The Mid-Infrared RR Lyrae
Period-Luminosity Relation
by
Meredith J. Durbin
Advisors
Dr. Victoria Scowcroft
Postdoctoral Research Associate
Carnegie Observatories
Dr. Alma Zook
Professor of Physics and Astronomy
Pomona College
Submitted to the faculty of
Pomona College
in partial fulfillment of the requirements for the degree of
Bachelor of Arts in Physics
Abstract
We present new period-luminosity relations for pulsating RR Lyrae variable stars derived
from data in IRAC channels 1 (3.6 µm) and 2 (4.5 µm). Our data set comprises the globular clusters ω Centauri (NGC 5139) and M54 (NGC 6715), and contains a total of 69 RR
Lyrae. We find slopes consistent with those of Madore et al (2013), Klein et al (2014), and
Dambis et al (2014)’s investigations of the period-luminosity relations in the WISE bands
W 1 (3.4 µm) and W 2 (4.6 µm). We also use metallicities from Sollima et al (2006) and
Rey et al (2000) to investigate metallicity effects on the period-luminosity relation, and
find no strong evidence for a metallicity term in either channel.
Acknowledgements
This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract
with NASA. Support for this work was provided by NASA through an award issued by
JPL/Caltech.
I have been blessed with the opportunity to work with many extraordinary people in my
time at Pomona. Here are my thanks to a few of them:
To Vicky Scowcroft, for your infinite patience with my failure to understand PSF photometry and aperture correction, and for being willing to extend your advising and assistance
past the summer of my official internship. I have yet to understand why you trusted me
with ω Cen, but I am very grateful that you did.
To Alma Zook, for your valuable insights and wonderful humor, and for being the first to
introduce me to astronomy research. Also for the kitten cam websites.
To Phil Choi, for always pushing me to think harder and do better, and for all the life
coaching. I’m truly sorry I spoiled How I Met Your Mother for you.
To Joseph Long, Claire Dickey, Dulcie Head, Zach Glassman, Annie Hedlund, Emily Yang,
and Catherine Wilka for the friendship, support, and commiseration that got me through
even the roughest patches of these last four years. I couldn’t have done it without you all.
And lastly, to Mom and Dad (and Kelvin, of course), for not letting the fact that you
didn’t understand a word of what I was doing ever stop you from supporting me in every
way, at every turn. I love you guys, and I owe you everything.
Table of Contents
1 Background: Distances & Variables
1.1 The Cosmic Distance Scale
1.1.1 Distance Determination
1.2 The Carnegie-Hubble Program
1.3 RR Lyrae Variables
1.3.1 Pulsation Mechanism
1.4 The Advantages of the Mid-Infrared
1.4.1 The Period-Luminosity-Color Relation
1.4.2 The Period-Luminosity-Metallicity Relation
1.5 This Thesis
1
1
2
3
4
6
9
9
11
12
2 Data, Reduction, and Calibration
2.1 Data
2.1.1 CCDs and the Point-Spread Function
2.1.2 Observations
2.2 Reduction & Photometry
2.2.1 Photometry Procedure
2.2.2 Calibration
2.3 Limitations
13
13
13
14
16
17
18
20
3 ω Centauri Period-Luminosity Relations
3.1 Background
3.2 Period-Luminosity Relations
3.2.1 3.6 µm
3.2.2 4.5 µm
3.3 Summary
23
23
24
26
30
34
4 Metallicity
4.1 Photometric Metallicities
4.2 Spectroscopic Metallicities
4.3 Combined Metallicities
4.4 Summary
35
36
39
41
43
5 M54
5.1 Background
5.2 Period-Luminosity Relation
5.2.1 3.6 µm
5.2.2 4.5 µm
45
45
45
45
47
6 Conclusions & Future Work
6.1 Conclusions
6.2 Future Work
49
49
49
Bibliography
51
Appendices
57
A Data Tables
A.1 ω Centauri
A.2 M54
B Light Curves
B.1 ω Centauri
B.1.1 3.6 µm
B.1.2 4.5 µm
B.2 M54
B.2.1 3.6 µm
B.2.2 4.5 µm
1
1
3
5
5
5
7
9
9
10
1
Chapter 1
Background: Distances & Variables
1.1
The Cosmic Distance Scale
One of the most fundamental problems of astronomy is the question of distance scales.
When observing a celestial object, be it a star, globular cluster, galaxy, or something more
exotic, and whether it be with our own eyes or through a telescope, how can we determine
whether the object is bright and far away or dim and nearby? The tools and practices of
astronomical measurement have come a long way since Greek astronomers’ first attempts
to calculate the distance to the Sun using shadows, but there is still much refinement to
be done.
Before answering the question of how distances can be determined, however, we must
ask why the question requires an answer. If the universe is isotropic and homogeneous as
current cosmological models postulate, the question of distance ought to be moot, as there
are no privileged positions within the universe.
The distance scale is important on both local and cosmological levels. Accurate
distances are crucial for characterizing the intrinsic properties of stars (luminosity, temperature, radius, etc.) and other celestial bodies, as the light we receive from them is
diminished according to the inverse square law. On a larger scale, distances are required
to understand the rate of cosmic expansion, H0 , defined as the relation between galaxy
distance and recessional velocity. A substantial part of modern cosmology hinges on the
value of H0 , including the nature of dark energy and the density parameters and curvature
of the Universe.
2
1.1.1
Distance Determination
When astronomers refer to the cosmic distance ladder, they typically are referring not so
much to distances themselves but to the ways in which distances are measured and over
what ranges different methods are useful.
3C 273
Coma
Cluster
ω Cen LMC
Betelgeuse
Sirius
0
M31
2
4
6
log d (light years)
8
10
Figure 1.1 The approximate viable ranges of some common distance calibration methods,
including markers for the measured distances of several well-known celestial objects. Each
bar can be thought of as a rung on the distance ladder. Although there appears to be a
substantial amount of overlap, galaxies with more than one indicator are difficult to find
(Zaritsky, Zabludoff, and Gonzalez 2013).
The bars labeled ‘RR Lyrae’, ‘Cepheids’, and ‘Type Ia Supernovae’ on the diagram
are all known as standard candles in astronomical parlance. A standard candle is essentially
an object with a known intrinsic luminosity (see Aaronson & Mould 1986 for more detailed
criteria), which is critical to distance calibration. The distance modulus µ is defined as the
difference between an object’s apparent and intrinsic magnitudes (m and M respectively):
µ=m−M
(1.1)
3
and is related to the distance d in parsecs by
µ = 5 log(d) − 5
(1.2)
Accurate magnitude measurement is thus critical to distance scale calibration.
RR Lyrae and Cepheids are two different types of variable stars, which are so named
because they undergo observable brightness oscillations. There are many classes of variables, but Cepheids and RR Lyrae are of particular importance for distance determination
due to the periodic nature of their variation. The existence of Cepheid variables has been
known since John Goodricke’s discovery of Delta Cephei in 1784, but it was not until the
beginning of the 20th century that Henrietta Swan Leavitt discovered a pattern among
classical Cepheid variables in the Large and Small Magellanic Clouds. In her work examining photographic plates from Harvard College Observatory, she noted that there appeared
to be a correlation between the stars’ periods and their average brightnesses. Her results
came to be known as the Leavitt Law, which demonstrates an empirical linear relationship
between Cepheid absolute magnitudes M and the logarithm of their periods P (Leavitt &
Pickering 1912):
M = a log P + b
(1.3)
This Cepheid period-luminosity (PL) relation has been crucial to the mapping of the Milky
Way and local group of galaxies. A similar period-luminosity relation has been found to
apply to RR Lyrae variables, which is the subject of this thesis.
1.2
The Carnegie-Hubble Program
The Carnegie-Hubble Program (hereafter CHP) is an effort led by the Observatories of the
Carnegie Institution of Washington to reduce the total systematic and statistical error in
H0 to ±2% or better, a notable advance from the 10% error of only a decade ago. This
4
goal requires above all a precise and accurate distance scale. The program is structured
such that calibrations are performed on Cepheids and RR Lyrae that are near enough in
the Milky Way to have measurable parallax values; the Hipparcos satellite has measured
parallaxes for 219 Cepheids (Benedetto 2002) and 125 RR Lyrae (Tsujimoto et al 1998).
Once the absolutely period-luminosity relations are ascertained, work is done on variables
that are too far for parallax to reach, and continued such that the distance ladder keeps
extending outward once each previous rung has been well-established.
The primary data source for the project at present is the Spitzer Space Telescope
post-cryogenic mission using the telescope’s Infrared Array Camera (IRAC). Future work
on galaxies too distant to be observed with Spitzer will make use of the James Webb Space
Telescope, which should be able to see objects 10 to 100 times more faint than the Hubble
Space Telescope can resolve.
The Carnegie RR Lyrae Program (CRRP) is a branch of the CHP focusing on
distance calibration with RR Lyrae. The RR Lyrae distance scale does not extend as far
as the Cepheid distance scale, but it is more precise than and rigorously independent of
the Cepheid scale, which makes it an excellent candidate for constraining systematic errors
in Cepheid distances as well as a valuable calibrator in its own right.
1.3
RR Lyrae Variables
RR Lyrae variables are helium-burning Population II stars that are more common but less
luminous than Cepheids. They are named after the first of their type to be observed, the
eponymous star RR Lyrae near the border of the constellations Lyra and Cygnus, which
was discovered by Williamina Fleming at Harvard Observatory (Pickering et al 1901). RR
Lyrae, along with Cepheids and δ Scuti variables, occupy a region of the HertzspringRussell diagram known as the instability strip, where certain post-main sequence stars
become unstable to radial pulsations. All RR Lyrae lie near the bottom of the instability
strip below Cepheids and above δ Scuti, with mean luminosities on the order of 50 L ,
5
mean effective temperatures on the order of 6000–7000 K, radii below 10 R , and masses
of approximately 0.6 − 0.8 M (Cacciari 2012).
Figure 1.2 Where most known types of variable stars occur on an HR diagram. The
instability strip is marked by two dashed lines, and the horizontal branch is marked by a
dash-dotted line. Plot adapted from Christensen-Dalsgaard (2004).
As depicted above, RR Lyrae occur where the instability strip and horizontal branch
intersect, and occupy a much smaller range of luminosities than Cepheids. Color-magnitude
diagrams of globular clusters often show a gap in the horizontal branch where RR Lyrae
ought to occur; although there is a range of possible RR Lyrae colors, this gap is not strictly
6
an intrinsic phenomenon. RR Lyrae require precisely timed observation cycles over several
hours to a day to determine their true magnitudes and colors, whereas many photometric
observations catch them at only one point in their pulsation cycles.
1.3.1
Pulsation Mechanism
RR Lyrae pulsate radially in either the fundamental harmonic (RRab), the first overtone
(RRc), or both (RRd), although the latter is relatively rare. This form of radial pulsation
is analogous to an organ pipe with one end open, with a node at the center of the star and
an antinode at the surface.
f0
f0
f1
f1
Figure 1.3 Left: the first two harmonics in a pipe open in one end. Right: the analogous
pulsation modes in star cross sections. The thin arc in the first overtone cross section is
analogous to the second node in the first overtone pipe.
RRab have periods typically ranging from 12 to 24 hours and light curves characterized by a sharp rise and slow decline; RRc have periods ranging from about 4 to 12
hours and nearly sinusoidal light curves (Cacciari 2012). There is still some debate as to
the exact nature of RR Lyrae pulsation, and a detailed pulsation model is a continual work
in progress, but there is consensus that the main radial pulsation mechanism is the same
as the one known to be present in Cepheids, the κ mechanism (Marconi 2009). κ is used
to denote the opacity of a star, which is defined as the change in intensity of a light wave
as it propagates through a gas. Opacity typically decreases with increasing temperature
7
according to Kramer’s Law:
κ∝
ρ
T 3.5
(1.4)
However, there are certain zones in some stars where this proportionality is reversed, known
as partial ionization zones, where the temperature of the zone is such that ionization
of hydrogen and/or helium may begin. In the HeII partial ionization zone, the double
ionization of helium occurs at a characteristic temperature of 4 × 104 K: He+ ↔ He2+ + e− .
In RR Lyrae and similar variables, radiation from the star’s core increases ionization in
this zone and thus the opacity of the layer, as the energy from the radiation is absorbed
by ionization. The radiation pressure pushes the layer outwards, and as it expands, the
layer cools until the helium begins to deionize. As deonization occurs, the opacity drops
and allows radiation through, decreasing the radiative pressure and allowing the partial
ionization zone to compress and begin the cycle over again.
The simplest case of radial pulsation (the fundamental mode) can be roughly modeled as a sound wave propagating from the center of the star, where the speed of sound is
related to pressure P and density ρ like so:
s
vs =
γP
ρ
(1.5)
The pulsational period Π is then:
2R
vs
r
ρ
= 2R
γP
Π=
(1.6)
(1.7)
With the (admittedly unrealistic) assumption of constant density (ρ = M/ 43 πr3 ), we can
approximate the pressure gradient as a function of radius with the boundary condition that
8
P = 0 at the surface R:
G( 4 πr3 ρ)ρ
dP
GM ρ
4
=− 2 =− 3 2
= − πGρ2 r
dr
r
r
3
Z r
4
− πGρ2 r
P (r) =
3
R
2
= πGρ2 (R2 − r2 )
3
(1.8)
(1.9)
(1.10)
Substituting this back into the period equation and integrating over all possible r, we
obtain:
Π = 2R
ρ
r
Z
2
2
2
3 πGρ (R
R
s
=2
=
s
=
dr
2
2
3 πGρ(R
0
r
− r2 )γ
− r2 )γ
(1.11)
(1.12)
3π
2Gργ
(1.13)
2π 2 R3
GM γ
(1.14)
This equation is relatively simple, but R and M are not easily measurable for most stars.
Luminosity L and temperature T are far easier to directly observe, so we can use the
Stefan-Boltzmann law L = 4πσT 4 R2 to find a period equation in terms of luminosity and
temperature:
s
Π=
2π 2
3/2
L
4πσT 4
GM γ
(1.15)
3
∝
L4
T3
(1.16)
This proportionality relation is best known as the period-luminosity-color (PLC) relation,
as a star’s temperature directly determines its color. It should be noted that this method
of deriving the PLC relation is far too simplified to be of any practical use, but it is
nevertheless an important demonstration of relationships between relevant quantities.
9
1.4
1.4.1
The Advantages of the Mid-Infrared
The Period-Luminosity-Color Relation
The PLC relation has been a subject of controversy in recent decades, mainly due to
investigations into the systematic effects of reddening. Reddening is caused by interstellar
dust grains, which absorb short wavelength radiation and re-emit it in longer wavelengths,
causing objects to appear cooler and dimmer than they intrinsically are. For Cepheids
and RR Lyrae, this results in distances that are systematically too high if reddening is
not accounted for in visual wavelengths. Madore and Freedman (1991) have demonstrated
the difficulty of decoupling reddening and intrinsic color and luminosity deviations from
an empirical PLC in visual wavelengths. They recommend using data in the near- to midinfrared to reduce systematic error, as the effects of reddening are dramatically reduced at
long wavelengths. Compared to V -band (0.55 µm) data, extinction in a 3.6 µm bandpass
is reduced by a factor of 14-17, and up to a factor of 43 in 4.5 µm (Freedman et at 2011).
This is the primary reason to use the mid-IR for distance calibration.
The PL relation is a projection of the three-dimensional PLC relation onto periodluminosity space. As there is a range of possible temperatures for a star of a given period,
the PL relation is expected to have some amount of intrinsic scatter. The magnitude of
this scatter is known to be wavelength-dependent; RR Lyrae luminosities are much less
sensitive to temperature in infrared bands than in optical bands. This results in reduced
PL relation scatter and thus more precise distances, which is another asset of the midIR. RR Lyrae in particular are superb distance indicators in this regime, with PL scatter
as low as ±0.03 mag (up to four times smaller than the Cepheid PL scatter in the same
wavelengths) and yielding distances precise to below ±2% for single stars (Spitzer Proposal
#90002).
10
Figure 1.4 Simulated RR Lyrae PL relations in the UBVRIJHK system showing the dramatically decreased scatter as one moves from the optical to infrared (Catelan, Pritzl, &
Smith 2004).
As seen above, there is a notable reversal of the sign of the slope of the PL relation
that occurs between the B and R bands. As one moves into the infrared, the dominant
factor in the PL slope changes from temperature, which decreases with increasing period, to
radius, which increases with increasing period. The PL slope is expected to asymptotically
approach the slope of the period-radius relation as one moves farther into the IR; Madore
et al (2013) find that the period-radius relation derived by Burki & Meylan (1986), which
gives an expected asymptotic PL slope of −2.60, is most consistent with their preliminary
results. Klein et al (2014) and Dambis et al (2014) both find slightly shallower slopes.
The changing slope and scatter of the PL relations as one moves into the infrared
cannot be explained by decreasing color dependence alone, however. The other dominant
factor in determining RR Lyrae luminosity is metallicity.
11
1.4.2
The Period-Luminosity-Metallicity Relation
There are two ways of denoting the metallic (non-hydrogen or helium) content of a star.
The first is simply the fraction of all elements Z other than H (X) or He (Y ) in a star:
Z =1−X −Y
(1.17)
Observationally speaking, Z is usually difficult to measure directly. Hence, a second way
of measuring metallicity is a calculation of a star’s numerical iron abundance relative to
the sun’s, which is obtainable via spectroscopy:
[Fe/H] = log
n(Fe)
n(H)
− log
?
n(Fe)
n(H)
(1.18)
In the case where the element distribution is close to solar, Z and [Fe/H] are related by:
[Fe/H] = log
= log
Z
X
Z
X
− log
?
Z
X
(1.19)
+ 1.61
(1.20)
?
Iron is used mainly for convenience, as it is easy to distinguish in the optical regime,
but other elements may be used in place of iron and/or hydrogen for specific abundance
measures.
Metallicity affects a star’s observed luminosity in the optical and near-IR bands due
to a phenomenon known as line blanketing, in which dense absorption lines from metals
absorb light in these bands and reemit it farther in the infrared. Thus, the RR Lyrae PL
relation in these wavelengths requires an extra term to account for metallicity:
M = a log P + b + c log Z
(1.21)
Theoretical models suggest that the metallicity dependence of the RR Lyrae PL relation
should decrease monotonically from the optical to the near-infrared (Catelan, Pritzl, &
12
Smith 2004, Bono et al 2001), and observational evidence corroborates this; previous investigations performed on WISE data suggest no obvious metallicity dependence in the
mid-IR PL relations (Madore et al 2013, Klein et al 2011). This is a third advantage of
the mid-IR, as it reduces the number of parameters required for distance calibration and
thus reduces possible sources of error.
Figure 1.5 Comparison of error sources in H0 between the optical HST Key Project and
the Carnegie-Hubble program (Freedman 2012).
1.5
This Thesis
The mid-IR has not previously been used for distance calibration despite the clear advantages, primarily because it is only observable from above the atmosphere and the technology to obtain the necessary data has only been made possible in the past few decades.
This thesis presents the first calibration of the slope and scatter of the mid-IR RR Lyrae
period-luminosity relation derived from Spitzer data, as well as an investigation of metallicity effects, using the globular clusters ω Centauri (NGC 5139) and M54 (NGC 6715).
13
Chapter 2
Data, Reduction, and Calibration
2.1
2.1.1
Data
CCDs and the Point-Spread Function
The instrument most commonly used for astronomical imaging today is a charge-coupled
device, or CCD, which consists of an array of silicon pixels on a chip. Upon exposure to
light of certain wavelengths, the incident photons excite valence electrons in the silicon into
the conduction band. To prevent recombination, a voltage is applied to the pixel such that
the electrons are trapped in a potential well for the duration of the exposure. The total
charge per pixel is then read out and mapped as an integer flux value onto the resulting
image.
All stars with the exception of the Sun are distant enough that they can be treated as
infinitesimal point sources in the context of CCD observations. The point-spread function
(abbreviated PSF, sometimes referred to as the point-response function or PRF) describes
the response of CCDs to light from point sources. In an ideal system where the only limiting
factor is diffraction, the PSF intensity distribution is described by a function known as the
Airy pattern:
I(θ) = I0
2J1 (x)
x
2
(2.1)
where I0 is the central (maximum) intensity, J1 is the first-order Bessel function, θ is the
angle between the aperture center and the observation point, and x is a dimensionless
quantity related to the focal length of the system. In data reduction software, the PSF is
14
commonly approximated as a Gaussian function.
Figure 2.1 Top left: an ideal Airy disk PSF. Top right: a Gaussian approximation of an
Airy disk. Bottom: surface plot of a sample star from a Spitzer image.
2.1.2
Observations
All data for this project is from the Warm Spitzer mission, taken with the Infrared Array
Camera (IRAC) in 3.6 and 4.5 µm. The observations were designed such that each 3.6
µm field captured as many RR Lyrae as possible, as the cameras for each bandpass are
slightly offset from each other and Spitzer is slightly more sensitive in 3.6 µm. Exposures
were taken in time series of 12 and timed according to the longest-period RR Lyrae in
each field, such that all RR Lyrae would be observed for at least one complete period.
Initial reduction was done by Dr. Victoria Scowcroft using the Spitzer-specific package
MOPEX (MOsaicker and Point source EXtractor), which includes alignment, stacking,
outlier rejection, and background smoothing. Photometry was done on the unstacked
frames when possible in order to obtain light curves, but in some cases the stacked images
15
were required in order to generate a viable PSF, which will be discussed in detail later on.
Figure 2.2 Color-inverted maps of ω Centauri (top) and M54 (bottom) in 3.6 µm, with RR
Lyrae circled in red (catalogs from Kaluzny et al 2004 and Sollima et al 2010).
For ω Centauri there are 3 fields in both bands, with 12 frames per field for a total
of 72 frames. For M54 there is 1 field in both bands with 12 frames for a total of 24 images.
The blank spaces are regions that were rejected due to saturation and scattering.
16
2.2
Reduction & Photometry
The aim of photometry is to accurately measure the apparent magnitudes of objects in a
CCD exposure. There are two kinds of photometry for point sources: aperture and PSF.
Aperture photometry is a process in which the total flux contained in a given aperture
radius is summed, and then the flux within a larger radius, the sky annulus, is subtracted
from that. The resulting value is converted to a magnitude by the equation:
m = m0 − 2.51212 log(N − hSi) + 2.51212 log dt
(2.2)
Here m is the apparent magnitude, m0 is a zero point defined as the magnitude
for which the detector receives one photon per second (typically calibrated using standard
stars in the field), N is the number of counts in the aperture radius, hSi is the average sky
brightness, and dt is the exposure time in seconds.
Aperture photometry is limited in that it can only be used in fields where there are
no sources within a sky annulus radius of each other. If the field is crowded, aperture photometry measurements will be contaminated by neighboring sources, and PSF photometry
must be used instead.
PSF photometry takes the concept of the point spread function previously described
and uses it to model the number of counts within an aperture radius by integrating under
the specified function rather than simply using summation. It then convolves the model
with selected stars, effectively subtracting them from the image and obtaining more accurate magnitudes for them in one swoop. Once a round of subtraction has been completed,
sources that were previously hidden may now be revealed, and the process may be repeated.
PSF photometry is advantageous for crowded fields in that it can obtain more accurate
magnitudes for a greater number of stars than aperture photometry.
17
Figure 2.3 Left: an aperture and annulus surrounding a star. Right: the residual light
after a model PSF was fitted to and subtracted from the same star.
2.2.1
Photometry Procedure
All photometry was performed using the DAOPHOT II: The Next Generation software
suite (Stetson 1987) within IRAF (Image Reduction and Analysis Facility). DAOPHOT is
designed to perform stellar photometry on crowded fields using PSF fitting.
Prior to beginning the data reduction, all images had to be converted from flux units
(MJy/sr) to counts (electrons per pixel) by multiplying by the exposure time and dividing
by the flux converstion factor. The conversion to counts is necessary for DAOPHOT’s
methods of noise detection to work correctly. As Spitzer magnitudes were originally calibrated using flux images, this will require recalibration of the DAOPHOT magnitudes later
on.
The first step for photometry was to detect as many stars as possible using the
daofind command. daofind uses Poisson statistics to recognize regions that are a certain
number of standard deviations brighter than the sky and noise background of the image.
A standard detection threshold of 3σ was used on this data.
Next, aperture photometry was performed on the identified stars using phot to
calculate background sky values and to provide an initial magnitude estimate to be used
and refined during PSF fitting. Photometry on these fields was performed using an aperture
radius of 3.600 , an annulus radius of 4.800 , and zero points of 18.672 and 18.188 mag for 3.6
18
and 4.5 µm respectively.
Once aperture photometry was completed, a model PSF was built for each frame
using a selection of bright, isolated stars to best approximate the frame’s ideal PSF. Typically 15-30 PSF stars were used per field. Stars were first chosen visually, and then accepted
or rejected based on how closely their surface plots resembled a Gaussian function. This
model PSF was then used in the allstar command to subtract out and calculate new
magnitudes for the stars for which phot had previously measured aperture photometry
magnitudes. The stars that were used to build the PSF were excluded from this round of
subtraction so that a new PSF could be made with them on the subtracted frame to ensure
minimal contamination of the PSF from other sources.
As is apparent in Figure 2.3, the PSF appears distinctly triangular; this is the
result of a Reuleaux triangle dither pattern applied during observations. It does not pose
a problem for photometry, as DAOPHOT computes a ‘look-up table’ of the difference
between the pixel values of the stars that are used to build the empirical PSF and the
values of the analytical Gaussian PSF.
The subtraction process revealed many stars that had been previously hidden behind
brighter foreground stars. To account for these stars, daofind was run on the subtracted
frame and phot was run on the original frame with the new coordinates. The resulting
output photometry file was then merged with the one for the original stars, and allstar
was then run again on the original frame using the merged photometry data. These last
three steps were repeated three times in order to subtract and obtain magnitudes for as
many stars as possible.
2.2.2
Calibration
The magnitudes obtained from the final allstar pass are as precise as they are likely to
be, but they are not accurate. IRAC magnitude calibrations were done with a 600 aperture
on images in flux units, but this photometry was done on images in counts units with a
3.600 aperture due to the crowding of the field. (In crowded regions, flux from neighboring
19
stars will affect magnitude calculations if the aperture and annulus radii are too large.) To
convert the instrumental magnitudes to the calibrated scale, an aperture correction must
be applied.
The images from the first epoch in each time series were used as reference frames
for the maps for which time series data was used. (There was no need for reference frames
when the stacked images were used.) The aperture correction was found by subtracting
the phot magnitudes of the PSF stars in the original flux image from the final allstar
magnitudes of the same stars in the corresponding counts image and taking the mean of
the difference:
∆map = mals,1 − mphot,1
(2.3)
∆map was then subtracted from the allstar magnitudes of all the stars in the
reference frame, which gave the corrected reference magnitudes, mcor,1 :
mcor,1 = mals,1 − ∆map
(2.4)
The IRAC Instrument Handbook contains tabulated aperture correction factor values for correcting magnitudes to the standard aperture radius (Carey et al 2012). To
calibrate to the standard aperture, mcor was converted to flux and multiplied by the correction factor c, where c = 1.12841 and 1.12738 for 3.6 and 4.5 µm respectively:
F cal,1 = c × 10−mcor,1 /2.512
(2.5)
Converting this calibrated flux back to magnitudes yields calibrated reference magnitudes mcal,1 .
Next, the offsets between epochs were found by subtracting the allstar magnitudes
of the PSF stars from mcal,1 and taking the mean for each epoch:
20
∆moffset,n = mcal,1 − mals,n
(2.6)
For all remaining allstar magnitudes, subtracting ∆moffset,n yields calibrated magnitudes mcal,n for that epoch.
The final step is to correct the magnitudes for location in the frame. The IRAC
CCDs do not have a uniform response across the array; there are variations in the spectral
response and solid angle per pixel that are not corrected in the initial field-flattening process
(Reach et al 2005). To account for these, the location correction is defined to be unity at
the center of the array and is mapped outward from there, allowing for a correction file
to be included with all frames. The correction values for each frame, f loc,n , were obtained
by running the getpix routine from wcstools on the correction files using the coordinates
from the final allstar files. The location-corrected flux is:
F cor,n =
10−mcal,n /2.512
f loc,n
(2.7)
The final calibrated apparent magnitudes are found by converting F cor,n back to
magnitudes.
For the time series data, the sources in each frame must then be matched to each
other. This is done with the standalone DAOPHOT package using the commands daomatch
and daomaster.
2.3
Limitations
The primary limiting factor in this data is crowding. For ω Cen, 77 RR Lyrae out of a
catalog of 192 (Kaluzny et al 2004) were rejected due to crowding. To decide which stars
to reject, a K-band image from the Magellan telescope at Las Campanas Observatory was
used, as it provided a full view of the entire cluster, and the most crowded regions were
more obvious than in the Spitzer data, although the bandpasses are close enough that they
21
are still comparable. Stars were rejected on a primarily visual basis.
For M54, the crowded core of the cluster is entirely contained in the Spitzer mosaics,
so there is no need to use Las Campanas images. Here 75 RR Lyrae out of a catalog of
144 were rejected due to central crowding or proximity to bright foreground stars.
Figure 2.4 Color-inverted images of ω Cen from Las Campanas Observatory with RR Lyrae
circled in green. The full catalog is on top, and the unrejected stars are on bottom.
22
Figure 2.5 Color-inverted Spitzer images of M54 with RR Lyrae circled in green. The full
catalog is on top, and the unrejected stars are on bottom.
23
Chapter 3
ω Centauri Period-Luminosity Relations
3.1
Background
ω Centauri is the largest globular cluster in the Milky Way, located at 13h 26m 47.28s,
-47◦ 280 46.100 (J2000), with an angular diameter of approximately 360 and a total mass of
approximately 4 × 106 M (D’Souza & Rix 2013). It is unique among Milky Way globular
clusters in that it contains multiple generations of stars, with an age difference of at least
2 Gyr from oldest to youngest, and a corresponding wide metallicity spread (Hughes 1999,
Stanford et al 2006). Several, most notably Majewski et al (1999) and Hilker et al (2000),
have taken this as evidence that ω Cen is not a globular cluster at all, but the core of an
ancient dwarf galaxy that was stripped of much of its mass due to tidal interaction with
the Milky Way.
ω Cen is ideal for constraining the RR Lyrae period-luminosity-metallicity relation,
as it contains 192 known RR Lyrae (Kaluzny et al 2004) with a metallicity range spanning
over 1.5 dex (Bono 2013, private communication.) A metallicity spread this wide is not
found in any other Milky Way globular clusters. As the RR Lyrae are all at roughly the
same distance from Earth (ω Cen is about 4800 pc from Earth, whereas its radius is only
about 26 pc), we can be reasonably confident that any apparent effects on the periodluminosity relation are due to metallicity rather than differences in distance or reddening;
the use of the mid-IR also ensures that any effects of interstellar or intracluster reddening
are minimized.
24
Figure 3.1 Left: ω Cen CMD with known RR Lyrae circled in red (RRab) & blue (RRc).
Right: histogram of RR Lyrae metallicity distribution (Bono 2014, private communication).
3.2
Period-Luminosity Relations
Our RR Lyrae sample in 3.6 µm totals 36 stars, 20 of which are RRab’s and 16 of which
are RRc’s. In 4.5 µm we have 19 RRab’s and 18 RRc’s for a total of 37 stars. Out of these,
25 appear in both fields; as mentioned in chapter 2, the IRAC cameras for each bandpass
are slightly offset from each other, and thus the fields do not overlap precisely.
It is common practice to convert the RRc periods to fundamental mode periods
(to “fundamentalize” them) using the ratio observed in double mode RR Lyrae, where
P1 /P0 = 0.74432 ± 0.00003 or log P0 = log P1 + 0.128 (Walker & Nemec 1996), such that
the types can be combined for a larger sample size, as done in Dall’Ora et al (2004).
However, there has been some debate as to the appropriateness of this method; Giuseppe
Bono (2013, private communication) has found differing slopes for the two types, and Klein
et al (2014) present separate relations for each type, arguing that the combination of the
types is physically inappropriate. We present both combined and separate PL relations here
for consideration, as well as relations that are both weighted and unweighted by individual
magnitude errors. We do not apply any reddening corrections; the K-band extinction AK
25
in the line of sight to ω Cen is approximately 0.043 (NED Galactic Extinction Calculator),
and Indebetouw et al (2005) calculated Aλ /AK extinction ratios for Spitzer bandpasses
where A[3.6] /AK = 0.56 ± 0.06 and A[4.5] /AK = 0.43 ± 0.08. Therefore, A[3.6] = 0.024, and
A[4.5] = 0.018, which are low enough that extinction corrections will have negligible impact
on the final result.
PL relations were fit using SciPy’s linregress and leastsq functions in the stats
and optimize packages. linregress was used for the unweighted case, as its output
includes the standard error of the estimate, which was necessary to calculate the errors in
the slopes and zero points. leastsq was used for the weighted case, as it is possible to
define a function that weights the dependent variable by individual errors in leastsq.
In the unweighted case, the uncertainties in slope and intercept for a function of
the form y = Ax + B were calculated as follows:
s
N
P
N
− ( x)2
s
P 2
x
P 2
P
σB = σy
N x − ( x)2
σA = σy
P
x2
(3.1)
(3.2)
where σy is the standard error of the estimate and N is the number of data points. In the
weighted case, the uncertainties were
s
σA =
P
P
w
P
wx2
w
P
− ( wx)2
(3.3)
w
P
wx2
P
wx2 − ( wx)2
(3.4)
s
σB =
P
P
where wi = 1/σi2 , the inverse square of each individual error in the y values (Taylor 1997).
26
3.2.1
3.6 µm
In 3.6 µm the weighted and unweighted combined PL relations of the form
m = A (±σA ) log P + B (±σB )
are as follows for ω Cen:
A
σA
B
σB
σ
Weighted
−2.50
0.09
12.38
0.02
0.10
Unweighted
−2.49
0.39
12.41
0.11
0.10
where the scatter σ is the standard deviation of the residuals, mobserved − mexpected . The
errors in the weighted fit parameters are significantly lower than the unweighted errors
primarily due to the minimization of influence of extreme (high- or low-period) points
when individual errors are weighted. See fig. 3.2 for the corresponding plots.
These slopes agree within uncertainty of the slopes of the W 1 (3.4 µm) absolute PL
relations derived by Madore et al (2013), Klein et al (2014), and Dambis et al (2014), and
the scatter agrees with that of Madore et al. Their respective results are as follows:
A
σA
B
σB
σ
Madore
−2.44
0.95
−1.26
0.25
0.10
Klein
−2.38
0.20
−1.113
0.013
—
Dambis
−2.381
0.097
−1.150
0.077
—
The zero points for these relations indicate absolute magnitudes, as opposed to the
apparent magnitude zero points we have obtained for ω Cen. It should be noted that Klein
et al present relations of the form M = A log(P/P0 )+B, where P0 is a period normalization
factor; here we have subtracted A log P0 from their derived zero points to give zero points
consistent with the form of the other relations, not taking into account error in A log P0 .
They also separate the RRab’s and RRc’s; here we use their RRab relations. Dambis
et al present two zero-point estimates, one based on statistical parallaxes and one based
on HST trigonometric parallaxes; here we use the HST parallax estimates for the sake of
consistency with Madore and Klein. Dambis et al also include a metallicity term, which
will be discussed later.
27
Weighted
± σ, σ =0.10 mag
± 2σ
RRLab
RRLc, fundamentalized
Apparent magnitude
12.5
13.0
13.5
0.5
0.4
0.3
log P (days)
0.2
0.1
0.2
0.1
Unweighted
± σ, σ =0.10 mag
± 2σ
RRLab
RRLc, fundamentalized
Apparent magnitude
12.5
13.0
13.5
0.5
0.4
0.3
log P (days)
Figure 3.2 ω Centauri PL relations in 3.6 µm with combined RRab’s and RRc’s.
28
Madore et al (2014, in prep) have found that the W 1 and 3.6 µm bandpasses are
similar enough to be compatible, and that it is acceptable to obtain distance moduli by
combining measurements from W 1 and 3.6 µm. Combining the weighted ω Cen zero point
here with the Dambis zero point gives a distance modulus of µ = 12.38 ± 0.02 − (−1.150 ±
0.077) = 13.53±0.08, which corresponds to a distance of 5082±203 parsecs. (Our weighted
and unweighted zero points are close enough that either is acceptable for our purposes here;
similarly, given that the Dambis and Klein results are in excellent agreement with lower
errors than Madore, our choice of which to use is insignificant.) This is slightly lower than
previous distance measurements using RR Lyrae in the near-IR (Del Principe et al 2006)
and the eclipsing binary OGLEGC17 (Thompson et al 2000), but higher than the distances
measured by dynamical modeling (Watkins et al 2013, van de Ven et al 2006).
When we separate the RRab’s and RRc’s and do not fundamentalize the RRc
periods, we find the following relations:
A
σA
B
σB
σ
RRab, weighted
−2.36
0.17
12.41
0.04
0.11
RRab, unweighted
−2.46
0.73
12.41
0.16
0.11
RRc, weighted
−2.71
0.16
11.96
0.07
0.10
RRc, unweighted
−2.33
0.64
12.18
0.28
0.10
In the weighted case there is a 2σ difference between the RRab and RRc slopes,
and well over a 5σ difference between the zero points. The differences are not so dramatic
in the unweighted case, although the errors are much higher. See fig. 3.3 for plots.
Klein et al find a W 1 RRc slope and zero point of −1.64 (±0.62) and −1.043 (±0.031)
respectively, which is appreciably different from both the RRc slopes found here. This is
may simply be an accident of small sample sizes; in both cases the sample size is under 20, and more data will likely resolve the discrepancies. Combining the weighted
ω Cen zero point with the absolute RRc zero point gives a distance modulus of µ =
11.96±0.07−(−1.043±0.031) = 13.00±0.08, which corresponds to a distance of 3981±159
pc and is well below even the lowest accepted measurements.
29
Weighted
± σ, σab =0.11, σc =0.10
± 2σ
Apparent magnitude
12.5
RRLab
RRLc, +1 mag
13.0
13.5
14.0
14.5
0.6
0.4
0.3
log P (days)
0.2
0.1
0.2
0.1
Unweighted
± σ, σab =0.11, σc =0.10
± 2σ
12.5
Apparent magnitude
0.5
RRLab
RRLc, +1 mag
13.0
13.5
14.0
14.5
0.6
0.5
0.4
0.3
log P (days)
Figure 3.3 ω Centauri PL relation in 3.6 µm with separate RRab’s and RRc’s.
30
3.2.2
4.5 µm
The weighted and unweighted PL relations for ω Cen in 4.5 µm are as follows:
A
σA
B
σB
σ
Weighted
−2.40
0.12
12.42
0.03
0.09
Unweighted
−2.43
0.36
12.42
0.10
0.09
See fig. 3.4 for plots. Previous calibrations for the WISE W 2 band (4.6 µm) find
the following:
A
σA
B
σB
σ
Madore
−2.55
0.89
−1.29
0.23
0.10
Klein
−2.39
0.20
−1.108
0.013
—
Dambis
−2.269
0.127
−1.105
0.077
—
Both 4.5 µm slopes are strongly consistent with Klein et al in this case. Our
weighted slope is 1σ from both Madore et al and Dambis et al’s values, albeit in opposite
directions. Combining the weighted ω Cen zero point with the Dambis zero point gives a
distance modulus of µ = 12.42 ± 0.02 − (−1.105 ± 0.077) = 13.53 ± 0.08, the same value as
found with the 3.6 µm weighted zero point.
Similar to Dambis et al, we find that the slope becomes shallower moving from
3.6 µm to 4.5 µm, which is contrary to the predictions made by Madore et al that the
period-luminosity slope will asymptotically approach the period-radius slope as one moves
farther into the infrared. Further study is required on this to determine whether this is
truly an intrinsic effect or simply a coincidence of the data sets. It should be noted that
in the case of Cepheid variables, Scowcroft et al (2011) have shown that in 4.5 µm there is
absorption due to a CO bandhead at 4.65 µm, which flattens the slope of the PL relation.
However, this effect is due to the low temperature of Cepheid atmospheres and disappears
in the hottest, shortest-period Cepheids, as the CO dissociates at temperatures above 6000
K (Monson 2012). As all RR Lyrae have effective temperatures higher than 6000 K, we
expect no such CO absorption. However, if it proves to be true that the RR Lyrae PL
31
Weighted
± σ, σ =0.09 mag
± 2σ
RRLab
RRLc, fundamentalized
Apparent magnitude
12.5
13.0
13.5
0.5
0.4
0.3
log P (days)
0.2
0.1
0.2
0.1
Unweighted
± σ, σ =0.09 mag
± 2σ
RRLab
RRLc, fundamentalized
Apparent magnitude
12.5
13.0
13.5
0.5
0.4
0.3
log P (days)
Figure 3.4 ω Centauri PL relations in 4.5 µm with combined RRab’s and RRc’s.
32
slope is intrinsically flatter in 4.5 µm than in 3.6, there may be reason to look for an effect
similar to the CO bandhead in RR Lyrae.
For separate RRab’s and RRc’s, we find:
A
σA
B
σB
σ
RRab, weighted
−3.10
0.25
12.27
0.05
0.10
RRab, unweighted
−2.77
0.75
12.35
0.16
0.10
RRc, weighted
−1.80
0.25
12.38
0.12
0.08
RRc, unweighted
−2.04
0.69
12.29
0.31
0.08
See fig. 3.5 for plots. Here, in contrast to the 3.6 µm relations, we see a 5σ difference
between the weighted slopes and a 2σ difference at most between the weighted zero points,
and it is the RRab slope that is considerably steeper in both fits. This time our RRc
slopes are consistent with Klein et al’s; they find a W 2 slope of −1.70 ± 0.62. However,
our slopes are strongly affected by the longest-period RRc, ID 47; removing it gives an
unweighted RRc slope of −2.39 ± 1.49 and a weighted slope of −2.33 ± 0.32. Similarly,
our RRab slopes are pulled down dramatically by the two shortest-period RRab’s, IDs 5
and 107; removing them gives an unweighted slope of −2.21 ± 2.10 and a weighted slope of
−2.43 ± 0.31. Again, more data is needed to properly characterize the slopes of each type.
Combining our weighted RRc zero point with Klein et al’s RRc absolute zero point
−1.057 ± 0.031 gives a distance modulus of 13.44 ± 0.12, or distance of 4875 ± 293 pc, which
is slightly low but consistent with previous measurements.
33
Weighted
± σ, σab =0.10, σc =0.08
± 2σ
Apparent magnitude
12.5
RRLab
RRLc, +1 mag
13.0
13.5
14.0
14.5
0.6
0.4
0.3
log P (days)
0.2
0.1
0.2
0.1
Unweighted
± σ, σab =0.10, σc =0.08
± 2σ
12.5
Apparent magnitude
0.5
RRLab
RRLc, +1 mag
13.0
13.5
14.0
14.5
0.6
0.5
0.4
0.3
log P (days)
Figure 3.5 ω Centauri PL relations in 4.5 µm with separate RRab’s and RRc’s.
34
3.3
Summary
There is overall insufficient data here to provide any insight towards the question of separating the stars by type; though the differences between the slopes of the RRab’s and RRc’s
are fairly dramatic, there is no consistency between bands, and it is not clear whether
the differences between slopes are intrinsic or primarily a product of outlier influence and
small data sets. Certainly when the types are combined it is not immediately apparent
that the slopes are or should be different, although the width of the PL relations may
contribute towards obscuring any difference there may be. The slopes and zero points are
most consistent with previous measurements when the types are combined.
The scatter in these PL relations is higher than we expect the intrinsic scatter in
these bands to be. Some factors that may contribute to this are residual crowding (despite
the most crowded stars having been removed), misidentification of stars in daomatch, and
image artefacts present in certain frames. Several of the light curves (see Appendix B)
have one or two data points which are up to half a magnitude brighter or dimmer than
the typical range of the rest of the stars; this could affect the mean magnitudes and lead
to increased scatter overall.
35
Chapter 4
Metallicity
To check for a metallicity term in the PL relation, we plot individual RR Lyrae metallicities
against their PL residuals. If there is a relationship between metallicity and the residuals,
as in the optical and near IR, this would indicate the presence of a metallicity term. In
this case, the residuals we have used are from the weighted, combined PL fits.
Before we examine the metallicity relationships, we check the quality of our data by
plotting the residuals against each other. We don’t expect a good deal of color variation
in this data, so the residuals should match each other fairly well.
0.2
∆[4.5]
0.1
0.0
0.1
0.2
0.2
0.1
0.0
∆[3.6]
0.1
0.2
Figure 4.1 Deviations from the PL relation in [3.6] vs. deviations in [4.5].
36
Running linear regression on this data gives a slope of 0.63(±0.29), which is 1σ from
the slope of unity that we expect, and zero point 0.01(±0.03). This may indicate a slight
systematic error in the photometry, but nothing too concerning (Scowcroft 2013, private
communication). The slight overabundance of negative residuals is a coincidence of the
fact that these are only the stars that appear in both channels, and not all the stars that
were used to calculate the PL relations.
We can now examine the metallicity relations. We present both photometric (Rey
et al 2000), spectroscopic (Sollima et al 2006), and combined metallicities for comparison;
while the Rey catalog contains 131 RR Lyrae metallicites as opposed to the 74 in Sollima,
spectroscopic metallicities are much more reliable than photometric metallicities, as photometric metallicities rely on certain assumptions about the chemical abundances of stars
whereas spectroscopic metallicities provide actual abundances. For stars appearing in our
sample, there is a metallicity spread of up to 1.19 dex using the Rey catalog, and 0.76 dex
using Sollima. When we combine the catalogs, we favor Sollima when both are available.
If there is any correlation between [Fe/H] and the PL residuals, we expect it to be
a linear one, consistent with the metallicity terms of order c log Z in the near-IR. Relations
were fit using linregress without weighting individual errors in either x or y. Although
there are relatively large errors in both variables, they are consistent enough that weighting
by errors would result in a fit heavily skewed by the few data points with low errors.
4.1
Photometric Metallicities
For photometric metallicity vs. residuals we find relations of the form
∆[λ] = A[Fe/H] + B
as follows (see figs. 4.2 and 4.3):
λ
A
σA
B
σB
σ
[3.6]
−0.06
0.05
−0.12
0.07
0.11
[4.5]
−0.10
0.04
−0.18
0.06
0.09
[3.6] − [4.5]
0.12
0.05
0.19
0.08
0.07
37
3.6 µm
± σ, σ =0.11
± 2σ
0.2
∆[3.6]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
1.4
[Fe/H]
4.5 µm
1.2
1.0
0.8
± σ, σ =0.09
± 2σ
0.2
∆[4.5]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
[Fe/H]
1.4
1.2
1.0
0.8
Figure 4.2 Photometric metallicity vs deviation from the period-luminosity relations.
38
[Fe/H]-Color
± σ, σ =0.07
± 2σ
0.2
[3.6]−[4.5]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
1.4
[Fe/H]
Period-Color
1.2
1.0
0.8
± σ, σ =0.08
± 2σ
0.2
[3.6]−[4.5]
0.1
0.0
0.1
0.2
0.6
0.5
0.4
0.3
log P (days)
0.2
Figure 4.3 Photometric metallicity vs. color and period vs. color.
0.1
0.0
39
The slopes are 1σ away from zero in 3.6 µm, and 2σ in 4.5 µm. These could be
construed as evidence for a weak metallicity relationship, but it should be noted that the
formal scatter in each case is of the same width as that of the PL relations themselves.
While it is uncommon to use the coefficient of determination R2 in astronomy, there is a
case for its relevance in this context. The values of R2 are 0.02 in 3.6 µm and 0.08 in 4.5
µm. While there is little consensus as to what a “good” value of R2 would be (particularly
in astronomy, where one often deals with data that has intrinsic scatter), an R2 of below
0.1 indicates that less than 10% of the data is explained by the fit, suggesting that the
evidence for a metallicity term is weak at best.
As an additional check for metallicity effects, we also examine the relationship
between period and color ([3.6] - [4.5]) (see fig. 4.3). Here we find:
[3.6] − [4.5] = −0.03 (±0.15) log P − 0.02 (±0.06), σ = 0.08
There is no evidence for a period-color relation by any metric; the slope and zero point are
both less than 1σ away from zero, and R2 = 0.002.
4.2
Spectroscopic Metallicities
Using the Sollima catalog of 74 spectroscopic metallicities, we find the following metallicityresidual relations:
λ
A
σA
B
σB
σ
[3.6]
−0.05
0.09
−0.08
0.14
0.09
[4.5]
−0.04
0.14
−0.07
0.06
0.09
[3.6] − [4.5]
−0.06
0.11
−0.11
0.18
0.08
All slopes here are within 1σ of zero, and are shallower than the corresponding ones
from photometric metallicities; however, this likely has more to do with the smaller sample
sizes (22 vs. 32 RR Lyrae in 3.6 µm, and 18 vs. 33 in 4.5 µm) than the improved accuracy
of the spectroscopic metallicities; see figs. 4.4 and 4.5.
40
3.6 µm
± σ, σ =0.09
± 2σ
0.2
∆[3.6]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
1.4
[Fe/H]
4.5 µm
1.2
1.0
0.8
± σ, σ =0.09
± 2σ
0.2
∆[4.5]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
[Fe/H]
1.4
1.2
1.0
0.8
Figure 4.4 Spectroscopic metallicity vs deviation from the period-luminosity relations.
41
[Fe/H]-Color
± σ, σ =0.07
± 2σ
0.2
[3.6]−[4.5]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
[Fe/H]
1.4
1.2
1.0
0.8
Figure 4.5 Spectroscopic metallicity vs color.
4.3
Combined Metallicities
We now combine both catalogs, preferring the spectroscopic metallicities when both are
available. We find:
λ
A
σA
B
σB
σ
[3.6]
−0.05
0.05
−0.10
0.07
0.10
[4.5]
−0.1
0.04
−0.17
0.08
0.09
[3.6] − [4.5]
0.06
0.05
0.09
0.08
0.08
These are more similar in slope and scatter to the photometric metallicity relations,
as there are several high-metallicity points from the photometric metallicities that affect
the fits strongly, particularly in 4.5 µm; see figs. 4.6 and 4.7.
42
3.6 µm
± σ, σ =0.10
± 2σ
0.2
∆[3.6]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
1.4
[Fe/H]
4.5 µm
1.2
1.0
0.8
± σ, σ =0.09
± 2σ
0.2
∆[4.5]
0.1
0.0
0.1
0.2
2.2
2.0
1.8
1.6
[Fe/H]
1.4
1.2
1.0
0.8
Figure 4.6 Combined metallicities vs deviation from the period-luminosity relations.
43
[Fe/H]-Color
0.2
[3.6]−[4.5]
0.1
0.0
0.1
0.2
± σ, σ =0.08
± 2σ
2.2
2.0
1.8
1.6
[Fe/H]
1.4
1.2
1.0
0.8
Figure 4.7 Combined metallicities vs color.
4.4
Summary
While this data does not completely rule out the possibility of a metallicity term in the
PL relations, the evidence in favor of one is not extremely compelling. Given that the
2σ width of each PL relation is about 0.2 mag, any metallicity effect must be within that
range, so we expect it to be intrinsically small. The fact that there is stronger evidence for
a metallicity term in 4.5 µm as opposed to 3.6 µm is consistent with the finding of a flatter
PL slope in 4.5 µm, and Dambis et al (2014) also find a slightly stronger metallicity relation
in 4.5 µm, but this is contrary to the predictions of Madore et al (2013) that metallicity
dependence should weaken and the PL slope should become steeper as one moves further
into the IR. Again, further investigations are required.
44
45
Chapter 5
M54
5.1
Background
Originally discovered in 1778 by Charles Messier, M54 (NGC 6715) was thought to be a
standard Milky Way globular cluster until 1994, when Siegel et al discovered that it was
actually part of the Sagittarius Dwarf Spheroidal Galaxy (sometimes referred to as the
Sagittarius Dwarf Elliptical Galaxy or Sag DEG). This places it at a distance of 27,000
parsecs or 87,000 light years from Earth. Like ω Centauri, it contains multiple stellar
populations and a comparable metallicity spread (Carretta et al 2010), and there is debate
as to whether it is actually the nucleus of Sag DEG, rather than an independent cluster.
Our sample of RR Lyrae for this cluster totals 21 stars, 20 of which appear in 3.6
µm and 5 of which appear in 4.5 µm. The catalog used is from Sollima et al (2010).
5.2
5.2.1
Period-Luminosity Relation
3.6 µm
In 3.6 µm the weighted and unweighted combined PL relations are as follows for M54:
A
σA
B
σB
σ
Weighted
−2.44
0.28
19.57
0.07
0.14
Unweighted
−2.46
3.86
19.58
0.93
0.14
These slopes are in good agreement with all results previously described, including
the ones for ω Cen. The problematic aspect for M54 is the zero point: combining the zero
46
Weighted
19.6
Apparent magnitude
19.8
± σ, σ =0.14 mag
± 2σ
RRLab
RRLc, fundamentalized
20.0
20.2
20.4
20.6
20.8
0.40
0.35
0.30
0.25
log P (days)
0.20
0.15
0.20
0.15
Unweighted
19.6
Apparent magnitude
19.8
± σ, σ =0.14 mag
± 2σ
RRLab
RRLc, fundamentalized
20.0
20.2
20.4
20.6
20.8
0.40
0.35
0.30
0.25
log P (days)
Figure 5.1 M54 PL relations in 3.6 µm with combined RRab’s and RRc’s.
47
point with Dambis et al gives a distance modulus of µ = 19.57 ± 0.07 − (−1.150 ± 0.077) =
20.72 ± 0.10, which translates to a distance of over 300,000 light years, or over twice
the accepted distance. We suspect that this is due to a simple error in photometry and
calibrations; reevaluation of the photometry process and calibration code has yielded no
obvious culprits thus far, but it is likely that it will be easily corrected once diagnosed.
Given that the period-luminosity relation and light curves appear reasonable, it is most
likely a matter of a constant offset rather than an intrinsic flaw in the photometry, and
that it does not merit particular concern at the present time (Scowcroft 2014, private
communication).
5.2.2
4.5 µm
In 4.5 µm only RRab stars were found. The weighted and unweighted PL relations are as
follows:
A
σA
B
σB
σ
Weighted
−0.17
1.28
20.0
0.25
0.11
Unweighted
0.95
76.82
20.25
15.51
0.11
Even with the removal of one relative outlier, the slopes only reach −1.76 at best.
Although Madore et al (2013) were able to derive a period-luminosity relation with only
four calibrator stars from the WISE preliminary data release, the expected higher scatter
in this data means that the smaller the sample size, the less representative it is likely to
be of the actual relation.
48
Weighted
± σ, σ =0.11 mag
± 2σ
19.6
Apparent magnitude
RRLab
19.8
20.0
20.2
20.4
0.26
0.24
0.22
0.20
log P (days)
0.18
0.16
0.14
Unweighted
± σ, σ =0.11 mag
± 2σ
19.6
Apparent magnitude
RRLab
19.8
20.0
20.2
20.4
0.26
0.24
Figure 5.2 M54 PL relations in 4.5 µm.
0.22
0.20
log P (days)
0.18
0.16
0.14
49
Chapter 6
Conclusions & Future Work
6.1
Conclusions
While this work does not settle the question of whether it is appropriate to fundamentalize
the slopes of RRc’s, and it does not provide conclusive evidence with regards to the nature of
the metallicity terms, it corroborates and supplements previously calibrated WISE periodluminosity slopes as well as previous distance measures for ω Cen, and lays the groundwork
for further inquiries into RR Lyrae with Spitzer data.
6.2
Future Work
The most important future developments for the Carnegie Hubble Program will undoubtedly involve the combined powers of the GAIA and JWST space telescopes. The parallax
measurements from GAIA should increase the number of parallaxes available for galactic
RR Lyrae substantially (Freedman 2011), and JWST, with its 0.100 resolution and ability
to see objects 10 to 100 times fainter than HST can see, will be able to resolve RR Lyrae at
much greater distances than we have seen previously and establish the infrared PL relations
with unprecedented accuracy and precision.
For the more immediate future, the next steps are to continue doing work of this
type on more clusters, and to further investigate the question of the metallicity term. This
could conceivably be done in a manner similar to that of Dambis et al (2014) in which they
use multiple clusters to derive a metallicity term based on the slopes of each cluster’s PL
relation vs. the mean cluster metallicity; it could also be done with individual RR Lyrae
50
metallicities in multiple clusters. The question of whether the PL slope in 4.5 µm is in fact
intrinsically shallower may also be worth investigating.
Other possibilities for future work might include the development of an algorithm
to search for outliers in the light curves and reject them in order to decrease PL scatter,
and a theoretical consideration of whether it is appropriate to fundamentalize the periods
of RRc’s and combine them with RRab’s in the PL relation.
51
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Appendices
57
1
Appendix A
Data Tables
A.1
ω Centauri
Periods and pulsational modes are from Kaluzny et al (2004), and all star IDs are from
OGLE. ∆[λ] is the deviation from the PL relation in each band, [Fe/H]p is the photometric
metallicity from Rey et al (2000), and [Fe/H]s is the spectroscopic metallicity from Sollima
et al (2006).
2
Table A.1. ω Centauri Data
ID
Type
m[3.6]
σm
m[4.5]
σm
P
∆[3.6]
∆[4.5]
[Fe/H]p
σp
[Fe/H]s
σs
3
4
5
9
10
11
12
13
14
15
18
20
21
23
30
33
34
40
44
45
47
49
50
51
54
56
58
59
64
66
67
68
82
94
95
97
102
103
107
115
117
120
121
122
169
274
291
357
ab
ab
ab
ab
c
ab
c
ab
c
ab
ab
ab
c
ab
c
ab
ab
ab
ab
ab
c
ab
c
ab
ab
ab
c
ab
c
c
ab
c
c
c
c
ab
ab
c
ab
ab
c
ab
c
ab
c
c
c
c
12.66
13.063
13.285
13.265
13.144
12.94
13.18
—
—
12.605
12.832
12.955
13.308
12.989
12.938
—
—
12.88
13.065
—
13.107
—
—
13.012
12.627
—
13.255
12.945
—
13.103
13.211
12.74
13.206
13.598
—
12.786
12.804
13.347
—
12.855
12.856
12.875
13.309
12.849
13.32
13.412
—
13.361
0.046
0.086
0.067
0.045
0.047
0.106
0.075
—
—
0.102
0.05
0.067
0.094
0.059
0.032
—
—
0.036
0.023
—
0.111
—
—
0.034
0.037
—
0.067
0.054
—
0.059
0.062
0.052
0.063
0.057
—
0.073
0.057
0.076
—
0.043
0.043
0.027
0.026
0.038
0.056
0.031
—
0.042
12.619
13.113
13.279
—
13.083
—
13.2
12.761
13.111
12.69
—
12.955
13.169
—
13.05
12.861
12.677
12.967
13.015
12.876
13.064
12.899
13.151
—
—
13.07
13.196
—
13.16
—
—
—
13.166
13.582
12.983
12.688
12.899
13.462
13.272
12.825
13.034
12.981
13.334
12.786
13.392
—
13.196
13.279
0.071
0.08
0.042
—
0.075
—
0.105
0.034
0.05
0.099
—
0.053
0.068
—
0.092
0.092
0.072
0.063
0.043
0.045
0.059
0.053
0.07
—
—
0.049
0.137
—
0.05
—
—
—
0.037
0.083
0.058
0.047
0.051
0.098
0.086
0.067
0.104
0.062
0.068
0.084
0.058
—
0.067
0.055
0.841
0.627
0.515
0.523
0.375
0.565
0.387
0.669
0.377
0.811
0.622
0.616
0.381
0.511
0.404
0.602
0.734
0.634
0.568
0.589
0.485
0.605
0.386
0.574
0.773
0.568
0.37
0.519
0.344
0.407
0.564
0.535
0.336
0.254
0.405
0.692
0.691
0.329
0.514
0.63
0.422
0.549
0.304
0.635
0.319
0.311
0.334
0.298
-0.091
-0.175
-0.184
-0.18
-0.018
0.061
-0.087
—
—
0.004
0.066
-0.047
-0.198
0.122
0.106
—
—
-0.004
-0.069
—
-0.26
—
—
-0.028
0.034
—
-0.114
0.149
—
-0.067
-0.209
0.001
0.041
-0.047
—
-0.005
-0.022
-0.077
—
0.027
0.143
0.158
0.045
0.025
-0.018
-0.083
—
0.016
-0.022
-0.21
-0.171
—
0.049
—
-0.101
0.075
0.015
-0.054
—
-0.033
-0.054
—
0.003
0.085
0.062
-0.075
-0.008
0.092
-0.201
0.043
-0.05
—
—
-0.064
-0.05
—
0.06
—
—
—
0.08
-0.044
0.069
0.113
-0.097
-0.194
-0.162
0.073
-0.025
0.062
0.016
0.105
-0.092
—
0.056
0.093
-1.54
-1.74
-1.35
-1.49
-1.66
-1.67
-1.53
-1.91
-1.71
-1.64
-1.78
—
-0.9
-1.08
-1.75
-2.09
-1.71
-1.6
-1.4
-1.78
-1.58
-1.98
-1.59
-1.64
-1.66
-1.26
-1.37
-1.0
-1.46
-1.68
-1.1
-1.6
-1.56
-1.0
-1.84
-1.56
-1.84
-1.92
-1.36
-1.87
-1.68
-1.39
-1.46
-2.02
—
—
—
—
0.05
0.05
0.08
0.06
0.1
0.13
0.14
0.0
0.13
0.39
0.28
—
0.11
0.14
0.17
0.23
0.0
0.08
0.12
0.25
0.31
0.11
0.19
0.21
0.12
0.15
0.18
0.28
0.23
0.34
0.0
0.01
0.2
0.11
0.55
0.37
0.13
0.11
0.11
0.01
0.25
0.06
0.13
0.18
—
—
—
—
—
—
-1.24
—
—
-1.61
—
—
—
-1.68
—
-1.52
—
-1.35
-1.62
-1.58
—
-1.62
-1.29
—
—
—
—
-1.84
-1.8
—
-1.91
—
—
—
-1.19
—
-1.71
—
—
-1.74
-1.65
-1.78
—
-1.64
—
-1.15
-1.83
-1.79
-1.65
—
—
-1.64
—
—
0.11
—
—
0.22
—
—
—
0.18
—
0.34
—
0.58
0.28
0.42
—
0.19
0.35
—
—
—
—
0.23
0.23
—
0.31
—
—
—
0.23
—
0.56
—
—
0.17
0.16
0.27
—
0.32
—
0.16
0.4
0.21
0.19
—
—
0.99
3
Table A.2.
A.2
M54 Data
ID
Type
m[3.6]
σm
m[4.5]
σm
P
∆[3.6]
∆[4.5]
5
7
15
29
31
32
34
36
39
40
41
47
49
50
63
76
83
87
94
122
131
ab
ab
ab
ab
ab
ab
ab
ab
ab
ab
ab
ab
c
ab
ab
ab
ab
ab
ab
ab
ab
20.292
20.111
19.968
19.934
20.05
20.266
20.566
19.979
20.092
19.926
20.156
20.266
20.475
20.046
20.166
20.069
20.154
20.427
—
20.307
20.019
0.056
0.044
0.035
0.049
0.059
0.067
0.047
0.086
0.071
0.056
0.06
0.051
0.059
0.04
0.044
0.055
0.039
0.074
—
0.048
0.044
—
20.011
—
—
19.993
—
—
—
20.075
—
—
—
—
—
—
—
—
—
19.96
20.264
—
—
0.058
—
—
0.055
—
—
—
0.066
—
—
—
—
—
—
—
—
—
0.035
0.057
—
0.579
0.594
0.587
0.59
0.647
0.519
0.503
0.599
0.6
0.586
0.618
0.507
0.331
0.564
0.59
0.706
0.578
0.555
0.656
0.654
0.634
-0.141
0.013
0.17
0.198
-0.016
0.002
-0.266
0.136
0.023
0.212
-0.073
0.027
-0.044
0.134
-0.035
-0.129
-0.001
-0.23
—
-0.285
0.036
—
0.027
—
—
0.039
—
—
—
-0.038
—
—
—
—
—
—
—
—
—
0.071
-0.233
—
M54
Periods and pulsational modes are from Sollima et al (2010), and IDs are from OGLE.
Author’s note: This table refuses to anchor properly. My apologies.
4
5
Appendix B
Light Curves
B.1
ω Centauri
B.1.1
3.6 µm
3
5
4
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
9
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
20
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
2.5
3.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
47
0.4
0.5
1.0
21
0.4
0.4
0.0
2.0
18
0.4
0.4
0.0
0.5
15
12
1.5
11
0.4
1.0
1.0
10
0.4
0.5
0.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
6
58
54
66
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5
1.0
67
1.5
2.0
2.5
3.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
1.0
1.5
2.0
2.5
3.0
0.5
1.0
103
0.4
0.2
0.2
0.0
0.0
0.2
0.2
0.4
0.0
0.4
0.0
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
169
0.4
0.5
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
102
0.4
0.5
0.5
97
94
1.5
82
0.4
0.5
1.0
68
0.4
0.4
0.0
0.5
0.5
1.0
1.5
7
B.1.2
4.5 µm
3
5
4
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5
1.0
10
1.5
2.0
2.5
3.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
30
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
1.5
2.0
2.5
3.0
0.5
1.0
40
1.5
2.0
2.5
3.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
47
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
1.5
2.0
2.5
3.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
50
0.4
0.5
1.0
49
0.4
0.4
0.0
3.0
45
0.4
0.5
0.5
44
0.4
0.4
0.0
2.5
34
0.4
1.0
1.0
33
0.4
0.5
2.0
21
0.4
0.4
0.0
0.5
15
14
1.5
13
0.4
0.5
1.0
12
0.4
0.4
0.0
0.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
8
58
56
64
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
117
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
1.5
2.0
2.5
3.0
0.5
1.0
122
1.5
2.0
2.5
3.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.5
3.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
357
0.4
1.0
1.0
263
0.4
0.5
2.0
121
0.4
1.0
0.5
120
0.4
0.5
1.5
115
0.4
0.5
1.0
95
94
0.4
0.4
0.0
0.5
2.0
2.5
3.0
0.5
1.0
1.5
9
B.2
M54
B.2.1
3.6 µm
5
7
15
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
29
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
34
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
1.5
2.0
2.5
3.0
0.5
1.0
40
1.5
2.0
2.5
3.0
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.5
1.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
3.0
1.5
2.0
2.5
3.0
1.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
63
0.4
0.4
0.0
0.5
50
49
2.5
47
0.4
0.5
1.0
41
0.4
0.4
0.0
2.0
39
0.4
1.0
0.5
36
0.4
0.5
1.5
32
0.4
0.5
1.0
31
0.4
0.4
0.0
0.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
10
76
87
83
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5
1.0
122
0.4
0.2
0.2
0.0
0.0
0.2
0.2
0.5
B.2.2
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
4.5 µm
7
31
39
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.2
0.2
0.2
0.4
0.0
0.4
0.0
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.5
1.0
94
0.4
0.2
0.2
0.0
0.0
0.2
0.2
0.5
1.0
1.5
1.5
2.0
2.5
3.0
2.0
2.5
3.0
122
0.4
0.4
0.0
0.5
131
0.4
0.4
0.0
1.5
2.0
2.5
3.0
0.4
0.0
0.5
1.0
1.5
0.5
1.0
1.5