 min window and the beginning of an eclipse. Next, we calculate the number of 30 min
data points that are required to cover the transit or eclipse duration. The data points at the
beginning and end of the series are omitted if they do not increase the signal-to-noise. This
changes the apparent duration of the eclipse. Finally, we compute the effective depth of the
transit or eclipse by averaging over all of the 30 min data points spanning the event. This
step can reduce the depth because some of the data points include time spent out of the
transit or eclipse.
For transits with durations shorter than 1 hour, the finite window of the FFIs causes the
apparent transit duration to be lengthened and the apparent transit depth to become more
shallow. However, the depths and durations of transits with longer durations are largely
unaffected. The effects of a finite window size on the uncertainties of transit parameters
derived from lightcurve fitting are analyzed in Price & Rogers (2014).
Our calculated detection threshold of 7.3 only ensures that no more than one statistical
false positive is detected among the 2 x 105 target stars. Since many more stars can be
searched for transits in the FFIs, the number of statistical false-positives will be much
greater than one if the same threshold is adopted.
1.7
Survey Yield
Having calculated the SNR for each eclipsing or transiting system, we determine that a
system is "detected" if the SNR > 7.3 in the phase-folded lightcurve and at least 2 transits
or eclipse events are observed. We thereby produce a simulated catalog of detected planets
and false positives.
Figure 1-18 shows the tallies for each class of planet and false positive. We show the
yield both from the 2 x 10 5 target stars and from the full-frame images. Five trials are used
61
10
I
5
37k
2x105 Target Stars
10
191k
193k
Full-Frame Images
17k
10
-
6 III
1910
Q3
10
300
10
101
Earths Super-Earths Sub-Neptunes
< 1.25Rjh
1.25 - 2Rs,
2 - 4RT
Giants
EBs
BEBs
HEBs
> 4R,
Figure 1-18: Mean numbers of planets and eclipsing binaries that are detected in the TESS
5
simulation. Results are shown for the 2 x 10 target stars that are observed with 2 min time
sampling as well as stars in the full-frame images that are observed with 30 min sampling.
to calculate the expected yields among the target stars; only one trial is used to calculate
the yields of the full-frame images. The statistical uncertainties in each category are simply
the Poisson error. However, the systematic uncertainties, which propagate from the planet
occurrence rates and various aspects of our simulated stellar population, are almost surely
larger than the statistical uncertainties.
Figure 1-19 is a sky map in ecliptic coordinates of the simulated detections from one
trial.
1.7.1
Transiting Planets
From five trials with the 2 x 105 target stars, we expect TESS to find 45 planets smaller
than 1.25 RD, 285 planets in the range 1.25-2 RT, 692 planets in the range 2-4 Re, and
135 planets larger than 4 Re. Table A. 1 presents the catalog of planets from one of these
five trials.
Figure 1-20 shows the distribution of detected planets plotted on the radius-period plane
in the same fashion that the input planet occurrence rates were plotted in Figure 1-8.
62
TESS Planet Detections
*0
*0
*
*
0
- *- *i~9~
~ *~*
~
**
-~
~
~)i,
d'
W~-~'
~
" Full-Frame Images
-
2x105 Target Stars
'o
0'
0-*
TESS Eclipsing Binary Detections
Figure 1-19: Sky maps of the simulated TESS detections in equal-area projections of ecliptic coordinates. The lines of latitude are spaced by 30', and the lines of longitude are spaced
by 60'. Top.-Planetdetections. Red points represent planets detected around target stars
(2 min cadence). Blue points represent planets detected around stars that are only observed
in the full-frame images (30 min cadence). Note the enhancement in the planet yield near
the ecliptic poles, which TESS observes for the longest duration. Note also that the inner 60
of the ecliptic is not observed. Bottom.-Astrophysical false positive detections, using the
same color scheme. For clarity, only 10% of the false positives detected in the full-frame
images are shown. (All other categories show 100% of the detections from one trial.) Note
the enhancement in the detection rate near the galactic plane, which is stronger for false
positives than for planets.
63
22-
22
--
-
4-
6
C
-e
4
2
2
1.25
1.25
0.8
0.5
"
0.8 -
0.5
0.8
2
X 10-3
50 145
[days]
Period
418
50 145
17
5.9
Period [days]
418
5.9
17
0
0.005 0.01
dN/dlog(R)
S4094
0.8
2
Figure 1-20: The distribution of detected planets in log(R,) - log(P) space. The shading
of the 2-d histogram is the same as in Figure 1-8. The sawtooth patterns in the radius and
period histograms are an artifact of the planet occurrence rates having coarse bin sizes in
radius and period, combined with the sensitivity of TESS favoring planets with larger radii
and shorter periods.
The top panel of Figure 1-19 maps the simulated planet detections in ecliptic coordinates. The target catalog favors stars near the ecliptic poles that receive a larger number of
pointings, so detections among the target stars (red points) are enhanced in these regions.
Otherwise, the target catalog prioritizes nearby stars, so the detections are nearly uniformly
distributed across the sky. The detections from stars that are only observed in the full-frame
images (blue dots) also show a strong enhancement near the galactic plane due to the vast
number of faint and distant stars around which giant planets can be detected.
Of the 331 planets smaller than 2 RE, a subset of 12 have a relative insolation S/Se
between 0.5 and 2, placing them within or near the habitable zone (HZ). Figure 1-21 shows
the distribution of S/Se for the simulated detections in the vicinity of the HZ as a function
of stellar effective temperature. Not surprisingly, nearly all of the HZ planets are found
around low-mass, cool stars (Tff < 4000 K). A subset of 4 HZ planets are found within
15' of the ecliptic poles, the optimal locations for observation with the James Webb Space
Telescope.
64
Planet radius:
e
5000-
4R(D
*2R,
*1R(D
0
*Single Thansit
4000
-
I
3000*
5
3
2
0.5
1
Insolation [S/S0 ]
0.2
0.1
Figure 1-21: Small planets in and near the habitable zone (HZ). The inner and outer edges
for a simple model of the HZ are shown with vertical dashed lines, where the insolation S
falls between 0 .5 5 e and 2 Se. More realistic models of the HZ are functions of both S and
Teff. Planets detected with a single transit, shown in gray, have longer periods and are more
likely to fall in the HZ.
The smallest planets will be of particular interest for mass measurement since there are
presently very few small (and potentially rocky) planets with measured masses and sizes.
Among the 45 simulated planets smaller than 1.25 Re, the median period is 2.4 days, and
the median stellar effective temperature is 3500 K. The median Ic magnitude is 11.1.
The degree of completeness of the TESS survey can be assessed by comparing the simulated planet detections against the total number of transiting planets on the sky (as discussed
in Section 1.4.1). Plotted in Figure 1-22 are the cumulative numbers of transiting planets
as a function of the limiting apparent magnitude of the host star. We make the comparison
for short-period planets around Sun-like and smaller stars for planets of different sizes as
well as small planets near the HZ.
In some cases, the SNR of a transit exceeds the threshold of 7.3, but only a single transit
is observed. The number of such single-transit detections in the simulations is 69. These are
not counted as detections in the tallies given above, but they are included in Figure 1-21 as
gray points. Although the periods of these planets will not be well-constrained using TESS
65
10
104
103
103
Ci2
R <2
2< R, <4
10
102
TESS
101
101
100
100
10
PCI
4
103
M
102
r, 102
0.5< S <2
101
101
100
100
5
9
7
11
5
9
7
11
IC
IC
plot the
Figure 1-22: Completeness of the TESS survey. For each category of planet, we
magnitude
apparent
cumulative number of transiting planets as a function of the limiting
K and
of the host star. Only planets with P < 20 days and host stars with Tff < 6500
transiting
all
for
distributions
the
show
R, < 1.3R® are considered. The colored lines
planets in the simulation; the black lines are for the simulated TESS detections.
66
data alone, and the probability of the "detection" being a statistical fluke is higher, it may
be worthwhile to conduct follow-up observations of these stars as a means of identifying
longer-period planets. The simulated single-transit detections have a median planet size of
~3 RE and a median orbital period of ~30 days.
1.7.2
False positives
Among the 2 x 10' target stars, TESS detects 1016 eclipsing binary systems along with the
transiting planets. These can be divided into the following cases:
1. Eclipsing Binary (EB): The target star is an eclipsing binary with grazing eclipses.
There are 258 detections of EBs.
2. Hierarchical Eclipsing Binary (HEB): The target star is a triple or quadruple system
in which one pair of stars is eclipsing. There are 432 detections of HEBs.
3. Background Eclipsing Binary (BEB): The target star is blended with a background
eclipsing binary. There are 327 detections of BEBs.
These tallies are also illustrated in Figure 1-18. The bottom panel of Figure 1-19 shows a
sky map of the astrophysical false positives in the same coordinate system as the top panel.
The surface density of false positives is a much stronger function of galactic coordinates
than the density of planet detections, for binary eclipses are deeper than planetary transits
and can be detected out to greater distances. The period and depth distributions of the
eclipsing binary population is discussed in Section 1.8.6.
1.8
Distinguishing False Positives from Planets
Experience has shown that the success of a transit survey depends crucially on the ability
to distinguish transiting planets from astrophysical false positives. Our simulations suggest
that for TESS, the number of astrophysical false positives will be comparable to the number
of transiting planet detections. In many cases, it will be necessary (or at least desirable) to
undertake ground-based follow-up observations to provide a definitive classification.
67
However, there will also be useful clues within the TESS data, even before any followup observations are undertaken, that a candidate is actually an eclipsing binary: (1) ellipsoidal variations, (2) secondary eclipses, (3) lengthy ingress and egress durations, or
(4) centroid motion associated with the eclipse events. In this section, we investigate the
prospects for using these four characteristics to identify false positives with TESS data
alone. Specifically, we determine the number of cases, summarized in Table 1.5, for which
any of these characteristics can be measured with an SNR of 5 or greater. This statistic
indicates that the information will be available to help make the distinction between planet
and false positive. The next step would be to combine all the measurable characteristics in a
self-consistent manner and attempt to arrive at a definitive classification. This is a complex
process which we have not attempted to model here.
1.8.1
Ellipsoidal Variations
The members of a close binary exert strong tidal gravitational forces on one another, causing their photospheres to deform into ellipsoids. These deformations lead to ellipsoidal
variations in the light curve. A model for these photometric variations was presented by
Morris & Naftilan (1993). Mazeh (2008) gave a simple expression for the dominant component, which has a period equal to half of the orbital period, and a semi-amplitude
AF 1
= 0.15
(15+ u)(1+T)
(3 U-)(1
q
1(3-I01)
R-- 1
ka
2
sini,
(1.24)
where R1 is the primary radius, a is the orbital distance, i is the orbital inclination, ui is
the linear limb-darkening coefficient,
T
is the gravity-darkening coefficient, and q is the
mass ratio. To estimate the amplitude of this effect for our simulated TESS detections, we
adopt an appropriate value of ul for each star using the tables of Claret et al. (2012) and
Claret et al. (2013), which come from the PHOENIX stellar models. For gravity darkening,
we use a value of T=
0.32 for all stars, which is thought to be appropriate for stars with
convective envelopes (Lucy, 1967).
The formal detection limits for ellipsoidal variations are quite low because the signal is
present throughout the entire light curve rather than being confined to eclipses of a narrower
68
-
-
10 5
10
.0
Planets"
Ce
4
-D
r0 164 aHE
*--0
-2
10
10
2
10
Period [days]
Figure 1-23: Ellipsoidal variations of the primary star among the simulated TESS detections. We consider the variations to be detectable if the semi-amplitude is greater than
10 ppm and the SNR exceeds 5 (horizontal dashed line). We also require the orbital period (twice the period of the ellipsoidal variations) to be shorter than 13.7 days (the TESS
spacecraft's orbital period; vertical dashed line). Short-period eclipsing binary systems
give larger ellipsoidal variations; a significant number of EBs and HEBs can be identified on this basis. Only a small number of BEBs would give rise to detectable ellipsoidal
variations. No planets produce detectable ellipsoidal variations.
69
duration. Since the period and phase are fixed from the observed eclipses, we model the
detection of the ellipsoidal variations as a cross-correlation of the lightcurve with a cosine
function of the appropriate period. If the fractional uncertainty in flux of each data point is
a-, and the total number of data points is N, then the SNR of ellipsoidal variations SNREV
is
SNREV =
E1
Do-d
(1.25)
Here, D denotes the dilution of the target star in the photometric aperture, which is defined
in Section 1.6.3. Due to this factor, ellipsoidal variations from BEBs are more difficult to
detect since their eclipses are usually more diluted than EBs and HEBs. The factor of N/2
arises from the RMS value of a cosine function.
It seems likely that correlated noise will prevent the detection limit from averaging
down to extremely low values as the duration of observations is extended. Somewhat arbitrarily, we require the semi-amplitude of the ellipsoidal variations to exceed 10 ppm, in
addition to the criterion SNREV > 5, to be counted as "detectable". We also require that
the orbital period of the binary, which is twice the period of ellipsoidal variations, is shorter
than one spacecraft orbit (13.6 days) out of concern that thermal or other variations of the
satellite will induce systematic errors with a similar frequency. Under these detection constraints, shown in Figure 1-23, ellipsoidal variations are detected for 36% of the eclipsing
binaries in the simulation. The majority of these are grazing-eclipse binaries rather than
HEBs or BEBs. The results are summarized in the second column of Table 1.5.
1.8.2
Secondary Eclipse Detection
Another key difference between eclipsing binaries and transiting planets is that the secondary star in a binary is more luminous than a planetary companion. This distinction is
somewhat blurred when comparing brown-dwarf and hot-Jupiter companions but is quite
clear between ordinary stars and lower-mass planets. If the two stars in a binary have nearly
the same surface brightness, then the depths of the primary and secondary eclipses will be
indistinguishable. In this case, the system might appear to be a planet with an orbital period
equal to half of the true orbital period of the binary. However, if the surface brightnesses
70
of the stars differ and both eclipses are detected with a sufficiently high SNR, then the
secondary eclipse can be distinguished from the primary eclipse and the system can be
confidently classified as an eclipsing binary.
To estimate the number of cases for which the primary and secondary eclipses are distinguishable, we identify the simulated systems for which signal-to-noise of the secondary
eclipses, SNR 2 , is> 5, and the SNR in the difference between the primary and secondary
eclipse depths, SNR 1- 2 , is also > 5. The latter quantity is calculated as
SNR 1- 2
=
J1 -2
where J1,2 denote the depths of the eclipses and
a1, 2
,1.26)
denote the noise in the relative flux
over the observed duration of each eclipse. Figure 1-24 shows the detectability of secondary
eclipses by plotting SNR-
2
versus SNR 2 . The secondary eclipse can be distinguished from
the primary eclipse for the systems that lie in the upper-right quadrant of the plot.
The results are also summarized in the third column of Table 1.5. A majority of the
false positives have detectable secondary eclipses that are distinguishable in depth from
the primary eclipses. The notable exceptions include the HEBs in which the eclipsing
pair consists of equal-mass stars (q ~ 1). In such cases, 6
~ 62 and it is impossible to
distinguish between primary and secondary eclipses. For the BEBs, the difficulty is that
the eclipse depths are often strongly diluted and the secondary eclipses are not detectable.
Most planets are too small and faint to produce detectable secondary eclipses in the TESS
bandpass. In the simulations, the fraction of detected planets with detectable secondary
eclipses is only 0.1%.
1.8.3
Ingress and Egress Detection
Eclipsing binaries can also be distinguished from transiting planets based on the more prolonged ingress and egress phases of stellar eclipses. As above, we adopt an SNR threshold
of 5 for the ingress/egress phases to be detectable. The average "signal" during ingress and
egress is half the maximum eclipse depth, and the "noise" is calculated for the combined
durations of ingress and egress. In order to ensure that the ingress/egress can be temporally
71
Planets
EBs
HEBs
BEBs
c
*
104
e
9
0
1i0
.0
0
2
0
0010 0
1CID0
0
0
0
o1
0
00
0
00
010
0
00o
-44
0
0
-200
10
1
10
10
10
SNR of Secondary Eclipse
Figure 1-24: Distinguishing secondary eclipses from primary eclipses based on TESS photometry. The vertical dashed red line shows where the secondary eclipses can be detected
at SNR 2 > 5. The horizontal dashed red line shows where the difference in eclipse depths
can be measured with SNR 1- 2 > 5. The filled points in the upper-right quadrant meet both
conditions, so the secondary eclipse can be distinguished from the primary eclipse. For
64% of the eclipsing binaries that TESS detects in the simulation it is possible to classify
them as false positives based on the TESS data alone.
72
resolved, we require the duration of the ingress or egress to be more than twice as long as
the duration of an individual data sample (2 min for the target stars and 30 min for the rest
of the stars).
Since transiting planets generally have ingrees or egress phases lasting a few minutes,
TESS will only be able to detect the ingress/egress for a small fraction (~ 10%) of transiting
planets observed with 2 min sampling. Only large planets observed in the 30 min. FFIs
would have resolvable ingress/egress. However, the the ingress/egress phases of eclipsing
binaries are more readily detectable.
We note that detection of the ingress/egress alone does not classify a signal as an eclipsing binary. One would next examine the period and shape of the eclipse signals to determine
whether the radius of the eclipsing body is consistent with the observed depth.
Figure 1-25 illustrates the detection of ingress/egress for planets and false positives.
The fourth column of Table 1.5 summarizes the results. Approximately 70% of the eclipsing binary systems that TESS detects among the target stars might be classified as false
positives by virtue of a lengthy ingress or egress duration. For stars that are only observed
at a 30 min cadence, this method is not as effective.
1.8.4
Centroid Motion
Another diagnostic of false positives, particularly background eclipsing binaries, is the
centroid motion that accompanies the photometric variations. If there are detectable shifts
in the centroid of the target star during transit or eclipse events, it is more likely that the
target is a blended eclipsing binary rather than a transiting planet or an eclipse of the target
star itself. Transits or eclipses of the target star can still have significant centroid motion if
another bright star is blended with the target.
With real data, one could interpret the amplitude and direction of the measured centroid
shift using the known locations of neighboring stars in order to determine the most likely
source of the photometric variations. This is a complicated process to simulate, so we simply investigate the issue of the detecting the centroid shift. As verified in our simulations,
the systems with detectable centroid shifts are much more likely to be false positives than
73
10 3
*
*~. ~
a
I
10
0
10
r
+
0
0
10 2
S
EBs
S
HEBs
BEBs
1
.
1 -3
,
I,, ,
100
10
Ingress/Egress Duration
10
[mini
Figure 1-25: Detectability of the ingress and egress phases of eclipses observed with TESS.
We require the time-averaged ingress/egress depth (half of the full depth) must be detectable with SNR > 5 from data obtained during ingress/egress (horizontal dashed line).
Also, we require the ingress/egress duration to be longer than the 2 min averaging time of
sample (vertical dashed line). Filled circles represent systems for which the ingress/egress
are detectable according to these criteria.
74
transiting planets.
We simulate the detectability of centroid shifts by calculating the two-dimensional centroid (center-of-light) of the target star, C, and C,, within the 8 x 8 synthetic images
described in 1.6.2. We calculate the centroids both during and outside of the loss of light
to find the magnitude and direction of the centroid shift. Next, we calculate the uncertainty
in the centroid acX and -cy, which stems from the photometric noise of each pixel. If
each pixel (i, 1) has coordinates (x, y), and its photometric noise relative to the total flux is
denoted by -ij,then the noise propagates to the centroid measurement uncertainty through
(i
=
-
C
a)2og
and
o
(y - CY) 2 ori.
=
(1.27)
i
i
In an analogous fashion to determining the optimal photometric aperture, we select the
pixels that maximize the signal-to-noise ratio of the centroid measurement. Finally, we
project the x and y centroid uncertainties in the direction of the centroid shift. The signalto-noise ratio of the centroid measurement is the magnitude of the centroid shift divided
by the centroid uncertainty projected in the direction of the centroid shift. We consider a
centroid shift to be detectable if the signal-to-noise is 5 or greater.
In practice, the centroid measurement uncertainty could be much larger if the spacecraft
jitter does not average down during the hour-long timescales of transits and eclipses. On
the other hand, monotonic drifts in the spacecraft pointing during a transit or eclipse are
less likely to impact the centroid measurement since the motion is common to all stars.
We find that centroid shifts can be detected for 30% of the BEBs and HEBs. These
results are illustrated in Figure 1-26 and summarized in column 5 of Table 1.5. The BEBs
have a higher fraction of detectable centroid shifts from the larger angular separations between the eclipsing system and the target star. Only 0.03% of planet transits produce a
detectable centroid shift.
1.8.5
Imaging
As shown in Table 1.5, the simulations suggest that blended eclipsing binaries are the type
of false positive that is most difficult to identify based only on TESS data. Assuming that
75
0%
10
E000
0
0
0
BE0
0
cP
0
09?
10-
ic -5
10
10
0
000
0
0
C
96
000
_-D
10
C0-2
0
0
HE~s0
0
10 2
0
03-
10-
Apparent Eclipse Depth
Figure 1-26: Measurement of the shift in the target star centroid during eclipses for various
types of detections. Eclipses from background binaries give the largest centroid shifts for
a given depth. If the TESS data permits a measurement of the centroid shift with SNR > 5,
we consider the shift to be detectable and plot it with a filled circle.
Table 1.5: Methods of distinguishing false positives from transiting planets.
EB
BEB
HEB
All FP
N
Ellip.
Sec. Ecl.
In/Egress
Centroid
258
327
432
1016
81.4
1.0
44.2
39.7
79.5
33.8
76.1
63.3
91.5
11.1
76.0
59.1
28.6
69.7
71.2
59.9
Any
98.6
75.0
93.0
88.6
3.8
12.6
0.9
55.5
Planets
< 4 Re
> 4 Re
1020
140
0.0
0.0
0.0
0.6
0.9
55.2
N: Mean number of each type of system that is detected.
The central four columns indicate the percentage of systems each with detectable
ellipsoidal variations, secondary eclipses, ingress and egress, and centroid motion.
"Any" indicates the percentage of systems for which one or more of these four
characteristics are detectable.
76
all of the false-positive tests described in the previous sections are applied, approximately
190 of the 1016 false positives would fail to be identified. The large majority (83%) of
these more stubborn cases are BEBs.
If archival images or catalogs do not reveal a system in the vicinity of a TESS target
star that is consistent with any measurable centroid motion, then additional imaging is
needed. An effective way to identify these BEBs is through ground-based imaging with
higher angular resolution than the TESS cameras. A series of images spanning an eclipse
could reveal which star (if any) is the true source of variations. Due to the large pixel
scale of the TESS optics, it will not be difficult to improve upon the angular resolution with
ground-based observations.
Even modest contrast and a well-sampled PSF can resolve
many ambiguous cases.
Figure 1-27 illustrates the requirements on angular resolution and contrast. For each
BEB, we have plotted the angular separation and the J-band magnitude difference between
the BEB and the target star. Natural-seeing images with 1" resolution would be sufficient
to resolve all of the simulated BEBs. In more difficult cases, adaptive optics might be
necessary to enable high contrast.
Figure 1-28 shows the photometric requirements to detect the planets as well as BEBs
and other eclipsing systems for which the TESS photometry cannot distinguish whether
the candidate is a false positive. We plot the eclipse depth against apparent system magnitude to indicate the photometric precision that is required of the facilities performing these
observations.
1.8.6
Statistical Discrimination
The false positives and transiting planets have significantly different distributions of orbital period, eclipse/transit depth, and galactic latitude. Therefore, the likelihood that a
given source is a false positive can be estimated from the statistics of these distributions
in addition to the characteristics described above that can be observed on a case-by-case
basis.
Figure 1-29 shows the distributions of apparent period and apparent depth of the eclipses
77
10
.
8-
0
Cn
0
10~
100
Angular Separation
1
[arcsecj
Figure 1-27: Magnitude differences and angular separations between BEBs and the target
star with which they are blended. Gray dots show the BEBs for which the TESS photometric
data already provides some evidence that the source is a false positive through ellipsoidal
variations, secondary eclipses, ingress/egress or centroid motion. Black dots are those for
which none of those effects are detectable; ground-based images spanning an eclipse might
be the most useful discriminant in such cases.
78
10
10 -
-
..
Planets
EBs
0.
-10
0
4
5
*
.
.
15
10
Resolved System Ic
0
HEBs
BEBs
20
Figure 1-28: Follow-up photometry of the TESS candidates, which are a mixture of planets
and astrophysical false positives. We only show the false positives that cannot be ruled out
from the TESS photometry, which are primarily BEBs. In order to show the photometric
precision that is required to detect a transit or eclipse, we plot the depth against apparent
magnitude. We assume that the BEBs are resolved from the target star (see Figure 1-27),
so the full eclipse depth and apparent magnitude of the binary are observable. A shot
noise-limited observation designed to detect most of the planets (dashed line) is sufficient
to detect the eclipsing binaries as well.
79
caused by transiting planets and false positives. Here, the "apparent period" is the period
one would be likely to infer from the TESS photometry; if the secondary eclipse is detectable but not distinguishable from the primary eclipse, one would conclude that the period is half of the true orbital period. The "apparent depth" takes into account the dilution
of an eclipse from background stars or, in the case of BEBs, the dilution from the target
star.
These populations are seen to be quite distinct. Eclipsing binary systems tend to have
larger depths and shorter periods than planets. Simply by omitting sources which have
eclipse/transit depths >5% or periods <0.5 days, approximately 79% of the false positives
among the target stars would be discarded.
The galactic latitude b of the target also has a strong influence on the likelihood that
a given source is a false positive. Figure 1-30 shows the fraction of detections that are
due to planets, BEBs, and other false positives as a function of galactic latitude. Only the
events with apparent depth <10% are included in this plot. For
Ibi
< 100, the density of
background stars is very high, and any observed eclipse is far more likely to be from a
BEB than any other kind of eclipse. For IbI > 20 , planets represent a majority over false
positives. A weaker dependence on galactic latitude is seen for grazing-eclipse binaries
and hierarchical eclipsing binaries.
1.9
Prospects for Follow-Up Observations
We now turn to the prospects for follow-up observations to characterize the TESS transiting
planets. As already discussed in Section 1.8.5, it is desirable to obtain transit lightcurves
of the planetary candidates with a higher signal-to-noise than the TESS discovery. The
photometry could be carried out with ground-based facilites or with upcoming space-based
facilities such as CHEOPS (Fortier et al., 2014). This data can be used to look for transit
timing variations and to improve our estimates of relative plantary radii.
Constraining the absolute planetary radii of the TESS planets will benefit from additional determinations of the radii of their host stars. Although beyond the scope of this
paper, interferometric observations will be possible for the brightest and nearest host stars.
80
e
detecti
Planet
P1lnu
detection raite
0.1
False-positive rate
as-eiiert
0.1
1
0.8
0.01
0.01-
0.6
0.4
0.001
0.001
-C4
0.2
0.1
10
1.0
10
1.0
0.1
100
100
Apparent Period [days]
Apparent Period [days]
Figure 1-29: The likelihood, shown in grayscale, that an eclipse observed with TESS is
a false positive or transiting planet based on its apparent period and depth. Left.-The
fraction of detections from five trials that are transiting planets; the planets from one trial
are plotted as red dots. Right.-The fraction of all eclipses that are due to false positives;
the red dots are individual false-positives.
Planets
BEBs
0.8-
-Other FPs
Ca
4 0.6-
0
~0.2
0
60
40
20
Abs. Galactic Latitude [deg]
80
Figure 1-30: The likelihood that an eclipse observed with TESS is a false positive or transiting planet based on its galactic latitude. Planets tend to be detected at higher galactic
latitude while background eclipsing binaries (BEBs) dominate detections at low galactic
latitude. Here, we consider all eclipses with an apparent depth <10%.
81
For this reason, we report the stellar radii and distance moduli (in the "DM" column) of
Table A. 1 so that angular diameters can be calculated.
Asteroseismology can also be used to determine the radii of host stars if finely-sampled,
high-precision photometry is available. Such data could come from the upcoming Plato
mission (Rauer et al., 2014) or from the TESS data itself. There is discussion of having
TESS record the pixel values of the most promising targets for asteroseismology with a
finer sampling than the 2 min. cadence.
Next, we turn to the follow-up observations that TESS is designed to enable: radialvelocity observations to measure a planet's mass and spectroscopic observations to detect
and characterize a planet's atmosphere.
1.9.1
Radial Velocity
The TESS planets should be attractive targets for radial-velocity observations because the
host stars will be relatively bright and their orbital periods will be relatively short. Both
of these factors facilitate precise Doppler spectroscopy. To evaluate the detectability of the
Doppler signal we assign masses to the simulated planets using the empirical mass-radius
relation provided by Weiss et al. (2013). For Rp < 1.5 Re, the planet mass AMp is calculated
as
M, = M
+ 0.614
[ 0.440
1
R,,
,
RD
(1.28)
and for Rp > 1.5RE, the mass is calculated as
0.93
Alp = 2.69M1
()
.
(1.29)
The masses of small planets are more realistically described as a distribution parametrized
with the planet radius (e.g., Rogers 2014) and not the simple one-to-one relationship used
here for convenience. For this reason, we publish the planet radii in Table A. 1 to allow other
masses to be calculated as our understanding of the mass-radus relationship improves.
From the masses calculated here, we then find the radial-velocity semiamplitude K,
which is reported in Table A. 1. Figure 1-31 shows K values of each planet detected in one
82
trial as a function of the apparent magnitude of the host star. Because of the short periods,
even planets smaller than 2 R@ will produce a radial-velocity semiamplitude K close to
1 m s-', putting them within reach of current and upcoming spectrographs.
P > 7 days
P < 7 days
M7
4. 1
7000
x GJ 1214b
~
*0
00
/
e'
*
Kep-48c 0 A
0
SKep-20b
.
/CoRoT-7b
HI 67658b
-d
.
x0
.
-
/
10[
1
.0o
6000
5000
0
X
0
-
1.0
4000
0
6*
0
Kep-l0b/
3000
6
8
10
IC
12
14
16
6
8
10
IC
12
14
16
Figure 1-31: Mass measurement of the TESS planets. The radial velocity semi-amplitude
is
K plotted against apparent magnitude for the TESS planets with RP < 3Re. The sample
split at the median period of 7 days, and open symbols indicate planets near the habitable
2
al.
zone with an insolation S < Se. We assume the mass-radius relation from Weiss et
(2013). Several well-known exoplanets are also shown with open circles for context: HD
97658b (Dragomir et al., 2013), CoRoT-7b (Hatzes et al., 2011), GJ 1214b (Charbonneau
et al., 2009), Kepler-20b and Kepler-48c (Marcy et al., 2014), and Kepler-10b (Dumusque
et al., 2014), which is plotted in blue for clarity.
1.9.2
Atmospheric Characterization
The composition of planetary atmospheres can be probed with transit spectroscopy. Such
measurements can be carried out with space-based or balloon-based facilities, or even from
ground-based facilities if the resolution is high enough to separate telluric features from
stellar and planetary features. The enhanced sensitivity of TESS to transiting planets near
the ecliptic poles will provide numerous targets for observations inside or near the continuous viewing zone of the James Webb Space Telescope. The prosepects for follow-up with
JWST have been detailed in Deming et al. (2009) and elsewhere. More specialized space
83
missions, including FINESSE (Deroo et al., 2012) and EChO (Tinetti et al., 2012), have
also been proposed to perform transit spectroscopy.
Here, we use the simulation results to explore the relative difficulty of transit spectroscopy of the TESS planets independent from the facility that is used. We compute a
figure-of-merit
6
H, which is the fractional loss-of-light from an annulus surrounding the
planet (with radius Rp) and a thickness equal to the scale height, H:
(1.30)
H
The scale height is calculated from
H =
kBTpR2
(1.31)
where Ai, is the planet mass and mp is the proton mass. We calculate the temperature of
the planet, Tp, assuming it is in radiative equilibrium with zero albedo and isotropic reradiation (see Eqn. 1.12). We assume a mean molecular weight
pt
of 2, which corresponds
to an atmosphere consisting purely of H 2 . However, an atmosphere with an Earth-like
composition would have
[
= 29, and a Venusian atmosphere would have A = 44. In such
cases, the atmospheric transit depth 6H is reduced by a factor of [p/2.
Figure 1-32 shows
3
H
for all of the detected planets in the simulation as a function of
the apparent magnitude of the host star. For a molecular species to be identifiable, one
must observe transits with a sensitivity on the order of
6
H
both in and out of the absorption
bands of that species. The detection of various species therefore depends on the depth of
the absorption bands and the spectral resolution used to observe them. The presence of
clouds and haze can reduce the observable thickness of the atmosphere.
1.10
Summary
We have simulated the population of transiting planets and eclipsing binaries across the
sky, and we have identified the subset of those systems that will be detectable by the TESS
mission. To do so, we employed the TRILEGAL model of the galaxy to generate a catalog
84
10
7000
*
GJ 1214b
-
I
0
,
n
.c
0
6000
0
55 Cnc. eu-
~XI
~
1*5
*O
*
4000
'
HD-97 "&b
3000
't
4
6
10
8
12
14
Ks
Figure 1-32: Feasibility of transit spectroscopy of the TESS planets. The transit depth
of one atmospheric scale height, assuming a pure H 2 atmosphere, is plotted against the
apparent stellar Ks magnitude. Atmospheric transit depths are lower by a factor of A/2
for other mean molecular weights. The points are colored by stellar Teff, and open symbols
2
indicate planets with an insolation S < SE. The dashed lines indicate the relative photoncounting noise versus magnitude, spaced by decades. Planets with R, < 3R( are shown
in addition to GJ1214b (Charbonneau et al., 2009), 55 Cancri e (Winn et al., 2011 a), and
HD97658b (Van Grootel et al., 2014).
of stars covering 95% of the sky. We adjusted the modelled properties of those stars to align
them with more recent observations and models of low-mass stars, the stellar multiplicity
fraction as a function of mass, and the J-band luminosity function of the galactic disk.
We then added planets to these stars using occurrence rates derived from Kepler. Then,
we modeled the process through which TESS will observe those stars and estimated the
signal-to-noise ratio of the eclipse and transit events.
The primary source of systematic uncertainty in our population of detected planets
arises from the planet occurrence rates. We also assumed that we can perfectly identify
the 2 x 105 best "target stars" for TESS to observe at the 2-min cadence. In reality, it
is difficult to select these stars since subgiants can masquerade as main-sequence dwarfs.
Parallaxes from Gaia could help determine the radii of TESS target stars more accurately,
and examining the full-frame images will help find planets transiting the stars excluded
from the 2-minute data.
85
The TESS planets will be attractive targets for follow-up measurements of transit properties, radial velocity measurements, and atmospheric transmission. Knowing the population of planets that TESS will detect allows the estimation of the follow-up resources that
are needed, and it informs the design of future instruments that will observe the TESS planets. The simulations provide fine-grained statistical samples of planets and their properties
which may be of interest to those who are planning follow-up observations or building
instruments to enable such observations. Table A. 1 presents the results from one trial of
the TESS mission. This catalog contains all the detected transiting planets from among the
2 x 10 5 target stars that are observed at a 2 min cadence.
We look forward to the occasion, perhaps within 5-6 years, when TESS will have completed its primary mission and we are able to replace this simulated catalog with the real
TESS catalog. This collection of transiting exoplanets will represent the brightest and most
favorable systems for further study.
86
Chapter 2
InGaAs Detectors for Near-IR Imaging
and Photometry
Sullivan, P. W., Croll, B., & Simcoe, R. A. 2013, PASP, 125, 1021
and
Sullivan, P. W., Croll, B., & Simcoe, R. A. 2014, Proc. SPIE, 9154, 91541F
Imaging and photometry at near-infrared (NIR) wavelengths remains far more challenging and expensive than at optical wavelengths. Beyond the silicon cutoff at A ~ lpm,
detector materials with smaller bandgap energies are required. This gives NIR detectors
higher intrinsic dark current than their silicon counterparts, which often makes cryogenic
cooling of the detector down to ~100 K necessary. Furthermore, the sensitivity of the detector to thermal emission from the telescope and instrument requires cold stops and careful
control of stray light. Charge-coupled devices (CCDs) are not traditionally fabricated from
NIR-sensitive materials, so the detector material must instead be hybridized onto a complementary metal-oxide semiconductor (CMOS) readout integrated circuit (ROIC) made from
silicon. The charge-to-voltage conversion takes place in each pixel of a CMOS detector;
since the pixel area is limited, the charge amplifier cannot be optimized to minimize the
noise and nonlinearity to the same degree as the amplifiers at the output of a CCD.
Despite these challenges, high-performance CMOS infrared detectors, like the Teledyne HAWAII family made from HgCdTe (Beletic et al., 2008), have revolutionized in-
87
frared astronomy over the past decade. The non-linearity of the pixel amplifier can, in most
cases, be corrected in post-processing. The higher read noise that CMOS detectors exhibit
over CCDs can be mitigated if the ROIC supports non-destructive readout during an exposure, enabling Fowler (Fowler & Gatley, 1990) or up-the-ramp (Chapman et al., 1990)
sampling. Still, the high cost of HgCdTe detectors and the cryogenics involved limit their
employment to the largest telescopes and to modestly-sized focal planes.
Indium Gallium Arsenide (InGaAs) has the potential to greatly reduce the cost of IR
instrumentation. InGaAs carries approximately half the cost per pixel of HgCdTe since
it is a widely-used material in the defense and telecommunications industries. The performance of domestically-produced InGaAs imaging arrays has dramatically improved in
recent years, and they are becoming available in sizes >1K x 1K pixels, making it possible
for InGaAs to complete with HgCdTe in some applications.
As a direct-bandgap semiconductor, InGaAs has high quantum efficiency up to a cutoff wavelength of 1.7 pm, which is set by lattice-matching to the substrate. The quantum
efficiencies of InGaAs and Si are plotted with atmospheric transmission in Figure 2-1. The
1.7 pm cut-off allows InGaAs to be used across the Y, J, and most of the H bands while
having little sensitivity to thermal emission from the telescope and the sky. While HgCdTe
can be tuned to the same cutoff wavelength, InGaAs has a much lower lower dark current
density-by a factor of approximately 100-than 1.7pm HgCdTe at the same temperature.
InGaAs detectors will probably never compete with HgCdTe for low-background instruments such as spectrographs or space-based instruments. However, the sky background
affecting broadband, ground-based imaging relaxes the requirements for detector performance. In order to achieve background-limited imaging performance, the detector needs to
be cooled to the point where the dark current per pixel is less than the sky surface brightness, and the read noise (after up-the-ramp processing) must be lower than the Poisson
noise from the dark current and the sky.
In this Chapter, we will first discuss two possible applications of InGaAs instrumentation in Section 2.1. In Section 2.2 , we will compare the operation of commercial InGaAs
arrays to more familiar HgCdTe arrays. Next, we will describe our efforts to bring InGaAs
imaging arrays into the realm of achieving background-limited imaging performance. We
88
0.9
0.8
0.7-
0.6-
0.5
0.40.3 -
---
Atrrospheric Transmission
---
LBL CCD OE
InGaAs
aE
-
0.2
-
0.1
S-0.6
0.8
1
1.2
1.4
Wavelength [pm]
1.6
1.8
Figure 2-1: Atmospheric transmission (Gordley et al., 1994) and representative quantum
efficiencies of fully-depleted silicon (Holland et al., 2003) and substrate-thinned InGaAs
(red).
began with the FLIR APS640C sensor, described in Section 2.3. While the read noise of
this device was acceptable (53 e- in sample-up-the-ramp), readout glow created a dark current floor of 840 e- s-1 pix-1 that limited the utility of the detector. We then tested the
newer FLIR API 121 sensor, described in Section 2.4, that shows promise towards the goal
of background-limited performance. We conclude by suggesting further improvements to
make the AP 1121 and larger-format sensors competitive in astronomical instrumentation.
2.1
Motivations for InGaAs Instruments
We will begin by discussing how InGaAs detectors are equally appropriate for instruments
on modest telescopes, where HgCdTe detectors are too costly, as well as for mosaics covering the focal planes of large telescopes, where the price per pixel drives the cost.
2.1.1
IR Transit Photometry
Modestly-sized telescopes are well-suited to studying exoplanets with NIR transit photometry. However, modem HgCdTe arrays are too costly for 1 m-class facilities, and commer-
89
cial InGaAs cameras are not usually desgined for scientific work.
Once the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2014) begins to
identify planet candidates around bright stars, there will be a need for high-contrast imaging
coupled with high-precision photometry to classify the candidates as bona fide planets or
eclipsing binary false-positives. For the planets, it will then be necessary to have lightcurves
with a higher signal-to-noise ratio for estimating the planet properties with greater accuracy.
Among the most interesting TESS planets are those near the habitable zone, which will have
M dwarf hosts emitting most of their flux in IR wavelengths. The M dwarfs that host planets
detectable with TESS will have J ~12. Many of the other TESS planets will be quite hot,
and it will be valuable to measure their secondary eclipses or even their full phase curves
in the NIR.
In addition, a survey of very low-mass stars and brown dwarfs for new exoplanets could
be carried out with InGaAs-equipped facilities. This would complement TESS since the
targets are too red for CCDs to be used. Such a survey requires dedicated telescopes, so the
cost of the detector must also be low to make the survey economical. This is particularly
true if multiple telescopes are employed to cover more targets.
Ground-based detection and characterization of exoplanets can only be carried out if
the host stars are bright, so the photon-counting noise from the star should dominate the
background or read noise. Correlated noise must not prevent the observation from reaching high precision when long timeseries are analyzed. In the following Sections, we will
demonstrate the stability of InGaAs sensors with long timeseries photometry both in the
laboratory and on the sky.
2.1.2
Wide-Field IR Transient Searches
For large telescopes, which are less sensitive to the price of individual detectors, the low
cost per pixel of InGaAs should still be attractive if many detectors are required to sample
a wide field at high spatial resolution.
A wide-field infrared telescope to follow-up gravitational wave triggers would complement the optical capabilities of the upcoming Zwicky Transient Factory and the Large
90
Synoptic Survey Telescope. Gravitational wave sources will be extra-galactic, so observing candidate host galaxies at J ~ 19 - 21 for an electromagnetic counterpart will require
background-limited observations across a wide field. Exposure times could last from tens
of seconds up to one minute.
For example, we consider a telescope with a 1 deg 2 field-of-view where the natural seeing is 0."8. Nyquist sampling of the seeing dictates a pixel scale of 0."4, so a camera would
need 65M pixels to cover 80% this field-of-view. This could be accomplished either with
32 FLIR 1920x1080-pixel InGaAs sensors, or with 4 4096x4096-pixel Teledyne HAWAII4RG detectors. At current prices, the FLIR detectors and electronics would cost $1.9M,
and the Teledyne detectors and SIDECAR electronics would cost $3.34M. One also has
to consider the higher construction and operating costs of a cryogenic instrument, which
is necessary for the Teledyne HgCdTe detectors. These costs would be unnecessary for an
InGaAs instrument if background-limited performance can be obtained with thermoelectric
cooling. We will show how this is possible with the AP 1121 sensor in Section 2.4.
2.2
Background
Like most astronomical CCDs, the HAWAII family of HgCdTe detectors are designed to
read out pixel values at a rate of 100 kHz. Depending on the size of the array, and the
number of outputs employed, it can take from one to tens of seconds to read out a frame.
This limits the duty cycle of observations, which is the proportion of observing time that is
spent collecting photons.
Commercially-availiable InGaAs detectors, on the other hand, are usually hybridized to
ROICs designed for video. This requires a frame rate of at least 10 Hz, so duty cycles near
100% can be obtained. These ROICs support pixel rates on the order of 10 MHz; since the
bandwidth is two orders of magnitude higher than a CCD, the read noise is roughly one
order of magnitude higher. However, the large number of frames that one obtains can be
leveraged to average the noise back down to a useable level.
The higher bandwidth of commercial InGaAs detectors is due, in part, to their use of
a capacitive transimpedance amplifier (CTIA) pixel architecture. This is compared to the
91
Vbias
VDD
Vbias
Reset
Read
Iphoto
Vout
Sample
Iphoto
Cint
Cint
Vout
-1117
-
Vref
+
Reset
R ad
Vreset
(a) Source Follower
(b) CTIA
Figure 2-2: A comparison of source follower and CTIA pixel amplifier architectures. Both
convert a photocurrent, Iphoto., into a voltage, Vout, with an integration capacitance, Cat.
For IR arrays, the photodiode is bump-bonded onto the silicon readout circuitry.
source follower architecture of the HAWAII family in Figure 2-2. Although CTIA architectures are more complex, which adds noise and increases the power consumption, they can
offer better linearity than a source follower amplifier. This is because a source follower accumulates the photocurrent on the gate capacitance of a single field-effect transistor (FET),
whereas a CTIA stores the accumulated photocurrent in the feedback loop of the amplifier.
The negative feedback helps maintain a linear response between the input and output of the
amplifier.
An additional benefit of the CTIA is that the reverse bias across the photodiode can be
minimized to limit the dark current. The bias remains fairly constant even as the photocurrent is integrated. The reverse bias of a photodiode attached to a source follower must be
maintained at higher voltage since the voltage of the FET gate rises as photocurrent is integrated. The amplifier response will otherwise become non-linear, or worse, the photodiode
can become forward-biased.
Another key difference between the HAWAII ROICs and off-the-shelf ROICs used with
92
InGaAs sensors is their exposure control. The HAWAII ROICs use a rolling electronic
shutter, where the exposure time is defined as the time between the reset and readout of
each pixel. Even if the exposure time is made to be uniform across the array, each pixel will
sample a slightly different phase across the frame. On the other hand, the ROICs offered
with most commercial InGaAs sensors have a global electronic shutter. At the beginning of
an exposure, all of the pixels are simultaneously reset; when it is time to read out a frame,
all of the pixel values are captured by a sample-and-hold circuit in each pixel. This circuit
shown downstream of the CTIA in Figure 2-2. The stored pixel values are then read from
the output of the ROIC one at a time. This is analogous to a CCD with a frame-store area.
For both the source follower and CTIA architectures, the act of resetting the capacitor
where the photocurrent is stored introduces another source of noise. This is known as
reset noise, or kBTC noise, since it is proportional to the temperature and capacitance.
Reset noise can be cancelled out by sampling the pixel at the beginning and end of an
exposure, known as correlated double-sampling (CDS). CDS can be performed either in
the pixel amplifier, using the sample-and-hold, or digitally, by reading out the two samples
separately. Digital CDS is also known as Fowler-1 sampling; reading out n frames at the
beginning and end of the exposure to further reduce the read noise is known as Fowler-n
sampling (Fowler & Gatley, 1990). If many frames can be read during an exposure, then
a least-squares fit to the accumulated charge over time, known as up-the-ramp sampling
(Chapman et al., 1990), offers the lowest noise. Since commerical InGaAs arrays have
high frame rates, up-the-ramp sampling is quite effective at minimizing the read noise.
The high frame rate also allows for a wide dynamic range. Bright sources what would
saturate in the full exposure time can be extracted from the first few frames of an up-theramp sequence, and faint objects can be extracted using all of the frames in the ramp. This
way, nearly all objects can be extracted with the same photon-counting noise.
For all of these reasons, we believe that fast CTIA architectures are actually appropriate
for NIR photometry and surveys.
93
Figure 2-3: The assembled camera shown in the laboratory. The chilled water lines are
visible behind the camera.
2.3
APS640C
In order to assess the astronomical utility of commercial InGaAs detectors, we first examined fully-assembled cameras available off-the-shelf. Specifically, we looked for units with
large cooling capacity to minimize dark current; low-noise, non-destructive readout; highresolution analog-to-digital conversion; and affordability. Such a camera was not available
on the commercial market, so we proceeded with constructing our own camera with these
requirements in mind. It was built around the APS640C InGaAs image sensor supplied by
FLIR Electro-Optical Components. This VGA-format detector has 640x512 pixels with a
size of 25 pm (Macdougal et al., 2011). To reduce cost, we acquired a "B"-grade sensor
which is allowed more inoperable pixels (especially in clusters) than the "A" grade, but
we took care to exclude the inoperable pixels from our observations and analyses. The
assembled camera is shown in Figure 2-3, and its design is described in the following subsections.
94
2.3.1
Design
Thermal Considerations
Domestically-produced InGaAs has reached dark current densities of 1-2 nA/cm2 at room
temperature. The dark current halves with every 7'C of cooling (Macdougal et al., 2011),
so we sought to minimize the operating temperature of the detector. The InGaAs detector
and its silicon ROIC are housed in an evacuated package with a thermo-electric cooler
(TEC) sandwiched between the ROIC and the housing. A standard laboratory closed-loop
water chiller holds the warm side of the TEC at
-
10 C, which is above the dewpoint during
favorable observing conditions. The TEC drives a AT of -50'C, allowing the detector to
operate down to -40'C. Although water cooling complicates the operation of the detector,
the heat exchanger can be moved far from the telescope, whereas a fan mounted to the
camera might introduce electrical or mechanical noise.
Analog-to-Digital Electronics
In order to allow -10
frames to be taken up-the-ramp during exposures lasting several
seconds, we digitized the analog output of the APS640C detector through one channel at
1 MHz. While using multiple detector outputs or a higher digitization rate could increase
the frame rate beyond the 2.7 Hz that we achieved, this design makes any instabilities in
the signal chain common to all pixels, simplifies the analog design, and eases the power
consumption of the ROIC and its cooling needs.
We connected the ROIC's analog output to a 16-bit analog-to-digital converter (ADC)
via an op-amp buffer. The ADC has a single-ended input and uses the successive-approximation
architecture. A 12-bit digital-to-analog converter (DAC) drives the bias inputs on the ROIC
and reference voltages in the ADC chain. Both the ADC and the DAC are tied to the same
low-noise, low-drift (3 ppm/0 C) voltage reference, which ensures the electronic stability of
the camera. Relatively few other parts are needed to operate the sensor.
95
Data Processing
Digital Design
Clocking of the detector, and the communication between the data converters and the host
computer, is handled by a Virtex Spartan 6 field-programmable gate array (FPGA). It is
housed on a daughter circuit board made by Opal-Kelly that provides access to the FPGA's
input/output pins as well as a Universal Serial Bus (USB) 2.0 interface; a library supplied
by Opal-Kelly supports data transfers between FPGA registers and data structures in the
host computer's software. The FPGA is powered by the computer via the USB cable.
While USB provides ample throughput, it can also introduce noise from the host computer's switching power supplies. To keep this noise from entering the analog portion of
the circuit board, and to break a potential ground loop, digital isolators are placed on the
lines connecting the ADCs, DAC, and detector to the FPGA.
The clocking patterns that drive the image sensor and ADCs are designed to be as
regular as possible to minimize thermal transients. In fact, the ADCs continuously convert
the detector output whenever the camera is powered on; when pixels are read out, the only
change is that the ADC values are piped through the USB interface rather than terminating
in the FPGA.
2.3.2
Detector Characterization
We measured the gain, read noise, and dark current from a series of dark and flat-field
frames. The flat-field images were illuminated with a broadband quartz lamp, and a Y band
filter was placed in front of the detector. We calculated the gain by following the approach
in Janesick (2001), where the increase in shot noise is examined against the increase in
signal from dark through illuminated frames. By assuming that the variance of the shot
noise is equal to the mean signal, and that the read noise does not increase with signal,
we measure the gain to be 0.6 e-/ADU. The read noise was obtained from extrapolation to
zero signal, which is 149 e- if one sample is taken per frame. (This is larger than the 70 especified by the manufacturer, but we later discovered a byte swap in our data processing
that effectively adds 256 ADU to the read noise.) Taking multiple samples per frame (either
96
lbU
..... ............
150 -- A..... ............
150A
A
U
140
-*
130
--
-
120 -
SnleR
Single
a
Read
Fowler-1 (dCDS)
Fowler-2
Up-The-Ramp
.-
C)
110
-.-.-.-...........
- . . .. ..-....
-.
80 0
...
.
S 1 00
-
2
70
10
10
Samples per Exposure
Figure 2-4: Multiple sampling during an exposure significantly reduces the effective read
noise of the APS640C. Filled markers indicate measured values, and open markers indicate
values predicted from (Offenberg et al., 2001) based on the Fowler-1 read noise. The singlesample read noise is far higher than prediced since reset noise is present. The exposure time
is fixed at 2.3 s for all cases.
in Fowler pairs or sampling up-the-ramp) cancels out the reset noise on the integration
capacitor and dramatically reduces the read noise, as shown in Figure 2-4. Ramps of 10
samples minimize the effective read noise to 54 e-, which is consistent with prediction from
the Fowler- 1 read noise value.
Dark current was also measured in this setup over temperature and is plotted in Figure
2-5. Although we expect the dark current to halve for every - 7C of cooling, we find
that it plateaus near the -20'C value of 840 e-/pixel/s. The dark current is even higher in
the corners and edges of the array, shown in Figure 2-5, which suggests that recombination
glow from the ROIC is the source. FLIR has acknowledged this issue and reduced the
ROIC glow in subsequent designs, as we will demonstrate below.
Under flat-field illumination, pixel-to-pixel non-uniformity is measured at approximately 4.0% among the operable pixels.
97
5000
0
0~
200
0
0
0
1000
Cu
0
r00
0
-5
-20
-15
-10
Temperature FC]
-25
500
-30
2500
2000
1500
1000
1
1
Dark Current [e- s pixel ]
3000
Figure 2-5: Left: Dark current versus detector temperature for the APS640C. The error
bars indicate the lo- pixel-to-pixel spread in the measured values. No further reduction in
dark current is possible past -20'C. Right: The spatial pattern of the dark current across
the array. Because it is stronger near the edges of the detector, we attribute the excess dark
current to ROIC glow.
2.3.3
Photometric Testing in the Laboratory
Setup
We obtained a long photometric time series with a simulated star field in the laboratory. In
a fashion similar to that of Clanton et al. (2012), a microlens array generates the ensemble
of stars. Here, a 50 pm pinhole is illuminated with a lamp and integrating sphere. The
microlens array re-images the pinhole into an 8x10 grid of simulated stars with a spacing
of 1.015 mm. This pattern is then re-imaged onto the detector with slight magnification,
which is shown in Figure 2-6.
The pinhole and microlens relay is attached to a linear
stage to allow us to move the image across the detector in the horizontal direction. To flatfield the detector, the pinhole and microlens array are removed to allow the output of an
integrating sphere to be imaged directly. Dark frames were also acquired with the shutter
4
on the lamp closed. The median flux per aperture is 2.36 x 10 e-/s, similar to a J~~9.1 star
imaged in the Y band with a 0.25 m effective aperture. The imaged stars have a full-width
half-maximum of 3 pixels. Individual exposures lasted 3.7 seconds, and a total of 9600
images were acquired over nearly 11 hours.
98
500
-
5W
0.12
40450
S
0.2
400-
400
50 --
50
0
0
S0.08
*
30300
250-
x 250
-
200 20D
50150
0.041
5
100-
100
50
100
0.02
so200
300
Pixels
400
500
1
10
600
200
300
Pixels
400
500
600
0
Figure 2-6: Left: Full-frame image of stars generated with the lenslet array. Right: Absolute value of the Pearson correlation between a simulated star at the center of the field (+)
and the other simulated stars; there is no clear trend across the array. The diagram omits
stars exhibiting non-Gaussian noise.
Measurement of Precision
We measure the precision in the photometric time series as the relative standard deviation
as a function of co-averaging time. Hence, we divide the timeseries
fi into
M bins of
duration r, calculate the mean within the bins, and find the standard deviation of the binto-bin means.
The bin means are denoted fj; each is calculated from
N(j+1)
(2.1)
fi,
fj =
i=Nj
where N =
T/AT
gives the number of samples for a sample period Ar. The relative
standard deviation over the bin means is then
-(T) = --
m
M -1
f
99
f
'22
(2.2)
where f denotes the mean of
fi over the
whole timeseries. Care is taken to ensure that the
same time samples are always used in the calculation by using bin lengths N that divide into
the total number of samples without remainder. In other words, the product MN is always
equal to the total length of the time series. For uncorrelated noise, we expect o-(T) c
-/
Reference Signal Subtraction
In addition to the photometry, we also obtain a set of five reference signals during the data
acquisition. Before and after each row of pixels is read out from the detector, the bias level
is sampled in a similar fashion to the "overscan" region of a CCD. Rows of blind pixels on
the top and bottom of the detector are also read out at the beginning and end of each frame,
which should track changes in the column amplifiers. The mean of the reference pixels at
the top and bottom of the detector constitute two of the reference signals. Furthermore,
we record the x and y centroid position of the photometric aperture. The centroids will be
coouple to the photometry through residual flat-fielding errors when the stars and apertures
are moving. The fifth reference signal is the background level that we subtract from each
frame. If the background was under- or over-subtracted at the location of the star, then
the there will be a residual photometric signal that is correlated or anti-correlated with the
background.
De-correlation of the reference signals and centroid position from the photometric timeseries has proved effective with Teledyne H2RG detectors (e.g., Sullivan et al. 2011 and
Clanton et al. 2012), and we apply the same basic approach here. A robust linear regression determines the fit coefficients
ch
between the measured photometric signal
fi
and the
reference signals RL:
f, =
+ c1 Rjj + c 2 R 2i +
...
(2.3)
The de-correlation estimates the true photometric timeseries f 1 by subtracting the reference
signals and centroid position from the measured photometric signal. The reference signals
are cross-correlated with one another, so we construct their principal components prior to
de-correlation so that the linear regression yields a unique solution. However, we do not
use PCA to reduce the dimensionality of the reference signal set, as was done with the
100
H2RG in Sullivan et al. (2011)
Results of Laboratory Photometry
We obtained a time series of 9600 frames in 10.8 hours. Of the 80 simulated stars, 72
produce noise near 1% per exposure. The other 8 apertures contained at least one pixel
exhibiting non-Gaussian noise, which is consistent with our finding that -0. 1% pixels show
popcorn, or burst, noise. Table 2.1 shows a breakdown of the calculated sources of noise in
the measurement. Read noise, dark current, and Poisson noise alone can account for 91%
of the measured noise.
Differential photometry is performed for each of the 72 target stars by selecting one
comparison star at random from the remaining 71 and finding the ratio of the flux. The
comparison stars were chosen at random; as Figure 2-6b shows, there is no obvious advantage (in terms of a higher Pearson correlation) to selecting comparison stars near the
target star. The precision of differential aperture photometry scales as T-1/2 as expected
for uncorrelated noise at 631 ppm hr1 / 2 with a standard deviation of t205 across the 72
apertures (Figure 2-7, left). The median flux per aperture was 2.36 x 104 e-/s. We repeated
the analysis with the mean of three comparison stars, which improves the precision to 534
204 ppm hri/ 2 . If we de-correlate the reference signals from the photometry, the precision
further improves to 483
161 ppm hr'/ 2
The effects of adding comparison stars and de-correlating the reference signals can also
be seen in the frequency domain. In Figure 2-7 (right), both white noise and 1/f (red) noise
are evident. Averaging over three comparison stars lowers their collective Poisson noise
and hence the white noise floor by 30%, but the 1/f component remains. The reference
signals can reduce the 1/f component, but they do not lower the white noise floor at higher
frequencies. In all cases, the 1/f noise becomes significant at timescales longer than one
hour, and approximately 0.5% of the total noise power is attributed to 1/f noise above the
white noise floor.
101
O
1Comparison Star
3Comparison Stars
o Reference De-correlated
-- - Poisson Limit
102
-D
-
-
I
......-
.0::: :
a..
....
1.
....
~....
1...
.... .
1a
10
1 Comparison Star
3 Comp;rson Star
Reference De-correlated
/f Scaling
10lo
10,
Averaging Length [Minutes]
10
10
10-3
Frequency [Hz]
0-
Figure 2-7: Left: Precision achieved with the APS640C on an ensemble of 72 simulated
stars. Differential photometry is performed between one target and one comparison star
(blue), three comparison stars (green), and three comparison stars with reference signal
de-correlation (red). The T-1/ 2 expectation scaled to the leftmost data point (with no averaging) is shown as the dashed line. Right: Noise power spectrum for the differential
photometry. Adding more comparison stars (green) reduces the white noise floor, and the
reference signal de-correlation (red) reduces the 1/f component (dashed line).
Table 2.1: Noise budget for a single exposure in laboratory and on-sky testing.
Noise Source Lab Photometry HD80606 Photometry
201
79
Number of pixels
766
480
[e-]
54
e-/pixel)
Read noise (at
794
498
Dark noise (at 840e-/pixel/sec) [e-]
0
113
Poisson noise [e-]
Scintillation noise [e-]
Quadrature sum of noise sources
/Tincrease for differential noise [e-]
Measured Noise [e-]
295
0
752
1063
1172
470
493
1301
1840
2740
Explained Noise
91%
67%
Sky noise [e-]
102
2.3.4
Testing on the Sky
Observations
In February 2013, we observed HD80606 and HD80607 with the 0.6 m telescope at MIT's
Wallace Astrophysical Observatory. Both stars are spectral type G5 (Feltzing & Gustafs-
son, 1998), with HD80606 at J = 7.73 and HD80607 at J = 7.80 Cutri et al. (2003).
Although HD80606 is known to host a planet (Naef et al., 2001), the transit (Fossey et al.,
2009) of HD80606b was not occurring during our observation. The observations were carried out in the Y band A ~ 1.05pm using a filter from Omega Optical. Guiding commands
were derived from the science frames; the tracking errors are plotted in Figure 2-8. Again,
we obtained 2200 images in 2.5 hours with an exposure time of 3.7 seconds. Flat-field
frames were obtained with a screen in the dome.
Sky Results
Many practical issues make testing on the sky more difficult than in the laboratory. At
our site in northern Massachusetts, the seeing was poor (at several arcseconds), requiring
photometric apertures with a radius of 8 pixels for our plate scale. Also, we estimate the
optical efficiency of the telescope at only 18% even after accounting for all other known
losses, including secondary obstruction and filter transmission. This reduces our effective
aperture to 25 cm. Atmospheric scintillation noise is also significant for a telescope diameter of D=60 cm and exposure time of rep=3 . 7 seconds. We estimate it at X=1.3 airmasses
from Young (1967) and Young (1993) with the wavelength A scaled from the V to Y band
using Dravins et al. (1997):
(9%) (X) 3 / 2 (Dcm)
2
/3 (2Te)-1/2 ( 0 .5
8
,mexp (2.4)
where z denotes the site altitude in km. Table 2.1 shows that, unlike with the laboratory
data, we can account for only 67% of the observed noise by considering the read noise and
dark current levels measured in the laboratory, Poisson noise expected from the stellar flux
and sky, and scintillation noise. Higher electronic noise at the telescope, underestimated
103
-
20
Differential Photometry
18Radial Tracking Erro
Reference De-Correlated
Position De-Correlated
---
10-2
--- - PSF Radius
16
14.
- -
.
Poisson Limit
12
'2
-
0
.10
C
CEO015
6
0.
4
0
50
100
Time [Minutesi
150
0
10
1
41
10
10*
Averaging Length [Minutes]
Figure 2-8: Left: Telescope tracking errors frequently exceeded the PSF width during observations of HD80606 and HD80607 (inset). Right: Precision achieved in differential
photometry with the APS640C.
background, and tracking errors could also contribute to the higher observed noise. The
laboratory testing showed that gross tracking errors, where the flux centroid strayed from
its mean location beyond the PSF half-maximum 56% of the time, can degrade the precision
by a factor of 2. Hence, a degradation by a factor of 1.5 can be accounted by the tracking
errors at the telescope, where the flux centroid strayed from the PSF half-maximum 35%
of the time.
Despite these issues, the differential photometry (Figure 2-8) yields a precision down
4
to 611 ppm in 45 minutes for a mean observed flux of 6.7 x 10 e-/s. Over long timescales,
de-correlating the reference signals (with or without the 2-dimensional centroid position)
1/ 2
. The actual valyields precision that is consistent with the Poisson limit of 415 ppm hr-
ues of the relative standard deviation fall below the Poisson limit for long averaging times,
but with only two stars there is large uncertainty in measuring the precision. Over 1-minute
timescales, the precision with the centroid position de-correlated shows 12% improvement
over the precision with only the reference signals de-correlated. This brings the precision
into agreement with the Poisson limit, indicating that the effects of tracking and flat-field
errors are overcome.
104
2.4
AP1121
Due to the excessive dark current of the APS640C detector, we next tested a newlydesigned detector from FLIR, the AP1 121. This sensor offers improved masking between
the ROIC and the pixel array to reduce its sensitivity to ROIC glow. Furthermore, the
AP 1121 shares the same pixel design as the larger 1920x 1080-pixel detector made by FLIR,
so many of our design decisions and results should also apply to this device.
2.4.1
Design Changes
We originally designed the AP 1121 camera electronics using similiar parts as the APS640C
electronics. A slightly faster ADC was employed with a sample rate of 1.33 MHz, giving
a frame rate of 3 Hz. However, testing revealed a high degree of nonlinearity. The nonlinearity ranged from 1-6%, increasing with the time delay between the sampling of a pixel
and its readout. After conferring with FLIR, we attributed this nonlinearity to leakage of
the sample-and-hold capacitors that form the electronic shutter. We minimized the impact
of this issue by running the detector at a higher frame rate of 25 Hz to minimize the time
delay between the pixel sampling and readout.
The higher frame rate required a complete re-design of the camera electronics. In order
to take full advantage of the 10 MHz pixel rate of the AP 1121 sensor, we changed the ADC
from a simpler single-ended part to a high-performance, differential ADC. This, in turn,
required a high-speed differential amplifier to drive the ADC inputs. The higher data rates
also required us to upgrade the Opal-Kelly FPGA module to one with a USB 3.0 interface
for higher throughput. The bit rate between the FPGA and the ADC increased to 200 MHz,
which is too fast for the digital isolators we used with the APS640C camera. Instead, we
isolate the camera from the computer by using an optical fiber connection between the two.
This is also needed because the cable length of USB 3.0 is limited to a few meters.
In order to reduce the demands on the computer, we only record every other frame from
the camera during exposures lasting longer than a few seconds. For shorter exposures, it is
advantageous to record every frame to in order to minimize the read noise.
105
4
--
Figure 2-9: The nonlinearity of the API 121 increases from 1% to 3% from the top to the
bottom of the array. Pixels at the bottom of the array have the longest delay between the
sampling of the pixel voltage and the readout.
2.4.2
Linearity
After constructing and testing the new camera electronics, we measured the linearity again
by taking many exposures up-the-ramp with steady flat-field illumination. Since the time
interval between exposures is ultimately set by the crystal oscillator on the FPGA module,
we assumed that an equal amount of photoelectrons should be collected from exposure to
exposure. A perfectly linear detector should then have a linear signal versus time. While
we have not quantified the stability of the lamp, we allowed it to equilibrate for several
hours and averaged 10 ramps to minimize the effects of any drifts in the lamp output.
We quantified the relative nonlinearity for each pixel over the lower 85% of the dynamic
range, or 40k ADU. First, we subtracted a linear fit to the signal (relative to 40k ADU)
versus time; after sorting the residuals, we found the difference between the 95th- and
5th-percentile residuals. The nonlineary is shown for each active pixel in the array in
Figure 2-9.
With the higher frame rate, the nonlinearity ranges from 1-3% per pixel. We find that
a third-order polynomial fit to the response curve is sufficient to reduce the nonlinearity to
the level of 0.4% per pixel.
106
60
X
X
Fowler
Up-The-Ramp
55
"5 50
z
_0
45
X
X
x
X
X
40
1
4
16
Number of Samples
64
Figure 2-10: Read noise for the API 121 sensor with various sampling schemes. The exposure time is 5 sec. for all methods.
2.4.3
Gain and Read Noise
As before, we examine the variance versus the signal for a series of dark and flat-field images to determine the conversion gain from the assumption that the variance follows Poisson statistics. However, we also correct the nonlinearity of each pixel before calculating
the signal and its variance. We find that the conversion gain is 1.17 e-/ADU.
By extrapolating the variance to zero signal, we calculate that the read noise is 59
e- with digital CDS (Fowler-1). Sampling the detector up-the-ramp gives a read noise
of 43 e- in a 5-second exposure consisting of 64 frames. We show the read noise for
other sampling schemes in Figure 2-10. Fowler-4 sampling gives lower noise unless ,>30
samples are available for up-the-ramp processing.
We can also calibrate the full-well capacity with the gain measurement.
Since our
nonlinearity correction is valid over 40k ADU, the full-well capacity is approximately 47k
e-. This gives a dynamic range of more than 3 decades, or 60 dB. Although we record
16-bit images, one could digitize the individual frames using only 12 bits with no adverse
effects.
107
2.4.4
Dark Current
Our previous experience with the AP640C detector had shown that dark current from ROIC
glow limited its performance. The API 121 detector has improved masking between the
readout and the photodiode array to reduce this source of background. Furthermore, the
API 121 has a smaller pixel size of 15 pm versus 25
jam.
We were also able to operate
the detector down to -50'C with the first-generation API 121 electronics. However, at the
higher speeds employed by the new electronics, the ROIC dissipates more power, and the
TEC can only bring the temperature down to -40'C. For that reason, we report dark current
measurements from the earlier electronics taken during a cold night (with an ambient temperature of ~0 C at Wallace observatory to minimize the contribution of thermal emission.
(The dark current measurements taken at -40'C are consistent between the two versions of
the electronics.)
We found that the dark current scales exponentially with temperature through -50'C,
halving with every 7TC of cooling as expected. The temperature scaling is shown in the left
and 87% of pixels have dark current below 200 e- s-
1
1
pix-
1
,
panel of Figure 2-11. The dark current at -50'C has a median value of 163 e- s-
pix-1. The full distribution of the
dark current is shown in the right panel of Figure 2-11.
For comparison to the sky surface brightness, we use the broadband sky measurements
from the FourStar cameral on the 6.5 Magellan Baade telescope. We adjust their measurements from the J1 and J bands to account for the slightly different bandpasses of the Y
band and MKO J band. Figure 2-11 shows that the dark current of InGaAs operated at
-50'C is well below the sky brightness in the J and H bands for a 1.0 m telescope. However, requiring sky-limited observations in the Y band give a more strict constraint on the
dark current. The Y-band sky scales to 220 e- s- 1 arcsec
2,
so sky noise should dominate
dark current noise for telescopes larger than 1.0 m. A larger telescope, or deeper cooling,
is advantageous if the pixels have a smaller projected area on the sky.
1
http://instrumentation.obs.camegiescience.edu/FourStar/calibration.html
108
.0.4
- - - HgCdTe
+ InGaAs
- Sky Brightness
0.35
0.3
x
- 10
-
0.25
0.2-
7
0.
-
105
0.02
VC 0.15
0
0.1
cc
--- -0.05
-
102
0
-10
-30
-20
Temperature r C]
-
~... . . . . .
-40
..0
9
-50
J~o
160
250
40
400
60
630
10
1000
Dark Current [e- s-1 pix-]
Figure 2-11: Left: Thermoelectric cooling can reduce the dark current of the AP 1121
below the sky surface brightness, which is plotted for the Y, J, and H bands. We have
assumed 1.0 m telescope with 1" pixels. The dark current of HgCdTe Beletic et al. (2008)
for the same cutoff wavelength and scaled to the same pixel size (red dashed line) is much
higher, so cryogenic cooling is required to reach the same level of dark current. Right: The
1
distribution of dark current per pixel at -50'C has a median value of 163 e- s-1 pix ; 87%
of pixels have dark current below 200 e- s-' pix-1.
2.4.5
Persistence
Next, we measured the persistence (or latent image) by measuring the excess dark current
of the detector immediately after the removal of illumination. We first exposed the detector
to a high flux level of 100k e- s-' for approximately one hour. Then, we removed the flux
with a shutter and obtained a series of 0.3 s exposures in CDS mode were then taken for
the next 1000 s. The timeseries is shown in Figure 2-12.
At our nominal operating temperature of -50'C (achieved with the old API 121 electronics), the detector persistence falls below the dark current level (-200 e- s-' pix-')
within 2 s. Within 20 s, which is comparable with the time to slew a telescope to a new
target, no persistence is detectable. Modestly bright targets should also not persist through
dithered images.
We repeated the experiment at -40'C and -30'C and found that the persistence is reduced at these higher temperatures. This is expected for persistence due to trapping sites;
at lower temperatures, photoelectrons can be trapped for a longer period of time.
The AP1 121 characterized here did not have its substrate removed, so we hypothesized
109
350
---
300
30 *C
-40 0C
-40 *C
-50
-250
200-
100-
-1500'
10
_50
''
'
'
' ' '
10
' '2
10
10
3
Time [s]
Figure 2-12: Persistence of the AP 1121 measured immediately after shuttering a bright
flat-field source.
that some, or all, of the persistence could originate in the substrate and not the InGaAs. We
tested this hypothesis by conducting the same test with an earlier version of the API 121
that had its substrate removed; we found no difference in the magnitude of the persistence.
While there are other advantages to removing the substrate (including blue-end response
and cosmic ray reduction), the persistence is unaffected since it appears to originate entirely
in the InGaAs itself.
2.4.6
Laboratory Photometry
Finally, we re-measured the photometric performance of the API 121 detector using the
new electronics and the same laboratory setup described in Section 2.3.3. In this case, we
obtained 7000 exposures with an integration time of 5 seconds. Each exposure consists of
64 frames sampled up-the-ramp. This is the same configuration used to measure the read
noise and dark current, so we can directly analyze their contribution to the photometric
noise.
4
The photon rate was 7.8 x 10 ph s-1, which corresponds to a J = 10.5 star observed
2
with a 1.0 m telescope with 50% efficiency (or an effective area of 0.4 m ). This photon rate
110
is approximately 4 times higher than our testing with the APS640C, so we should expect
the photon-counting noise to be halved.
For reference signals, we use the medians of the x and y centroids across all of the
stars, the dark current and background flux that was subtracted from each frame, and the
median flux across all 63 stars. This allows us to de-correlate trends in the photometry due
to common-mode shifts in the lenset array, over- or under-subtraction of the dark current,
and variations in the lamp that illuminates all of the stars.
We performed the aperture photometry on all 63 stars of a 9x7 grid. The aperture
sizes were chosen empirically to maximize the signal-to-noise ratio, and they range from
27 to 30 pixels. The dark current and background flux have a combined median value of
443 e- pix-
1
s-1. Together with the read noise of 43 e- pix- 1 , the relative noise in each
photometric aperture should be 1.6 x 10-3 per 5-second exposure.
As before, we compute the noise as the standard deviation in the mean of binned subsets of the timeseries. With a bin size of 1 exposure, the relative noise is 1.7 x 10-3. This is
only 6% higher than the quadrature sum of the known sources of noise computed above.
T-i/
2
.
As the bin length r increases, we expect the binned standard devation to fall as
Figure 2-13 shows the noise plotted against bin length. The mean photometric precision
across the 63 stars is 68
19 ppm hr'/ 2 , and it is consistent with a r-1/ 2 scaling up to the
3-hour bin length considered here.
2.5
Discussion
We now return to the two motivations for InGaAs instrumentation described in Section 2.1:
transit photometry of J e 12 M dwarfs and searches for extragalactic transients. In Table 2.2, we compare the modes of operation for these two cases, assuming that the performance characteristics of the AP1121 sensor will carry over to the 1920x1080 sensor.
We also assume that
1 second between exposures is spent writing the data to disk; this
still allows a high duty cycle in both cases (80% for transit photometry and 95% for faint
transient detection).
For bright sources, the InGaAs detector performs quite well. Photometry of a J e 12
111
10-3-
CL
.2 io-4
9105
0
.
10-1
1
.2.. .. ,..: :
"
10 -6
0.
100
101
102
Averaging Time [minutes]
Figure 2-13: Stability of the AP 1121 with photometry in the laboratory. Individual lines
show the relative noise of 63 apertures; the error bars show the mean and 1 o- spread in
the relative noise. The mean noise averages down consistently with the expectation for
uncorrelated noise, which is shown by the red dashed line.
exoplanet host star reaches a precision of 1.7 x 10-
4
hrl/ 2 . An hour-long transit of a 1.5 RI
planet around an 1.5RD mid-M dwarf would be detected at a signal-to-noise ratio (SNR)
of 50. This level of precision would allow one to precisely determine the time of transit,
duration of transit, and the duration of ingress and egress from fitting the lightcurve.
Observing fainter sources quickly becomes more challenging. In Table 2.2 we consider
a J = 18 source and a 2.5 m telescope as an intermediate case. The noise is dominated by
the J sky background for the dark current we measure at -40'C. A SNR of 10 is reached in
just over one minute. Observing a J = 20 galaxy with a 2.5 m telescope is only practicable
in the Y band, where the sky background lower more than threefold. Furthermore, we
assume that the detector is operated at -60'C; we extrapolate the dark current to a value
of 105 e- s-
pix-
1
from the ratio of dark current at -40'C to -50*C. Photometry would
reach a SNR of 3.5 in 10 minutes, or a relative precision of 0.3 mag. In 20 minutes, the
SNR reaches 5, and the precision is 0.2 mag.
If a survey for IR transients is conducted with 32 of the 1920x1080-pixel FLIR detectors, one also has to consider the data volume. To keep the frame rate similar to that of the
112
Table 2.2: Observational modes for InGaAs cameras.
Transit Photometry
Telescope:
Aperture [m]
Pixel Scale
Signal and noise:
Sky-limited Detection
1.0
1."0
2.5
0."4
2.5
0."4
12
18
20
Target J mag.
Bandpass of Observation
J
J
Y
0.16
0.16
0.10
Target flux [ph s1]
Pixels in photometric aperture
2.0 x 10 4
9
80
9
8
9
Read Noise [e- pix- 1 exp- 1]
43
43
43
250 (-40 C)
250 (-40-C)
105 (-60-C)
750
750
220
0.3
4
1.8
23
5.7
72
0.16
0.83
0.032
0.013
0.29
0.12
AA/A
Dark Current [e- pix
1
s-1]
Sky Background [e- pix- 1 s-]
Exposure times:
To overcome read noise [s]
To reach 50% full well [s]
Photometric Precision:
In 1 exposure to 50% full well:
In 10 minutes:
In I hour:
0
4.6 x10-
3
4.2x10-4
1.7 x10- 4
640x512-pixel AP1 121 we tested, 8 outputs from each detector would be simultaneously
digitized. Even at the minimum 12-bit resolution, the data rate is 4 GB/s. Therefore, it is
desirable to perform the up-the-ramp processing in real time (either in the camera FPGAs,
graphics processors, or software), which effectively compresses the data rate to 20 MB/s.
Over 500 GB would still be accumulated over the course of a night.
On the other hand, not every frame has to be recorded during long up-the-ramp exposures. More than 1200 frames could be recorded during the 72-second exposure time we
consider for Y band observations. However, we have shown that the sample-up-the-ramp
noise appears to asympotote to ~ 40e- with >100 frames. One should consider the tradeoffs of only recording one of every few frames in this application; the data rate would scale
directly with its duty cycle.
113
2.6
Conclusions
We currently have an operational InGaAs camera with the following characteristics, which
no commercially-available product can offer simultaneously:
" A format of 640 x 512 pixels
" Read noise of 43 e- pix-
1
with up-the-ramp processing
" Nonlinearity less than 3%, which is correctable to 0.4%, across a well capacity of
47k e" Low persistence on the timescales required to dither or slew to new targets
" Uncorrelated photometric noise on timescales up to 3 hours
" A dark current of 255 e- pix- 1 s-
1
at -40'C, below the J sky for a 1.0 m telescope
with 1" pixels
" A cost per pixel <3 cents
However, the original goal of sky-limited performance can only be met in the Y band if
the detector is cooled to -50'C, or as we assumed in Table 2.2, -60'C. Our current camera
configuration cannot operate below -40'C, so it is desirable to re-package the detector with
an additional stage (or stages) of thermoelectric cooling. Such repackaging should also
include a cold shield around the detector held at ~0 C to limit its sensitivity to thermal
emission.
A larger detector format is also desirable. FLIR uses the same pixel architecture as
the AP 1121 in their newly-designed 1920x 1080-pixel detector, so many of our test results
should carry over to this device. To keep the data volume of these devices manageable,
real-time up-the-ramp processing should be implemented, and it is not necessary to record
every frame during a long exposure.
With these relatively straightforward improvements, InGaAs detectors can offer a relatively inexpensive route towards sky-limited imaging and photometric performance across
wavelengths from 0.85 to 1.65 pm.
114
Appendix A
Tables
115
Table A. 1: Catalog of simulated TESS detections. This catalog is based on one realization of the
Monte Carlo simulation using 2 x 10 5 target stars that are observed with a 2 min cadence.
a
6
Rp
[0]
[0]
0.682
1.146
1.560
2.005
2.620
3.407
4.883
4.980
6.073
7.441
84.329
57.854
-12.056
-72.624
-9.737
9.235
56.926
-22.130
-38.646
-79.559
[Re]
1.96
2.30
1.10
2.72
2.79
3.69
1.32
1.84
1.72
2.88
7.806
-44.121
1.97
7.821
8.312
8.799
9.066
9.128
9.880
10.304
10.578
12.246
13.330
13.442
13.697
15.894
16.947
18.375
18.395
18.526
20.034
38.740
-41.907
-10.533
-32.566
-74.305
-29.778
22.675
66.188
34.334
-15.628
25.899
46.963
-54.826
15.821
-71.580
20.203
47.602
-44.771
1.96
2.64
1.85
1.78
3.65
3.81
3.04
2.18
1.72
5.20
2.08
1.69
4.26
2.39
2.21
2.49
1.35
4.71
P
[days]
4.68
11.04
4.60
6.23
6.14
18.84
1.03
2.08
1.00
29.85
6.04
0.96
4.15
15.30
8.91
4.69
20.14
3.82
7.40
2.54
5.79
11.03
2.41
14.66
6.99
9.56
9.24
2.30
19.50
S
[Se]
1753.9
43.0
11.4
11.2
313.6
16.7
2369.4
692.0
605.0
14.7
2.8
491.5
15.2
19.9
60.6
109.1
146.5
118.8
4.3
609.4
1762.6
1.1
16.5
51.1
8.0
66.4
387.8
21.8
325.0
K
[m/s]
1.43
1.82
1.10
3.96
2.21
2.63
1.74
2.58
3.53
1.65
5.87
4.22
4.77
1.54
1.68
4.10
1.86
3.61
4.07
2.13
3.08
7.18
5.01
2.84
3.62
1.91
1.56
3.22
1.98
R*
[Ro]
1.79
0.81
0.36
0.43
1.15
0.70
0.99
0.89
0.66
0.79
0.19
0.63
0.39
0.69
0.75
0.71
1.41
0.72
0.30
0.87
1.55
0.18
0.27
0.87
0.40
0.78
1.39
0.31
1.56
Teff
[K]
7047
5023
3501
3658
5957
4954
5754
5508
4649
5383
3241
4427
3552
4809
5156
4944
6699
4742
3396
5834
8260
2958
3365
5585
3564
5370
6714
3435
7980
V
7.45
8.26
13.11
14.43
9.49
10.29
8.60
10.39
12.02
10.11
16.24
12.33
13.85
10.75
10.27
12.01
8.14
11.64
15.87
9.42
8.21
15.10
15.51
10.90
14.33
11.00
8.91
14.47
8.10
Ic
7.08
7.29
10.95
12.43
8.84
9.27
7.90
9.62
10.84
9.30
13.73
11.01
11.74
9.67
9.34
10.99
7.69
10.56
13.58
8.75
8.09
12.77
13.18
10.15
12.23
10.19
8.46
12.23
7.92
J
6.81
6.62
9.73
11.28
8.39
8.60
7.41
9.06
10.06
8.72
12.34
10.16
10.55
8.95
8.73
10.31
7.35
9.78
12.29
8.27
7.95
11.02
11.88
9.60
11.03
9.60
8.13
10.98
7.75
K,
6.66
6.10
8.95
10.48
8.08
8.02
7.05
8.64
9.39
8.27
11.57
9.42
9.76
8.33
8.21
9.73
7.14
9.15
11.52
7.91
7.91
10.24
11.11
9.20
10.25
9.15
7.93
10.21
7.69
DM
4.85
2.15
2.35
4.45
5.10
3.75
3.70
5.00
4.90
4.35
3.35
4.75
3.40
4.00
4.15
5.50
4.75
4.85
4.45
4.30
5.95
1.55
3.80
5.55
3.95
5.20
5.50
3.25
5.70
Dil.
1.00
1.00
1.01
1.00
1.00
1.00
1.01
1.01
1.00
1.00
1.00
1.01
1.01
1.00
1.00
1.04
1.00
1.00
1.25
1.00
1.00
1.00
1.10
1.00
1.00
1.00
1.01
1.01
1.00
log1 0
oV
-4.65
-4.94
-4.66
-3.55
-4.95
-4.61
-4.46
-4.64
-4.83
-4.87
-3.61
-3.59
-3.61
-5.13
-4.80
-4.86
-4.24
-4.68
-3.66
-4.72
-4.52
-4.33
-3.53
-4.09
-3.45
-4.91
-4.48
-3.61
-4.68
SNR
8.5
26.1
7.4
17.2
14.7
32.4
7.4
10.5
8.5
22.9
12.0
9.8
19.8
8.8
8.9
25.0
12.5
20.2
9.7
10.6
26.5
20.5
8.5
28.1
9.8
8.8
8.0
8.0
26.6
6.11
11.16
20.799
73.741
2.18
21.356
-25.761
2.58
21.454
22.856
-72.359
41.389
1.33
2.49
5.81
23.278
23.410
-12.103
-44.933
2.59
1.95
23.929
24.069
24.080
24.490
26.146
26.422
27.917
28.457
28.707
28.784
29.024
29.116
-85.708
-0.917
-72.601
1.006
-55.294
-1.928
-52.969
20.909
-53.940
-14.506
-75.143
74.117
1.95
1.43
1.43
1.98
6.41
2.70
2.73
2.73
1.64
1.92
2.53
1.70
9.32
11.61
5.18
29.370
-65.268
1.30
29.632
29.682
-57.058
-19.023
1.74
3.23
29.825
29.890
-9.126
-29.956
2.34
1.43
29.914
29.947
30.058
30.334
31.025
31.747
-56.009
-55.155
-14.954
-44.361
-45.531
28.937
1.87
2.70
1.71
2.48
2.52
1.69
31.847
21.089
2.19
32.108
32.404
32.691
33.589
33.876
51.020
-34.710
-43.949
-13.701
4.115
1.89
2.39
4.15
2.40
20.03
8.52
3.48
0.83
4.15
30.93
13.89
22.61
17.49
7.80
6.94
3.56
2.26
2.48
35.22
10.45
17.55
24.44
2.17
25.77
11.55
10.05
9.35
2.30
9.57
5.39
18.46
18.00
10.14
4.12
1.72
4.38
1.21
3.51
2.15
2.81
1.39
0.28
0.73
0.39
0.80
0.35
6730
3370
4764
3564
5248
3480
9.11
16.14
10.11
14.18
11.03
13.56
8.67
13.80
9.03
12.08
10.17
11.38
8.34
12.50
8.27
10.89
9.55
10.15
8.14
11.72
7.65
10.11
9.06
9.37
5.70
4.50
3.40
3.80
5.15
2.70
1.02
1.00
1.00
1.01
1.00
1.01
-4.56
-4.90
-5.13
-3.61
-4.63
-3.77
3.29
3.35
3.07
3.85
3.30
3.55
3.20
3.41
1.47
0.40
0.33
0.71
0.37
0.83
0.35
0.33
0.34
0.83
3560
3448
4934
3509
5117
3473
3431
3452
5117
14.69
14.52
12.33
14.37
10.10
14.92
15.96
15.17
10.10
12.58
12.29
11.28
12.21
9.18
12.72
13.70
12.95
9.18
11.38
11.04
10.59
10.99
8.52
11.48
12.43
11.69
8.52
10.60
10.27
10.00
10.21
8.02
10.70
11.65
10.92
8.02
4.30
3.45
5.75
3.70
4.15
4.05
4.85
4.15
4.15
1.26
1.00
1.07
1.00
1.00
1.00
1.01
1.00
1.00
-3.60
-4.30
-4.63
-3.59
-4.88
-3.67
-3.20
-3.66
-4.87
711.9
10.3
3.57
3.07
2.44
1.34
0.94
0.30
0.72
0.76
0.89
0.83
3430
5082
5035
5929
5117
13.08
10.91
12.00
10.00
10.10
10.85
10.00
11.07
9.35
9.18
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247.9
5.4
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3227
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7.9
9.2
17.0
31.8
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74.3
19.4
13.4
7.5
11.2
12.9
9.4
22.5
7.6
517.9
8.7
8.2
16.9
12.9
8.7
42.4
17.5
20.2
8.9
11.9
10.8
7.4
8.3
13.4
7.8
33.5
25.5
51.6
9.4
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1.56
1.42
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1.54
1.67
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250.1
30.3
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0.90
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2.6
264.6
1070.9
965.8
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0.7
29.4
162.2
283.5
6675.8
33.6
38.0
53.3
94.4
2.4
6.0
8.8
72.6
36.6
147.8
1.6
33.3
2.9
62.7
8.5
11.3
10.2
8.2
8.4
10.5
15.4
12.2
10.1
11.0
8.5
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47.3
36.1
9.9
46.0
34.6
8.7
16.2
9.4
23.5
8.0
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7.6
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267.328
267.341
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24.4
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222.5
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16.4
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119.1
3.6
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424.6
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239.6
3522.2
3.50
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30.6
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29.7
2.6
6.3
7.6
570.8
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260.7
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21.8
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9.70
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9.84
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1.28
1.38
0.77
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2.5
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67.3
2531.9
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2.6
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24.2
9.3
41.4
14.4
17.2
77.6
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8.5
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0
2
0
0
2
0
1
0
1
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1
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1
1
1
0
0
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3474
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3582.0
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221.5
385.7
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1430.2
865.8
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0
1
0
1
1
0
1
0
0
0
0
1
0
1.30
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2.15
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0.72
0.46
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0.81
48.65
7.9
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98.3
11.5
8.6
8.9
183.4
26.3
7.9
94.1
22.3
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7.4
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3.55
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8.14
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255.9
12.5
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3321
5610
14.67
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10.82
10.08
9.49
5.35
3.44
3.15
1.76
1.15
0.43
1.17
6012
3611
6053
8.91
14.41
9.35
8.28
12.36
8.74
7.83
11.19
8.31
7.52
10.39
8.01
4.55
4.30
5.10
4.23
2.64
5.89
1.99
2.07
2.48
0.25
0.69
0.25
0.75
0.69
0.70
3328
4813
3345
4853
4767
4842
16.18
11.57
13.91
11.95
11.82
10.08
16.29
10.16
13.57
15.38
7.59
8.68
11.96
11.89
9.90
13.80
10.46
11.56
10.91
10.72
9.06
13.85
9.35
11.46
13.02
6.97
7.98
10.88
10.73
8.84
12.47
9.73
10.25
10.17
9.99
8.32
12.50
8.77
10.26
11.27
6.53
7.47
10.19
9.96
8.15
11.71
9.11
9.49
9.58
9.35
7.72
11.73
8.31
9.48
10.49
6.24
7.11
9.59
9.29
7.56
4.20
4.75
1.95
5.40
5.00
3.40
3.95
4.30
3.10
1.85
3.80
3.65
5.30
4.85
3.25
9.78
8.76
8.02
7.42
3.20
9.19
9.78
8.32
8.76
7.70
8.02
7.22
7.42
3.40
3.20
10.58
10.37
14.44
9.82
9.82
9.69
12.38
9.04
9.28
9.20
11.21
8.47
8.88
8.85
10.41
8.04
5.30
5.50
4.25
4.35
9.16
10.24
11.40
8.39
9.30
10.30
7.84
8.64
9.53
7.42
8.11
8.91
3.70
4.15
4.70
9.03
11.83
8.19
10.76
7.59
10.06
7.13
9.46
3.25
5.15
12.42
11.19
10.40
9.71
5.15
118.7
7.9
38.4
13.0
22.3
2.0
334.3
25.9
17.5
426.2
270.0
20.0
42.6
58.2
4.98
0.23
3294
2.64
3.01
6.84
2.15
1.97
2.63
2.88
2.17
0.76
0.39
0.18
1.46
0.94
0.71
0.67
0.70
5370
3547
2951
6039
5728
4903
4668
4922
45.6
51.6
2.41
3.60
0.73
0.84
4842
5236
126.9
170.4
89.1
1.28
2.18
0.73
0.91
4842
5546
1.96
0.99
5808
2.0
51.8
89.9
2.47
0.41
3597
2.87
0.87
5458
1.43
0.85
5508
24.9
63.9
102.6
1.98
0.80
5070
2.58
0.74
4753
0.70
0.81
5309
2.73
5.19
0.70
0.65
4878
4590
62.5
26.7
7.6
33.7
9.8
9.8
10.2
7.9
38.2
7.5
15.4
47.3
8.6
16.6
10.0
11.9
12.2
8.4
12.2
17.8
21.5
26.2
67.3
7.6
13.7
10.9
10.4
31.3
9.9
18.9
14.6
8.3
14.2
44.4
0
4.20
1.36
1.00
1.00
-3.49
-4.73
-4.42
3.05
1.00
-4.17
15.3
28.0
10.27
10.09
10.70
9.58
6.82
10.08
6.13
8.95
9.60
2.70
3.45
3.25
3.70
3.75
4.45
2.95
2.35
3.10
6.73
4.15
1.08
1.00
1.02
1.00
1.00
1.00
1.00
1.00
1.02
1.00
-3.53
-4.42
-3.66
-3.43
-3.83
-3.61
-4.41
-3.49
-3.64
-4.86
9.9
11.3
11.8
8.5
205.4
8.5
52.4
15.6
14.5
14.4
351.500
351.604
8.489
-62.281
1.23
2.69
1.37
20.49
12.4
22.1
5.14
1.77
0.14
0.77
3141
14.45
12.49
11.01
10.21
1.25
5309
11.51
10.68
10.07
9.60
5.60
353.058
353.395
78.671
78.649
2.76
2.58
7.55
4.48
5.9
114.3
4.34
3.11
0.36
0.70
3491
14.96
12.78
11.55
10.77
4924
9.60
8.59
7.90
7.32
355.560
355.739
355.965
356.232
357.152
357.956
358.073
358.612
358.941
-11.557
12.595
-10.331
38.125
-41.159
-45.862
-20.043
-13.185
77.704
1.29
1.71
1.66
1.60
11.72
2.34
4.85
1.79
1.93
0.82
2.30
2.99
1.43
2.13
7.13
20.48
12.36
12.10
12.79
11.51
7.60
11.99
6.82
10.95
11.58
11.04
10.87
11.45
10.38
7.14
10.90
6.41
9.73
10.38
2.50
0.25
0.35
0.24
0.45
1.11
0.49
1.03
0.36
0.37
1.28
14.74
14.28
15.16
13.48
8.26
13.89
7.36
13.11
13.71
9.565
4.73
4.16
4.71
3.84
12.22
3.10
3.12
4.15
3.78
1.61
3330
3487
3305
3686
5916
3762
6324
3501
3530
359.428
59.1
27.2
9.5
90.2
1191.3
13.1
80.8
24.0
15.5
371.7
6982
7.58
7.19
6.89
2.63
3.77
9.75
17.7
12.2
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