A Dissertation entitled
A Variability Study of Y Dwarfs: A Spitzer Space Telescope Program
by Jesica Lynn Trucks
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Physics with concentration in Astrophysics
Dr. Michael Cushing, Committee Chair
Dr. S. Thomas Megeath, Committee Member
Dr. Rupali Chandar, Committee Member
Dr. Richard Irving, Committee Member
Dr. Stanimir Metchev, Committee Member
Dr. Cyndee Gruden, Interim Dean College of Graduate Studies
The University of Toledo August 2019 Copyright 2019, Jesica Lynn Trucks
This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of A Variability Study of Y Dwarfs: A Spitzer Space Telescope Program by Jesica Lynn Trucks
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Physics with concentration in Astrophysics The University of Toledo August 2019
I present the results of a search for variability in 14 Y dwarfs consisting 2 epochs of observations, each taken with Spitzer for 12 hours at [3.6] immediately followed by
12 hours at [4.5], separated by 122–464 days and found that Y dwarfs are variable.
We used not only periodograms to characterize the variability but we also utilized
Bayesian analysis. We found that using different methods to detect variability gives different answers making survey comparisons difficult. We determined the variability fraction of Y dwarfs to be between 37% and 74%. While the mid-infrared light curves of Y dwarfs are generally stable on time scales of months, we have encountered a few that vary dramatically on those time scales. When we combined the variability frac- tions of L and T dwarfs with our variability fractions of Y dwarfs the results support the standard paradigm which suggests that clouds are responsible for the observed variability because of the cloudy→cloud free→cloudy nature of the LTY spectral se- quence. We have determined the rotation periods of 5 Y dwarfs ranging from 2.44 hours to 8.42 hours, with two additional tentative periods. We also considered the oblateness of Y dwarfs as rotation period is one factor in its calculation. One of our targets WISE 0359−54 has a very small period so we showed that depending on its’ mass it could have an oblateness comparable to that of Jupiter or Saturn. We also determined that with its short, consistent period across all four light curves, it would make a perfect candidate for future variability studies with JWST. Interestingly we iii found that our survey failed to detect any small amplitude variability (< 0.5%) even though it is sensitive down to 0.2%, but is not due to observational bias. We con-
firmed that the maximum amplitude increases as a function of spectral type. We find that the average [4.5]/[3.6] amplitude ratio is less than unity which suggests that hot spots may be the physical mechanism of the observed variability. As [3.6] and [4.5] probe similar atmospheric layers unsurprisingly we find no phase changes in the light curves for WISE 0359−54.
iv To my dad Ronald L Trucks, who has always taught me to reach for the stars. Acknowledgments
I would like to say thanks to my advisor Michael Cushing who worked with me on this project. I would also like to say thanks to Kevin Hardegree-Ullman for his work on this project. I would like to say thanks to my parents and family who have supported me throughout the long journey to get to this point. 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.
vi Contents
Abstract iii
Acknowledgments vi
Contents vii
List of Tables ix
List of Figures xi
1 Introduction and Background Material 1
1.1 Introduction ...... 1
1.2 What is a brown dwarf? ...... 2
1.3 Theory of substellar objects: Interiors ...... 3
1.4 Theory of substellar objects: Atmospheres ...... 8
1.5 L, T, and Y dwarfs ...... 13
1.5.1 L dwarfs ...... 13
1.5.2 T dwarfs ...... 14
1.5.3 Y dwarfs ...... 15
1.6 Previous variability studies ...... 16
1.7 Conclusion ...... 18
2 The Search for Variability 22
2.1 Motivation ...... 22
vii 2.2 Observations and Data Reduction ...... 23
3 Data Analysis 30
3.1 Variability Analysis ...... 30
3.1.1 A Visual Inspection ...... 30
3.1.2 Variability Fractions ...... 31
3.1.2.1 Bayesian Analysis ...... 32
3.1.2.2 Periodogram Analysis ...... 37
3.1.3 Variability Fraction Confidence Intervals ...... 41
3.2 Discussion ...... 44
3.2.1 Variability Fractions ...... 44
3.2.2 Rotation Periods ...... 46
3.2.3 Semi-Amplitudes ...... 49
3.2.4 Phases ...... 52
4 Conclusions and Future Prospects 54
4.1 Conclusions ...... 54
4.2 Future Prospects ...... 55
A Bayesian Parameter Tables 75
viii List of Tables
1.1 Field brown dwarf observablesa...... 3
1.2 Variability Fraction of L and T dwarfs...... 18
2.1 Y Dwarf Targets ...... 25
3.1 ∆ BIC Values ...... 35
3.2 Periods and Semi-Amplitudes for Variable-Source Models ...... 36
3.3 Y Dwarf Variability Summary ...... 40
3.4 Survey Completeness ...... 42
3.5 Rotation Periods of Y Dwarfs ...... 46
A.1 WISEJ0146+4234 ...... 76
A.2 WISEJ0350−56 ...... 77
A.3 WISEJ0359−54 ...... 78
A.4 WISEJ0410+15 ...... 79
A.5 WISEJ0535−75 ...... 80
A.6 WISEJ0713−29 ...... 81
A.7 WISEJ0734−71 ...... 82
A.8 WISEJ1405+55 ...... 83
A.9 WISEJ1541−22 ...... 84
A.10 WISEJ1639−68 ...... 85
A.11 WISEJ1738+27 ...... 86
A.12 WISEJ1828+26 ...... 87
ix A.13 WISEJ2056+14 ...... 88
A.14 WISEJ2220−36 ...... 89
x List of Figures
1-1 Evolution of luminosity (in L ) of isolated solar-metallicity red dwarf stars and substellar-mass objects versus age (in years). Adapted from Burrows
et al. (2001)...... 4
1-2 Evolution of the central temperature (Tc) in Kelvin of isolated solar- metallicity red dwarf stars and substellar-mass objects versus age (in
years). Adapted from Burrows et al. (2001)...... 5
1-3 Evolution of the radius (R) in gigacentimeters of isolated solar-metallicity
red dwarf stars and substellar-mass objects versus age (in years). Adapted
from Burrows et al. (2001)...... 7
1-4 Equilibrium composition of a gas at solar metallicity as a function of tem-
perature for two different pressures (chemical models Bailey and Kedziora-
Chudczer (2012); Figure from Bailey (2014))...... 8
1-5 Plot of the abundance of elements as a function of atomic number. The
bubbles contain molecules/atoms/condensates that play an important role
in substellar atmospheres. Adapted from Burrows et al. (2001) ...... 9
1-6 Temperature Pressure profiles of various temperature objects. Equi-abundance
curves for the dominate forms of C and N. Condensate curves illustrating
where in the atmosphere specific molecules will form...... 10
1-7 Illustration of different cloud layers at different temperatures (Lodders 2004). 11
xi 1-8 Low temperature model spectra over plotted with their blackbody curve
(Spectra references; Marley et al., 2002; Saumon and Marley, 2008; Morley
et al., 2014; Hauschildt et al., 2003)...... 12
1-9 Left: Red optical spectral sequence of the optical L dwarf sequence. Right:
Near-infrared spectral sequence of the near-infrared L dwarf sequence.
Figure from Cushing (2014)...... 19
1-10 Left: Red optical spectral sequence of the optical T dwarf sequence. Right:
Near-infrared spectral sequence of the near-infrared T dwarf sequence.
Figure from Cushing (2014)...... 20
1-11 Left: J and H band spectra of Y0, Y1, and Y2 spectral standards. Figure
from Cushing (2014)...... 21
2-1 [3.6] (blue) and [4.5] (red) light curves for the Y0 dwarf WISE 0359−54,
taken on January 4, 2013 (top) and April 13, 2014 (bottom). The light
curves observed on January 4, 2013 (top) shows periodic variability at
both wavelengths with a semi-amplitude of 4.2% and 2.8% with periods of
2.41 hours and 2.45 hours respectively. The light curves observed on April
13, 2014 (bottom) shows periodic variability at both wavelengths with a
semi-amplitude of 2.5% and 2.2% with a common period of 2.44 hours. . 27
2-2 Normalized IRAC [3.6] (blue) and [4.5] (red) photometry for the fourteen
Y dwarfs in our sample. The epoch 1 data is found in the left panel and
the epoch 2 data is found in the right panel. For display purposes only,
outliers have been removed as described in §3.1.2.1. Note that the scale
of the ordinate is different for [3.6] and [4.5] light curves...... 28
xii 2-2 Normalized IRAC [3.6] (blue) and [4.5] (red) photometry for the fourteen
Y dwarfs in our sample. The epoch 1 data is found in the left panel and
the epoch 2 data is found in the right panel. For display purposes only,
outliers have been removed as described in §3.1.2.1. Note that the scale
of the ordinate is different for [3.6] and [4.5] light curves...... 29
3-1 Epoch 1 periodograms for the 14 Y dwarfs in our sample. The grey dashed
lines give the 5% (95% confidence) False Alarm Levels which give the power
levels above which we would expect to see power less than 5% of the time
under the null hypothesis of pure gaussian noise (no variable signal). Any
object with power in its periodogram above this level is considered variable. 38
3-2 Epoch 2 periodograms for the 14 Y dwarfs in our sample. The grey dashed
lines give the 5% (95% confidence) False Alarm Levels which give the power
levels above which we would expect to see power less than 5% of the time
under the null hypothesis of pure gaussian noise (no variable signal). Any
object with power in its periodogram above this level is considered variable. 39
3-3 Posterior distributions for the variability fraction f for each epoch and
each band. 68% and 95% central credibility intervals are indicated in light
grey and dark grey, respectively...... 43
3-4 Spitzer variability fractions for L and T dwarfs (Metchev et al., 2015)
and the Y dwarfs (this work). The points without error bars are the raw
fractions (number of variable objects/total sample) while the points with
error bars have been corrected for survey sensitivity limits. The error bars
for the L and T dwarfs are 95% confidence limits while the error bars for
the Y dwarfs are centered 95% credibility intervals...... 45
xiii 3-5 Periods (left) and Semi-amplitudes (right) for a sample of L, T, and Y
dwarfs as a function of spectral type. The data come from the compilation
of Crossfield (2014), the survey of Metchev et al. (2015), and this work. . 47
3-6 Model oblateness values as a function of rotation period for a body with
the effective temperature of WISE 0359−54 (left) and a hypothetical T0
dwarf (right). Also shown are the oblateness values of Jupiter and Saturn
and the oblateness (0.44) beyond which an n = 1 polytrope becomes
rotationally unstable. The observed rotation period of WISE 0359−54 of
2.44 hours is shown as a vertical line in both panels...... 50
3-7 Reproduction of Figure 9 from Metchev et al. (2015) with the Y dwarfs
from this work included. We have not included the upper limits in the
original figure...... 52
3-8 Reproduction of Figure 10 from Metchev et al. (2015) with the Y dwarfs
from this work included...... 53
4-1 A surface map of Luhman 16B showing large scale cloud inhomogeneities.
Each projection is time stamped over the rotation period of the object
which is 4.9 hours.(Figure from Crossfield et al. 2014) ...... 56
4-2 Model spectra of a 450 K Y dwarf plotted with JWST Instrument wave-
length coverage...... 58
xiv Chapter 1
Introduction and Background
Material
1.1 Introduction
In this chapter, we review some of the information about brown dwarfs necessary
to understand the focus of this dissertation, the search for and characterization of
photometric variability in Y dwarfs. In §1.2 we discuss the definition of a brown
dwarf and a historical perspective on the theory of their existence. In §1.3 we give an
overview of the basic interior physics of these objects citing work from a collection
of ongoing research done by theoretical modelers. We then move on to summarize
the theory of substellar atmospheres in §1.4. From there we move on to discuss the properties and discoveries of the three spectral classes of brown dwarfs: L, T, and Y in §1.5. Then in §1.6 we provide an overview of the previous searches for variability in brown dwarfs, and the requirements needed for detection. We conclude (§1.7) by
summarizing the state of brown dwarf variability research and motivating the need
for the research presented in this dissertation.
1 1.2 What is a brown dwarf?
A star is defined by its ability to stabilize its luminosity for a stretch of time by burning hydrogen in its core by the following process:
+ 6 p + p → d + e + νe Tcrit & 3 × 10 K, (1.1)
3 5 d + p → He + γ Tcrit & 6 × 10 K, (1.2)
3 3 4 6 He + He → He + p + p Tcrit & 6 × 10 K, (1.3)
where Tcrit is the critical temperature for fusion to begin. During a star’s main sequence phase, 100% of its luminosity is derive from this nuclear fusion. A brown dwarf is an object that is unable to balance the energy radiated from its surface with nuclear fusion generated at its core. The lower mass limit of the main sequence, or the hydrogen-burning minimum mass (HBMM), is about 0.075 M , or about 75 MJup, for objects with solar metallicity, while objects with lower metallicities the lower limit approaches 90 MJup (Saumon et al., 1994). Typical values of the observables of a field brown dwarf (age = 5 Gyr) can be found in Table 1.2.
Interest in the physics of the objects that lie at the bottom and below the Main
Sequence began in the early 1960s with Kumar (1963) and Hayashi and Nakano
(1963). Kumar theoretically inferred the existence of brown dwarfs when he realized
“there exists a limiting mass below which a contracting star cannot reach the main- sequence stage because the temperature and density at the center are too low for hydrogen-burning to start” (Kumar, 1963).
While the high mass limit of brown dwarfs is well defined at M ≈ 0.075 M , the lower mass limit of brown dwarfs is still up for debate. A proposed lower limit is 13
MJup because below that deuterium burning does not occur (see §1.3), incidentally it is also thought to be near the limit for direct collapse of an interstellar cloud (Basri,
2 Table 1.1. Field brown dwarf observablesa
Parameter Range
−4 −8.5 Luminosity (L/L ) 10 log(L/L ) − 10 log(L/L ) Effective Temperature (Teff) 150 K − 1800 K Mass (M) 1MJup − 75MJup Radius (R) 0.75 RJup − 1.25 RJup Density (hρi) 0.1 g cm−3 − 100 g cm−3
aFor a brown dwarf with an age of 5 Gyr.
2000). As much of the physics attributed to brown dwarfs remain the same for gas
giant planets, one can make an argument that would push this limit lower into the
range of their masses. There is some evidence that suggests low mass objects form via
different mechanisms: brown dwarfs forming from interstellar cloud collapse versus
planets forming via accretion in disks. This may provide a method of distinguishing
the low mass end, although determining how an object formed is a difficult process
fraught with uncertainty which can be avoided by using a mass-based definition (Basri,
2000). In this work, we adopt the mass range of a brown dwarf to be from 0.075 M down to 1 MJup.
1.3 Theory of substellar objects: Interiors
Brown dwarfs have predominantly molecular atmospheres, with a metallic hydro- gen/helium mixture in their fully convective interiors. They are brightest when they form and then continually cool “like dying embers plucked from a fire” (Burrows et al.,
2001). Figure 1-1 shows the evolution of the luminosity of isolated objects of solar metallicity with masses ranging from 0.3 MJup (the approximate mass of Saturn) to
0.2 M (an M dwarf), where the color gradient is indicative of the mass with those
3 of the lowest masses in red to those of the highest masses in blue. The bifurcation of
stars and brown dwarfs can be seen between 108.3 − 109.8 years. At lower metallicities
for the same mass range, the luminosity gap widens at an earlier age and is more
pronounced (Burrows et al., 2001).
0
−2 211 MJ
) 13 M −4 J sun 73 MJ
1 MJ (L/L 10
Log −6
0.3 MJ
−8
−10 106 107 108 109 1010 Age (yr)
Figure 1-1 Evolution of luminosity (in L ) of isolated solar-metallicity red dwarf stars and substellar-mass objects versus age (in years). Adapted from Burrows et al. (2001).
Figure 1-2 shows that radiative losses from the surface will cause the core tem- perature Tc and density to increase due to the negative specific heat of an ideal gas in hydrostatic gravitational equilibrium, and for an object of sufficient mass, this will eventually lead to the thermonuclear power equalling the total luminosity output, i.e.
6 a star. For brown dwarfs the core temperatures will never reach the Tcrit = 3 × 10 K required for the thermonuclear power to balance surface losses before the core becomes partially electron degenerate (Burrows et al., 2001). At this point, these ob-
4 jects continue to cool leading to the rise and fall seen in Tc in Figure 1-2. Figure 1-2 also shows that around the HBMM the core temperature will decrease slightly before
6 stabilizing. For solar metallicity, the Tc can be as low as 3 × 10 K, where the edge
mass is 0.075 M .
7·106
211 MJ 6·106
5·106
4·106 (K) c T 3·106
73 MJ 2·106
1·106
0 106 107 108 109 1010 Age (yr)
Figure 1-2 Evolution of the central temperature (Tc) in Kelvin of isolated solar- metallicity red dwarf stars and substellar-mass objects versus age (in years). Adapted from Burrows et al. (2001).
While brown dwarfs are characterized by their inability to generate sufficient tem-
peratures and pressures in their cores for thermonuclear fusion to balance surface
losses, they can have some thermonuclear processes of hydrogen, deuterium, and
lithium down to masses of ∼13 MJup. Albeit these processes are only temporary and partially successful in balancing the energy lost at the surface. Objects more massive than ∼ 0.065 M will burn lithium isotopes for a time via the reaction (Burrows
5 et al., 2001),
7 4 4 6 p + Li → He + He Tcrit & 2 × 10 K. (1.4)
Objects more massive than ∼ 13 MJup will burn deuterium for a time via the reaction (Burrows et al., 2001),
3 5 p + d → γ + He Tcrit & 4 × 10 K. (1.5)
Deuterium burning is the cause of the early bumps in Figure 1-1 at high luminosi- ties, but the low abundances of deuterium never leads to a deuterium-burning main sequence.
Figure 1-3 shows that at early times the radius decreases monotonically as a function of mass. While for a single mass the radius is always decreasing as a function of age. However, at later times the dependence on mass upon radius inverts and the more massive substellar objects have smaller radii. Perhaps most importantly,
Figure 1-3 shows that for masses of 0.3 to 70 MJup the radii of older brown dwarfs are independent of mass to within about 30% of 1 RJup (Burrows et al., 2001). Burrows and Liebert (1993) derived power law relations between the properties of substellar mass objects (i.e. L, Teff, M, t, and R) which are shown in Equa-
−2 tions 1.7−1.10, where g is the surface density in cm s and κR is the average atmo- spheric Rosseland mean opacity (Burrows et al., 2001).
6 40
35
30
25
20 Radius (Gcm)
15
10 1 RJup
5 M. Cushing, U of Toledo 106 107 108 109 1010 Age (yr)
Figure 1-3 Evolution of the radius (R) in gigacentimeters of isolated solar-metallicity red dwarf stars and substellar-mass objects versus age (in years). Adapted from Burrows et al. (2001).
9 1.3 2.64 0.35 −5 10 yr M κR L ∼ 4 × 10 L −2 2 −1 (1.6) t 0.05M 10 cm g 9 0.32 0.83 0.088 10 yr M κR Teff ∼ 1550 K −2 2 −1 (1.7) t 0.05 M 10 cm g g 0.64 T 0.23 M ∼ 35 M eff (1.8) J 105 1000 K g 1.7 1000 K2.8 t ∼ 1.0 Gyr 5 (1.9) 10 Teff 105 0.18 T 0.11 R ∼ 6.7 × 104 km eff (1.10) g 1000 K
7 1.4 Theory of substellar objects: Atmospheres
Important topics in the theory of substellar objects are atmospheric chemistry and
abundances. The low atmospheric temperatures (Teff < 2500 K) of brown dwarfs allow for condensates to form at various depths throughout their atmospheres. Figure 1-
4 shows the mass fraction of a solar metallicity gas as a function of temperature,
separating the material into ions, atoms, gas-phase molecules and solid and liquid
condensates. At temperatures below 3500 K atoms begin to form molecules, while
noble gases remain atoms at all temperatures. At temperatures below 2000 K, the
gas begins to form condensates that become a large fraction of the mass. At lower
pressures this begins to happen at even lower temperatures. The chemistry depicted in Atmospheres of Exoplanets and Brown Dwarfs 3 f f
Figure 3. Equilibrium composition of a gas with solar elemental abun- Figure 2. Radius and surface gravity (log g in cgs units) as function of mass Figure 1-4 Equilibriumdances as a composition function of temperature of a gas at attwo solar different metallicity pressures using as a the function of tem- for the models of Baraffe et al. (2003) at ages of 1 Gyr, 5 GYr and 10 Gyr. perature for twochemical different model pressures of Bailey & Kedziora-Chudczer (chemical models (2012 Bailey). and Kedziora-Chudczer that orbit stars are generally agreed to be designated(2012); Figure as from Bailey (2014)). planets. There is less consensus on how to refer to object The number of ultracool dwarfs1 has increased rapidly below this mass limit that do not orbit a star. While these over the years since the recognition of the first brown dwarfs are sometimes referred to as ‘free-floating planets’ (Lucas & in 1995. Most of the objects have come from deep surveys Roche 2000;Delormeetal.2012)ithasalsobeenargued such as the Sloan Digital Sky8 Survey (SDSS — Fan et al that such objects should not be referred to as planets but 2000;Hawleyetal.2002)andtheCanada-FranceBrown as ‘sub-brown dwarfs’ or some other designation (see Boss Dwarfs Survey (CFBDS — Delorme et al. 2008a; Albert et al. 2003;Basri&Brown2006,foradiscussionofthe et al. 2011)andparticularlyfrominfraredsurveyssuchasthe issues involved in this controversy). Deep Near-Infrared Sky Survey (DENIS — Delfosse et al. The electron degeneracy in the cores of brown dwarfs 1997;Mart´ın, Delfosse, & Guieu 2004), the 2 Micron All results in their radius varying little with mass as can be seen Sky Survey (2MASS — Kirkpatrick et al. 2000; Burgasser in Figure 2.Allbrowndwarfs(exceptatveryyoungages) et al. 2002, 2004), and the UKIRT Infrared Deep Sky Survey have radii not far from 0.1 R or about 1 Jupiter radius. (UKIDSS — Pinfield et al. 2008;Burninghametal.2010, ⊙ 2 Aconsequenceofthisisthatsurfacegravity(g GM/R ) 2013). 2 = varies with mass from more than 1 000 m s− (log g 5 The most recent additions have come from the Wide-field 2 = in cgs units) to around 30 m s− for Jupiter mass objects as Infrared Survey Explorer (WISE — Wright et al. 2010). This shown in the lower panel of Figure 2. Earth orbiting NASA mission surveyed the entire sky at four Brown dwarfs are objects whose atmospheric composition wavelengths (3.4, 4.6, 12 and 22 µm). The first of these is dominated by molecular gas, as opposed to atoms and ions wavelengths probes a deep CH4 absorption band in brown in the case of hotter stars. This is apparent from Figure 3 dwarfs. WISE has proved effective in identifying the coolest which shows the chemical equilibrium composition of a solar brown dwarfs. It has led to the discovery of many T dwarfs composition gas (using the abundances of Grevesse, Asplund (Kirkpatrick et al. 2011;Maceetal.2013) and to the first Y &Sauval2007). It shows the division of the material by dwarfs (Cushing et al. 2011; Kirkpatrick et al. 2012;Tinney mass fraction into ions, atoms, gas-phase molecules and solid et al. 2012). or liquid condensates as calculated by the chemical model Other recent discoveries from WISE are that of a binary of Bailey & Kedziora-Chudczer (2012). It can be seen that brown dwarf (Luhman 2013)andanextremelycoolbrown molecules become dominant over atoms for temperatures dwarf (Luhman 2014)bothatdistancesofaround2pc.WISE below about 3 500 K. Helium and other noble gases persist J104915.57 531906.1 (also known as Luhman 16) consists as atoms at all temperatures, but other elements are mostly of an L7.5–L8− primary and T0.5–T1.5 secondary (Burgasser, in the form of molecules. Below about 2 000 K condensed phases start to appear, and become a significant fraction of 1Ultracool dwarf is a name normally used for objects with spectral type later the material. At lower pressures, as shown in the lower panel, than about M7, which could potentially be brown dwarfs, but could also the pattern is similar but shifted to lower temperatures. be stars
PASA, 31, e043 (2014) doi:10.1017/pasa.2014.38
http://journals.cambridge.org Downloaded: 13 May 2015 IP address: 131.183.160.171 Figure 1-5 shows which elements/compounds are prevalent in substellar atmospheres.
After hydrogen (H2) and helium, comes the dominate C/N/O bearing molecules in the form of H2O, CO, CH4,N2, and NH3. Next we come to the elements that make up the solid and liquid condensates starting with Mg and Si that bind with O to form
Mg/Si/O compounds like forsterite (Mg2SiO4) and enstatite (MgSiO3), followed up by many other compounds bearing heavier elements. They have molecular-rich atmospheres
0 H H2 SiO(g) He H2O Ca2MgSiO7 FeH, Fe(c) CaMgSi2O6 pyroxenes, MgAl2O4 −2 olivines CO, CH4 MgSiO4 O Mg2SiO3 H2S C NH4SH −4 N Fe Mg Si N2, NH3 S CrH Na Al Ar −6 Ca Cr Al2O3 P TiO log abundance MgAl2O4 Na, NaCl Ti Ca2Al2SiO7 K Na2S V −8 VO Li K, KCl K2S CaH Ca2Al2SiO7 −10 PH3, PN Ca2MgSi2O7 Li, LiCl P4O6 CaMgSi2O6 LiOH
0 10 20 30 Atomic Number
adapted from Burrows (2001, RvMP, 73, 79) Figure 1-5 Plot of the abundance of elements as a function of atomic number. The bubbles contain molecules/atoms/condensates that play an important role in substel- lar atmospheres. Adapted from Burrows et al. (2001)
Figure 1-6 shows us what happens in these atmospheres as we move to colder and colder temperatures. This figure shows temperature pressure profiles (hereafter
T/P) for example objects at various temperatures: the Sun (5778 K), 3000 K, 1800
K, 1000 K, 400 K, and 128 K. It also shows equi-abundance curves for NH3/N2 and
CH4/CO, where NH3 and CH4 dominate on the left and N2 and CO dominate on the right. Also shown are condensate curves that represent where a condensate will form in the atmosphere when crossed by a T/P profile of an object of a given Teff. Above
9 1800 K, CO and N2 are the dominate forms of C and N, but moving down through the atmosphere will encounter the condensation points of Al2O3, Ca4Ti3O10, Fe, and
MgSiO3. Moving on to 1000 K this is where the gas-molecules become interesting. Starting deep in the atmosphere and up the T/P profile it will pass the point, ∼ 10 bar, where the dominant form of carbon-bearing molecule becomes CH4 rather than
CO, while N is still predominantly N2. Moving up in the atmosphere the temperature decrease until condensates of Na2S, ZnS, and KCl are able to form, and finally the point where NH3 becomes the dominant form of N gas. At even lower temperatures,
H2O and NH3 condensates begin to form.
0.001
NH /N CH /CO } 3 2 4 10
{Fe} Sun O 3 Ti 4 } 3 0.010 {Ca } 3 O {MgSiO 2 O} 2 {Al {H {KCl} } 0.100 3 {NH {ZnS} S} P (bar) 2 {Na
1.000
M (3000 K)
10.000 Jupiter (128 K) Y (400 K) T (1000 K) L (1800 K)
100 1000 10000 T (K)
Figure 1-6 Temperature Pressure profiles of various temperature objects. Equi- abundance curves for the dominate forms of C and N. Condensate curves illustrating where in the atmosphere specific molecules will form.
In the standard paradigm it is believed that the scores of condensate species that form in the atmospheres of these ultracool dwarfs (with spectral types M7 and
10 later) gravitationally settle and form clouds (see Figure 1-7). Figure 1-8 shows model spectra for five objects of varying Teff: 3500 K, 2400 K, 1500K, 1000 K, and 400 K, along with their blackbody curves. Here the reddened nature of the 1500 K spectra at near-infrared wavelengths indicates a dusty/cloudy atmosphere. At 1000
K the spectrum becomes bluer at near-infrared wavelengths indicating that between these some unknown mechanism is clearing the atmosphere of dust/clouds. Then the 400 K the spectrum reddens again indicating the reappearance of a dusty/cloudy atmosphere. The model atmospheres that include simple cloud models do indeed provide a reasonably good match to the observed spectral energy distribution of the
2400 K to 700 K dwarfs (Tsuji, 2005; Cushing et al., 2008; Stephens et al., 2009;
Witte et al., 2011)1. These models generally assume a uniform cloud coverage but the heterogeneous nature of the clouds on our giant planets (e.g., Westphal, 1969) belies this simplifying assumption.
Figure 1-7 Illustration of different cloud layers at different temperatures (Lodders 2004).
1Tremblin et al. (2015) and Tremblin et al. (2016) suggested that condensate clouds are not required in order to match the observations of brown dwarfs. 11 12 H2O H2O CO Teff = 3500 K
H2O 2400 K 10 1500 K 1000 K K 400 K 8 FeH TiO H2O + Constant ν 6
CH4 CH4 CH4 4
Normalized f CH CH4 4 CIA H NH CH4 2 3 2
NH3? NH3? NH3? 0 1 2 3 4 5 6 7 8 9 10 Wavelength (µm)
Figure 1-8 Low temperature model spectra over plotted with their blackbody curve (Spectra references; Marley et al., 2002; Saumon and Marley, 2008; Morley et al., 2014; Hauschildt et al., 2003).
12 1.5 L, T, and Y dwarfs
Early classification schemes of the low-mass end of the main sequence were formu-
lated in the visible domain (3,900−7000 A),˚ used the strength of TiO bands to deter- mine sub-types. The advent of detectors that collect light at wavelengths longer than the visible spectrum moved the classification scheme red-ward to red optical wave- lengths (6,000−10,000 A),˚ where TiO and VO band strengths are used to determine
M0−M9 sub-types of M dwarfs. The development of near-infrared (λ = 1 − 2.5 µm)
detectors in the late 1980s and early 1990s led to the discovery of the first brown dwarf
(Rebolo et al., 1995). This led to the need for a spectral classification scheme describ-
ing these very low-mass objects. In general, any star or brown dwarf with a spectral
type later than M6 is considered an ultracool dwarf. GD 165B (Becklin and Zucker-
man, 1988), Gliese 229B (Nakajima et al., 1995), and WISEPA J182831.08+265037.8
(Cushing et al., 2011) are the archetypes of the L, T, and Y spectral classes.
1.5.1 L dwarfs
The archetype of the L dwarfs GD 165B was discovered by Becklin and Zuckerman
(1988) as a low-temperature companion to a white dwarf star. Kirkpatrick et al.
(1993) took first red optical spectrum of this object that showed the TiO and VO
bands of late-type M dwarfs were missing. Therefore at the time they were unable
to classify the object as it was unique. However, later discoveries of similar late–type
objects (Kirkpatrick et al., 1997; Ruiz et al., 1997; Delfosse et al., 1997) lead to a new
classification subsequently termed “L” by Martin et al. (1997).
Kirkpatrick et al. (1999) defined a L dwarf sequence in the red optical (6300 −
101000 ˚a),consisting of nine distinct subclasses L0 to L8. A spectral sequence is
described by behavior in a variety of spectral features. The L dwarf sequence (see
Figure 1-9) is determined via weakening TiO and VO bands, strengthening CrH
13 and FeH bands, as well as strengthening resonance lines of K I, Rb I, and Cs I.
Kirkpatrick et al. also devised spectral indices that measure the slope (redness) of the
spectrum and the strengths of various absorption features. The peak of the spectral
energy distributions of L dwarfs is in the near-infrared (see Figure 1-8). Therefore
a classification scheme at these wavelengths is ideal. Geballe et al. (2002) proposed
a near-infrared classification scheme consisting of subclasses ranging from L0 to L9.
Near-infrared spectra of L dwarfs exhibit deep bands of H2O, bands of CO, strong alkali lines of Na I and K I, and FeH band heads. Reid et al. (2001) were the first to
show a smooth spectral sequence formed by the near-infrared spectra when ordered by
their Kirkpatrick et al. (1999) red optical spectral types. The classification scheme
for L dwarfs there is still in debate and no unified scheme has been decided upon
(Kirkpatrick et al., 2010).
1.5.2 T dwarfs
Nakajima et al. (1995) discovered the archetype of the T dwarfs Gliese 229B as a
companion to the M1 dwarf Gliese 229A during a search for brown dwarf companions
to nearby stars. The first near-infrared spectrum taken by Oppenheimer et al. (1995)
showed a number of strong absorption bands, similar to those seen in Jupiter, at-
tributed to CH4 and collisionally induced H2 absorption. Oppenheimer et al. (1998) presented a spectrum of Gliese 229B from 0.837 µm to 5 µm which in addition to previous spectra showed strong Cs I lines and broad H2O absorption bands. The spectral features of H2O, CH4, and collisionally induced H2 are the trademark of T dwarfs.
Burgasser et al. (2006b) unified the classification schemes from Geballe et al.
(2002) and Burgasser et al. (2003) to create scheme for T dwarfs with subclasses from
T0 to T8 (see Figure 1-10). The T dwarf spectral sequence is marked by increasing absorption of H2O and CH4. The near-infrared classification scheme is the primary
14 method for classifying T dwarfs due to the requirement of large telescopes to obtain red optical spectra of T dwarfs. Nevertheless a red optical scheme does exist created by Burgasser et al. (2003) consisting of only four spectral standards To2, To5, To6, and To8, where the o indicates an optical spectral type.
1.5.3 Y dwarfs
The coldest T dwarfs have estimated effective temperatures of 750 K (e.g. Bur- gasser et al., 2006a; Saumon et al., 2007) which leaves a 600 K gap between the coolest
T dwarfs and Jupiter. This leads to a question: What would the spectrum of a brown dwarf with Teff < 700 K look like and would these objects require a new spectral class to classify them?
The confirmation that cooler brown dwarfs existed came with the discovery of
WD 0806−661B (Luhman et al., 2011) and CFBDSIR J145829+101343B (Liu et al.,
2011), but these objects are too faint and too close to their companion for spectro- scopic follow-up, respectively. It took the launch of the Wide-field Infrared Survey
Explorer (WISE, Wright et al., 2010) to find a bright enough single object for follow- up spectroscopy, and WISE J182831.08+265037.7 (hereafter WISE J1828+2650) be- came the archetype Y dwarf. Cushing et al. (2011) identified six brown dwarfs along with WISE J1828+2650, whose spectra seemed to be of a type beyond the latest
T dwarf. WISE J173835.533+273259.0 (hereafter WISE J1738+2732) was chosen as the Y0 standard due to absorption on the blue wing of the H band from NH3. Kirkpatrick et al. (2012) extended the spectral sequence by identifying a Y1 standard
WISE J035000.32−565830.2 (hereafter WISE J0350−5658), noting that ratio of the peak of the H band to the J band is higher than in WISE J1738+2732. This leads to the prototype Y dwarf WISE J1828+2650 which is classified as ≥Y2 due to the equal
flux heights of the J and H bands. The overall sequence of Y dwarfs (see Figure 1-11) is marked by the narrowing of the J band peaks and slow progression of the J/H
15 band peak flux ratio to unity.
1.6 Previous variability studies
As noted in §1.4, the atmospheres of brown dwarfs span the temperatures of the condensation points of various chemical compounds, leading to the formation of var- ious cloud decks. Shortly after the discovery of brown dwarfs it was suggested that weather patterns on the surface of brown dwarfs could result in rotation-based vari- ability (Tinney and Tolley, 1999). Clearly with heterogeneous cloud coverage comes the possibility that the integrated light intensities of ultracool dwarfs may vary due to the modulation of the cloud structures by the rotation period of the object (Ackerman and Marley, 2001) and/or changes in the clouds structures themselves2. For compar-
ison, the Jovian planets of our solar system show complex weather patterns on their
surface, but observations of them as unresolved point sources show rotation based
variability of 1-2% to 4% in the visible region for Neptune and Jupiter, respectively
(Simon et al., 2016). Gelino and Marley (2000) showed that Jupiter exhibits a 4.78
µm peak-to-trough intensity variation of 0.2 mag over its approximately 10 hour ro-
tation period and the first detections of photometric variability in early-type L dwarfs
in the I -band were ascribed to the evolution of the cloud structures (Bailer-Jones and
Mundt, 2001; Gelino et al., 2002). Various other ground based surveys conducted in
the red-optical (e.g., Gelino et al., 2002; Koen, 2003, 2005) also resulted in detectable
variability. A notable early success was the detection of variability in the L dwarf bi-
nary Kelu-1AB with a period of 1.8 hours at 860 nm by Clarke et al. (2002), followed
by later spectroscopic observations Clarke et al. (2003) that showed variations in Hα
emission at the same period. The push to cooler objects comes with surveys done in
2Magnetic activity can also result in variability for the hottest ultracool dwarfs but it is generally believed that the atmospheres of the cooler brown dwarfs are too neutral for magnetic fields to couple to the gas and produce magnetic spots (Gelino et al., 2002; Mohanty et al., 2002)
16 the near-infrared (e.g., Enoch et al., 2003; Koen et al., 2004; Khandrika et al., 2013;
Radigan et al., 2014; Buenzli et al., 2014) and mid-infrared (e.g., Morales-Calder´on
et al., 2006; Metchev et al., 2015). Morales-Calder´onet al. (2006) was the first to make
the jump to a stable space-based platform, by monitoring three late-L dwarfs with
the Inrared Array Camera (IRAC; Fazio et al., 2004) on board Spitzer Space Tele-
scope (Werner et al., 2004). Buenzli et al. (2012) obtained time-resolved simultaneous
observations of a known variable T6.5 dwarf 2MASS J22282889−431026 consisting
of near-infrared spectra and mid-infrared photometry and confirmed the 1.4 h period
for most wavelengths. The spectroscopic observations were taken with Wide Field
Camera 3 (WFC3) on board Hubble Space Telescope (hereafter HST ) over the wave- lengths 1.1 − 1.7 µm during six consecutive orbits spanning 8.58 h, and photometry was taken in [4.5] with IRAC on Spitzer for 5.75 h.
These variability studies show that photometric variability is ubiquitous among
brown dwarfs, and that this activity and variability could be due to magnetic fields
and/or weather (dust formation and condensation) (Bailer-Jones, 2002; Mohanty
et al., 2002; Burgasser et al., 2011; Metchev et al., 2015; Cushing et al., 2016; Leggett
et al., 2016). Buenzli et al. (2012) uncovered a phase shift between wavelengths where
the lag increased with increasing atmospheric altitude, suggesting that brown dwarf
atmospheres are not only heterogeneous horizontally but vertically as well. Radigan
et al. (2014) show that objects at the L/T transition have a high variability fraction.
The exact mechanism for the transition is currently unknown but one theory is that
the iron-, magnesium, and silicon-bearing clouds present in L dwarfs break-up at this
point in the cooling sequence leaving T dwarf atmospheres clear. Therefore the high
variability fractions observed in these objects are presumably due to the breakup of
clouds. Models from Morley et al. (2012, 2014) suggest that Y dwarfs have NaS, KCl
and H2O clouds in their atmospheres indicating that Y dwarfs may be variable. This illustrates that observations of L and T dwarfs and models of Y dwarfs are consistent
17 Table 1.2. Variability Fraction of L and T dwarfs.
Paper SpT Total Number Fraction Filters Range Observed Variable
Bailer-Jones and Mundt (2001) L0-L5 10 7 70% I Gelino et al. (2002) L0-L4.5 18 7 39% Ic Koen (2003) L0-T2.5 12 3 25% Ic Enoch et al. (2003) L2-T5 9 3 33% Ks Koen et al. (2004) L0-T8 18 6 33% J, H, K Koen (2005) L1-T6 12 3 25% Ic Morales-Calder´onet al. (2006) L5-L8 3 2 66% [4.5] Khandrika et al. (2013) L3.5-T8 15 4 27% J, K0 Radigan et al. (2014) L4-T9 57 9 16% J Buenzli et al. (2014) L5-T6 22 6 27% WFC3 Metchev et al. (2015) L3-T8 44 21 48% [3.6],[4.5] with the standard paradigm discussed in §1.4 stating that the nature of the LTY spectral sequence is cloudy→cloud free→cloudy.
1.7 Conclusion
This chapter has provided the framework on which this research was conducted.
While considerable theoretical and observational research has been done, there are still numerous questions about the nature of brown dwarfs that have yet to be answered, and they include but are not limited to: How does convection behave in a brown dwarf and what role does it bear in abundance observations?, and Do brown dwarfs form heterogeneous clouds and have complex weather patterns?
18 Right:
14 8 L0 L0 12 L1 L1 L2 6 10 L2 L3
8 L3 L4 + Constant λ
L4 4 L5 19 6 L5 L6 L7 4 Normalized f Normalized L6 2 L8 2 L7 L9 L8
0 0 Red optical spectral sequence of the optical L dwarf sequence. 7000 8000 9000 10000 1.0 1.5 2.0 Wavelength (Å) Wavelength (µm) Left: Near-infrared spectral sequence ofCushing the (2014). near-infrared L dwarf sequence. Figure from Figure 1-9 8 T0 8 T1 6 T2 To0 6 T3 T 2 o T4 + Constant
λ 4 20 To5 4 T5
T6 To6 2 Normalized f T7 2 To8 T8
To9 T9 Red optical spectral sequence of the optical T dwarf sequence. 0 0
7000 8000 9000 10000 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Left: Wavelength (Å) Wavelength (µm) Near-infrared spectral sequence of the near-infrared T dwarf sequence. Figure Figure 1-10 Right: from Cushing (2014). Ultracool Objects: L, T, and Y Dwarfs 129
4
3 Y0
Y1
+ Constant 2 λ
1
Normalized f ≥Y2
0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 Wavelength (µm)
Fig. 6 J -andFigureH -band 1-11 Left: spectralJ and sequence H band spectra (R of130) Y0, Y1, of the and tentative Y2 spectral Y0 standards. and Y1 spectralFigure standards (Cushing etfrom al. 2011 Cushing; Kirkpatrick (2014). et al. 2012),! and the Y2 dwarf WISE J1828 2650 (Cushing et al. 2011). Spectra are normalized to unity and offset with" constants (dotted linesC). Note that the ratio of the peak fluxes at 1.27 and 1.58 mapproachesunitywithlatersubtype
With a prototype Y dwarf in hand, Cushing et al. (2011)attemptedtoidentify the T/Y boundary. The T spectral sequence had already been extended to T10 based on the extrapolation of certain spectral ratios beyond T8 (Burningham et al. 2008; Lucas et al. 2010). In addition to21 WISE 1828 2650, Cushing et al. (2011) also identified six additional brown dwarfs whoseC spectra appeared later than UGPS 0722–05 based on the width of the J -band peak. Using the MK Process (in particular precept #2, cf. Sect. 3), UGPS 0722–05 was selected as the T9 spectral standard based on its near-infrared spectrum. WISE J173835.53 273259.0 (hereafter WISE 1738 2732) was then chosen as the tentative Y0 spectralC standard C because it exhibited absorption (both visually and by the NH3-H index of Delorme et al. 2008a)onthebluewingoftheH-band peak that was tentatively ascribed to NH3. Bochanski et al. (2011)andSaumon et al. (2012)haveidentifiedweak NH3 features in higher resolution (R 6,000) spectra of UGPS 0722–05 over this wavelength range suggesting the correct! carrier of the absorption has indeed been identified. Kirkpatrick et al. (2012)extendedtheYsequencebyidentifying WISE J035000.32–565830.2 as the tentative Y1 spectral standard. They also noted that the peak flux in the H band relative to the peak flux in the J band is higher than in WISE 1738 2732 presaging the equal flux heights in WISE 1828 2650 which is currently classifiedC as Y2. C Table 2 lists the two" tentative Y-dwarf spectral standards and Fig. 6 shows the current Y spectral sequence. Y dwarfs are currently classified over the 1.1–1.7 m wavelength range due to the difficulty of obtaining spectra at other wavelengths (see also Sect. 8). The sequence is marked both by a narrowing of the J -band peak, Chapter 2
The Search for Variability
2.1 Motivation
In §1.6, we gave a summary of the searches for variability in L and T dwarfs.
In contrast to the L and T dwarfs, the cooler Y dwarfs have not been studied in as much detail because they are difficult to observe with high precision at red optical and near-infrared wavelengths given how intrinsically faint they are these wavelengths
(MH > 20 mag; Kirkpatrick et al., 2012). Leggett et al. (2016) did observe WISEPA J173835.53+273259 in the Y and J bands for nearly 5 hrs over two different nights but could not identify any statistically significant variability. Mace (2015) was the
first to search for mid-infrared photometric variability in the Y dwarfs using All-
WISE single-frame photometry (Cutri et al., 2013) and found no obvious variability due to the large single-frame photometric uncertainty of ∼0.2 mag. Cushing et al.
(2016) presented the first detection of photometric variability in the Y dwarf WISEPC
J140518.40+553421.4 using Spitzer at [3.6] and [4.5]. They observed the Y dwarf at two different epochs separated by 149 days and found variability at [4.5] with a semi- amplitude1 of 3.5% and a period of 8.5 hrs. Moreover, they found that the light
1The definition of “amplitude” varies in the literature. In this and our previous work (Cushing et al., 2016; Leggett et al., 2016), we define the amplitude, 2A, to be the difference between the peak and trough height (i.e. peak-to-trough) and the semi-amplitude, A, to be half the difference
22 curves changed between epochs suggesting a change in the underlying cause or causes
of the variability. Variability at the same wavelengths and at similar semi-amplitudes
has also been detected in multi-epoch observations of two other Y dwarfs WISEPA
J173835.53+273259 (Leggett et al., 2016), and WISE J085510.83−071442.5 (Esplin
et al., 2016). The former shows periodic variability only at [4.5] while the latter shows
periodic variability in both bands in the first epoch but more irregular variability in
the second epoch.
The Spitzer observations of WISEPC J140518.40+553421.4 and WISEPA
J173835.53+273259.0 were both part of a larger 550 hr Spitzer Space Telescope Ex- ploration Science program to survey fourteen Y dwarfs for photometric variability and in this paper we present the results from the entire sample. In §2, we describe the observations and how we reduced the data while in §3, we discuss how we determine
which objects are variable and compute variability fractions. Finally, in §4 and 5, we
discuss in detail the results of our variability analysis and then summarize the take
away points of our Y dwarf survey.
2.2 Observations and Data Reduction
Our Spitzer campaign (program 90015, PI: M. Cushing) was designed to search for
and characterize photometric variability of the fourteen spectroscopically confirmed
Y dwarfs known at the time of the proposal’s submission in 2012. A log of the
observations can be found in Table 2.1; hereafter we abbreviate WISE sources as
WISE HHMM+DD. One object, WISE 0410+42 was originally classified as Y0 by
Kirkpatrick et al. (2012) but was later found to be a T9/Y0 binary by Dupuy et al.
(2015). We conducted our observations using IRAC at both [3.6] and [4.5]. Observing
was conducted in “staring mode” whereby the target is kept close to the same position,
between the peak and trough height.
23 in our case the so-called “sweet spot”, on the array in order to minimize the effect that variations in quantum efficiency across an individual pixel has on the photometry (i.e. the pixel phase effect; Reach et al., 2005). Each Y dwarf was observed continuously for 24 hours—12 hours in [3.6] followed by 12 hours in [4.5]. The duration of 12 hours was motivated by four factors: 1) the rotation periods of L and T dwarfs measured from photometric variability were all below 10 hours (Bailer-Jones, 2005;
Crossfield, 2014), 2) the upper limit to the rotation periods of T dwarfs from v sin(i) measurements is 12.5 hours (Zapatero Osorio et al., 2006), 3) the rotation period of
Jupiter is ∼10 hours, and 4) the recommended maximum duration for staring mode by the Spitzer Science Center is 12 hours. The observations were repeated again between 122 and 464 days later in order to search for changes in the light curves on timescales of months.
24 Table 2.1. Y Dwarf Targets
Epoch 1 Epoch 2 Object Spectral Type Ref Spitzer AOR # Date Spitzer AOR # Date ([3.6],[4.5]) ([3.6],[4.5])
WISE J041056.66+423410.0AB Y0 1 (47160064,47172864) 2013 Apr 19 (47167744,47163392) 2013 Oct 11 WISE J035000.32−565830.2 Y1 1 (47147264,47147776) 2012 Dec 11 (47172352,47168256) 2013 Oct 08 WISE J035934.06−540154.6 Y0 1 (47170560,47165952) 2013 Jan 04 (47160832,47173632) 2014 Apr 13 WISEPA J041022.71+150248.5 Y0 2 (46713088,46713856) 2012 Nov 15 (47169024,47164160) 2013 Nov 03 25 WISE J053516.80−750024.9 ≥Y1 1 (50483200,50482688) 2014 May 25 (50483968,50483456) 2014 Sep 29 WISE J071322.55−291751.9 Y0 1 (47169280,47164672) 2013 Jun 18 (47158784,47171072) 2014 Feb 07 WISE J073444.02−715744.0 Y0 1 (47158528,47172096) 2013 Apr 28 (47166464,47162368) 2013 Aug 26 WISEPC J140518.40+553421.4 Y0.5 pec? 3 (47166208,47162624) 2013 Mar 22 (47173888,47169792) 2013 Aug 17 WISEPA J154151.66−225025.2 Y1 3 (47159808,47173120) 2013 May 02 (47168000,47163648) 2013 Oct 06 WISEA J163940.84−684739.4 Y0 pec 4 (47165184,47161088) 2013 Jun 09 (47172608,47168768) 2013 Oct 27 WISEPA J173835.53+273259.0 Y0 2 (47174144,47170048) 2013 Jun 29 (47164416,47159296) 2013 Oct 29 WISEPA J182831.08+265037.8 ≥Y2 2 (46988032,46987776) 2012 Nov 28 (47157760,47170304) 2013 Dec 14 WISEPC J205628.90+145953.3 Y0 2 (47171584,47166720) 2013 Aug 19 (47161600,47174400) 2014 Jan 17 WISE J222055.32−362817.5 Y0 1 (47171840,47166976) 2013 Aug 13 (47162112,47174656) 2014 Jan 30
Note. — References for spectral types: (1) Kirkpatrick et al. (2012), (2)Cushing et al. (2011)(3)Schneider et al. (2015),(4)Tinney et al. (2012) We reduced the data in a similar manner to Cushing et al. (2016), with a few
minor differences in centroiding and background subtraction. We began our data
reduction by converting the Basic Calibrated Data (BCD) frames generated by the
Spitzer Science Center using version S19.2.0 of the IRAC science pipeline from units
of MJy sr−1 to total electrons. We did not apply the high-resolution gain map of the sweet spot pixel or the pixel phase correction because our targets sometimes fell up to two pixels from the sweet spot. In order to identify the centroid position of our targets in each frame, we used the 2D Gaussian centroiding function from the photutils Python package. As a initial guess for the routine, we used the centroid position determined from a median stack of the entire time series. We then used the IDL aper routine with the EXACT keyword set to measure the total number of electrons within a 2 pixel radius of the centroid position. In some cases there are
background sources close to our target and so instead of using an annulus around our
target to determine the background level, we identified a region on the frame devoid of
sources to compute the median background level within a 7 pixel radius. We checked
that the selection of the background region did not affect the intensity or shape of
each light curve. Clear outliers in the light curves, defined as data points where the
total number of electrons exceed the median intensity level by more than 50 times
the median absolute deviation (MAD=|median{x-median(x)}|, were then removed
before finally dividing by the median intensity level. We note that one Y dwarf,
WISE 1639−68, is located ∼3.3 (400) and ∼4.5 (500) pixels away from the J=14.90
star 2MASS 16394085−6847446 in epoch 1 and 2 respectively and as such, its light
curve may be slightly contaminated by light from this star.
The normalized light curves of the Y0 dwarf WISE 0359−54 are shown in Figure 2-
1. The light curves not only exhibit clear variability at [4.5], but also a small fraction
of bad data points. Figure 2-2 shows the light curves of all 14 Y dwarfs. For display
purposes only, we have identified and removed the bad data points similar to those
26 [3.6] 2013 Jan 04 [4.5]
1.2 1.1
1.0 1.0
0.8 0.9
2014 Apr 13
1.2 1.1 Normalized Flux 1.0 1.0
0.8 0.9
0 5 10 15 20 25 Time (hours)
Figure 2-1 [3.6] (blue) and [4.5] (red) light curves for the Y0 dwarf WISE 0359−54, taken on January 4, 2013 (top) and April 13, 2014 (bottom). The light curves observed on January 4, 2013 (top) shows periodic variability at both wavelengths with a semi- amplitude of 4.2% and 2.8% with periods of 2.41 hours and 2.45 hours respectively. The light curves observed on April 13, 2014 (bottom) shows periodic variability at both wavelengths with a semi-amplitude of 2.5% and 2.2% with a common period of 2.44 hours.
27 Figure 2-2 Normalized IRAC [3.6] (blue) and [4.5] (red) photometry for the fourteen Y dwarfs in our sample. The epoch 1 data is found in the left panel and the epoch 2 data is found in the right panel. For display purposes only, outliers have been removed as described in §3.1.2.1. Note that the scale of the ordinate is different for [3.6] and [4.5] light curves. seen in Figure 2-1 as described in §3.1.2.1.
28 Figure 2-2 Normalized IRAC [3.6] (blue) and [4.5] (red) photometry for the fourteen Y dwarfs in our sample. The epoch 1 data is found in the left panel and the epoch 2 data is found in the right panel. For display purposes only, outliers have been removed as described in §3.1.2.1. Note that the scale of the ordinate is different for [3.6] and [4.5] light curves.
29 Chapter 3
Data Analysis
3.1 Variability Analysis
3.1.1 A Visual Inspection
Inspection of the light curves shown in Figure 2-2 reveal periodic and semi-periodic variability at the few percent level in many of the Y dwarf light curves. By eye, we estimate a variability fraction of 21% (3/14) and 42% (6/14) at [3.6] and [4.5] in epoch 1 and 29% (4/14) and 42% (6/14) at [3.6] and [4.5] in epoch 2. The [3.6] and
[4.5] photometric bands sample similar pressures levels in the atmospheres of Y dwarfs
(Cushing et al., 2016), and so the most likely explanation for the differences in the by- eye variability fractions between bands is that we are not as sensitive to variability at
[3.6] because the [3.6] and [4.5] exposure times are the same and Y dwarfs are fainter at [3.6] by ∼2.5 mag. We defer the computation of a more quantitative estimate of the variability fractions until §3.1.2.
One of the goals of our survey was to search for changes in Y dwarf light curves on timescales of months in order to investigate whether the light curves are stable or not.
While many of the light curves show subtle changes between the two epochs, three
Y dwarfs exhibit dramatic changes to their light curves between epochs. The first,
WISE 0713−29, show a clear signature of variability at [3.6] in epoch 1 but the second-
30 epoch [3.6] light curve obtained 234 days later shows no evidence of this variability.
A second Y dwarf, WISE 1405−55 which was discussed in detail in Cushing et al.
(2016), exhibits both changes in the shape of the light curves and in the amplitudes
of the variation. The first-epoch observations only show low-level variability at [4.5]
while the both the [3.6] and [4.5] light curves are variable in the second epoch. In
addition, the shape and amplitude of the [4.5] light curve has changed dramatically
becoming more sinusoidal and having a larger amplitude. The third Y dwarf, WISE
1639−68, shows the same pattern of changes between epochs that is exhibited by
WISE 1405−55. As noted in §2.2, WISE 1639−68 is located ∼4.500 away from the
J=14.90 star 2MASS 16394085−6847446 and as such, its light curve may be slightly contaminated by light from this star. However, any contamination is unlikely to cause the dramatic changes seen between epochs. Taken as a whole, Spitzer Y dwarf light curves are generally stable over timescales of months, but in some cases can show dramatic variations.
3.1.2 Variability Fractions
To provide a more rigorous estimate of the variability fraction in our sample, we use two different methods. The first is described in §3.1.2.1 and is a Bayesian method inspired by the work of Bailer-Jones (2012) wherein we use the Bayesian Information
Criterion (hereafter BIC; Schwarz, 1978) to identify which of three different variability models (a constant, a sine curve, and a double sine curve) fit the data the best. The second is described in §3.1.2.2 and uses the more traditional periodogram analysis to identify periodic signals in time series data (VanderPlas, 2018).
31 3.1.2.1 Bayesian Analysis
Our Bayesian analysis starts with the assumption that our data arise from the
generative probabilistic model
Di = Mm,i + , (3.1)
where Di is a random variable for the number of electrons detected at time ti, is ran- dom variable with a mean of zero and a variance of σ2 that accounts for measurement uncertainty, and Mm,i is one of the following three models:
M1,i = C, (3.2) 2π M = A sin t + φ + C, and (3.3) 2,i P i 2π 2π M = A sin t + φ + A sin 2 t + φ + C, (3.4) 3,i 1 P i 1 2 P i 2
where C is a constant, and A, P , φ are the semi-amplitude, period, and phase of a sine
curve. We selected these three models because visually they seemed to encompass
the variety of variability seen in our light curves.
If we denote the parameters of a given model as θ (e.g., θ=[A, P , φ] for model
M1]), then we can determine the joint probability distribution of the model param-
eters given our N observations d = [d1, d2, d3, ··· , dN ] using Bayes’ theorem
p(θ|d) ∝ L(d|θ)p(θ), (3.5)
where p(θ|d) is the posterior distribution, p(θ) is the prior function, and L(d|θ)
is the likelihood. We assume that the random variable follows a normal distribution
with a mean of zero and a variance of σ2, and so given that the data points are
32 independent, we can write the likelihood function for the mth model as,
N " N 2 # 1 X [di − Mm,i(θ)] Lm(d|θ, σ) = √ exp − . (3.6) 2 2σ2 2πσ i=0 However as noted in §2.2, our data exhibit outliers and so we must account for these points in our likelihood function. We therefore assume a combined generative model wherein the good data points are generated from the probability density func- tion given by Equation 3.6 while the outliers (or “bad”) data points are generated
2 from a normal distribution with a mean of Ybad and a variance of σbad (Hogg et al., 2010). The likelihood for our data set of N observations is then given by,
N " 2 ! 2 !# Y 1 − Pbad [di − Mm,i] Pbad [di − Ybad] Lm (d|θ) = √ exp − + exp − , (3.7) 2 2σ2 p 2 2σ2 i=0 2πσ 2πσbad bad where Pbad is the probability that a data point is bad (see Hogg et al. (2010) for derivation of this equation).
We sample the joint posterior distribution using Goodman & Weare’s Affine In- variant Markov Chain Monte Carlo (MCMC) Ensemble Sampler implemented by the
Python program emcee (Foreman-Mackey et al., 2013). We used 100 walkers to ini- tialize the ensemble sampler along with 10500 steps, and discarded 50,000 samples from the initial burn-in leaving a total sample set of 1×106. We also used uniform (or
“uninformative”) priors for all parameters of the generative probabalistic models. We
then used the integrated autocorrelation time to determine if our MCMC chain has
run long enough for each of the parameter chains to converge (Hogg and Foreman-
Mackey, 2018). Following the blog post1 of Daniel Foreman-Mackey, we calculate the
autocorrelation time at 10 points along our sampling chain. If each parameter reaches
N > 50τ, where N is the number steps along the chain and τ is the autocorrelation
1https://dfm.io/posts/autocorr/
33 time at that step then the model is considered converged. As noted by Hogg and
Foreman-Mackey (2018), this test is really only a heuristic test of convergence and so we also visually inspected the marginalized posterior distributions generated by the corner routine and identified 5 additional models that fail to converge.
We then identify which of the converged models best fit our data by assigning a score to each model and then comparing the scores to determine the best fit. As our score we use the BIC which is defined as
BIC = −2Lmax + k log(n), (3.8)
where Lmax is the maximized likelihood function, k is the number of parameters in the model, and n is the number of data points. The latter term penalizes the BIC for overfitting by the increase in the number of model parameters. We compare two models, M1 and M2 by calculating ∆ BIC12=BIC1−BIC2. The ∆BIC12 values provide the evidence to favor M2 over M1: if ∆ BIC12 is 0 − 2 then the difference is not worth mentioning, 2−6 is positive evidence for M2, 6−10 is strong evidence for M2, and any value greater than 10 is very strong evidence for M2 (Neath and Cavanaugh, 2011). Table 3.1.2.1 gives the ∆BIC values between the constant and sine models and the sine and double sine models. The prior distributions used in the MCMC analysis and the 16, 50, and 84 percentile values of the parameters of the best fitting models are given for each Y dwarf in Tables A.4 to A.14 in the Appendix. With the best
fitting model in hand, we can now identify the outliers in the data following Hogg et al. (2010) whereby we compute the ratio of the second term in Equation 3.7 to the first term (the ratio of the “bad” part of the likelihood to the “good” part of the likelihood) and identify data points as outliers if this ratio exceeds 0.2.
We identify a source as variable if the ∆BIC value between the constant and sine model and or the sine and double sine model is greater than 6 and the results are
34 Table 3.1. ∆ BIC Values
Epoch 1 Epoch 2 [3.6] [4.5] [3.6] [4.5] Object c−sa s−sd c−s s−ds c−s s−ds c−s s−ds
WISEJ0410+42 ························ WISEJ0350−56 ······ 101.3 ··· 6.1 8.9 219.8 ··· WISEJ0359−54 77.6 ··· 228.2 ··· 14.3 ··· 166.1 ··· WISEJ0410+15 ······ 132.7 ········· 224.9 ··· WISEJ0535−75 ························ WISEJ0713−29 136.4 18.5 21.8 ········· 24.4 0.9 WISEJ0734−71 ··· ··· ··· ··· ··· ··· −0.2 ··· WISEJ1405+55 ······ 76.4 56.8 146.4 ··· 715.0 34.1 WISEJ1541−22 ························ WISEJ1639−68 ······ 50.9 51.9 69.4 ··· 333.2 112.6 WISEJ1738+27 9.3 ··· 169.5 0.4 ······ 59.9 28.3 WISEJ1828+26 ························ WISEJ2056+14 ························ WISEJ2220−36 ·················· 18.4 ···
ac=constant, s=sine, ds=double summarized in Table 3.3. We find a variability fraction of 21% (3/14) and 50% (7/14) at [3.6] and [4.5] in epoch 1 and 29% (4/14) and 57% (8/14) at [3.6] and [4.5] in epoch
2. The model periods and the semi-amplitudes of the variable sources are given in
Table 3.2. The double sine models have semi-amplitudes for each sine curve and so we computed a single semi-amplitude for these models using a Monte Carlo technique.
We generated 10,000 models by drawing randomly from normal distributions for the model parameters P , A1, A2, φ1, φ2, and C. The semi-amplitude is given by half the difference between the crest and trough heights and the values given in Table 3.2 are the mean and standard deviation of the 10,000 values.
35 Table 3.2. Periods and Semi-Amplitudes for Variable-Source Models
Epoch 1 Epoch 2 [3.6] [4.5] [3.6] [4.5] Object P (hours) A (%) P (hours) A (%) P (hours) A (%) P (hours) A (%)
+0.5 +0.2 a +0.9
36 WISEJ0350−56 ······ 11.4−0.4 1.07 ± 0.09 6.2−0.1 4.0±0.4 14.0−0.7 1.55 ± 0.09 +0.02 +0.06 WISEJ0359−54 2.41−0.03 4.2±0.4 2.45±0.01 2.8 ± 0.1 2.44−0.05 2.5±0.5 2.44 ± 0.02 2.2 ± 0.2 +0.09 WISEJ0410+15 ······ 7.8 ± 0.2 0.92 ± 0.07 ······ 8.2 ± 0.2 1.59−0.08 a +0.07 WISEJ0713−29 10.1 ± 0.3 4.0±0.2 6.5 ± 0.2 0.58 ± 0.09 ······ 3.15−0.06 0.44 ± 0.07 +0.4 a +0.3 +0.06 a WISEJ1405+55 ······ 9.1−0.3 1.08±0.06 8.4−0.2 3.7 ± 0.3 8.42−0.05 3.52±0.04 +0.09 a a WISEJ1639−68 ······ 6.32−0.08 0.68±0.04 6.8 ± 0.2 1.6 ± 0.2 6.69 ± 0.05 1.75±0.06 +0.10 a WISEJ1738+27 4.8 ± 0.2 1.7 ± 0.3 5.91−0.09 1.3 ± 0.09 ······ 6.09 ± 0.09 1.08±0.07 WISEJ2220−36 ·················· 3.23 ± 0.08 0.7 ± 0.1
aThe light curve is fit best by a double sine curve. The semi-amplitude is derived as described in the text. 3.1.2.2 Periodogram Analysis
The second method we used to identify variable sources is to construct peri- odograms of the light curves using the LombScargle routine in the astropy.stats library of Python. We first removed data points wherein the normalized intensity exceeded unity by more than ±5× the median absolute deviation of the light curve
(removes between 1.2% and 6.9% of the light curve) to ensure those outliers did not skew the results. We then set our frequency grid by taking the inverse of an evenly spaced array of periods ranging from 1.5 to 24 hours with a step of 0.1 hours. Figures
3-1 and 3-2 show the epoch 1 and 2 periodograms for the 14 Y dwarfs, respectively.
Also shown are the 5% (95% confidence) False Alarm Levels (computed using the
Baluev (2008) approximation) which give the power levels above which we would ex- pect to see power less than 5% of the time under the null hypothesis of pure gaussian noise (no variable signal). Any object with power in its periodogram above this level is considered variable and a summary of which Y dwarfs are variable in each band and epoch is given in Table 3.3. We find a variability fraction of 50% (7/14) and 57%
(8/14) at [3.6] and [4.5] in epoch 1 and 36% (5/14) and 64% (9/14) at [3.6] and [4.5] in epoch 2.
37 10-1 10-2 10-3 -4 10 [3.6] WISE 0146+42 [4.5] 10-5 10-1 10-2 10-3 10-4 WISE 0350 56 10-5 − 10-1 10-2 10-3 10-4 WISE 0359 54 10-5 − 10-1 10-2 10-3 10-4 WISE 0410+15 10-5 10-1 10-2 10-3 10-4 WISE 0535 75 10-5 − 10-1 10-2 10-3 10-4 WISE 0713 29 10-5 − 10-1 10-2 10-3 10-4 WISE 0734 71 10-5 − 10-1 10-2 10-3 10-4 WISE 1405+55 10-5 10-1 10-2 10-3 Periodogram Power 10-4 WISE 1541 22 10-5 − 10-1 10-2 10-3 10-4 WISE 1639 68 10-5 − 10-1 10-2 10-3 10-4 WISE 1738+27 10-5 10-1 10-2 10-3 10-4 WISE 1828+26 10-5 10-1 10-2 10-3 10-4 WISE 2056+14 10-5 10-1 10-2 10-3 10-4 WISE 2220 36 10-5 − 0 3 6 9 12 15 18 21 24 0 3 6 9 12 15 18 21 24 Period (hours) Period (hours)
Figure 3-1 Epoch 1 periodograms for the 14 Y dwarfs in our sample. The grey dashed lines give the 5% (95% confidence) False Alarm Levels which give the power levels above which we would expect to see power less than 5% of the time under the null hypothesis of pure gaussian noise (no variable signal). Any object with power in its periodogram above this level is considered variable.
38 10-1 10-2 10-3 -4 10 [3.6] WISE 0146+42 [4.5] 10-5 10-1 10-2 10-3 10-4 WISE 0350 56 10-5 − 10-1 10-2 10-3 10-4 WISE 0359 54 10-5 − 10-1 10-2 10-3 10-4 WISE 0410+15 10-5 10-1 10-2 10-3 10-4 WISE 0535 75 10-5 − 10-1 10-2 10-3 10-4 WISE 0713 29 10-5 − 10-1 10-2 10-3 10-4 WISE 0734 71 10-5 − 10-1 10-2 10-3 10-4 WISE 1405+55 10-5 10-1 10-2 10-3 Periodogram Power 10-4 WISE 1541 22 10-5 − 10-1 10-2 10-3 10-4 WISE 1639 68 10-5 − 10-1 10-2 10-3 10-4 WISE 1738+27 10-5 10-1 10-2 10-3 10-4 WISE 1828+26 10-5 10-1 10-2 10-3 10-4 WISE 2056+14 10-5 10-1 10-2 10-3 10-4 WISE 2220 36 10-5 − 0 3 6 9 12 15 18 21 24 0 3 6 9 12 15 18 21 24 Period (hours) Period (hours)
Figure 3-2 Epoch 2 periodograms for the 14 Y dwarfs in our sample. The grey dashed lines give the 5% (95% confidence) False Alarm Levels which give the power levels above which we would expect to see power less than 5% of the time under the null hypothesis of pure gaussian noise (no variable signal). Any object with power in its periodogram above this level is considered variable.
39 Table 3.3. Y Dwarf Variability Summary
Bayesian Information Criterion False Alarm Probability Epoch 1 Epoch 2 Epoch 1 Epoch 2 Object [3.6] [4.5] [3.6] [4.5] [3.6] [4.5] [3.6] [4.5]
WISE J0410+4234AB N N N N N N N N WISE J0350−5658 N Y Y Y Y Y Y Y WISE J0359−5401 Y Y Y Y Y Y Y Y 40 WISE J0410+1502 N Y N Y Y Y N Y WISE J0535−7500 N N N N N N N N WISE J0713−2917 Y Y N Y Y Y N Y WISE J0734−7157 N N N N N N N N WISE J1405+5534 N Y Y Y N Y Y Y WISE J1541−2250 N N N N N Y N Y WISE J1639−6847 N Y Y Y Y Y Y Y WISE J1738+2732 Y Y N Y Y Y N Y WISE J1828+2650 N N N N N N N Y WISE J2056+1459 N N N N Y N Y N WISE J2220−3628 N N N Y N N N Y Fraction 3/14 (21%) 7/14 (50%) 4/14 (29%) 8/14 (57%) 7/14 (50%) 8/14 (57%) 5/14 (36%) 10/14 (71%) 3.1.3 Variability Fraction Confidence Intervals
To provide confidence intervals for the variability fractions we derived in the pre- vious two sections, we follow the Bayesian formalism originally devised by Lafreni`ere et al. (2007) to compute exoplanet occurrence rates but which has recently been used by Vos et al. (2019) to compute brown dwarf variability fractions. If we define f to be the fraction of objects in our survey that exhibit variability with semi-amplitudes and periods in the interval [0.5%, 10%]∩[1.5 hrs, 20 hrs] in a given epoch and in a given band, then we can use Bayes’ theorem to derive the posterior probability distribution
14 for f given the data {di}i=1, where di = 1 means variability was detected in the light curve of the ith object and di = 0 means variability was not detected in the light curve of the ith object (see Table 3.3 wherein Y=1 and N=0).
The probability of detecting variability in the ith object is fpi, where pi is the probability that such variability would be detected in the semi-amplitude and period interval [0.5%, 10%]∩[1.5 hrs, 20 hrs] given the observations. It then follows that the probability of not detecting variability in the ith target is (1 − fpi). Since the probability of detecting variability in a given target is independent of the other targets,
14 the likelihood function for the entire dataset {di}i=1 can be written as,
14 14 Y 1−di di L {di}i=1|f = (1 − fpi) (fpi) . (3.9) i=1 The posterior distribution for f is then given by Bayes theorem,
14 14 L ({di}i=1|f) p (f) p f|{di} = . (3.10) i=1 R 1 14 0 L ({di}i=1|f) p (f) df
In order to evaluate the posterior distribution, we require knowledge of pi for each target in each band and at each epoch. As noted above, pi is the probability that variability would be detected in the ith target given the observations. We compute this value for each light curve in a Monte Carlo fashion wherein we generate 10,000
41 Table 3.4. Survey Completeness
Epoch 1 Epoch 2 Object [3.6] [4.5] [3.6] [4.5]
WISE J0410+4234AB 0.76 0.94 0.76 0.95 WISE J0350−5658 0.70 0.96 0.68 0.97 WISE J0359−5401 0.72 0.91 0.71 0.92 WISE J0410+1502 0.88 0.99 0.84 0.96 WISE J0535−7500 0.73 0.97 0.67 0.96 WISE J0713−2917 0.85 0.98 0.89 1.00 WISE J0734−7157 0.77 0.89 0.69 0.95 WISE J1405+5534 0.84 0.99 0.84 0.92 WISE J1541−2250 0.89 0.99 0.86 0.99 WISE J1639−6847 0.90 1.00 0.92 0.98 WISE J1738+2732 0.81 0.98 0.80 0.98 WISE J1828+2650 0.85 1.00 0.81 1.00 WISE J2056+1459 0.93 1.00 0.92 1.00 WISE J2220−3628 0.78 0.98 0.77 0.97
variable light curves with random phase shifts, semi-amplitudes, and periods and then
determine pi by computing the fraction of these light curves identified as variable. For purely computational efficiency, we choose to use periodograms to identify which sources are variable. We generated sine curves with phase shifts, semi-amplitudes, and periods drawn from random distributions given by U(0, 2π), U(0.5,10), and U(1.5, 20).
Noise for each curve was generated by drawing randomly from a normal distribution
given by N (0, σ2), where σ is the standard deviation of the best fitting model for that light curve. The pi values for each target in each band and epoch is given in Table
3.4. With the pi values in hand, we can now evaluate the posterior distribution for f in each band and epoch using Equations 3.9 and 3.10 and the results are shown in
Figure 3-3; 68% and 95% central credibility intervals are indicated in grey.
42 0.045 Epoch 1, [3.6] Epoch 1, [4.5] 0.040
33 47 62 77 90 33 45 59 71 81 0.035
0.030
0.025
0.020
0.015
0.010 Probability Density 0.005
0.000
0.045 Epoch 2, [3.6] Epoch 2, [4.5] 0.040
15 25 37 54 70 46 59 74 83 91 0.035
0.030
0.025
0.020
0.015
0.010 Probability Density 0.005
0.000 0 20 40 60 80 100 0 20 40 60 80 100 Variability Fraction (%) Variability Fraction (%)
Figure 3-3 Posterior distributions for the variability fraction f for each epoch and each band. 68% and 95% central credibility intervals are indicated in light grey and dark grey, respectively.
43 3.2 Discussion
3.2.1 Variability Fractions
The Bayesian and periodogram analyses find that 1) a significant fraction of the
Y dwarfs are variable and 2) the observed Y dwarf variability fraction is larger at
[4.5] than at [3.6]. However, the periodogram analysis finds that 30 of the 56 light curves are variable, 8 of which the Bayesian analysis does not. This is primarily because the periodograms often show increasing power to longer periods indicating long-term variability which the Bayesian analysis does not capture. The discrepant variability fractions underscores the fact that whether variability is detected or not may depend heavily on the technique used and as a result, comparisons between variability fractions derived from different surveys can be challenging. The inferred variability fractions shown in Figure 3-3 all indicate high variability fractions ranging from 37% up to 74%. The [3.6] fractions disagree at the 1σ level while the [4.5] fractions are formally consistent.
With variability fraction credibility intervals in hand, we can compare our results to previous surveys in order to place our Y dwarf survey into context. The survey that is most similar to ours is the aforementioned Metchev et al. Spitzer Exploration
Science program which surveyed 44 L3–T8 dwarfs at [3.6] and [4.5]. Their survey included a 14 hr observation of each dwarf at [3.6] followed by a 7 hr observation at [4.5] and so while not identical to our observing strategy, it is similar enough that our results can be compared. Figure 3-4 shows the variability fractions for the two different surveys. While the values are consistent within the uncertainties, the point estimates for the fractions show a clear drop for the T dwarfs. While it is generally assumed that variations in the horizontal and/or vertical cloud structures gives rise to the observed variability, it has proven difficult to prove definitively (e.g.,
Esplin et al., 2016). In broad terms, the drop in the variability fraction in the T
44 Epoch 1 [3.6] 100 Epoch 2 [4.5]
80
60
40 Variability Fraction (%) 20 Metchev et al. (2015) This Work
L T Y Spectral Type
Figure 3-4 Spitzer variability fractions for L and T dwarfs (Metchev et al., 2015) and the Y dwarfs (this work). The points without error bars are the raw fractions (number of variable objects/total sample) while the points with error bars have been corrected for survey sensitivity limits. The error bars for the L and T dwarfs are 95% confidence limits while the error bars for the Y dwarfs are centered 95% credibility intervals.
dwarfs actually supports the idea that clouds are the primary cause of the observed
variability. In the standard paradigm, L dwarfs become progressively redder at near-
infrared wavelengths due to the presence of silicate and liquid iron condensate clouds.
The early to mid-type T dwarfs are blue because these clouds are disrupted by some
as-yet unknown mechanism. In the late-type T dwarfs and Y dwarfs, additional
condensate clouds composed of KCl, and Na2S and even water ice form (Marley, 2000; Morley et al., 2012). The cloudy→cloud free→ cloudy nature of the spectral
sequence is supported by the high→low→high variability fraction in the L, T , and
Y dwarfs.
45 Table 3.5. Rotation Periods of Y Dwarfs
Object Rotation Period (hrs) Input Epochs and Bandsa
Secure Periods WISE 0359−54 2.44±0.01 E1 [3.6], E1 [4.5], E2 [3.6], E2 [4.5] WISE 0410+15 8.0±0.14 E1 [4.5], E2 [4.5] WISE 1405+55 8.42±0.06 E2 [3.6], E2 [4.5] WISE 1639−68 6.69±0.05 E2 [3.6], E2 [4.5] WISE 1738+27 6.01±0.07 E1 [4.5], E2 [4.5] Tentative Periods WISE 0350−56 12.0±0.4 E1 [4.5], E2 [4.5] WISE 2220−36 3.23±0.08 E2 [4.5]
aE1=epoch 1, E2=epoch 2
3.2.2 Rotation Periods
Five Y dwarfs show clear periodic variation in at least 2 light curves and have model periods that are consistent within 2σ. We therefore assume that the rotation periods of these Y dwarf are given by the model periods. The weighted means of the model periods, which range from 2.44 to 8.42 hours, are given in Table 3.2.2. Two additional Y dwarfs, WISE 0350−56 and WISE 2220−36 show periodic variations but we cannot be sure they correspond to their rotation periods. In the former, the
+0.5 +0.9 model derived periods of 11.4−0.4 and 14.0−0.7 are clearly inconsistent while in the latter, variability with a period of 3.23 hours is seen in only one band. We designate the periods of these two Y dwarfs in Table 3.2.2 as tentative. Three of the light curves for a third Y dwarf, WISE 0713−29, are fit by sine curves with disparate periods (10.1, 6.5, and 3.15 hours) and so we do not include it in Table 3.2.2. The disparate periods underscore the fact even though a model may describe a given data set well, it may not have any physical meaning.
The left panel of Figure 3-5 shows the periods of L, T, and Y dwarfs as a function
46 25 100.0 Secure, This Work This Work Tentative, This Work Metchev et al. (2015) Metchev et al. (2015) Crossfield (2014) 20 Crossfield (2014)
10.0
15
10 1.0 Period (hours) Semi-Amplitude (%) 5
2M0050 0.1 W0359
2M0030 2M2228 0 L0 L5 T0 T5 Y0 L0 L5 T0 T5 Y0 Spectral Type Spectral Type
Figure 3-5 Periods (left) and Semi-amplitudes (right) for a sample of L, T, and Y dwarfs as a function of spectral type. The data come from the compilation of Crossfield (2014), the survey of Metchev et al. (2015), and this work. of spectral type. The sample is derived from the compilation of Crossfield (2014), and the survey of Metchev et al. (2015), and this work. The rotation periods of the 7 Y dwarfs fall within the range of rotation periods of the L and T dwarfs which suggests the rotation evolution of Y dwarfs is most likely similar to that of L and T dwarfs.
One object, WISE 0359−54, has a rotation of period of only 2.44 hours and so joins a small list of cool brown dwarfs with very short rotation periods including 2MASS
J22282889−4310262 at 1.43 hrs (T6; Clarke et al., 2008), 2MASS J00501994−3322402 at 1.55 hrs (T7; Metchev et al., 2015), and potentially the Y0 dwarf WISE 2220−36 at 3.23 hrs (this work).
Rotation period is one factor that determines the oblateness of a body; the shorter the rotation period the more oblate an object is. The oblateness f of a body is defined to be
47 f = 1 − Re/Rp, (3.11)
where Re is its equatorial radius and Rp is its polar radius. Jupiter and Saturn have oblatenesses of 0.065 and 0.11, respectively. The interiors of brown dwarfs are well modeled by a polytrope with n between 1 and 1.5 (Burrows and Liebert, 1993) and the expression for the oblateness of a stable polytropic gas configuration under hydrostatic equilibrium is given by,
2 Ω2R f = C e , (3.12) 3 g
where C is a constant that depends on the polytropic index, and Ω, Re, and g are the angular frequency, equatorial radius, and surface gravity of the body, respectively
(Chandrasekhar, 1933).
Since brown dwarfs lack a stable internal energy source with which to stabilize
surface radiative losses, they slowly cool and contract over time. Their radii, surface
gravities, and periods are therefore a function of time which means their oblateness is
also a function of time. The left panel of Figure 3-6 shows model predictions for the
oblateness of three brown dwarfs with the effective temperatures of WISE 0359−54
(420–450 K; Leggett et al., 2017) as a function of rotation period. The radii and
surface gravities corresponding to the effective temperature range 420–450 K were
obtained using the Sonora solar metalicity cloudless evolutionary models of Marley
et al. (in prep)2 and we assume an n = 1 polytrope for which C = 1.1399. Also
shown are the oblatenesses of Jupiter and Saturn and the oblateness (0.44) at which
an n = 1 polytrope becomes unstable. It is clear that with a rotation period of
2.44 hours (shown as a vertical line), WISE 0359−54 is nowhere near the point of
becoming unstable but depending on its mass and age, it could have an oblateness
2https://zenodo.org/record/2628068.
48 similar to that of Jupiter or even Saturn.
Y dwarfs with short rotation periods like WISE 0359−54 are the ideal field test cases with which to study the effects of oblateness on the observed properties of brown dwarfs including the polarization of light that can arise from rotating oblate bodies with dust in their atmospheres (Sengupta, 2003; Sengupta and Kwok, 2005;
Sengupta and Marley, 2010; Marley and Sengupta, 2011) and the impact the oblate- ness would have on atmospheric dynamics. To first order the radii of all brown dwarfs are equal and so for a fixed polytropic index and angular frequency, the oblateness is
2 inversely proportional to the mass since g ∝ M/Re . By virtue of their low effective temperatures (Teff < 500 K), Y dwarfs are constrained by interior physics to be less massive than about 35 MJup (e.g., Saumon and Marley, 2008) and so all else being equal, they should exhibit more oblateness than the on-average more massive L and
T dwarfs. For example, the right panel of Figure 3-6 shows model predictions for the oblateness of brown dwarfs with Teff=1350–1450 K (T0). At the same rotation period of WISE 0359−54, the oblateness of this object is far below that of Jupiter even at an age of 300–400 Myr. Of course field L dwarfs with even shorter periods like the L7 dwarf 2MASSW J0030300−145033 at 1.38 hours (Enoch et al., 2003) may show significant oblateness but as shown in Figure 3-6, there is a paucity of L dwarfs with such short rotation periods. The search for additional Y dwarfs (e.g. CWISEP
J193518.59−154620.3, Marocco et al., 2019) and the measurement of their rotation periods should therefore be of paramount importance.
3.2.3 Semi-Amplitudes
The right panel of Figure 3-5 shows the variability semi-amplitudes as a function of spectral type for a sample of L, T, and Y dwarfs (Crossfield, 2014; Metchev et al.,
2015, this work). The Y dwarf semi-amplitudes do not stand out in this diagram but interestingly, we did not find any small-amplitude (< 0.5%) variables even though our
49 Unstable Unstable T0 Dwarf
(Teff=1350−1450 K) Saturn Saturn 0.100 Jupiter Jupiter
5 MJup (300−400 Myr)
5 M (300−400 Myr) 0.010 Jup 15 MJup (4−6 Gyr) Oblateness
W0359
(Teff=420−450 K)
24 M (8−10 Gyr) 15 M (4−6 Gyr) 0.001 Jup Jup
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Rotation Period (hr) Rotation Period (hr)
Figure 3-6 Model oblateness values as a function of rotation period for a body with the effective temperature of WISE 0359−54 (left) and a hypothetical T0 dwarf (right). Also shown are the oblateness values of Jupiter and Saturn and the oblateness (0.44) beyond which an n = 1 polytrope becomes rotationally unstable. The observed rotation period of WISE 0359−54 of 2.44 hours is shown as a vertical line in both panels.
50 survey is sensitive down to at least 0.2% (i.e. our survey completeness is near unity
at [4.5]; see Table 3.4). This continues a trend set by the Metchev et al. survey which
has a similar sensitivity to ours and which fails to detect low-amplitude variability
in late-type T at [3.6] or [4.5]. Indeed with the exception of the T5 dwarf 2MASS
J23391025+1352284 which only shows a linear change in intensity over a 40 min
time frame in the near infrared (Buenzli et al., 2014), no dwarfs with spectral types
later than T5 have been found to be variable with an amplitude less than 0.5%. In
contrast, numerous L dwarfs and early-type T dwarfs have been observed to vary
with amplitudes between 0.1 and 0.5%. The paucity of objects in the lower right
hand corner of the diagram therefore appears to be real and is not an observational
bias. However, the reason for this dearth of low-amplitude variables remains unclear.
Metchev et al. (2015) noted that the maximum observed amplitude at [3.6] and
[4.5] increases with decreasing spectral type. Beyond a spectral type of T3, however,
the trend is driven by a single T6 dwarf, 2MASS J22282889−4310262. With our sample of Y dwarfs, we can test whether this trend is real and actually continues to later spectral types and cooler effective temperatures as 2MASS 2228−43 and
Jupiter’s 20% variability at 4.78 µm suggests. Figure 3-7 shows the amplitude (not the semi-amplitude) as a function of spectral type for the L and T dwarfs in the
Metchev et al. sample and the Y dwarfs in our sample along with the line (in the log-linear space) that Metchev et al. found to represent the envelope of maximum variability. The Y dwarfs all fall on or below the extension of the Metchev et al. line with WISE 0350−56, WISE 0359−54, WISE 0713−29, and WISE 1405+55, falling nearly on it. Our observations therefore confirm the trend continues to at least a spectral type of early Y. Interestingly, all 4 of the largest amplitudes are from [3.6] light curves even though there are more [4.5] variable light curves. The exact physical mechanism or mechanisms responsible for the trend however, remain unknown at this time.
51 [3.6] [4.5] 10.0
1.0 Amplitude (%)
0.1 L5 T0 T5 Y0 Spectral Type
Figure 3-7 Reproduction of Figure 9 from Metchev et al. (2015) with the Y dwarfs from this work included. We have not included the upper limits in the original figure.
Figure 3-8 shows the ratio of the [4.5] to [3.6] semi-amplitudes as a function of
spectral type for L, T, and Y dwarfs from Metchev et al. (2015) and this work.
Metchev et al. found the mean amplitude ratio to be 1 with a standard deviation of
0.7. The mean of this ratio derived from Y dwarf light curves with 1.1±1.1 which
is consistent with the LT dwarf value. Cushing et al. (2016) noted that the near
unity ratio for WISE 1405−55 ruled out hot spots as a possible cause of the observed
variability because the Morley et al. (2014) hot spots models have ratios less than
unity. While a detailed study is beyond the scope of this work, the larger sample of
Y dwarfs suggest that clouds account for the observed variability.
3.2.4 Phases
Buenzli et al. (2012) found that the 1.1–1.7 µm spectroscopic observations with the Hubble Space Telescope and [4.5] photometric observations with Spitzer showed variability with a common period but with phases that were a function of wavelength
(see also Yang et al., 2016). Since different wavelengths probe different layers of an at-
52 5 Metchev et al. (2015) This Work, Epoch 1 [3.6] & [4.5] fit 4 This Work, Epoch 2 only [4.5] fit
3
2
A[4.5]/A[3.6] 1
0
1 L5 T0 T5 Y0 Spectral Type
Figure 3-8 Reproduction of Figure 10 from Metchev et al. (2015) with the Y dwarfs from this work included.
mosphere, they found that the phase lags increased with decreasing pressure level (i.e.
higher in the atmosphere). Their most plausible interpretation involves a heteroge-
neous deep cloud controlling the near-infrared variability with horizontal temperature
variations at higher pressure levels driving the longer wavelength variability. No such
joint observations have been conducted to date on Y dwarfs but we do have one Y
dwarf in our sample whose light curves are fit with sine curves in both bands and both
epochs, with which we can explore whether phase variations are present between the
[3.6] and [4.5] bands. WISE 0359−54 has the shortest period in our sample at 2.44 hours and shows strong variability in all four light curves. The phases of the best fit sine curves are all consistent within their uncertainties suggesting no phase difference exists between [3.6] and [4.5]. This is perhaps not surprising as Cushing et al. (2016) has shown that the [3.6] and [4.5] bands probe very similar, low-pressure layers of a
Y dwarfs atmosphere. WISE 0359−59 is therefore a perfect candidate for variability studies with the James Webb Space Telescope because its NIRSpec instrument covers the 0.7–5 µm wavelength range in a single exposure.
53 Chapter 4
Conclusions and Future Prospects
4.1 Conclusions
We performed a search for variability in Y dwarfs using 2 epochs of observations each taken with Spitzer for 12 hours at [3.6] immediately followed by 12 hours at [4.5] separated by 122–464 days and found that Y dwarfs are variable.
• Our variability fractions implies that between 37% and 74% of Y dwarfs are
variable.
• While the mid-infrared light curves of Y dwarfs are generally stable on time
scales of months, we have encountered a few that vary dramatically on those
time scales.
• When combined with variability fractions of L and T dwarfs from Metchev et al.
(2015), our variability fractions of Y dwarfs support the standard paradigm
(see §1.4, §1.6 and §3.2) that clouds are responsible for the observed variability
because of the cloudy→cloud free→cloudy.
• We have determined the rotation periods of 5 Y dwarfs ranging from 2.44 hours
to 8.42 hours, with two additional tentative periods.
54 • We also considered the oblateness of Y dwarfs as rotation period is one factor
in its calculation. One of our targets WISE 0359−54 has a very small period so
we showed that depending on its’ mass it could have an oblateness comparable
to that of Jupiter or Saturn. We also determined that with its short, consistent
period across all four light curves, it would make a perfect candidate for future
variability studies with JWST.
• Interestingly we found that both Metchev et al. survey and ours failed to
detect any small amplitude variability (< 0.5%) among late-T and Y dwarfs,
even though our survey is sensitive down to 0.2%. This dearth is therefore not
due to observational bias but from an as yet unknown cause.
• Our observations confirm the Metchev et al. determination that the maximum
observed amplitude increases as a function of spectral type at [3.6] and [4.5]
continues to at least the early Y spectral type.
• We find that the average [4.5]/[3.6] amplitude ratio is greater than unity which
suggests that clouds may be the physical mechanism of the observed variability.
• As [3.6] and [4.5] probe similar atmospheric layers unsurprisingly we find no
phase changes in the light curves for WISE 0359−54.
• Different methods of detecting variability gives different answers making survey
comparisons difficult.
4.2 Future Prospects
The future of variability studies of brown dwarfs and extrasolar planets is impor- tant for understanding the atmospheres of these objects. Doppler imaging of brown dwarfs and extrasolar planets will be possible with the high resolution (R ∼ 100,000)
55 spectroscopy mode that is anticipated for the European Extremely Large Telescope
mid-infrared imager and spectrograph METIS similar to the work of Crossfield et al.
(2014) seen in Figure 4-1.
Figure 4-1 A surface map of Luhman 16B showing large scale cloud inhomogeneities. Each projection is time stamped over the rotation period of the object which is 4.9 hours.(Figure from Crossfield et al. 2014)
JWST will be revolutionary for the study of ultracool dwarfs. JWST possesses an
assortment of coronagraphs ideal for the study variability in relatively wide (> 500)
extrasolar planet/brown dwarf companions. At this time, very few brown dwarfs and
no directly imaged extrasolar planets have been characterized at wavelengths beyond
5 µm. JWST will have two instrument that will revolutionize our understanding of
ultracool atmospheres: NIRSpec and MIRI (see Figure 4-2). The NIRSpec instrument
capable of obtaining spectra in 0.6–5.3 µm covering the near infrared portion of
the spectrum. The MIRI instrument provides coverage of mid-infrared wavelengths
from 4.9 to 28.8 µm. It is capable of imaging using 9 broad band filters covering a
wavelength range of 5.6 to 25.5 µm. MIRI has spectroscopy capabilities as well using two modes: (1) a low spectral resolving power from 5 to 12 µm including both slitless
56 and slitted options; (2) a medium spectral resolving power integral field unit from
4.9 to 28.8 µm. As Y dwarfs emit most of their radiation in the mid-infrared, this will revolutionize the study of Y dwarfs leading to a possible classification scheme in the mid-infrared. The near- and mid-infrared spectroscopy and photometry from
JWST will allow for models to be tailored to spectra not only in the near-infrared but also in the mid-infrared. With JWST we will also useful for performing brown dwarf variability studies using time-series spectroscopy similar to a previous study using HST by Buenzli et al. (2014).
57 iue42Mdlsetao 5 wr lte ihJS ntuetwave- Instrument JWST with plotted dwarf Y K 450 a coverage. of length spectra Model 4-2 Figure
T −2 102 eff=450 K, g=316 m 4A (R~1500) fsed=nc 3C (R~2100) 4C (R~1400) 100 3B (R~2000) 4B (R~1600) 3A (R~2500) 10−2 2C (R~2800) NIRCam (R=1500) 2B (R~2700)
Jy) at 10 pc −4
µ 10 NIRISS (R=150)
( 2A (R~2800) λ f NIRISS (R=700) NIRSpec (R=1000/2700) 1C (R~3000) 10−6 NIRSpec (R=1000/2700) 1B (R~3200) NIRSpec (R=1000/2700) MRS1A (R~3200) NIRSpec (R=100) MIRI (R~100) 10−8
0.5 F1000W F1280W MIRI F770W F1500W 0.4 F560W F1130W F1800W F2100W 58 0.3 F2550W 0.2 Transmission 0.1 0.0
F356W 0.8 NIRISS F090W F115W F200W F277W F444W F150W 0.6
0.4 F380M F140M
Transmission F480M 0.2 F158M F430M 0.0 0.7 0.6 NIRCam F356W F200W F444W 0.5 F150W F115W F277W 0.4 F090W 0.3 F070W F430M F182M F360M Transmission 0.2 F140M F300M F480M 0.1 F162M F210M F250M F335M F410M F460M
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74 Appendix A
Bayesian Parameter Tables
75 Table A.1. WISEJ0146+4234
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 0.999±0.003 Standard deviation σ U(0, 1) 0.056±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.11 Bad data standard deviation σbad U(0, 4) 0.56−0.08 +0.14 Bad data mean Ybad U(0.5, 3.5) 1.52−0.13 Epoch 1 [4.5] constant model parameters Constant C U(C > 0) 0.9989±0.0009 +0.0008 Standard deviation σ U(0, 1) 0.0183−0.0007 Bad data probability Pbad U(0, 1) 0.6±0.01 +0.03 Bad data standard deviation σbad U(0, 4) 0.17−0.02 Bad data mean Ybad U(0.5, 3.5) 1.18±0.04 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 0.994±0.003 Standard deviation σ U(0, 1) 0.055±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.10 Bad data standard deviation σbad U(0, 4) 0.54−0.08 Bad data mean Ybad U(0.5, 3.5) 1.4±0.1 Epoch 2 [4.5] constant model parameters Constant C U(C > 0) 0.9993±0.0009 Standard deviation σ U(0, 1) 0.0182±0.0007 Bad data probability Pbad U(0, 1) 0.06±0.01 +0.04 Bad data standard deviation σbad U(0, 4) 0.20−0.03 Bad data mean Ybad U(0.5, 3.5) 1.12±0.04
aU(a, b) denotes a uniform distribution over the range a to b. bThe values reported correspond to the 16th, 50th, and 84th percentiles of the marginalized posterior distribution.
76 Table A.2. WISEJ0350−56
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 1.000±0.003 Standard deviation σ U(0, 1) 0.066±0.003 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.12 Bad data standard deviation σbad U(0, 4) 0.60−0.09 +0.2 Bad data mean Ybad U(0.5, 3.5) 1.4−0.1 Epoch 1 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 1.07±0.09 +0.5 Period P (hours) U(0, 20) 11.4−0.4 +25 Phase φ (degrees) U(−90, 450) 165−24 Constant C U(0.8, 1.2) 0.9992±0.0006 Standard deviation σ U(0, 1) 0.0122±0.0005 Bad data probability Pbad U(0, 1) 0.07±0.01 Bad data standard deviation σbad U(0, 4) 0.13±0.02 Bad data mean Ybad U(0.5, 3.5) 1.14±0.03 Epoch 2 [3.6] double sine model parameters Semi-Amplitude 1 A1 (%) U(0, 0.5) 2.5±0.5 Semi-Amplitude 2 A2 (%) U(0, 0.5) 2.2±0.5 +0.2 Period P (hours) U(0, 20) 6.2−0.1 +16 Phase 1 φ1 (degrees) U(−90, 450) 7−15 +23 Phase 2 φ2 (degrees) U(−90, 450) 37−22 Constant C U(0.8, 1.2) 0.994±0.004 Standard deviation σ U(0, 1) 0.071±0.003 Bad data probability Pbad U(0, 1) 0.02±0.01 +0.3 Bad data standard deviation σbad U(0, 4) 0.9−0.2 Bad data mean Ybad U(0.5, 3.5) 2.1±0.3 Epoch 2 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 1.55±0.09 +0.9 Period P (hours) U(0, 20) 14.0−0.7 +25 Phase φ (degrees) U(−90, 450) 39−22 +0.0008 Constant C U(0.8, 1.2) 0.9975−0.0009 Standard deviation σ U(0, 1) 0.0121±0.0005 Bad data probability Pbad U(0, 1) 0.07±0.01 +0.03 Bad data standard deviation σbad U(0, 4) 0.16−0.02 Bad data mean Ybad U(0.5, 3.5) 1.13±0.03
aU(a, b) denotes a uniform distribution over the range a to b.
bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
77 Table A.3. WISEJ0359−54
Model Parameter Priora Valueb
Epoch 1 [3.6] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 4.2±0.4 +0.02 Period P (hours) U(0, 20) 2.41−0.03 Phase φ (degrees) U(−90, 450) 293± 11 Constant C U(0.8, 1.2) 0.999±0.003 Standard deviation σ U(0, 1) 0.056±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 Bad data standard deviation σbad U(0, 4) 0.7±0.1 Bad data mean Ybad U(0.5, 3.5) 1.8±0.2 Epoch 1 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 2.8±0.1 Period P (hours) U(0, 20) 2.45±0.01 +16 Phase φ (degrees) U(−90, 450) 315−17 Constant C U(0.8, 1.2) 0.9973±0.0008 Standard deviation σ U(0, 1) 0.0197±0.0008 Bad data probability Pbad U(0, 1) 0.10±0.02 +0.04 Bad data standard deviation σbad U(0, 4) 0.29−0.03 Bad data mean Ybad U(0.5, 3.5) 1.15±0.05 Epoch 2 [3.6] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 2.5±0.5 +0.06 Period P (hours) U(0, 20) 2.44−0.05 +24 Phase φ (degrees) U(−90, 450) 356−23 Constant C U(0.8, 1.2) 1.000±0.003 +0.003 Standard deviation σ U(0, 1) 0.061−0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 Bad data standard deviation σbad U(0, 4) 0.6±0.1 +0.2 Bad data mean Ybad U(0.5, 3.5) 1.4−0.1 Epoch 2 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 2.2±0.2 Period P (hours) U(0, 20) 2.44±0.02 +20 Phase φ (degrees) U(−90, 450) 347−19 Constant C U(0.8, 1.2) 1.001±0.001 Standard deviation σ U(0, 1) 0.0202±0.0008 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.05 Bad data standard deviation σbad U(0, 4) 0.27−0.04 Bad data mean Ybad U(0.5, 3.5) 1.18±0.06
aU(a, b) denotes a uniform distribution over the range a to b.
bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
78 Table A.4. WISEJ0410+15
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 0.996±0.002 Standard deviation σ U(0, 1) 0.037±0.001 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.09 Bad data standard deviation σbad U(0, 4) 0.36−0.06 Bad data mean Ybad U(0.5, 3.5) 1.5±0.1 Epoch 1 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 0.92±0.07 Period P (hours) U(0, 20) 7.8±0.2 Phase φ (degrees) U(−90, 450) 135±0.2 Constant C U(0.8, 1.2) 0.9968±0.0005 +0.0004 Standard deviation σ U(0, 1) 0.0089−0.0003 Bad data probability Pbad U(0, 1) 0.07±0.01 Bad data standard deviation σbad U(0, 4) 0.13±0.02 Bad data mean Ybad U(0.5, 3.5) 1.10±0.03 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 0.999±0.002 Standard deviation σ U(0, 1) 0.041±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.09 Bad data standard deviation σbad U(0, 4) 0.45−0.07 Bad data mean Ybad U(0.5, 3.5) 1.5±0.1 Epoch 2 [4.5] sine model parameters +0.09 Semi-Amplitude A (%) U(0, 0.5) 1.59−0.08 Period P (hours) U(0, 20) 8.2±0.2 Phase φ (degrees) U(−90, 450) 222±17 Constant C U(0.8, 1.2) 0.9948±0.0006 +0.0005 Standard deviation σ U(0, 1) 0.0111−0.0004 +0.02 Bad data probability Pbad U(0, 1) 0.07−0.01 Bad data standard deviation σbad U(0, 4) 0.14±0.02 Bad data mean Ybad U(0.5, 3.5) 1.10±0.03
aU(a, b) denotes a uniform distribution over the range a to b.
bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
79 Table A.5. WISEJ0535−75
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 0.999±0.003 Standard deviation σ U(0, 1) 0.054±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 Bad data standard deviation σbad U(0, 4) 0.6±0.1 Bad data mean Ybad U(0.5, 3.5) 1.5±0.2 Epoch 1 [4.5] constant model parameters Constant C U(C > 0) 1.0002 ±0.0008 Standard deviation σ U(0, 1) 0.0161±0.0006 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.04 Bad data standard deviation σbad U(0, 4) 0.17−0.03 Bad data mean Ybad U(0.5, 3.5) 1.13±0.04 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 0.991±0.004 Standard deviation σ U(0, 1) 0.076±0.003 Bad data probability Pbad U(0, 1) 0.06±0.01 Bad data standard deviation σbad U(0, 4) 0.8±0.1 Bad data mean Ybad U(0.5, 3.5) 1.8±0.2 Epoch 2 [4.5] constant model parameters Constant C U(C > 0) 0.9993±0.0007 Standard deviation σ U(0, 1) 0.0141±0.0006 +0.02 Bad data probability Pbad U(0, 1) 0.08−0.01 +0.03 Bad data standard deviation σbad U(0, 4) 0.19−0.02 +0.04 Bad data mean Ybad U(0.5, 3.5) 1.10−0.03
aU(a, b) denotes a uniform distribution over the range a to b. bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
80 Table A.6. WISEJ0713−29
Model Parameter Priora Valueb
Epoch 1 [3.6] double sine model parameters Semi-Amplitude 1 A1 (%) U(0,.5) 3.4±0.2 Semi-Amplitude 2 A2 (%) U(0,.5) 1.3±0.2 Period P (hours) U(0, 20) 10.1±0.3 Phase 1 φ1 (degrees) U(−90, 450) 275±8 +19 Phase 2 φ2 (degrees) U(−90, 450) 349−17 Constant C U(0.8, 1.2) 0.999±0.002 Standard deviation σ U(0, 1) 0.031±0.001 Bad data probability Pbad U(0, 1) 0.06±0.01 +0.07 Bad data standard deviation σbad U(0, 4) 0.42−0.06 Bad data mean Ybad U(0.5, 3.5) 1.42±0.09 Epoch 1 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 0.58±0.09 Period P (hours) U(0, 20) 6.5±0.2 Phase φ (degrees) U(−90, 450) 269±31 Constant C U(0.8, 1.2) 0.9994±0.0007 Standard deviation σ U(0, 1) 0.0128±0.0005 +0.02 Bad data probability Pbad U(0, 1) 0.07−0.01 +0.02 Bad data standard deviation σbad U(0, 4) 0.10−0.01 Bad data mean Ybad U(0.5, 3.5) 1.04±0.02 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 0.997±0.002 Standard deviation σ U(0, 1) 0.031±0.001 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.05 Bad data standard deviation σbad U(0, 4) 0.24−0.03 Bad data mean Ybad U(0.5, 3.5) 1.24±0.06 Epoch 2 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 0.44±0.07 +0.07 Period P (hours) U(0, 20) 3.15−0.06 +42 Phase φ (degrees) U(−90, 450) 179−44 Constant C U(0.8, 1.2) 0.9996±0.0005 Standard deviation σ U(0, 1) 0.0093±0.0003 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.03 Bad data standard deviation σbad U(0, 4) 0.12−0.02 Bad data mean Ybad U(0.5, 3.5) 1.06±0.03
aU(a, b) denotes a uniform distribution over the range a to b.
bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
81 Table A.7. WISEJ0734−71
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 0.997±0.003 Standard deviation σ U(0, 1) 0.056±0.002 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.10 Bad data standard deviation σbad U(0, 4) 0.44−0.08 Bad data mean Ybad U(0.5, 3.5) 1.7±0.1 Epoch 1 [4.5] constant model parameters Constant C U(C > 0) 0.9997±0.0009 +0.0008 Standard deviation σ U(0, 1) 0.0175−0.0007 +0.02 Bad data probability Pbad U(0, 1) 0.09−0.01 Bad data standard deviation σbad U(0, 4) 0.22±0.03 Bad data mean Ybad U(0.5, 3.5) 1.15±0.04 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 1.000±0.003 +0.003 Standard deviation σ U(0, 1) 0.057−0.002 +0.02 Bad data probability Pbad U(0, 1) 0.06−0.01 +0.07 Bad data standard deviation σbad U(0, 4) 0.40−0.05 +0.10 Bad data mean Ybad U(0.5, 3.5) 1.33−0.09 Epoch 2 [4.5] constant model parameters Constant C U(C > 0) 0.9991±0.0009 Standard deviation σ U(0, 1) 0.0171±0.0007 +0.02 Bad data probability Pbad U(0, 1) 0.09−0.01 +0.03 Bad data standard deviation σbad U(0, 4) 0.19−0.02 Bad data mean Ybad U(0.5, 3.5) 1.11±0.03
aU(a, b) denotes a uniform distribution over the range a to b. bThe values reported correspond to the 16th, 50th, and 84th percentiles of the marginalized posterior distribution.
82 Table A.8. WISEJ1405+55
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 0.998±0.002 +0.002 Standard deviation σ U(0, 1) 0.038−0.001 Bad data probability Pbad U(0, 1) 0.06±0.01 +0.06 Bad data standard deviation σbad U(0, 4) 0.36−0.05 +0.09 Bad data mean Ybad U(0.5, 3.5) 1.36−0.08 Epoch 1 [4.5] double sine model parameters Semi-Amplitude 1 A1 (%) U(0,.5) 0.71±0.07 Semi-Amplitude 2 A2 (%) U(0,.5) 0.58±0.07 +0.4 Period P (hours) U(0, 20) 9.1−0.3 +32 Phase 1 φ1 (degrees) U(−90, 450) 95−28 +58 Phase 2 φ2 (degrees) U(−90, 450) 349−49 Constant C U(0.8, 1.2) 1.0005±0.0005 Standard deviation σ U(0, 1) 0.0092±0.0004 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.03 Bad data standard deviation σbad U(0, 4) 0.11−0.02 Bad data mean Ybad U(0.5, 3.5) 1.06±0.03 Epoch 2 [3.6] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 3.7±0.3 +0.3 Period P (hours) U(0, 20) 8.4−0.2 Phase φ (degrees) U(−90, 450) 201±9 Constant C U(0.8, 1.2) 1.008±0.002 Standard deviation σ U(0, 1) 0.036±0.001 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.08 Bad data standard deviation σbad U(0, 4) 0.42−0.06 Bad data mean Ybad U(0.5, 3.5) 1.4±0.1 Epoch 2 [4.5] double sine model parameters Semi-Amplitude 1 A1 (%) U(0, 0.5) 3.51±0.07 Semi-Amplitude 2 A2 (%) U(0, 0.5) 0.47±0.07 +0.06 Period P (hours) U(0, 20) 8.42−0.05 Phase 1 φ1 (degrees) U(−90, 450) 203±5 Phase 2 φ2 (degrees) U(−90, 450) 302±13 Constant C U(0.8, 1.2) 0.9870±0.0005 Standard deviation σ U(0, 1) 0.0094±0.0004 Bad data probability Pbad U(0, 1) 0.06±0.01 +0.04 Bad data standard deviation σbad U(0, 4) 0.24−0.03 Bad data mean Ybad U(0.5, 3.5) 1.10±0.05
aU(a, b) denotes a uniform distribution over the range a to b.
bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
83 Table A.9. WISEJ1541−22
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 0.998±0.002 Standard deviation σ U(0, 1) 0.029±0.001 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.08 Bad data standard deviation σbad U(0, 4) 0.31−0.06 Bad data mean Ybad U(0.5, 3.5) 1.33±0.09 Epoch 1 [4.5] constant model parameters Constant C U(C > 0) 0.9995±0.0005 Standard deviation σ U(0, 1) 0.0107±0.0004 Bad data probability Pbad U(0, 1) 0.05±0.01 Bad data standard deviation σbad U(0, 4) 0.11±0.02 Bad data mean Ybad U(0.5, 3.5) 1.11±0.03 Epoch 2 [3.6] constant model parameters +0.002 Constant C U(C > 0) 0.997−0.001 Standard deviation σ U(0, 1) 0.029±0.001 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.05 Bad data standard deviation σbad U(0, 4) 0.23−0.04 Bad data mean Ybad U(0.5, 3.5) 1.30±0.06 Epoch 2 [4.5] constant model parameters Constant C U(C > 0) 0.9997±0.0005 +0.0004 Standard deviation σ U(0, 1) 0.0098−0.0003 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.02 Bad data standard deviation σbad U(0, 4) 0.09−0.01 Bad data mean Ybad U(0.5, 3.5) 1.09±0.02
aU(a, b) denotes a uniform distribution over the range a to b. bThe values reported correspond to the 16th, 50th, and 84th percentiles of the marginalized posterior distribution.
84 Table A.10. WISEJ1639−68
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 1.003±0.002 Standard deviation σ U(0, 1) 0.030±0.001 Bad data probability Pbad U(0, 1) 0.02±0.01 +0.08 Bad data standard deviation σbad U(0, 4) 0.21−0.05 +0.09 Bad data mean Ybad U(0.5, 3.5) 1.21−0.08 Epoch 1 [4.5] double sine model parameters Semi-Amplitude 1 A1 (%) U(0,.5) 0.42±0.05 Semi-Amplitude 2 A2 (%) U(0,.5) 0.38±0.05 +0.09 Period P (hours) U(0, 20) 6.32−0.08 +16 Phase 1 φ1 (degrees) U(−90, 450) 53−15 +30 Phase 2 φ2 (degrees) U(−90, 450) 73−28 Constant C U(0.8, 1.2) 0.9994±0.0003 +0.0003 Standard deviation σ U(0, 1) 0.0067−0.0002 Bad data probability Pbad U(0, 1) 0.02±0.01 +0.04 Bad data standard deviation σbad U(0, 4) 0.11−0.03 Bad data mean Ybad U(0.5, 3.5) 1.07±0.04 Epoch 2 [3.6] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 1.6±0.2 Period P (hours) U(0, 20) 6.8±0.2 Phase φ (degrees) U(−90, 450) 44±11 Constant C U(0.8, 1.2) 0.999±0.001 Standard deviation σ U(0, 1) 0.0222±0.0009 Bad data probability Pbad U(0, 1) 0.07±0.01 +0.04 Bad data standard deviation σbad U(0, 4) 0.26−0.03 Bad data mean Ybad U(0.5, 3.5) 1.19±0.05 Epoch 2 [4.5] double sine model parameters Semi-Amplitude 1 A1 (%) U(0,.5) 1.52±0.05 Semi-Amplitude 2 A2 (%) U(0,.5) 0.62±0.05 Period P (hours) U(0, 20) 6.69±0.05 Phase 1 φ1 (degrees) U(−90, 450) 37±7 +14 Phase 2 φ2 (degrees) U(−90, 450) 16−15 Constant C U(0.8, 1.2) 0.9992±0.0004 Standard deviation σ U(0, 1) 0.0067±0.0003 +0.02 Bad data probability Pbad U(0, 1) 0.08−0.01 +0.02 Bad data standard deviation σbad U(0, 4) 0.12−0.01 Bad data mean Ybad U(0.5, 3.5) 1.07±0.02
aU(a, b) denotes a uniform distribution over the range a to b.
bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
85 Table A.11. WISEJ1738+27
Model Parameter Priora Valueb
Epoch 1 [3.6] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 1.7±0.3 Period P (hours) U(0, 20) 4.8±0.2 Phase φ (degrees) U(−90, 450) 42±21 Constant C U(0.8, 1.2) 0.996±0.002 Standard deviation σ U(0, 1) 0.045±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.08 Bad data standard deviation σbad U(0, 4) 0.43−0.06 Bad data mean Ybad U(0.5, 3.5) 1.5±0.1 Epoch 1 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 1.30±0.09 +0.10 Period P (hours) U(0, 20) 5.91−0.09 Phase φ (degrees) U(−90, 450) 135±18 Constant C U(0.8, 1.2) 0.9997±0.0005 Standard deviation σ U(0, 1) 0.0117±0.0005 Bad data probability Pbad U(0, 1) 0.05±0.01 Bad data standard deviation σbad U(0, 4) 0.12±0.02 Bad data mean Ybad U(0.5, 3.5) 1.12±0.03 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 0.998±0.002 Standard deviation σ U(0, 1) 0.043±0.002 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.09 Bad data standard deviation σbad U(0, 4) 0.44−0.07 Bad data mean Ybad U(0.5, 3.5) 1.4±0.1 Epoch 2 [4.5] double sine model parameters Semi-Amplitude 1 A1 (%) U(0,.5) 0.62±0.08 Semi-Amplitude 2 A2 (%) U(0,.5) 0.66±0.08 Period P (hours) U(0, 20) 6.09±0.09 Phase 1 φ1 (degrees) U(−90, 450) 216±19 +32 Phase 2 φ2 (degrees) U(−90, 450) 12−34 Constant C U(0.8, 1.2) 0.9989±0.0006 Standard deviation σ U(0, 1) 0.0107±0.0004 +0.02 Bad data probability Pbad U(0, 1) 0.08−0.01 +0.02 Bad data standard deviation σbad U(0, 4) 0.12−0.01 Bad data mean Ybad U(0.5, 3.5) 1.08±0.02
aU(a, b) denotes a uniform distribution over the range a to b.
bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
86 Table A.12. WISEJ1828+26
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 1.001±0.002 Standard deviation σ U(0, 1) 0.040±0.02 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.2 Bad data standard deviation σbad U(0, 4) 0.6−0.1 Bad data mean Ybad U(0.5, 3.5) 1.6±0.2 Epoch 1 [4.5] constant model parameters Constant C U(C > 0) 0.9992±0.0005 Standard deviation σ U(0, 1) 0.0092±0.0003 Bad data probability Pbad U(0, 1) 0.05±0.01 Bad data standard deviation σbad U(0, 4) 0.07±0.01 Bad data mean Ybad U(0.5, 3.5) 1.07±0.02 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 0.999±0.002 Standard deviation σ U(0, 1) 0.044±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 Bad data standard deviation σbad U(0, 4) 1.0±0.2 Bad data mean Ybad U(0.5, 3.5) 1.2±0.2 Epoch 2 [4.5] constant model parameters Constant C U(C > 0) 0.9996±0.0005 Standard deviation σ U(0, 1) 0.0102±0.0004 +0.02 Bad data probability Pbad U(0, 1) 0.07−0.01 +0.02 Bad data standard deviation σbad U(0, 4) 0.09−0.01 Bad data mean Ybad U(0.5, 3.5) 1.08±0.02
aU(a, b) denotes a uniform distribution over the range a to b. bThe values reported correspond to the 16th, 50th, and 84th percentiles of the marginalized posterior distribution.
87 Table A.13. WISEJ2056+14
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 0.998±0.001 Standard deviation σ U(0, 1) 0.0220±0.0009 Bad data probability Pbad U(0, 1) 0.06±0.01 Bad data standard deviation σbad U(0, 4) 0.18±0.03 +0.05 Bad data mean Ybad U(0.5, 3.5) 1.16−0.04 Epoch 1 [4.5] constant model parameters Constant C U(C > 0) 0.9997±0.0004 Standard deviation σ U(0, 1) 0.0082±0.0003 Bad data probability Pbad U(0, 1) 0.04±0.01 Bad data standard deviation σbad U(0, 4) 0.07±0.01 Bad data mean Ybad U(0.5, 3.5) 1.09±0.02 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 1.000±0.001 Standard deviation σ U(0, 1) 0.0226±0.0009 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.04 Bad data standard deviation σbad U(0, 4) 0.23−0.03 +0.06 Bad data mean Ybad U(0.5, 3.5) 1.23−0.05 Epoch 2 [4.5] constant model parameters Constant C U(C > 0) 1.0000±0.0004 +0.0004 Standard deviation σ U(0, 1) 0.0080−0.0003 +0.02 Bad data probability Pbad U(0, 1) 0.08−0.01 Bad data standard deviation σbad U(0, 4) 0.08±0.01 Bad data mean Ybad U(0.5, 3.5) 1.05±0.02
aU(a, b) denotes a uniform distribution over the range a to b. bThe values reported correspond to the 16th, 50th, and 84th percentiles of the marginalized posterior distribution.
88 Table A.14. WISEJ2220−36
Model Parameter Priora Valueb
Epoch 1 [3.6] constant model parameters Constant C U(C > 0) 1.001±0.002 Standard deviation σ U(0, 1) 0.048±0.002 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.12 Bad data standard deviation σbad U(0, 4) 0.50−0.08 Bad data mean Ybad U(0.5, 3.5) 1.2±0.1 Epoch 1 [4.5] constant model parameters Constant C U(C > 0) 0.9981±0.0007 Standard deviation σ U(0, 1) 0.0133±0.0005 Bad data probability Pbad U(0, 1) 0.07±0.01 Bad data standard deviation σbad U(0, 4) 0.12±0.02 +0.03 Bad data mean Ybad U(0.5, 3.5) 1.10−0.02 Epoch 2 [3.6] constant model parameters Constant C U(C > 0) 0.997±0.003 Standard deviation σ U(0, 1) 0.053±0.002 Bad data probability Pbad U(0, 1) 0.05±0.01 +0.07 Bad data standard deviation σbad U(0, 4) 0.34−0.06 Bad data mean Ybad U(0.5, 3.5) 1.4±0.1 Epoch 2 [4.5] sine model parameters Semi-Amplitude A (%) U(0, 0.5) 0.7±0.1 Period P (hours) U(0, 20) 3.23±0.08 Phase φ (degrees) U(−90, 450) 306±48 Constant C U(0.8, 1.2) 1.0004±0.0008 Standard deviation σ U(0, 1) 0.0158±0.0006 Bad data probability Pbad U(0, 1) 0.04±0.01 +0.04 Bad data standard deviation σbad U(0, 4) 0.18−0.03 Bad data mean Ybad U(0.5, 3.5) 1.14±0.05
aU(a, b) denotes a uniform distribution over the range a to b. bThe values reported correspond to the 16th, 50th, and 84th per- centiles of the marginalized posterior distribution.
89