A Theory of Exoplanet Transits with Light Scattering

4.5 MB·

by  Tyler D. Robinson

A Theory of Exoplanet Transits with Light Scattering
Tyler D. Robinson1,2,3
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA
95064, USA
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1 [email protected]
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2

b
ABSTRACT
e
F

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Exoplanet transit spectroscopy enables the characterization of distant worlds,
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and will yield key results for NASA’s James Webb Space Telescope. However,
]
P transit spectra models are often simplified, omitting potentially important pro-
E
cesses like refraction and multiple scattering. While the former process has seen
.
h
recent development, the effects of light multiple scattering on exoplanet tran-
p
- sit spectra has received little attention. Here, we develop a detailed theory of
o
r exoplanet transit spectroscopy that extends to the full refracting and multiple
t
s
scattering case. We explore the importance of scattering for planet-wide cloud
a
[ layers, where the relevant parameters are the slant scattering optical depth, the

2 scattering asymmetry parameter, and the angular size of the host star. The lat-
v
4 ter determines the size of the “target” for a photon that is back-mapped from
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an observer. We provide results that straightforwardly indicate the potential im-
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5 portance of multiple scattering for transit spectra. When the orbital distance is
0
smaller than 10–20 times the stellar radius, multiple scattering effects for aerosols
.
1
with asymmetry parameters larger than 0.8–0.9 can become significant. We pro-
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7 vide examples of the impacts of cloud/haze multiple scattering on transit spectra
1
: of a hot Jupiter-like exoplanet. For cases with a forward and conservatively scat-
v
i tering cloud/haze, differences due to multiple scattering effects can exceed 200
X
ppm, but shrink to zero at wavelength ranges corresponding to strong gas ab-
r
a
sorption or when the slant optical depth of the cloud exceeds several tens. We
conclude with a discussion of types of aerosols for which multiple scattering in
transit spectra may be important.
1Sagan Fellow
2University of California, Santa Cruz, Other Worlds Laboratory
3NASA Astrobiology Institute’s Virtual Planetary Laboratory
– 2 –
1. Introduction
Transit spectroscopy (Seager & Sasselov 2000; Brown 2001; Hubbard et al. 2001) is cur-
rently the leading technique for studying exoplanet atmospheric composition. Following the
discovery of the first exoplanet atmosphere (Charbonneau et al. 2002), transit observations
have enabled the characterization of a number of different exoplanets for atmospheric molec-
ular species, clouds, and/or hazes (Pont et al. 2008; Swain et al. 2008; Sing et al. 2009; Bean
et al. 2010; Fraine et al. 2014; Knutson et al. 2014b). Recent observational results have
shown that hot Jupiter transit spectra demonstrate a complex continuum from clearsky to
heavily clouded conditions (Sing et al. 2016; Stevenson 2016), and that cloudiness remains
a key factor into the regime of lower mass planets (Line et al. 2013; Kreidberg et al. 2014;
Knutson et al. 2014a).
The field of exoplanet transit spectroscopy will be revolutionized with the anticipated
launch of NASA’s James Webb Space Telescope (JWST) in 2018 (Gardner et al. 2006).
Over the course of the design five year mission for JWST, the observatory is expected to
provide in-depth observations of many tens of transiting exoplanets (Beichman et al. 2014).
Some of these observations will probe planets in the poorly understood 2–4 Earth mass
regime (Deming et al. 2009; Batalha et al. 2015). Excitingly, JWST may even be capable
of characterizing temperate Earth-sized planets (Kaltenegger & Traub 2009; Cowan et al.
2015; Barstow et al. 2016), though the ability to conduct such studies will depend largely on
how well systematic noise sources can be constrained (Greene et al. 2016).
As the quality of transit spectrum observations continues to improve, so should models
of exoplanet transits. Thus, certain processes initially thought to be of second-order impor-
tance should be revisited and possibly added to modeling tools. For example, atmospheric
refraction, which was initially shown to be unimportant for the case of certain hot Jupiters
(Hubbard et al. 2001), has recently been shown to be critically important for understand-
ing some terrestrial exoplanet spectra (Muñoz et al. 2012; Bétrémieux & Kaltenegger 2014;
Misra et al. 2014; Bétrémieux & Kaltenegger 2015) and, possibly, gas giant transit spectra
(Dalba et al. 2015; Bétrémieux 2016). Additionally, refraction can affect the shape of an
exoplanet transit lightcurve (Hui & Seager 2002; Sidis & Sari 2010).
Beyond refraction, another process that has seen little study with regards to exoplanet
transits is light multiple scattering. Hubbard et al. (2001) used a plane-parallel Monte Carlo
scattering model to determine the significance of a molecular Rayleigh scattering “glow”
around the limb of a transiting exoplanet. For the case of HD 209458b, these authors
found that the contribution of Rayleigh scattering glow to the transit lightcurve is negligible.
Similar conclusions were reached by Brown (2001), who used analytic arguments to deduce
that multiple isotropic scatterings could have only a small impact on transit depths.
– 3 –
Since the study of Hubbard et al. (2001), scattering opacity in exoplanet transits has
largely been treated as being equivalent to absorption opacity (e.g., Irwin et al. 2008; Line
etal.2013;Waldmannetal.2015). Thisequivalencecannotalwayshold. Forexample, avery
strong forward and conservatively scattering aerosol would only weakly attenuate a beam
following a straight-line path from a stellar disk, passing through an exoplanet atmosphere,
and traveling towards a distant observer. This issue was plainly demonstrated by De Kok &
Stam (2012), who used a Monte Carlo scattering model to study the transmission of a pencil
beam through a cloud or haze layer with variable optical thickness and whose scatterers had
asymmetry parameters of either 0, 0.9, or 0.98 (their Figure 3).
In the work that follows, we do not seek to outline specific definitions for “hazes” or
“clouds.” In general, though, we use the former to refer to aerosols that form aloft and
grow due to coagulation during the sedimentation process, thereby developing a distribution
that may look something like Titan’s tholin haze, which has a number density profile de-
scribed(roughly)byanexponentialwithaconstantscaleheightinTitan’supperatmosphere
(Tomasko et al. 2008). For clouds, we envision an aerosol distribution formed by lifting a gas
to its condensation point, thereby developing a distinct cloud base with an overlying cloud
deck whose thickness is controlled by mixing processes (see, e.g., Ackerman & Marley 2001).
Advances in exoplanet and brown dwarf cloud models have been reviewed recently by Mar-
ley et al. (2013). The most popular tools include the Eddysed model (Ackerman & Marley
2001), where an upward diffusion of condensible vapor balances a downward sedimentation
of aerosol particles, which has been successfully used to study both cloud and haze processes
in transiting exoplanet atmospheres (Morley et al. 2013). Additionally, the sophisticated
microphysical cloud treatments of Helling et al. (2001) continue to see development and
application to transiting exoplanets (Lee et al. 2016).
Thus, the question of the relative importance of scattering, refraction, and absorption
remains largely unexplored for a wide portion of exoplanet parameter space. Such studies
have been hindered by the lack of available simulation tools, and also by a lack of a single,
coherent theory of exoplanet transit spectroscopy. Significant advances can be made by
developing such a theory, and by building models that implement these ideas.
In this paper, we construct a theory of exoplanet transit spectroscopy that spans key
fundamental integral relations through to its efficient, vectorized implementation. Simpli-
fications of the theory cover the geometric (i.e., straight-line) limit (that is currently used
in computationally-demanding spectral retrieval models; e.g., Line et al. 2012; Benneke &
Seager 2012; Barstow et al. 2012; Lee et al. 2014; Waldmann et al. 2015; Morley et al. 2016)
and cases that add refraction. In general, though, we emphasize the “full physics” scenario,
where a three-dimensional Monte Carlo model is used to incorporate scattering effects.
– 4 –
Below, we introduce the concept of the “path distribution”, which effectively separates
the paths of photons (or rays) through a transiting exoplanet atmosphere from gaseous
and/or aerosol absorption effects. As the former tends to vary weakly in wavelength, while
the latter can vary strongly with wavelength, our approach is computationally efficient.
Following the presentation of these ideas, we validate our implementation of the theory
against a number of trusted simulation tools. We then give examples of the path distribution
for a variety of different model atmospheres. Finally, we use our tools to present the first
simulated transit spectra of hot Jupiters with multiply-scattering clouds.
2. Theory and Model Description
Transit spectra typically probe low density regions of exoplanetary atmospheres. Here,
molecular absorption lines are relatively narrow, as the effects of pressure broadening are not
dominant over Doppler broadening (as is the case in the deep atmosphere). Thus, the gas
opacity can vary by orders of magnitude over narrow spectral ranges. By contrast, gaseous
refractiveindexesaswellasgasandaerosolscatteringpropertiestendtovarysmoothly(and,
sometimes, weakly) in wavelength. Since refraction and scattering, which are more compu-
tationally intensive processes to simulate, influence the path of a photon (or ray) through an
atmosphere, significant advances can be made by outlining a technique that separates the
processes which influence photon trajectories from those which influence absorption.
We introduce the concept of the ray (or photon) atmospheric path distribution, P (h),
b
where b is the impact parameter of the ray and h is altitude in the planetary atmosphere.
We define the path distribution such that P (h)dh is the linear distance traversed by the ray
b
at altitudes between h and h+dh. While P is dimensionless, it can be thought of as having
b
units of km linear distance per km vertical distance. The geometry of these parameters, for
a simple case, is shown in Figure 1. The Appendix contains additional discussion on the use
of the path distribution.
In transit spectroscopy, the essential quantity is the attenuation of a ray along its path,
as this distinguishes the opaque portions of the atmosphere (that block light from the stellar
disk) from the transparent portions. In 1-D atmospheric structure models, which are the
variety most commonly used for exoplanets, the atmospheric opacity is only a function of
altitude (or pressure) and wavelength. Here, then, the optical depth along the path of a ray
can be computed using the path distribution via,
(cid:90) ∞
τ = α (h)P (h)dh , (1)
λ,b λ b
0
where α is the atmospheric extinction (in units of inverse distance), and the integral can
λ
– 5 –
be taken to infinity as the extinction is zero at very large altitudes. Since the layer vertical
differential optical depth is defined as dτ (h) = α (h)dh, this expression demonstrates that
λ λ
the path distribution can also be thought of as the linear optical depth encountered between
τ and τ + dτ . Similar extensions exist for quantities like the column mass and number
λ λ λ
densities. Note that Equation 1 is critical, as the strongly wavelength-dependent extinction
is separated from the ray path. Using the standard definition of transmission,
t (b) = e−τλ,b , (2)
λ
a simple transit spectrum in the pure absorbing limit (i.e., where all optical depth is taken
as absorption optical depth) can then be computed by considering the light transmitted
through concentric annuli on the planetary disk, each at their own impact parameter,
(cid:18)R (cid:19)2 2 (cid:90) ∞
p,λ
= [1−t (b)]bdb , (3)
R R2 λ
s s 0
where R is the wavelength-dependent planetary radius, R is the stellar radius, and
p,λ s
(R /R )2 is the transit depth.
p,λ s
In practice, model planetary atmospheres are defined on an altitude (or pressure) grid,
and Equation 3 is computed using a sum over a collection of impact parameters (and as-
suming that the planetary disk is opaque below some fiducial radius, R , which is either at
p
the surface or deep in the atmosphere). In this case, the path distribution is a matrix, P ,
i,j
where P ∆h is the path traversed through atmospheric layer ‘j’ (whose thickness is ∆h )
i,j j j
by a ray whose impact parameter is b . At any given wavelength, the transmission is now a
i
sum over N atmospheric layers,
lay
   
N N
lay lay
(cid:88) (cid:88)
tλ,i = EXP− αλ,jPi,j∆hj = EXP− ∆τλ,jPi,j , (4)
j=1 j=1
where we have used the definition of the layer vertical differential optical depth, ∆τ =
λ,j
α ∆h . A similar expression was used by Bétrémieux (2016, their Equations 5 and 7),
λ,j j
where these authors work in numerically-computed column number densities and molecular
opacities, as compared to our dimensionless quantities. Note that Equation 4 can be easily
written in matrix notation as,
t = 1−a = EXP(−∆τ ·P) , (5)
λ λ λ
where we have defined the absorptivity vector, a . The pure absorption transit spectrum
λ
derives from a sum over N impact parameters,
r
(cid:32) (cid:33)
(cid:18)R (cid:19)2 1 (cid:88)Nr
p,λ = R2 +2 [1−t ]b ∆b , (6)
R R2 p λ,i i i
s s i=1
– 6 –
where ∆b is the thickness of the impact parameter gridpoint. Continuing with the matrix
i
notation, if we define a vector of annulus areas as A = 2πb ∆b , then the transit spectrum
i i i
can be written simply as,
(cid:18)R (cid:19)2 1 (cid:18) 1 (cid:19)
p,λ = R2 + a ·A . (7)
R R2 p π λ
s s
We briefly note that the transit spectra expressions given in Equations 3, 6, and 7 are
in the pure absorption limit, and assume that all rays trace back to the stellar disk, and
that the star has uniform surface brightness. While these are common assumptions when
computing transit spectra for model atmospheres, we will now discuss the more general
multiple scattering case. The Appendix contains brief details about computing the path
distribution in the geometric limit or in the case that refraction is considered.
2.1. Multiple Scattering Path Distributions and Transmissions
Cases that include clouds, especially strongly forward scattering clouds, require three-
dimensional radiative transfer treatments to compute accurate transit spectra, and are well
suited to Monte Carlo models. Here, the path distributions for a number of photons, N , are
p
used to derive an average transmission for a grid of impact parameters. The individual path
distributions are determined using a Monte Carlo approach, and only consider the scattering
optical depths (τ = ω˜ τ , where ω˜ is the single-scattering albedo, and a sub-script ‘s’
λ,s 0 λ,e 0
indicates “scattering” while ‘e’ indicates “extinction”) since only τ affects the path of rays
λ,s
through the atmosphere. Given the path distribution, P , for photon ‘m’, the transmission
m
for this photon is,
t = EXP(−∆τ ·P ) , (8)
λ,m λ,a m
where ∆τ is a vector of the wavelength-dependent layer differential absorption optical
λ,a
depths, and the transmission averaged over all photons is simply,
(cid:80)Np t
t¯ = m=1 λ,m . (9)
λ
N
p
Execution of our Monte Carlo model follows a standard approach. For a given impact
parameter, b , a photon is directed into the atmosphere along the −xˆ direction in Figure 1,
i
and then tracked through the atmosphere. As this photon originated from the direction of
the observer, we are performing a so-called “backward” Monte Carlo simulation. The optical
distance the photon travels, either initially or following a scattering event, is determined by
– 7 –
randomly sampling the scattering optical depth distribution, with,
(cid:90) τs
ξ = f(τ)dτ , (10)
0
where ξ is a random number between zero and one, and f(τ)dτ is the probability that a
photon scatters between τ and τ +dτ. As f(τ)dτ = e−τdτ, we have,
τ = −ln(1−ξ) . (11)
s
For whichever layer (centered at h ) the photon is currently in, the pathlength corresponding
j
to τ is,
s
s = τ /α , (12)
s s,j
whereα isthelayerscatteringextinctioncoefficient. Givenapathlength, aphotonlocation
s,j
(r = xxˆ +yyˆ +zzˆ), and trajectory (µˆ = µ xˆ +µ yˆ +µ zˆ), the photon position is updated
x y z
according to x → x+µ s, y → y +µ s, and z → z +µ s.
x y z
Depending on the size of s, the photon either experiences a scattering event within layer
j, in which case s/∆h is added to the path distribution for this photon (i.e., P ), or the
j m
photon exits the layer before a scattering event occurs. For the latter, s must be compared
to the distance to the layer boundaries. Given the trajectory of the photon, and its current
(cid:112)
radial position, r = x2 +y2 +z2, the altitude of the photon along its path is,
(cid:113)
h(s)+R = r(s) = (x+µ s)2 +(y +µ s)2 +(z +µ s)2 . (13)
p x y z
Thus, the quadratic equations (in s),
h(s ) = h ±∆h /2 , (14)
j j j
give the distances to the layer boundaries. When a photon travels to a layer boundary
before a scattering event, s /∆h is added to P , the cumulative optical depth experienced
j j m
by the photon us updated as τ → τ + α s , and the photon is passed to the appropriate
s,j j
layer (either j−1 or j+1). This process of randomly generating τ and passing the photon
s
through sequential layers is repeated until the photon either exits the atmosphere, or reaches
the lower boundary of the model and is considered absorbed. (Note that, as with any transit
spectrum model of gaseous worlds, the lower boundary must be set deep enough to not
influence the simulated spectrum.) For photons that exit the atmosphere, the position and
trajectory at exit determine whether or not the photon will intersect the stellar disk along a
straight-line trajectory.
– 8 –
When a scattering event occurs, a scattering angle must be sampled from the scattering
phase function and the photon trajectory must then be updated. Assuming that the scat-
tering phase function, P, is only a function of the cosine of a single angle, µ = cos(θ), then
a randomly sampled scattering deflection angle is determined via,
(cid:90) µ
ξ = P(µ(cid:48))dµ(cid:48) , (15)
−1
and an azimuthal scattering angle, φ, is sampled uniformly, with,
φ = 2πξ . (16)
As the scattering angles are referenced from the propagation direction, a transformation
must be done to convert the initial propagation direction, µˆ, and the scattering angles into a
newpropagationdirection, µˆ(cid:48). Standardexpressionsexistforcompletingthistransformation
(e.g., Witt 1977, their Equations 22 and 23).
Refraction can be included in the Monte Carlo simulation. As the photon moves along
s, the path can be sub-divided, and a curvature applied to each smaller ∆s. At the photon
height, the curvature, r , is computed according to Equation C1, and a deflection angle for
c
the photon is determined from the refractive portion of Equation C4 (i.e., ∆θ = ∆s/r ).
r c
The change in trajectory is referenced from µˆ and occurs in either the local upward or
downward direction (depending on the sign of r ). A new set of direction cosines is found
c
by requiring,
µˆ ·µˆ(cid:48) = cos(∆θ ) , (17)
r
(µˆ ×rˆ)·µˆ(cid:48) = 0 , (18)
and
|µˆ(cid:48)| = 1 , (19)
wherethefirstequationimpliesthatthetrajectorychangesthrough∆θ , thesecondequation
r
forces the trajectory change to be either locally up or down, and the final equation ensures
that the new direction of travel is a unit vector.
The computational efficiency of the Monte Carlo approach to multiple scattering in
transit spectra outlined above can be increased in several ways. First, for a given impact
parameter, the Monte Carlo routine need only be executed if the straight-line scattering
optical depth is a substantial fraction of the absorption optical depth. (We choose a conser-
vative cutoff at τ < 10−3τ .) Also, as suggested by De Kok & Stam (2012), the number
λ,s λ,a
of photons used in the Monte Carlo simulation at a given impact parameter should be in-
fluenced by the straight-line transmission—these authors recommend the use of 105 photons
in transparent conditions and up to 108 photons in opaque conditions. However, De Kok &
– 9 –
Stam (2012) used a traditional Monte Carlo radiative transfer approach, where photons are
lost to absorption while passing through the atmosphere, thus driving the need for very large
numbers of photons in their simulations. As we have separated absorption from scattering in
our model, (nearly) all of the photons we simulate in our Monte Carlo exit the atmosphere,
so that we only require 104–106 photons. Even more dramatic efficiency gains can be made
by noting that, since our Monte Carlo model only considers scattering optical depths, these
simulations need only be recomputed if either the layer differential scattering optical depths
or asymmetry parameters change significantly. Thus, the individual path distribution ma-
trices for each of the photons in our simulations can be saved and reused over a wide range
of wavelengths in a high-resolution spectral grid.
Of course, running N transmission calculations (i.e., using Equation 8) at every wave-
p
length in a high-resolution grid becomes computationally unfeasible if the grid is large. Here,
runtime reductions of factors of 10–100 can be straightforwardly made by computing the an-
alytic Jacobian of Equation 9, ∂t¯ /∂∆τ , with,
λ λ,a
(cid:18) ∂t¯ (cid:19) 1 (cid:88)Np ∂t (b ) 1 (cid:88)Np
λ λ,m i
= = − P ·t (b ) . (20)
m,i,j λ,m i
∂∆τ N ∂∆τ N
λ,a i,j p m=1 λ,a,j p m=1
This Jacobian can be used to rapidly adapt t¯ to changes in ∆τ , and our tests indicate
λ λ,a
that this approach is accurate with up to 25–50% variations in level-dependent absorption
optical depths. Since a set of path distributions, transmissions, and Jacobians apply to
wavelengths where the atmospheric optical property profiles are similar, the Monte Carlo
approach outlined above is best implemented within a spectral mapping model (West et al.
1990; Meadows & Crisp 1996).
Wenotethatthemanyefficiencygainsoutlinedabovestemfromourseparatetreatments
of scattering (which influences the path distributions) and absorption (which influences the
path-derived transmissions and Jacobians). So far as we know, this treatment is a novel
approachtoMonteCarloradiativetransfer. Additionally,ourapproachtomultiplescattering
canbeappliedbeyondexoplanettransits, and, forexample, couldbeusedtostudyscattering
effects in occultation observations (i.e., an extension of De Kok & Stam 2012).
2.2. Generalized Transit Spectra
The transit spectra expressions given in Equations 3, 6, and 7 are primarily useful in
the geometric limit, since rays in refracting or scattering models are not guaranteed to trace
back to the stellar disk. To efficiently model spectra in these latter cases, we generalize the
– 10 –
concept of the vector of annulus areas, A. Key variables used in the discussion below are
visualized in Figure 2.
Ultimately, a transit spectrum is determined by comparing intensities integrated over
the range of solid angles influenced by the planet (Ω ) in the case when the planet is present
p
versus when only the star is considered (with corresponding intensities I and I , respec-
p,λ s,λ
tively). Thus,
(cid:82) (cid:82)
(cid:18)Rp,λ(cid:19)2 = ΩpIs,λdΩ− ΩpIp,λdΩ , (21)
(cid:82)
R I dΩ
s Ωs s,λ
where Ω is the range of solid angles corresponding to the stellar disk. Given the large
s
distance D between Earth and any exoplanetary system, so that dΩ = dA/D2, and taking
the stellar intensity to be normalized such that,
(cid:82)
I dΩ
I = Ωs s,λ , (22)
0,λ
Ω
s
then,
(cid:82) (cid:82)
(cid:18)Rp,λ(cid:19)2 = ApIs,λ/I0,λdA− ApIp,λ/I0,λdA . (23)
R πR2
s s
The first integral in the numerator of Equation 23 is simply a stellar limb darkening
law integrated over the portion of the planetary disk that overlaps the stellar disk. Using a
polar coordinate system centered on the planetary disk, and letting d be the distance from
the center of the star to the center of the planet, this integral is then,
(cid:40)
(cid:90) I 0 , d ≥ R +R +h
s,λ s p t
dA = (24)
I (cid:82)Rp+ht(cid:82)θ0(r) Is,λ(µ )rdθdr , d < R +R +h ,
Ap 0,λ r0 −θ0(r) I0,λ s s p t
where µ = µ (r,θ) is the cosine of the angle of incidence on the stellar disk at coordinates
s s
r and θ, and
(cid:40)
0 , d ≤ R
s
r = (25)
0
d−R , d > R ,
s s
(cid:40)
π , d ≤ R −R −h
s p t
θ (r) = (26)
0 d2+r2−Rs2 , d > R −R −h .
2dr s p t
However, as limb darkening is accounted for in standard transit observation data reduction
procedures (e.g., Mandel & Agol 2002; Kreidberg 2015), one may wish to omit the limb
darkening law from the first integral in the numerator of Equation 23, leaving an analytic
integral that can be performed by considering the geometry of two overlapping disks.