Exoplanetary System WASP-3: Measurement of
the Projected Spin-Orbit Angle and Evidence for
Stellar Activity
MASSACHUSMS INS
OF TECHNOLOGN
JUL 0 7 2009
by
Anjali Tripathi
LIBRARIES
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2009
@ Anjali Tripathi, MMIX. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
ARCHIVES
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Department of Physics
May 21, 2009
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Professor Joshua N. Winn
Department of Physics
Thesis Supervisor
Accepted by .........................
Professor David E. Pritchard
Senior Thesis Coordinator, Department of Physics
Exoplanetary System WASP-3: Measurement of the
Projected Spin-Orbit Angle and Evidence for Stellar Activity
by
Anjali Tripathi
Submitted to the Department of Physics
on May 21, 2009, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
In this thesis, I present new spectroscopic and photometric observations of WASP-3,
a transiting extrasolar planetary system. From spectra obtained during two transits,
I use the Rossiter-McLaughlin effect in a simplified physical model to determine the
projected spin-orbit angle between the planetary orbital axis and the stellar rotation
axis, A = 2 .91 4 degrees. I use the new photometric data to refine the system parameters. Additionally, I present evidence for stellar activity, as the spectra obtained on
UT June 19, 2008 reveal an increase in radial velocity to a peak magnitude of 75 m s- 1
larger than the expected orbital velocity. Such an anomaly was not observed during
a subsequent transit. I find that a good fit to the radial velocity measurements requires omitting the anomalous data and adding a large stellar jitter of 11 m s - 1 to
the measurement uncertainties. The resulting planetary mass, Mp = 2.14:006 MJ
differs from previously reported measurements which found Mp = 1.76-0. M . Together, these observations provide evidence for a region of stellar activity on WASP-3,
and the observed anomaly suggests that material in this region exhibited a coherent
horizontal motion of appoximately 10 km s-' across the stellar surface at the limb.
Thesis Supervisor: Professor Joshua N. Winn
Title: Department of Physics
Acknowledgments
This thesis would not have been possible without the gracious support of a number of people. I would like to thank Professor Joshua Winn for his time, patience,
and guidance over the past year and for introducing me to the wide world of exoplanets. For their assistance and daily encouragement, I am grateful to the faculty,
staff, and graduate students of the sixth floor of building 37, particularly Professors
Chakrabarty and Hughes, as well as Joshua Carter, N. Madhusudhan, John Rutherford, and Michael Louis Stevens. I would also like to thank my peers, Josiah Schwab
and Katherine de Kleer, for engaging in helpful discussions, even in the midst of writing their own theses.
For the reduction of radial velocities, work with their analyses, and support via email,
I would like to thank Dr. John Johnson. Additional thanks go to the rest of the observing and data reduction team, including Dr. Geoff Marcy, Dr. Andrew Howard,
Katherine de Kleer, and the professional observers at the Fred L. Whipple Observatory. Given that some of the data herein was taken using the W.M. Keck Observatory,
I wish to recognize and acknowledge the very significant cultural role and reverence
that the summit of Mauna Kea has always had within the indigenous Hawaiian community. It was most fortunate that there was the opportunity to conduct observations
from this mountain.
Lastly, I have sincerely appreciated the warmth and support of my friends and family,
who have always made it possible for me to do the things I love, particularly scientific
research. For keeping me safe late nights in the office, either with her company or
that of her bike, I thank Cathy Zhang, and for getting me up in the mornings and
keeping me going throughout the day, I thank my loving parents.
Contents
1
Introduction
13
.
.............
.
.....
Transiting Extrasolar Planets
1.2
Photometric Transit Parameters ...................
1.3
The Rossiter-McLaughlin Effect and Spin-Orbit Alignment ......
16
1.4
WA SP-3 . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .. . .
18
1.5
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
..
2 Observations and Data Reduction
3
13
1.1
14
21
2.1
Photometric Measurements
.......................
2.2
Radial Velocity Measurements ...................
21
...
23
Transit Time Fitting and Determination
25
3.1
Model Details: Eccentricity and Limb Darkening . ...........
25
3.2
Photometric Parameter Set
3.3
Photometric Fitting Using Amoeba . ..................
28
3.4
Markov Chain Monte Carlo (MCMC) Method . ............
29
3.5
Photometric MCMC fitting
30
3.6
Characterization of Time Correlated Error . ..............
3.7
Determination of Ephemeris ..............
. ..................
....
.......................
30
........
31
4 Parameter Fitting and Determination
4.1
Model ......
4.2
Data Set and Uncertainties . ..................
........
27
35
.................
35
.....
36
4.3
Joint Fit to Radial Velocities and Photometry . ............
38
5
Results
41
6
Evidence for Stellar Activity
51
6.1
Radial Velocity Observations and Results Suggestive of Stellar Activity
51
6.2
Starspot model ..............................
53
6.3
Observational consequences
6.4
Underlying physics ...................
7
Conclusions
...................
....
........
57
60
63
A Tables
65
B Figures
67
List of Figures
1-1
Diagram of an exoplanet transit and accompanying light curve . . ..
15
1-2
Spectral Changes due to the Rossiter-McLaughlin Effect
17
1-3
Radial velocity signal from the Rossiter-McLaughlin Effect ......
18
2-1
Radial velocity measurements in time . .................
24
3-1
May 15 and June 10 Photometry with Best Fit Model .........
33
3-2
Transit timing plot ............................
34
5-1
Probability distributions for parameters used in the 11 parameter model 44
5-2
Probability distributions for physical system parameters (11 parameter
.......
m odel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .
5-3
Photometry and RV data with best-fit result from the 11 parameter
m odel . . . . . . . . . . . . . . . . . . .
5-4
45
. . . . . . . . . . . .. . .
46
Best fit to RV data with the 11 parameter model for the full orbital
phase. Open circles are those that were not included in the fit. ....
47
5-5
Probability distributions for parameters used in the 12 parameter model 48
5-6
Probability distributions for physical system parameters (12 parameter
m odel) . . . . . . . . . . . . . . . . . . . . .
5-7
49
Photometry and RV data with best-fit result from the 12 parameter
model
6-1
. . . . . . . . . . . . .
...........
..
...... ..............
....
Starspot position during observations . .................
B-i Probability distributions for mid-transit times and airmass parameters
50
58
67
10
List of Tables
3.1
WASP-3 Mid-transit times
.......................
5.1
System Parameters of WASP-3
.....................
42
6.1
Individual transit Rp/R, values .....................
59
A.1 HIRES Doppler Shift Measurements of WASP-3 . ...........
32
65
12
Chapter 1
Introduction
In this thesis, I provide measurements of physical parameters of the transiting exoplanetary system WASP-3.
This chapter offers an introduction to the material,
beginning with a description of exoplanets, particularly transiting systems. Additionally, I discuss the parameters that I will present in this thesis, including the
projected spin-orbit angle, and the system under consideration, WASP-3. Finally, I
provide an overview of this thesis.
1.1
Transiting Extrasolar Planets
Extrasolar planets, or exoplanets, are planets orbiting stars far beyond the limits of
the Solar System. With the study of exoplanets still in its youth (one of the first exoplanets, 51 Peg b was detected in 1995 [26]), much remains to be learned in this field.
The study of exoplanets is a key element to understanding the evolution of the Solar
System and to searching for life outside the Earth or for other potentially habitable
regions. For each of these pursuits, it is crucial to have accurate characterizations of
exoplanet parameters .
The majority of exoplanets, including some of the earliest detections, were discovered by radial velocity surveys that measured the line-of-sight component of the
wobble of a star due to its motion around the center of mass between it and an orbiting planet. While this method continues to be used, one of the most promising means
of detecting and characterizing exoplanets is to observe them transiting their parent
stars. A planet is known to transit its parent star if it is observed crossing the face of
its parent star nearly edge-on, as shown in Figure 1-1. Just as one can deduce much
information by observing an eclipsing binary, so too can one derive a wealth of information by observing a transiting exoplanet [7, 37]. Photometric observations giving
the amount of starlight received from the host star in and out of transit can provide
an indication of the relative sizes of the planet and star, among other parameters.
Furthermore, spectroscopic observations, which yield radial velocity measurements,
made in and out of transit can offer insight into the spin-orbit alignment of the system. In this thesis, I seek to determine parameters that come out of both of these
types of observations.
1.2
Photometric Transit Parameters
As shown in Figure 1-1, as a planet crosses the face of its parent star in projection,
there is a concomitant decrease in the relative flux of the system. By measuring
the amplitude and timing of this loss of light, it is possible to determine parameters
characterizing the planetary orbit around the star. If I consider both the star and
planet to be spherical with radii R, and R, respectively, the decrease in relative flux,
or transit depth 6 will be proportional to the ratio of the areas of the planet and star,
or (Rp/R,) 2 . Thus, the relationship between the relative observed flux, transit depth,
and radius ratio is
=1
1- 6
Out-of-transit (OOT)
1-
(R)
2
In-transit
For the sake of simplicity, I have omitted the cases of transit ingress and egress,
when the planet is only partly covering the limb of the star, from this expression.
Additionally, here and throughout the remainder of the thesis, I have assumed that
the planet has zero surface intensity, so it only blocks the starlight that we observe.
Thus, I ignore the possibility of the loss of reflected starlight when the planet travels
behind the star, known as an occultation.
Flux
Time
T>
I
Figure 1-1 Diagram of a transit and an accompanying light curve, from [37].
The times of importance during a transit are the mid-transit time tc, the planetary
orbital period P, the approximate transit duration T, and the partial duration T, or
approximate ingress/egress time, as denoted in Figure 1-1. In turn, these parameters,
combined with the transit depth 5, uniquely determine some of the parameters of
the planetary orbit, such as the impact parameter normalized to the stellar radius,
b, and the inclination i relative to the reference plane. Below I provide approximte
expressions that relate the transit parameters (ta, T, T, 6 ) and the physical parameters
of the system (Rp/R, b, i) when the orbital eccentricity is zero, which is common
of transiting exoplanets [23]. For the derivation of these expressions or relationships
between other parameters, see [6, 37].
Rp/RS ,. VI,
b
i
arccos
I - Tv ,
212nb
(1.2)
(1.3)
(1.4)
where n is the mean motion, or 27/P. For all of these expressions, I have assumed
that the star has a surface of constant intensity, with a relative out-of-transit flux of
1. However, in reality, the observed intensity of the star is less near the limb, an effect
known as limb darkening, because our viewing geometry of the star is such that we
actually see fewer layers of the atmosphere at the limb of the star. Thus, corrections
must be made to account for the change in surface intensity due to this effect, as
detailed in Section 3.1. Each of the photometric parameters described above can be
determined from the photometry of the transit, which can then be converted to other
physical parameters.
1.3
The Rossiter-McLaughlin Effect and Spin-Orbit
Alignment
One of the unique properties that can be determined from radial velocity data obtained during a transit is the projected spin-orbit angle of the system. As the star
in the system rotates, its spectral lines are Doppler broadened because one limb of
the planet is rotating towards the observer, causing a blueshift, while the other limb
is rotating away from the observer, causing a redshift. When the planet transits the
star, it passes over the stellar surface, blocking some part of the stellar surface at
each instant. As it crosses the stellar surface, it thus blocks some of this red or blue
shift that would otherwise be seen by the observer and, thus, leads to the removal
of part of a spectral line, as seen in Figure 1-2. The resulting change in the radial
velocity signal (where the velocities are approximated by the mean of the spectral
profile, even when anomalous) is known as the Rossiter-McLaughlin (RM) effect and
is a telling indication of the spin-orbit alignment of the system.
The spin-orbit alignment is parameterized by A, the projected angle between the
stellar rotation axis and the planetary orbital axis. A = 0 is indicative of alignment
between the two axes. So long as a transit does not occur directly over the equator
of the star (i = 900), a time series of the radial velocity measurements during transit
can reveal A, as seen in Figure 1-3. While a range of A are shown in Figure 1-3,
Figure 1-2 The geometry of the Rossiter-McLaughlin (RM) effect and the changes it
produces in the shape of a spectral absorption line, from [13]. Note that when the
change in line shape cannot be resolved, the center of the line is observed to move,
yielding a Doppler velocity.
A has been near zero for most observedexoplanetary systems [11], with the notable
exception of XO-3, which has A > 350 [16, 39].
A fit to the radial velocity data during transit, using the expression for the RM
amplitude given later in Section 4.1, is used to determine A as well as the line-ofsight rotational velocity of the star, v sin i. Out-of-transit radial velocity also gives
information about the orbital parameters of the system, including the mass of the
planet Mp (which requires that a stellar mass and inclination be assumed) and the
relative velocity of the system y. The relation of Mp to the orbital radial velocity
signal is seen in the definition of the orbital semi-amplitude, K. In the case of zero
eccentricity, K is defined as
K-
27G 1/3
Mp sini
.
(M + MP) 2/ 3
P
(1.5)
(1.5)
Thus, from radial velocity measurements during transit, we can determine the spinorbit alignment, as well as other system parameters.
7
60
E
40
20
0
.
.--.
-
-
-
60
40
20
60
E
.
S-60
-20
-40
.
0 -60
-60
-1
0
Time [hr]
1
2
-2
-1
0
Time [hr]
40
20
0
O
-20
-40
----
-0
-20
-40
-2
E
1
2
-2
-1
0
Time [hr]
1
2
Figure 1-3 Radial velocity signal from the RM effect for various values of A, from
[13]. The solid line is based on a model that includes limb-darkening for the host
star, while the dotted line does not include limb darkening. The dashed line shows
the orbital contribution to the radial velocity signal.
1.4
WASP-3
In this thesis, I focus on the recently discovered transiting exoplanet system, WASP31. In their discovery paper, Pollacco et al.
(2008) announced that WASP-3 is
home to one of the hottest exoplanets yet discovered [29]. Using both photometric
and spectroscopic data, the discoverers found that the WASP-3 system comprises a
196033 K, 1.7680.8 Mj planet transiting an F 7-8V parent star, with an apparent
V magnitude of 10.485 [31]. In addition, they found that the semi-major axis is a
mere 0.0317 + 055 AU. The combination of the planet's mass, nearly that of Jupiter,
and its semi-major axis, less than the distance between the Sun and Mercury, make
it completely any planet within our own Solar System. In our Solar System, such
massive planets are found with significantly larger semi-major axes; Jupiter's semimajor axis is 5.2 AU [9]. As a result, WASP-3 is an interesting candidate for study.
WASP-3 is particularly well-suited to measurements of the projected spin-orbit
angle, A. As previous radial velocity measurements of the system out-of-transit reveal,
WASP-3 is a rapid rotator. Its line-of-sight stellar rotational velocity, v sin i = 13.4 ±
1Regarding the nomenclature: In theory, WASP-3 references the host star of the exoplanet system
and WASP-3b references the planet. In practice, however, both the system and the planet are
referred to as WASP-3, where the meaning is derived at from context.
1.5 km s-
1
[29], is larger than that of many exoplanetary systems. As a result of
this large velocity, WASP-3 was expected to have a considerable contribution to the
radial velocity signal from the Rossiter-McLaughlin effect. It is for this reason that it
was chosen for spectroscopic observation. Determination of A for this system is also
interesting because it can help in understanding planetary migration theories. Given
WASP-3's large size and close-in orbit, understanding its migration and why similar
such planets do not exist in our own Solar System is of great interest.
1.5
Overview
As a transiting exoplanet system, WASP-3 offers the unique opportunity to study
properties of both the planet and its parent star, as well as the relative spin-orbit
alignment, given by the projected spin-orbit angle A. The latter quantity is of particular interest because WASP-3 is hotter and has a smaller semi-major axis than
planets of comparable mass in our own solar system. In this thesis, I use new data
from the Fred Lawrence Whipple Observatory (FLWO) and the W.M. Keck Observatory to study WASP-3. From this data, I provide updated parameters for the system,
determine its projected spin-orbit angle, and offer evidence for possible stellar activity.
In this thesis, I will describe how the aformentioned parameters were determined
for WASP-3. Chapter 2 describes the observation details and data reduction for the
photometry and spectroscopy of WASP-3. Chapter 3 describes the general method,
including the Markov Chain Monte Carlo method, used for fitting multiple nights of
photometry to determine the ephemeris. Chapter 4 describes the model and procedure
used to jointly fit the radial velocity and photometric data to find system parameters.
Chapter 5 provides the results of this fitting. In Chapter 6, I provide evidence for
stellar activity and suggest avenues for continuation of this work. Chapter 7 offers
the conclusions of this work.
20
Chapter 2
Observations and Data Reduction
In this chapter, I provide details about the observing conditions and methods used to
obtain the photometric and spectroscopic data that were used in this study. For the
photometric data presented, I include a description of the reduction of optical images
taken at the Fred L. Whipple Observatory to photometry. For the spectroscopic data
presented, I detail how raw spectra were reduced to Doppler shift radial velocity
measurements.
2.1
Photometric Measurements
We observed' two partial transits and one complete transit on UT 2008 May 15, June
10, and June 21, respectively, using Keplercam on the 1.2m telescope at the Fred
Lawrence Whipple Observatory (FLWO) on Mount Hopkins in Arizona. Observations
were taken through a Sloan i-band filter and spaced approximately 33 seconds apart,
with exposure times of 20 seconds each. Observations on the first night (May 15)
span from mid-ingress until after the transit ended. The second night of observations
(June 9) began mid-transit and again lasted until after the end of the transit. On
the third night (June 21), observations spanned the complete transit, with out-oftransit measurements made both before and after the transit. While some clouds
were seen on the first night of data collection (May 15), the subsequent observations
1
Gil Esquerdo and Mark Everett at the Fred L. Whipple Observatory made these observations.
were taken in clear conditions. Throughout each night of observations, images were
taken through air masses ranging from 1.0-1.3, 1.0-2.2, and 1.2-1.3, respectively.
Raw images from these three nights were then reduced and aperture photometry
was performed2 . Image calibration was achieved using standard IRAF3 procedures
for bias subtraction and flat-field division. To convert the calibrated images to flux
counts, aperture photometry was performed on WASP-3 and 6-20 nearby, comparison
stars using standard IRAF procedures. An aperture radius of 8, 13, and 11 pixels
was used for each night of data, respectively. We then performed differential photometry to determine the change in relative flux over time and to produce a final light
curve. We produced a comparison light curve by averaging the light curves of the
nearby comparison stars, as per [19]. The light curve of WASP-3 was divided by the
comparison light curve to correct for systematic trends that occurred throughout the
observations and then normalized such that the mean out-of-transit flux was unity.
To characterize the uncertainties in our data, we calculated the quadrature sum
of the scintillation and Poisson, or shot, noise of the target star using the formulas
of [42, 10].
We found that this contributed an average RMS of 8 x 10- 4 , which
was nearly half of the standard deviation of the out-of-transit flux of 1.1 x 10- 3,
1.6 x 10- 3 , and 1.5 x 10- 3, respectively. We used the latter as the uncertainties for
our flux measurements.
As described later in Section 3.1, I corrected the data for the effects of air mass.
Since air mass varied as a function of time, which in turn affected the measured
flux, I divided each of our light curves by a linear function of air mass when fitting
the system for other parameters. The light curves for the first two nights, with this
correction for correlated air mass are presented in Figure 3-1.
2
Katherine de Kleer carried out both of these steps.
Image Reduction and Analysis Facility - software for performing astronomical data reduction
and photometry
3
2.2
Radial Velocity Measurements
We observed the transits of UT 2008 June 19 and 21 using the High Resolution Echelle
Spectrometer (HIRES) on the Keck I 10m telescope at the W.M. Keck Observatory
[36]. We also observed the system out-of-transit on UT 2008 July 27. Of these three
nights of observations, only the second night had simultaneous FLWO photometry,
which spanned the full transit of WASP-3. This photometry was taken to investigate
possible recurrence of an anomalous Doppler shift that was observed on the first night
of radial velocity observations. This anomaly is discussed in Section 6.1.
Our data collection followed the same procedure as that of Johnson et al. (2008)
in their HIRES measurements of HAT-P-1 [22] . The spectrograph was set up in
the same configuration as used for the California planet search [20, 25]. Calibration
of the instrumental response and wavelength scale was achieved using the red crossdisperser and the I2 absorption cell . The slit width was set by the 0.85" B5 decker,
and the typical exposure times ranged from 3 to 5 minutes, giving a resolution of
about 60,000 at 5500
A
and a signal-to-noise ratio of approximately 120 pixel- 1
Doppler shifts were derived from the data using the algorithm of Butler et al. (1996)
with subsequent improvements [4]. For a given spectrum, measurement errors were
derived from the weighted standard deviation of the mean among the solutions for
individual 2 A spectral segments, with typical measurement errors ranging from being
6.1 to 10.0 m s- 1 , with a median of 7 m s-1 . Doppler shifts are given in Table A.1
and plotted in Figure 2-1.
200
S
00
+
0
-
0
-
0i
0
0
-+
0
-100
-
-200
*
I
D
I I II |
I I
40
I
60
50
HJD-2,454,600
70
8
Figure 2-1 The radial velocity measurements obtained on UT 2008 June 19 (filled
circles), June 21 (diamonds), and July 27 (crosses)
Chapter 3
Transit Time Fitting and
Determination
In this chapter I seek to determine the ephemeris, or transit time parameters, for
WASP-3 using three nights of transit photometry. The parameters that I seek, the
mid-transit times of each transit and the system's orbital period, are needed to determine the system parameters, a process which I describe in the next chapter. Measurement of transit times is also important because any non-periodicity in the timing
might be due to gravitational perturbations from hitherto-undetected additional planets or satellites [1], [18]. Thus, the transit times being determined in this chapter serve
two functions: to help understand potential companions in the system and for use in
further fitting to determine the ultimate system parameters that I seek.
3.1
Model Details: Eccentricity and Limb Darkening
To determine the ephemeris, I fit the photometry to a simplified model of a transit.
Based on the findings of the discovery paper, Pollacco et al. (2008), and Madhusudhan
and Winn (2009), who find that the system's eccentricity is less than 0.098 to within
95.4 confidence [23], I assume that the eccentricity of the planet's orbit is zero. This
assumption simplifies analysis of the system and enables me to use the equations
given in Section 1.2.
The model that I use for fitting the photometry takes into account limb darkening
on the stellar surface. As previously noted, limb darkening changes the flux observed
during transit. I incorporated limb darkening's effect on flux using the exact expressions given by Mandel & Agol (2002) that were based on a quadratic limb darkening
law
I = 1 - ul(1 - p) - U2(1 10
)2,
(3.1)
where the left-hand side of the equation is the relative intensity of the star, P is the
cosine of the angle between the line of sight and the normal to the stellar surface,
and ul and u 2 are limb darkening coefficients [24].
I sought to determine ul and
u 2 to parameterize the system. However, since these parameters were ill-constrained
by the data, I placed a priori constraints on the limb darkening parameters. The
first of these was to impose a penalty when fitting if the limb darkening coefficients
ever became unphysical for a transiting system, such that physical limb darkening
coefficients obey
ul
> 0,
(3.2)
0 < ul + u2
< 1,
(3.3)
as given by [5]. These conditions require the brightness of the star to monotonically
decrease from the stellar center to the limb. Because ul and u 2 are highly correlated
with one another, I instead used the optimal, uncorrelated, mixed limb darkening
coefficients, u' and u', given by Pal (2008) as
u'
=ulcos¢
+u
2 sin
u2 =-ulsine + u 2COSq,
,
(3.4)
(3.5)
where ¢ is the rotation parameter that optimizes the mixed limb darkening coefficients
based on the impact parameter, b, and the planet-star radius ratio, Rp/R, [28]. For
this work, I used ¢ = 390 because it optimizes the parameter set for b = 0.448,
the value given for WASP-3 in [14]. Returning to the fact that the limb darkening
coefficients, including the mixed quantities, were relatively unconstrained by the data,
I addressed this issue by fitting for only one of two parameters, u' because it was
better constrained than u'. I fixed u' = 0.0690 based on the values, ul = 0.2737
and u 2 = 0.3104, given by the Phoenix model of limb darkening. These values were
determined for a star observed in the Sloan i band with an effective temperature of
6400K, surface gravity (log g) of 4.25 cm s - 2 , solar metallicity and a microturbulence
velocity of 2 km s- 1 [8].
3.2
Photometric Parameter Set
As discussed in Section 1.2, a transit can be parameterized in terms of physical or
transit-parameters, and the two are easily converted to one another. As a result, for
the purposes of rapid fitting, I sought to use a set of uncorrelated parameters. Based
on the work of Carter et al. (2008) and verified in my testing, I used the relatively
uncorrelated parameter set: the duration of the transit as defined in Section 1.2
(T), the impact parameter b, the planet-star radius ratio Rp/Rs, and the first mixed
limb-darkening coefficient u'.
Particular to each night of data, there were three additional parameters required
to characterize the system: the mid-transit time (t,) and the airmass parameters (k
and C'). As mentioned in Section 2.1, I included the air mass parameters to remove
the effects on the flux values due to air mass generally increasing as a function of
time. The air mass parameters, k and C, are related to the flux by
fo(t) = fe(t) x Ce- kz(t),
(3.6)
where fo is the observed flux, f, is the air mass corrected flux, and z is the air mass.
However, since k and C are highly correlated, the parameters that I used in my fit
were k and C', which was defined as
C' = C- (mk + b),
(3.7)
where values for m and b were chosen so that k and C were relatively uncorrelated.
Thus, I fit the data to a model that was a function of 7 parameters, where 4 were
night independent and 3 were not. Note that the correlation of these parameters was
considered low because the correlation coefficient between any two of them was less
than 0.7.
3.3
Photometric Fitting Using Amoeba
The procedure used to fit the photometry involved several steps. The first of these
was to optimize all of the data for b, T, Rp/R,, u', k, C', and t, using IDL's 1 builtin downhill simplex minimization routine, amoeba [27]. As a quick and surprisingly
robust method, I used this routine to minimize the X2 statistic and thus fit the
photometry to a model based on the equations and assumptions previously described.
I defined X 2 for this procedure as
N
( ffi,obs -
fi,calc
2
(3.8)
where Nf is the number of data points, fi,obs are the observed fluxes, ai are their
uncertainties as described in Section 2.1, and fi,cal, are the fluxes calculated for a
given set of input parameters. As input, I used values from the most recent WASP-3
publication [14]. The resulting parameter values from the fit yielded x 2 /Nd.o.f = 1.02,
where Nd.o.f. was the number of degrees of freedom in the system, suggesting a good
fit to the data. This procedure thus allowed me to quickly find parameters that gave
a good fit to the FLWO data, with existing parameter values from [14] as the initial
conditions.
'Interactive Data Language - plotting and analysis software commonly used in astronomy
3.4
Markov Chain Monte Carlo (MCMC) Method
To obtain a better characterization of the uncertainties associated with parameter
values, I subjected data to a Markov Chain Monte Carlo (MCMC) routine in IDL
[32, 12]. For the MCMC, I input a set of initial parameter values, set a desired number
of steps for which the MCMC should add links to the chain, and then allowed the
computer to randomly pick one parameter to vary at each step, a procedure known
as the Gibbs sampler and outlined in Tegmark et al. (2004) and Ford (2005). Thus,
at any step of the sampling, only one parameter's value changed from the previous
step, while the rest were unchanged. This sampling method allows parameter space
to be searched in orthogonal directions. At each step, the random parameter that
was chosen to change, or jump was done so according to the Metropolis-Hastings
acceptance criteria. Jumps were immediately accepted if they resulted in a X2 value
less than the previous x2, but they were accepted with probability if the new X2 was
greater. To have a jump accepted with probability means that we assumed Gaussian
distributions and only accepted jumps when
e - Ax 2 > X,
(3.9)
where X is a pseudorandom number uniformly distributed between 0 and 1 and AX2
is the difference of the new and old X2 values. The difference between the new and
old values of a parameter at each step was given by the characteristic length scale, or
jump size for that parameter. The exact value of each jump was given as the jump
size multiplied by a number randomly drawn from a Gaussian distribution. The jump
sizes were chosen for parameters so that acceptance rates were (40 + 5)%. For the
initial parameter values for the MCMC, I used the best-fit parameter values output
by fits from the amoeba routine. The number of steps used in an MCMC was set so
that chains were fully converged, typically 1-2 million steps per parameter.
3.5
Photometric MCMC fitting
After finding parameter values that were a good fit to the photometry, as based on X 2
minimization with amoeba, I carried out an MCMC on the data, as described above.
To decrease the run time of the procedure, I held the night-dependent parameters (tc,
k, C') at their initial values, as output by amoeba, while allowing the other parameters
to vary. Such a division of the parameter-estimation problem was used because there
was little correlation between these three parameters and the other four parameters
being fit.
3.6
Characterization of Time Correlated Error
As previously mentioned, jump sizes depended on a pseudorandom number drawn
from a Gaussian distribution. This distribution was chosen to represent the data
as having Gaussian, time-uncorrelated noise. However, it was necessary to test this
assumption. Thus, following the MCMC that was just described, I evaluated the 0
statistic, given in [38] as
S= a (mn
r
1) 1/ 2
'
/2
(3.10)
where al is the standard deviation of the unbinned residuals between the best-fit light
curve given by the MCMC and the data. The other variables in 3 are dependent on
the rebinning of the residuals into m bins with - n points in each, so that an is the
standard deviation of the binned residuals. For these purposes, I binned the data
into intervals ranging from 10-30 minutes (nearly the timescale of ingress or egress)
and calculated
/
/3 for
each of these bin sizes. The median result that I found was
= 0.8, 1.3, 1.7 for each of the nights of data respectively. This result suggested that
the errors associated with each flux measurement should be increased by a factor of
/
for the second and third transit observations. I did not change the uncertainties
for the first night's measurements because
/3 <
1 cannot be truly accurate, as the
correlations should increase the parameter errors rather than reduce them. Thus I
made the assumption that 3
1 for the first night.
With the reweighted errors, which were increased to represent the effects of timecorrelated noise in the data, I conducted a second MCMC, following the same procedure as before. The resulting parameter values and their uncertainties were saved
for use in other MCMCs. The values of interest were the medians of the resulting
probability distributions, when the distributions were nearly Gaussian. For b, whose
distribution was asymmetric, I instead reported the mode of the data.
3.7
Determination of Ephemeris
To determine the night-dependent parameters tc, k, C', I held the night-independent
parameters (b, T, Rp/RS, u ) fixed at their values derived from the previous MCMC.
With those parameters fixed, I again conducted an MCMC in the same manner as
before, but with the different parameters and the reweighted errors (which included
, as previously defined). The resulting distributions are given in Figure B-1 in Appendix B. The fit corresponding to the data used is given in Figure 3-1 As before, the
parameter values determined were established as the median of the distributions.
With the mid-transit times t, determined for each transit from the MCMC method,
I sought to determine the new ephemeris: transit period and day offset. To do so, I
combined the times from the FLWO data with those in the literature, [29], [14]. For
all of the mid-transit times, I determined the epoch of using the ephemeris of [29].
With the epoch, I then applied a linear fit to the mid-transit times as a function of
epoch, which yielded a refined ephemeris of
Tc(E) = Tc(O) + EP,
(3.11)
where E is the epoch, Te(0)=2454605.55945 ± 0.00014 HJD and P=1.846836 ±
0.000002 days. The errors quoted here are those that result from the linear regression fit that I did. Using the values themselves, though, I determined the residuals
between the observed t, and those of the linear fit, shown in Figure 3-2. For this fit,
Table 3.1.
Epoch
-250
-2
0
12
18
59
WASP-3 Mid-transit times
Reference
Mid-transit time [HJD]
2454143.8503
2454601.86571
2454605.55956
2454627.72176
2454638.80348
2454714.52210
±
±
+
±
±
±
0.0004
0.00021
0.00035
0.00052
0.00041
0.00036
Pollacco et al. (2008)
This work
Gibson et al. (2008)
This work
This work
Gibson et al. (2008)
the goodness of fit was assessed using the x 2 statistic, where x 2 was computed as
2=6
i=1
tc,obs
- tc,fit
O'tc,bs
(3.12)
With six data points and two fit parameters, I found x 2 = 9.54, which is a X2 = 2.39
per degree of freedom. Although this x 2 suggests a poor fit, it appears that this
result is mainly due to the presence of two outliers. Given that error estimation can
be difficult in the presence of possible correlated noise and the small number of data
points used, strong conclusions regarding the presence of transit timing variations
cannot be made.
However, this large value of x 2 makes it advisable to increase
the errors of the timing parameters by
2-.39.
Thus, the final values I find are
Tc(0)=2454605.55945 + 0.00022 HJD and P=1.846836 ± 0.000003 days. I used this
ephemeris in subsequent fitting.
1.005
1.000
0.995
%0
0.990
0.985
0.980
-
0.05
0.00
-
0.15
0.10
HJD-2454601.81014
0.006
0.004
0.002
0.000
-0.002
-0.004
-0.006
1.005
r.,
1.000
0.995
0.990
0.985
000
n
I
25.92
•
'''''''
'''
''
25.94
25.96
25.98
HJD-2454601.81014
26.00
26.02
25.94
25.96
25.98
HJD-2454601.81014
26.00
26.02
26.04
0.006
0.004
0.002
0.000
-0.002
-0.004
-0.006
''
'''''''''''
25.92
''''
26.04
Figure 3-1 Photometry for first two nights (May 15, 2008 and June 10, 2008) with
best fit model and residuals.
r-
I
C
C)
-2
-3
-300
I
-200
I
-100
Epoch
I
100
Figure 3-2 Residuals from a linear ephermeris fit to the transit times given in Table
3.1.
Chapter 4
Parameter Fitting and
Determination
In this chapter I describe the process of jointly fitting the photometry and radial
velocity data to determine system parameters of WASP-3, including the projected
spin-orbit angle, which is a main result of this thesis. The data are fit to somewhat
simplified models of the system again using a Markov Chain Monte Carlo method to
estimate the statistical uncertainties in the determined parameters.
4.1
Model
To jointly fit the photometry and radial velocities, I used the same model of a planet
on a circular orbit around a star with limb-darkening that was previously used for
the photometry fitting to determine the ephemeris. While the photometric aspects
of this model have already been discussed, what follows is a description of the radial
velocity implications of this model.
Contributions to radial velocity variation come from the orbital motion and the
Rossiter-McLaughlin (RM) effect, described in Section 1.3. As a result, I employed
a Keplerian model with four free parameters, of which two were from the RM effect
(the projected spin-orbit angle A and the line-of-sight stellar rotational velocity v sin i)
and the other two were from the system's orbital motion (the planetary mass Mp,
which derives from the orbital semi-amplitude K and the systemic velocity 7). For
this model, I continue to assume zero eccentricity, and I use a mid-transit time and
a constant period as determined by the linear ephemeris previously given by the
photometry.
While the orbital contribution to the radial velocity signal can be determined
by considering the physics of the system, the contribution from the RM signal is
less straight-forward. Based on the template spectrum of Procyon (an F5 star with
Teff = 6500K, Fe/H=0, vsini = 3.1 km s- 1 ) [2], synthetic spectra of WASP-3 were
produced' and then fit 2 , as per [40], to arrive at the approximate, analytic expression
for the RM signal
AVRM
=
=
P=
VP
-y,
Rs
{1.51 - 0.44(v/10000.0) 2 m s - 1,
v sin i,
(4.1)
(4.2)
where vp is the subplanet velocity (the covered rotational velocity of the star), which
depends on vsin i and the projected distance between the planet and the projected
rotation axis xp.
4.2
Data Set and Uncertainties
Since I seek to determine the project spin-orbit angle A of the system, I used both
the radial velocity data available as well as the photometry. As described in Chapter
2, HIRES data and FLWO photometry were obtained simultaneously on UT June
21, 2008. Given this joint observation and the fact that this was the only night of
photometry that spanned the full transit, I included only this photometry in the
following fits; the other two nights of photometry were not included. However, I used
radial velocity data from all three nights of HIRES observations, allowing a larger
portion of the orbital phase to be studied.
I rescaled the uncertainties for this data. For the photometry, this was achieved by
1
This
2
process was carried out by Dr. John Johnson
Spectral fitting was completed by Prof. Joshua Winn
fitting the photometry using a downhill simplex (amoeba) X2 minimization routine to
a model that employed 7 parameters: {tc, b, Rp,/R, T, u', k, C'}, in the same manner
as in Section 3.3. Given that there were 522 data points in this photometric time
series and there were 7 parameters, I sought to obtain Xf = Nd.o.i. = 515 for this
fit, where X) is defined as before. Thus, the uncertainty of each photometric point
was decreased from a = 0.00148 to a = 0.00141, yielding xf = Nd.0 .f.. The error
was uniformly changed to this new value because the error had previously been set
uniformly to be the standard deviation of the out-of-transit flux. Upon making this
correction to the data, the residuals between the best fit model from theamoeba fitting
procedure and the data were examined. Using the same procedure as in Section 3.6,
the effect of time correlated error for this transit was determined (here unlike before,
the fit being used for reference was a fit to only one night of photometry, not to all
three nights of photometry) such that 0 = 1.16.
Uncertainties for the radial velocity were also updated. Unlike the photometry,
uncertainties in the radial velocity were adjusted by adding a constant term in quadrature with the existing uncertainties (which differed among measurements). This additive term, known as the jitter,was chosen such that a fit to the out-of-transit radial
velocity data yielded XRV,orb = Nd.o.f., based upon the definition
2
XRV,orb =
NRVOOT
RVi,obs - RVi,calc
RV(4.3)
(4.3)
Without the inclusion of jitter, X 2 was significantly larger, suggesting the presence of
systematic errors that were not modeled. I added the jitter term in quadrature with
the existing measurement uncertainty because I assumed that the jitter acts as excess
uncorrelated, Gaussian noise. In general, jitter is often attributed to astrophysical
processes such as spots and flows. I will revisit the issue of jitter in Section 6.1 but
empirically derived jitter estimates can be found in [41]. For now, it is important
to remember that I added nearly 11 m s- 1 of jitter to each of the radial velocity
measurements.
A model for the orbital component of the radial velocity signal was used since
I excluded any points that occurred during transit, which would otherwise require
the addition of the Rossiter-McLaughlin effect. The orbital model was used so as
to introduce as few assumptions into the fit as possible. This model required two
parameters (the mass of the planet Mp and the velocity offset (yi)) and t,, i fixed at
the values determined in the previous fit to the June 21 photometry using amoeba.
The period previously determined was also used. For the 12 out-of-transit points
Nd. o .f. was 10.79
fit with the two parameters, the jitter required to obtain X2RV,orb
SS-1
Given the observation of the three outstanding radial velocity measurements obtained on the first night (UT June 19 2008), this process was repeated with the
addition of another offset parameter 72,3 - 71 which was used to give radial velocities
from the second and third nights and offset relative to the first night of data. As I will
describe in Chapter 6, such an offset would be needed to account for radial velocities
contributed by a stellar feature that was visible on one night (the first night) and
not on others (the second and third night). With the same procedure as for the two
parameter model, I determined the jitter resulting from this three parameter model
to be slightly larger, 10.98 ms -
1
. This uncertainty as well as the others was used
in subsequent fitting. Before proceeding, it should be noted that the jitter resulting
from either the two or three parameter fit was nearly 11 m s-
1
, larger than the
expected value given by Wright (2005) [41].
4.3
Joint Fit to Radial Velocities and Photometry
To determine the system parameters as constrained by both the radial velocity data
and photometry, I simultaneously fit both data types. The photometry was fit using a
7 parameter model as before (Section 3.2). Note that unlike the fits used to determine
the ephemeris, I fit for the night-dependent parameters (mid-transit time and air mass
coefficients) at the same time as the other system parameters because there were fewer
data points to work with.
The radial velocity was fit using the approximate analytic RM model given by
Equation 4.1 added onto an orbital signal.
Thus the radial velocity fit required
the parameters: K, yl, A,v sin i and the optional fifth parameter 72,3 - 71. As seen
in Equation 4.1, the RM signal requires a depth 6. 6 can be determined using the
routine of Mandel and Agol, which requires the photometric parameters. Thus, unlike
in the fit used to determine the uncertainties, this radial velocity fit also allowed the
photometric parameters to vary. Thus the radial velocity fit actually depended on
the parameters: {K, yi, A,v sini,tc, b, T, Rp/R, u'} and the optional offset parameter
72,3 -
71-
The radial velocity and photometry fits were related using a joint X2 minimization
scheme, where X 2 was given by
Xt2ot = X
where x
+
X2f
="
RVi,obs -
X Xrv +RV
i
R V ,c
al
i=
fi,obs
-
i,calc(4.4)
(4.4)
and X2 were determined by fits to the radial velocity data and photometry,
respectively. Note the inclusion of 3 in the photometric component.
With this setup, the general process for fitting the data was very similar to that
of the photometry. I first fit the data using IDL's amoeba procedure to determine
best-fit values for all of the parameters in question. In so doing, I determined that it
was necessary to omit the three anomalous red-shifted radial velocities that occurred
near egress on the first night of data, June 19. Inclusion of these three points yielded
X v = 6.71 Nd.o.f, whereas their omission yielded x~ = 1.59 Nd.o.f for the 11 parameter
model. Similarly, for the 12 parameter model, including the points yielded x,
=
5.11 Nd.o.f, while their exclusion yielded x~ = 1.40 Nd.o.f. Given that these three
points are the largest outliers (as seen in Figure 5-3) and occur sequentially in time,
they appear to suggest physical characteristics that are not in the model. Thus the
model must not include these points or be expanded. The first option is chosen for
this study, leaving the second as the subject for future work, with a start in the model
proposed in a later chapter.
To fully characterize the parameter values and uncertainties from the data, I
employed an MCMC. The initial chain was made up of the best-fit values given by
the immediately preceding fit using amoeba. The number of links used in the MCMC
was chosen to be several million, so that there all parameter value jumps were fully
converged to their final probability distributions. This process was completed for
both the 11 and 12 parameter models, where the difference is the inclusion of an
offset term for the different nights of data. Results of this process are given in the
following section.
Chapter 5
Results
In this chapter, I provide values for system parameters of WASP-3 determined by the
fitting procedure outlined in the previous chapter. These values and their uncertainties are derived from the probability distributions that result from the MCMCs used
for fitting. These distributions are provided in this chapter, as well as comparisons
between existing values and those that I determine.
Results are provided for both the 11 and 12 parameter models, where the extra
parameter is a velocity offset for the second and third nights of RV data relative to
the first night. The probability distributions for parameters used in the 11 parameter
are given in Figure 5-1. The probability distributions for physical parameters derived
from the 11 parameters in our model are given in Figure 5-2.
An examination of the probability distribution shows that the majority of the distributions are nearly symmetric normal distributions. For each of these distributions,
the value that I quote is the median of the distribution. For noticeably asymmetric
distributions (b, v sin i, A), I instead quoted the mode of the distribution. Uncertainties are given as the difference between the median or mode (depending on which
was quoted) and the upper and lower 68.3% confidence levels. The resulting values
are given in Table 5. Additionally the best fit models to the photometric and radial
velocity data are shown in Figure 5-3. The best fit to the radial velocity using this
model for the full orbit is given in Figure 5-4.
Similarly, the distributions for the 12 parameter model and its best fit to the data
Table 5.1.
System Parameters of WASP-3
Value (12 parameter model)
Parameter
Value (11 parameter model)
Transit Ephemeris (Model independent)
Orbital Period (days)
Midtransit time (HJD)
1.846836 +0.000003
2454605.55945 +0.00022
Photometric Transit Parameters
00 16
0.1004
Pi /,• -0.0016
Rp/R
0.1007+00015
-- 0.0016
+ 0 067
0.445
V. -t_0.186
+2.19+278
0.4590.049
85.0.95
. 83
2.722+0. 39
U1
2.718+0
0.30039
41
0.308.0
0•30-0.043
Ut2~
0. 034
+
0.3381
2
•
--0.035
0.30.045
0.307 +.0 038
•
-0.040
+ 0 030
0.337
•
-0.032
b
(deg)
i (deg)
Total transit duration (hr)
Ingress/Egress time (hr)
-0.242
8505-0.76
Radial Velocity Parameters
14.861.
+ 63
vsin i (km s-1 )
A (deg)
M (M )+0.07
s )
T2,3 - '1(m s -
'71
72,3(m
-
1
s-1
15.561.68
, Q +4.75
'-2.76
2.14+o.o6
S97 8.
5
-1.95
-0.06
.,V_-0.07
8
36.05
20.9.64
-3.98
)
2469+2.64
2
20.96-5.11
are provided in Figures 5-5, 5-6, 5-7.
The first salient result is that A is within 2-3 standard deviations of 00, depending
on the fitting model (although the results do overlap). Such a finding suggests spinorbit alignment between the stellar rotation axis and the projected orbit normal.
For the most part, the results obtained from the distributions agree with previous
parameter values to within a few standard deviations [14, 29]. It should be noted that
my uncertainties are generally larger than those in the most recent study because I
tried to account for the time-correlated error, using the 13 statistic.
One interesting difference between these results and previous studies is that of
the planetary mass determined by radial velocity.
1.76+0 8 M
1
j
[29].
The discoverers found Mp =
Here, however, I determined Mp = 2.05 +007M for the 12 pa-
rameter model and 2.1480.6 M for the 11 parameter model. This yields a mean
difference of 0.34Mj between the published value and the values that I determined,
which is approximately equal to four standard deviations. Such a large difference requires further consideration of the system's physics. It should be noted, however, that
this change is indicative of a change in K, the orbital semi-amplitude and, thus, what
appears to be the acceleration of the system. It is not indicative of the planet changing mass in time. I will consider this discrepancy in the following section, whereby it
can be explained as a possible indicator of starspots.
1.2x10'
1.2x10
1.0x10'
1.0x10
8.0x 104
8.0x10'
6.0x 10'
6.0x10'
4.0x10'
4.0x10'
2.0x104
2.0x10'
I
0
1.9
2.0 2.1 2.2 2.3
M, [M,]
2.4
15
20
25
30
Vsys [m s-']
35
1.2x105
1.2x10'
I
1.0x10'
I
-60
-40
-20
0
20
Lambda [deg]
40
, -' -
2.0x10'
1.0x10
1.5x10'
I
S
8.0x10'
8.0x 10
6.0x104
6.0x104
.
.
1.0x 10'
4.0x104
S4.0x104
I
2.0x10'
I
5.0x104
I
2.0x104
0
0.0
0
15
20
vsini[km s"]
1.2x1I
25
0.80250.80300.80350.80400.8045
Tc [HJD-2,454,638
1.2x10
I
s
1.0x10'
1.0x10
8.0x110
I
8.0x104
8.0x1'04
6.0x1l 0
I
6.0x 10'
6.0x10'
4.0x104
4.0x104
2.0x1 0
0.098 0.100
T [days]
i
0
0.096 0.098 0.100 0.102 0.104
Rp/R*
0.102
1.2x1i
04
1.0x1 0I
1.2x10"
C 8.0x1 04
8.0x10
6.0xc
6.0x10
4.0x1
4.0x10
2.0x1c
2.0x10
b
s
1.0x10
0.6
I
2.Ox 104
2.0x104
0-
0.096
0.4
1.2x 10'
1.OxIC
4.0x1I
0.2
I
I
I
I
0
0.2
0.3
0.4
0.5
ul mixed
0.6
4
4
4
4
0
-0.002).0000.0020.0040.0060.008
0
0.00000.00020.00040.0006
fOn
Figure 5-1 Probability distributions for parameters included in the 11 parameter
model.
1.2x10
4
I
6.0x10
I
4.0x104
o 5.0x104
4
IL
4.0x104
0..
82
84
86
i (deg]
88
1.2xO
.
1.0x10
S1.5x10s
l1.xl10
" x
.
a
0
2.60 2.65 2.70 2.75 2.80 2.85 2.90
tJV-tJ [hr]
90
I1.2x10
5
I
5
1.0x10
"2
8.0x104
,
4
a. 2.0x10'
C
..
.
.......
0.17 0.18 0.19 0.20 0.21 0.22
R../o
2.0x10.
1.Ox10
8.010
1.0x10
S6.0x10
0
1.2x10'
.
I
860x104
a 2.0x10
I
1.5x10 '
I:
1.0x10
I
8.0x10
6.0x10
6.OxlO4
6.0x104
,.Oxl
1
5.0x104
a.
CL2.Ox10
0.1.
14
I
I
16
18 20 22 24 26 28
tJI-tJ [min]
2.0x10I
0.
0 1,
0.2 0.2 0.2 0.3 0.4 0.4
ul
0.20 0.25 0.30 0.35 0.40 0.45
u2
Figure 5-2 Probability distributions for physical system parameters derived from the
11 parameter model.
o
200
150
$,
100
100
0
li
50
r SAAA o
-100
0
A4;
0
**
S- --
A
-200
0. 40
A
0.45
0.50
Phase
0.55
0.60
-50
0.40
0.45
0.55
0.60
0.006
1.005
0.004
*S
1.000
0.995
00
*
0
( *v,
0.002
* .
*
4
0
0.000
-0.002
-. . •
.0
jW
0.990-
-0.004
0.985
0.40
0.50
Phase
0.45
, ,
0.50
Phase
0.55
0.60
-0.006
0.40
*
.
0.45
0.50
Phase
0.55
0.60
Figure 5-3 Top: Best fit to RV data at left, with residuals at right. The symbols are
as follows: Filled circles - night 1 data, filled triangles - night 2 data, open circles -
data not included in the fit. Bottom: At left, best fit to photometry, jointly fit with
the RV data shown above. At right are the fit residuals.
400
I
200 -
\I
E
0
-200 -
-400
,
0.0
0.2
, ,
,
0.6
0.4
0.8
1.0
Phase
Figure 5-4 Best fit to RV data with the 11 parameter model for the full orbital phase.
Open circles are those that were not included in the fit.
1.2xl 0
1.2x 10
5
1.0x10
1.0x10
U,
"
8.0x104
.
6.0x10'
~
6.Ox 10'
o 4.0x104
aI
,4
C2.0x10
o 4.0x10 4
4
2.0x10
8.0x 104
I
I
O
0
1.8
1.9
2.0 2.1
M, [MJ]
2.2
1.4x1
5
1.2x1 0s
00'5 1-'-1.Ox 105
'.:
30
2.3
,,-I--,......
40
Vsys [m s']
-20
1.2x105
1.2x105
1.0x10
I
1.0x10
I
S8.0x104
8.0x1l
50
. 6.0x104
:
I
5
6.0x104
b 4.0x104
m 4.0x10
4
. 2.0x10
2.0x10
2.0xl
60
8.0x10
6.0x1 04
4.0x1l0
0
20
40
Lombdo [deg]
4
4
0
10
0
12 14 16 18 20 22 24
vsini [km s ']
10
20
30
Offset [m s"']
0.80250.80300.80350.80400.8045
Tc [HJD-2,454,638]
40
5
1.2x10
2.0x1l
1.0x10'
I
I
1.5x1
8.0x104
05
.'
1.Oxl
6.0x10'
4.0x10'
,
5.0x1 0'0
.1I
2.0x10
4
O0
0.096
0.0 0.1 0.2 0.3 0.4 0.5 0.6
1.4x10
1.2x10
s
1.2x10I
s
s
1.0x10
1.0x105
1.0x10
S8.0x10
s
1.2x10
0.0960.0980.1000.1020.1040.106
Rp/R..
0.098 0.100 0.102
T [days]
I
I
I
I
S8.0x104
4
6.0x104
S6.0x10
o 4.0x10
S4.0x10
0 a
0.2
S20x104
:I
.
0.3
0.4
0.5
ul mixed
I
0.6
0.
-0.002).0000.0020.0040.0060.008
S4.0x10
a'
2.0x1
0
0.00000.0002 0.00040.0006
fOn
Figure 5-5 Probability distributions for parameters included in the 12 parameter
model, which included a velocity offset for the second and third nights of RV data.
1.2x10l
1.5x10
5
1.2x10s
:
4
1.0x10
1
s
1.0x10
S8.0x104
6.0x10
5
I
z,1.0x10o
8.0x10
5
4
6.0x 104
I
4
D 4.0x10o
5.0x10l
0
0
4.0x10
2.0x10'
0.17 0.18 0.19 0.20 0.21 0.22
R*/o
1.2x10
'
S8.oxio4
I
,
4
90
0
2.60 2.65 2.70 2.75 2.80 2.85
tJV-U [hr]
5
.
1.0x10
o
8.0x10"
4.0x10
o
002
4
M 2.0x10
C 2.0x10
J0
14
88
6.0x10
I
4
o
86
i [deg]
4
a 4.0x104
0
I
I'
S6.0x10
84
1.2x10'
5
z1 l.0x10
82
16
-0
18 20 22 24 26
[min]
tJI-J
28
0.2
0.3
ul
0.4
0.20 0.25 0.30 0.35 0.40 0.45
u2
Figure 5-6 Probability distributions for physical system parameters derived from the
12 parameter model.
200
150
100
o
100
0 o
0
50
-100
0
-200
o
- ----- - -
-
*0
0
g
S
00
0.45
0.50
Phase
0.55
0.60
0. 40
0.45
-
-
S,
rO~
-50
0. 40
- -
00
0.50
Phase
0.55
0.60
0.006
1.005
0.004
1 .000
*,
*
0
0.002
0g
0.000
0.995
-0.002
c.**
do
0.990
-0.004
0.9851 0.40
0.45
0.50
Phase
0.55
0.60
-0.006
0. 40
0.45
0.50
Phase
**
0.55
0. 60
Figure 5-7 Top: Best fit to RV data at left, with residuals at right. The symbols are as
follows: Filled circles - night 1 data, filled triangles - night 2 data, open circles - data
not included in the fit, diamonds - night 2 data adjusted by the offset parameter and
included in the fit. Unadjusted night 2 data was not included in the panel showing
the residuals. Bottom: At left, best fit to photometry, jointly fit with the RV data
shown above. At right are the fit residuals.
Chapter 6
Evidence for Stellar Activity
In this chapter, I provide evidence for activity on the surface of the star in the WASP-3
system. I find such evidence particularly in the radial velocity measurements, but not
in other observations. To understand the physics of the system, I provide a model of an
active stellar region, with different possible peculiar velocities and surface intensities,
that could explain the lack of observation in these other measurements.
6.1
Radial Velocity Observations and Results Suggestive of Stellar Activity
From radial velocity measurements and subsequent analysis, the main evidence for
stellar activity is:
1. Three anomalous radial velocity measurements taken on UT 2008 June 21
2. Large stellar jitter needed to fit the radial velocity data well
3. Value of K changing in time
Given this summary, I will now elaborate on each of these points.
On the first night of HIRES observations, an anomalous Doppler shift was observed, as shown in Figure 5-4 (denoted by unfilled circles). Consisting of three points,
residuals between the fit and data show in Figures 5-3 and 5-7 that the anomaly has
a peak amplitude approximately 150 m s-
1
greater than the RM signal, or 80 m
s- 1 greater than the orbital, out of transit signal. The time between the first and last
anomalous points is approximately 33 minutes. A similar radial velocity anomaly was
not seen in later nights of HIRES data.
The anomalous radial velocity is believed to be a genuine feature of the system
observed and not a result of systematic errors. The three consecutive spectra responsible for the anomaly did not show any noticeably different characteristics from the
other HIRES spectra that we obtained. While the moon was full on the evening of
the anomalous observations, other measurements made that same night did not show
similar deviations from their expected values. Additionally, I find this anomaly to be
physical because these three measurements occurred consecutively, were the largest
outliers (see Figures 5-3, 5-7), and had a time between the first and last anomalous
measurements comparable with that of the planet crossing a feature on the star. This
time was equal to almost twice the characteristic timescale (the ingress or egress time)
of the system. In particular, this timescale is the most compelling indicator that the
anomaly arose from stellar activity and not other sources, such as instrumental error.
Beyond this anomaly, another indicator of stellar activity is the large stellar jitter
needed to fit the radial velocity data. When the radial velocities were initially fit
(described in Section 4.3), I failed to achieve a good fit to the data with either the
11 or 12 parameter model, even when the anomalous data points were removed.
For example, before adjusting the uncertainties on the radial velocity data, the 11
parameter model yielded X, = 1.6 3Nd.0 .f with the anomalous points included and
X
=
0,
0.93Nd.o.f
without. Thus, to yield X, = Nd. o.f. for the radial velocity data,
I had to add a velocity jitter term of nearly 11 m s- 1 . The predicted jitter from
systematic sources for a star like WASP-3, with B - V < 0.6, is r, = 4.4 ± 2.2 with
60% confidence levels [41]. The jitter term that I used was well beyond these limits,
suggesting that another source of jitter, such as stellar activity, was present. The
issue of jitter addresses the issue of the effects that stellar activity would cause across
different nights of observations.
The issue of whether or not velocity offsets are needed when fitting different nights
of data also raises the potential for stellar activity. As I showed earlier in Section
4.1, the RM signal depends on both the brightness contrast caused by the planet's
transit 6 and the stellar rotational velocity hidden by the planet v,. Thus, a region on
the star that had a different brightness or velocity, as compared to the surrounding
stellar surface, would affect the amplitude of the RM signal. As the star rotated, it
would cause this region to be visible or hidden, thus changing the amplitude of the
RM signal. If this were the case for our system, WASP-3, then it should be necessary
to fit different nights of radial velocity data with an offset parameter for each night.
However, since the observed anomaly occurred near egress, it seems that the region
was rotated out of view for both of the subsequent nights of measurement, as I will
explore in Section 6.3. Thus, only one offset should be needed for fitting the later
nights of RV data. As I showed in Chapter 5, the results of fitting the data with and
without this offset were comparable. Although this finding does not yield conclusive
evidence for the need for a velocity offset, it also does not conclusively exclude the
need for a velocity offset.
The fourth suggestion of stellar activity arises from time scales even longer than
individual measurements or different nights of data. The value that I determined for
the mass of the planet,M, = 2.050:07Mj from the 12 parameter fit, is larger than that
determined in the discovery paper, Mp = 1.7680.8Mj [29]. Measurements discussed
in the discovery paper were collected nearly a year before our measurements. Over
the course of a year, one would expect patterns of surface activity to change for an
active star [15]. As a result of this changing pattern of stellar activity, there would
be a change in the velocity offsets and the resulting apparent acceleration observed,
which in turn affects K.
6.2
Starspot model
A planet transiting an active region, such as a starspot, on the surface of WASP-3
could explain the radial velocity anomaly noted above, as well as the other indicators
of stellar activity. To produce the aforementioned effects, a starspot would have to
characterized by a photometric or velocity contrast, as compared to the surrounding
stellar surface. As a means of better understanding the characteristics needed of
such a spot, in this section I develop a simplified model of a planet transiting a star
with a spot. Before continuing, it should be noted that for the system of WASP-3, I
assume that the spot is near the limb of the star because the observed radial velocity
anomaly immediately precedes egress. This assumption, however, does not affect the
mathematical results that I develop below; it only changes the physical interpretation.
I allow the spot to be a disk parameterized by a radius, R 2 , a constant surface
intensity 12 (defined as the received flux per unit emitted area), and a constant flow
velocity, V2, relative to the systemic velocity. I assume that this spot is on the
surface of a star that has a radius R 1, a constant surface intensity, Ii, and a constant
photospheric velocity directly under the spot, V1, relative to the systemic velocity .
With these parameters, both the photometric and kinematic changes induced by the
starspot can be determined.
First, I characterize the photometric change in flux caused by the spot. I consider
the flux difference between two times: (1) Out of transit (OOT) when the spot is
on the visible face of the star and (2) at the moment when the spot is obscured by
the planet, centered directly behind it in projection. Taking into account the surface
intensities of both the spot and the star, the flux at each of these times is given by
f(1) = rR I 1 + xR (I 2
TrR211 -
f(2 ) =
1
Rp
7R 211
rR'I + xR(I
(6.1)
- 1,),
> R2
(6.2)
2 - I,) - 7RRI
2
Rp < R 2
The absolute change in flux , Af between the out of transit and covered spot phases
is given by the difference of these two expressions. However, since I only consider
relative fluxes in, the more interesting quantity is the change in flux relative to the
out of transit flux, f( 2 ). Given that I found RI/R
1
. 0.1 from earlier fitting, I can
make the approximation that the radii of the starspot and planet are much less than
the stellar radius, i.e. R 1 > R 2, Rp. Thus, the relative change in flux is
f
-
(2)
/
+
2
R,
> R2
R(6.3)
<
(p
2
It is also possible to determine the spot's effect on the mean line-of-sight stellar
rotational velocity, relative to the orbital motion of the system. I am interested in
this quantity, particularly its difference between the covered spot and out of transit
scenarios, because it could reproduce the peak amplitude of the velocity anomaly
relative to the orbital signal. To determine this difference, I calculate the mean lineof-sight stellar rotational velocity at times (1) and (2), as described above.
Notationally, these times are given as subscripts on the quantity V (mean line-ofsight stellar rotational velocity). Here I am considering V, not vsini because I am
assuming sin i = 1. This assumption is based on the earlier result that A is near
zero, so it is likely that the true, not projected, spin-orbit angle is small and, thus,
sin ist, - siniorbit . 1 [11]. Additionally, I make the approximation that the velocities
that I consider (i and V2) are constant in their respective regions. Furthermore, I
approximate V, as a constant over the range of spot positions covered during transit
since. This approximation is needed because the spot will have rotated between the
start and end of transit, but only be covered by the planet in the latter case (as a
means of matching the case of our observed anomaly).
Using these approximations, I can determine which velocities were covered on the
photosphere, taking into account the size of the emitting (spot) or occulting (planet or
spot) body, its relative intensity (0 for the planet, 12 for the spot), and any emitted or
occulted velocities. Together, these parameters allow one to determine the anomaly
introduced into the spectra, which translates into a mean velocity shift, similar to
that seen in the case of the RM effect, Figure 1-2. Expressions for V at times of
interest are given as
(1) -
2 (2
55
-1
,
(6.4)
-
R
R
R22(;+
R2
2=
(6.5)
<R2
(
S2
R, <
V2
2
From these, it is possible to determine the difference in velocity, Av that we seek.
Unlike the case of the flux change, I am interested in the absolute difference of velocities because the observables are the absolute velocities and their differences, not the
relative quantities.
R
R
(2) (1)
2
-21R21
-
(2
R <fR2
2
(6.6)
With Equations 6.3 and 6.6, I can find the range of parameters which could produce the velocity anomaly noted. I tune the model properties to match the parameters
of the spike and transit, so that Av r 75 ± 13 ms-
1
and RI/R
1 40.1. With these
values and the assumptions that the spot is the same size as the planet (R 2 = Rp) and
the contrast between the spot and stellar intensity is 1 (12/I1 = 1), I determine that
the starspot should have the velocity v2 = -7.5 km s- 1 . A negative velocity (which
represents movement towards the Earth) is required because covering a blue shifted
velocity allows the planet's transit to yield a red shifted signal. It's worth noting that
given the spot characteristics that I have assumed, the resulting photometric signal
would not show any changes in relative flux as a result of the spot.
If I instead treat the starspot as having an intensity equal to the average sunspot
intensity contrast of 12/I1 = 0.68 as given by [33], I find that f 2 = -11.03 km sAdditionally, these conditions yield a relative change in photometric flux,
1
A-
fOOT
0.0068, or a relative flux increase during transit of 0.0032, roughly a third of the
transit depth. The amplitude of this signal is large enough that it would have been
detectable if we had gathered photometry during the anomalous event.
Conversely, if I consider that the spot has 12 > 1, then f2 would be smaller than
either of the previous calculations. Physically this scenario corresponds to a spot
that erupts hot, bright material. The brightness of this material would then also
cause a change in the photometric signal. From this case and the previous two, it
appears that without photometry simultaneous to the radial velocity anomaly, I have
significant freedom in adjusting the parameters of a possible starspot.
6.3
Observational consequences
Since I have only used anomalous radial velocities to place limits on the spot's properties, here I examine the rest of the data (photometry and other spectra) to look
for signs of any anomalies caused by a possible spot. The first topic I explore is that
of the stellar rotational period. As the star rotates, the spot moves across the face
of the star. As a result, for half of the star's rotation period, a spot will be hidden
on the side of the star not facing Earth. Since a planet transiting the face of the
star with a hidden starspot should yield no anomalies, we would only expect to see
anomalies when the spot is visible. To determine where the spot was at the time of
previous observations, I calculated the following approximate stellar rotation period
using expressions for the rotational velocity of the star and its radius
Prot
21flR,
2
v
2R
=
2R
ini sini <0.
vsini
- vsini
4.46 _0 ."664 [12 parameter model]
4.26:
74
days,
[11 parameter model]
(6.7)
where I used the line of sight rotational velocity (v sin i) that I previously determined
(Table 5) and the stellar radius (R* = 1.31 0.12 Re) given in [29]. The inequality
exists in our expression because sin i < 1.
The rotational periods that I find suggest that the star rotates approximately 90
during transit and approximately 10 during the anomaly, so I can consider the starspot
to have changed position across different nights, not over the course of individual
transits. To consider the effects of rotation on the position of the starspot at the
time of our observations, I plot the the azimuth of the spot on the surface of the star
as a function of time, with the observations times overlaid in Figure 6-1. Given the
range of rotational periods permitted, there are some for which the spot was in view
during other observations, including the FLWO photometry, and some for which it
was not. A plot of the latter scenario, which occurs for Prot=5.17 days, is given in
Figure 6-1. With this period, I find that the spot would not have been on the visible
face of the star during FLWO photometry, but it might have been visible during RISE
photometry used in [14].
I
*
*
x
*
*
*
A
*
*
300
*
Observation
FLWO
HIRES (Night 1)
HIRES (Night 2)
HIRES RV Anomaly
RISE Midtransit
SOPHIE (notshownhere)
HIRES (Night 3)
Potential RISE anomaly
RISE transit limits
250
200
ANOMALY VISIBLE
IS150
ANOMALY NOT VISIBLE
100-
I
oI
I
-40
-20
I
40
020
60
1
Days from Anomaly
Figure 6-1 The azimuth of the starspot observed on the first night of RV data is given
as a function of time, assuming a stellar rotational period of 5.17 days. The anomaly
(blue) is shown at a time and azimuth of zero. Different observations of the system
are indicated by different colored symbols. The red region indicates where the spot is
on the far side of the star. Note that this regions begins several degrees above zero.
Within the two nights of RISE data, I note two peculiar features. First, on Sept.
4, 2008 almost two months after the anomaly, I notice a tantalizing detection near
egress that looks like a possible spike in photometry from 0.015 to 0.023 in phase, or 22
minutes. Given that it occurred near egress, I looked to see the expected orientation
of the star at the time. With the rotation period of 5.17 days, this spike appeared very
shortly after the same azimuth of the star as when the anomaly was originally seen.
Table 6.1.
Individual transit Rp/R, values
Date [UT 2008]
RP/RS
May 15
June 09
June 20
0.1001 ± 0.0003
0.1013 + 0.0008
0.1005 + 0.0004
This potential offset and the fact that the main spike in data was chiefly based on only
two data points leaves this as an interesting, but ambiguous note. Second, on May
18, 2008, nearly one month before the anomaly, the 5.17 day rotation period predicts
that the spot was on the visible side of the star. However, there are no noticeable
anomalies in the photometric data presented. Such a non-detection presents limits on
theories of our starspot. I can explain the non-detection as possible evidence that the
spot had not yet formed, that it previously had less of a brightness contrast with the
stellar photosphere. Alternatively, the spot may have traveled on a trajectory that
was not parallel to the planetary transit. Such a hypothesis, however, is disfavored
by our measurement of A consistent with 0. As a final thought on this subject, if I
discount the potential spike in the later (Sept. 4) observations, this non-detection
could be indicative of the supposed spot not having a photometric contrast, only a
velocity contrast.
Light curves from our FLWO data did now show any clear photometric anomalies.
To examine this more carefully, I refit the photometry for each night separately with
only one free parameter, the planet to star radius ratio Rp/R, which is related to
the transit depth by 6 = (Rp/R,) 2 . The results of these fits are given in Table 6.1,
where the medians and standard deviations have been quoted. The results show that
the transit depths across different nights are consistent to within 0.0001. The lack of
significant depth variations is consistent with the scenario expected from a 5.17 day
stellar rotation period, where the anomaly was on the far side of the star.
Physically one expects a star with spots to have multiple spots. As such, I also
looked to see what bounds our photometry would give for other potential spots. Since
the anomaly lasts at least 33 minutes, I rebinned the photometric fit residuals into
15.84 minute intervals. This yielded a
'
4 x 10- 4 , leaving an approximate 2a upper
limit bound on the flux of a potential star spot, f < 8 x 10- 4 . Given that spots on
the sun have f< 7 x 10-6, this value is not sufficient to exclude possible starspots
on the far side of the spot. Together with the lack of transit depth variations, this
result does not exclude our hypothesis of a spot on WASP-3, and it potentially helps
to support the rotational period under consideration. Another test of this hypothesis
would be to compare the transit depth between each of the RISE nights separately
and also with each of the FLWO nights.
Spots are indicative of stellar activity, which can be evidenced though signals in
the observed spectra. Since starspots are regions of intense magnetic activity, I would
expect to see a change in the emission strength of spectral lines that are sensitive to
magnetism. Ca II H & K are emission lines that arise from the chromosphere of a
star and are highly representative of stellar magnetism [15]. As a result, the ratio of
the flux of the line cores of Ca II H & K relative to nearby continuum regions in all
of our HIRES spectra was examined. The time series of these ratios did not yield any
significant patterns, not even on the different faces of the star, suggesting that stellar
activity, if any, did not change greatly throughout the course of our observations.
This result does not exclude starspots, but it leaves the topic inconclusive. Thus,
follow up studies of the line spectra need to be made, including a potential search for
rotational modulation of spectra, which we would expect from a stellar feature.
6.4
Underlying physics
Thus far, I have produced an internally consistent description capable of explaining
our data. Here, I seek to compare physical realities with the parameters needed of our
starspot model. I begin by considering its peculiar velocity. In calculating possible
starspot velocities relative to orbital motion, I encountered values of
~
10 km s-
I compare this velocity to the approximate soundspeed of the star, cs. I calculate
the soundspeed of the star by assuming an ideal gas law equation of state, with a
chemical composition identical to the sun's (69.37% H and 30.63% He, from[9]) for a
mass of 1.24 M [29] yielding
Cs
(P
)1/2
p
kT)1/2
5
1.381 x 10-16 erg K- 1 6400 K
m
3
3.196 x 10-24 g
6.79 km s-1
=m
(6.8)
This value of c, suggests that the starspot velocity needed would be supersonic. A
supersonic velocity is unexpected, particularly because it introduces shocks and nonthermal radiation, including X-rays and radio. Such consequences provide avenues
for future study in other bands. Here, however, I am mainly interested in showing if
it possible for a star to emit supersonic flows horizontally across the stellar surface,
so that we would view the flows as traveling towards us. In considering this and other
physical requirements of our spot, I will use the sun as reference. WASP-3, an F7-8V
star, is more active, larger, and faster-rotating than the Sun. On the sun, supersonic,
horizontal flows are thought to be found in the penumbra of starspots, through a
phenomenon known as Evershed flows [3, 33]. While no direct analogs have been
found on other stars, this does leave room for the possibility of supersonic flows on
WASP-3. Traditionally, Evershed flows have velocities that are several km s- 1 , which
is somewhat smaller than what is required of our starspot if it is to be dark. However,
since the Sun is smaller and slower than WASP-3, one can hypothesize the existence
of similar flows with larger velocities on WASP-3. The need for supersonic velocities
can be foregone altogether if I suppose that the active region is brighter than the
surrounding stellar surface. This is a possibility that should be further explored in
future studies but is not covered here.
I also consider the size of our needed starspot, or active region. On the sun, spots
can be thousands and a few tens of thousands of km in diameter. If I assume that
the spot is the size of WASP-3b, the planet in the system, then the spot would need
to be - 90, 000km in diameter. Since WASP-3 is larger than the sun and other stars,
including rapid rotators, have been found with much larger spots [17, 21, 30], a spot
the size of the planet, R 2 = 1.31Rj appears to be possible.
While I have thus far considered starspots, I have not conclusively excluded other
possibilities for an active stellar region on the star. Supergranulation is a potential
alternative. Those on the sun are known to have radii that are approximately 3 x
107 [33] and velocities that are a few km s- ' [15]. Although supergranulation has
not been directly observed on other stars, there is evidence for supergranulation
on other stars.
Past studies of ( Bootis A gave evidence for a region of intense
stellar activity characterized by supergranulation or large horizontal flows [34, 35].
Termed a starpatch, this region of activity provides us with a case requiring similar
characteristics.
Two differences exist between the active region on WASP-3 and
supergranulation. First, supergranules on the sun last for approximately 10 minutes,
while spots can have longer lifetimes of forming in days, living for weeks-months and
then decaying faster than they form [33]. For our anomaly of 33 minutes and other
signals seen, the longer lifetime of spots is preferred. Second, while supergranules
have velocities that can nearly reach the magnitude that we seek, they are usually
closely surrounded by similar downward velocities.
In examining these and other
physical possibilities, I find that much is left to be clarified about WASP-3 regarding
spots or other active regions.
Chapter 7
Conclusions
In this thesis, I have presented both new and refined system parameters for the
exoplanetary system WASP-3. Of these parameters, the projected spin-orbit angle
was determined to be A = 2.86+4.75 degrees using an 11 parameter model and A =
5.97+8.63 degrees using a 12 parameter model, which differed from the 11 parameter
model by the addition of a velocity offset between the first night of radial velocity
data and the next two nights of data. Although these models yield slightly different
values, they are both consistent with 0 degrees, or projected spin-orbit alignment, to
within two to three standard deviations.
Such a small spin-orbit angle is interesting to consider in the context of planetary
migration theories. For systems like WASP-3, it is thought that the planet formed
with good alignment at approximately 5AU and then migrated in toward the star, to
reach its present distance of 0.03 AU. That WASP-3 has a small A today, suggests
that whatever the mechanism for inwards migration, it either did not perturb the
alignment or it drove the system back into alignment through a damping process.
Other system parameters which I determined were found to agree with previously
published values, with the exception of the planetary mass, which I determined to
be Mp = 2.05+807 M and M = 2.14o.o0 M for the 11 and 12 parameter models
respectively. These values are both four standard deviations larger than the published
value of M
= 1.761
+00 M [29]. This difference in mass, which I attribute to a change
in the apparent amplitude of the radial velocity's orbital signal and not to a change
in the planet's physical parameters, is one of several pieces of evidence for stellar
activity in this system. The existence of three redshifted radial velocities near egress,
as observed on UT June 19, 2008, which do not fit well in any of the models tested
here further suggest possible evidence of stellar activity. The large > 10 m s- 1 radial
velocity jitter needed to achieve a radial velocity fit to the orbital component of the
data is also suggestive of possible stellar activity, such as starspots.
Herein I have provided a possible model of star spots and proposed approximate
values of velocity shifts and photometric flux changes. Much work remains to examine
the spectra of these measurements for particular features, such as additional line
asymmetries.
Furthermore, analysis of more data, particularly photometry taken
close in time to the observed radial velocity shift, possibly from the RISE instrument
[14], may assist in getting a better understanding of the system. While much remains
to be done to better characterize the system, WASP-3 still offers the unique possibility
of using transits to probe the velocity structure of active regions on a star.
Appendix A
Tables
Table A.1:
HIRES Doppler Shift Measurements of
WASP-3
Heliocentric Julian Date
Radial Velocity
Measurement Uncertainty
2454636.830729
164.5045
6.5110
2454636.875822
104.2086
6.9670
2454636.896840
85.1936
7.0284
2454636.906053
159.2427
6.8073
2454636.915509
178.6607
6.0711
2454636.927616
140.0019
6.8060
2454636.937153
119.4519
6.8356
2454636.948275
53.7028
6.4906
2454636.956296
6.0193
6.1814
2454636.966944
-39.5166
7.6074
2454636.977998
-79.1501
7.0201
2454636.988090
-60.5185
8.0584
2454636.999039
58.0289
7.8611
2454637.011146
-2.2918
6.1403
2454637.057199
-91.8294
7.9118
2454637.087350
-100.8228
6.6638
Heliocentric Julian Date
Radial Velocity
Measurement Uncertainty
2454638.813310
-78.0447
7.7209
2454638.822639
-125.7164
6.8934
2454638.833692
-126.6211
7.6951
2454638.842581
-129.9433
7.0491
2454638.851794
-57.0043
7.3201
2454638.862257
-52.3533
7.3249
2454638.875023
-57.6522
7.0961
2454638.887928
-45.1826
6.5875
2454639.003206
-156.3164
6.8839
2454674.737824
-66.0818
7.9923
2454674.741238
-44.0116
10.0245
2454674.865822
64.8930
8.8225
2454674.991887
185.0540
9.1326
Appendix B
Figures
Epoch 12
Epoch -2
I
'
.t
6x10
I
I
I
6x104
Epoch 18
I
I
t 6x10'
4
• 4x10
4x10
2
I
2x10 4
4x10
I
2x104
2x104
I
SI
I
,
.
0
0
.
1.8650
5
1.8660
1.8655
1.8660
Tc [HJD-2,454,600]
Epoch -2
8x1C
*
I
6x10
3.802 i:'
38.802
ft
1.8665
27.720
27.722
27.723
27.721
Tc [HJD-2,454,600]
,..
8x 14
'
27.724
Epoch 12
3803
..
38.04
38.804
38.803
Tc [HJD-2,454,600]
38.805
Epoch 18
*
O
4
OxIV
6x10
I
6x10
4
4x10
4
4
Z' 4x10
S4x10
4
2
2x10 (3
e
2x10
4
I
I
A
0.0020
0.0040
k
0.0080
0.0060
I
I
6x104
.-
4x10
- .
OxlU
.
4
t
6x10
,
1.0060
1.0080
18
4
2x104
2x10
1.0040
fO
6x10
Epoch
Z 4x10
4
S2x104
-4
4
4
4x10
1.0020
-0.0040.0020.000 0.002 0.004 0.006 0.008
k
Epoch 12
I.
I
4
0000
1.0000
0.010
k
Epoch -2
8x10
0.005
0.000
-0.005
0.995
-
.
. . r ,,
1.000
1.005
fO
.
1.010
:4
0.9960.9981.0001.0021.0041.0061.0081.010
fO
Figure B-i Probability distributions for mid-transit times and airmass parameters, k
and C (here noted as fo). Lines indicate the medians.
68
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