MASSACHUSETTS INSTITUTE
OF EECHNOLOLGY
JUN 08 2015
LIBRARIES
A Discrete Forward-Modeling Method for Characterizing
Occultation Lightcurves of Tenuous Planetary Atmospheres
by
Ho Chit Siu
S.B., Aerospace Engineering with Information Technology
Massachusetts Institute of Technology, 2014
Submitted to the
Department of Earth, Atmospheric, and Planetary Sciences
in Partial Fulfillment of the Requirements for the Degree of
Master of Science
at the
Massachusetts Institute of Technology
June 2015
@Massachusetts Institute of Technology 2015.
All rights reserved.
Signature redacted
Author
_
Ho Chit Siu
Department of Earth, Atmospheric, and Planetary Sciences
Certified by
Signature redacted
May 8, 2015
--
Signature red acted
Professor Richard P. Binzel
Thesis Supervisor
Certified by
Dr. Amanda S. Bosh
Certified by
Signature redacted
Signature redacted
Thesis Supervisor
Dr. Michael J. Person
Thesis Supervisor
Accepted by
Robert D. van der Hilst
Schlumberger Professor of Earth Sciences
Head, Department of Earth, Atmospheric, and Planetary Sciences
2
Contents
Abstract
5
I
9
Introduction
1
The Stellar Occultation Method
9
2
Tenuous Planetary Atmospheres
11
3
Project Overview
15
II
Modeling Approach
18
1 Overview
18
2
Model Assumptions
18
3
Constructing an Atmosphere Model
21
4
Generating Lightcurves
25
III
Model Testing
39
1
Clear Atmospheres
39
2
Atmospheres with Haze
42
3 Potential Solutions to the Model Discrepancy
45
4
46
IV
Current Model Capabilities
Lightcurve Characterization
50
1
Goals and Testing Approach
50
2
Half-Light Radius
53
3 Minimum Normalized Flux
57
4 Slope at Half-Light
59
5
61
Application to Lightcurve Analysis
6 Full Pairwise Comparison of Lightcurves
7
Haze Effects on Near-Minimum-Flux Lightcurve Features
3
80
83
V
Discussion and Future Work
87
1
Implications for Lightcurve and Uncertainty Characterization
87
2
Future Work
88
Appendix: Coordinate Conventions
92
A The Source Plane
93
B The Planet Plane
94
C The Shadow Plane
94
Appendix: Software Documentation
96
A Introduction
96
B Quick Start Guide for Fitting
96
C Atmosphere Generation
97
102
D Lightcurve Generation
108
References
4
A Discrete Forward-Modeling Method for Characterizing
Occultations Lightcurves of Tenuous Planetary Atmospheres
by
Ho Chit Siu
S.B., Aerospace Engineering with Information Technology,
Massachusetts Institute of Technology, 2014
Submitted to the
Department of Earth, Atmospheric, and Planetary Sciences
in Partial Fulfillment of the Requirements for the Degree of
Master of Science at the Massachusetts Institute of Technology
Abstract
We present a discrete numerical approach for forward-modeling lightcurves
from stellar occultations by planetary atmospheres. Our discrete approach provides a way to arbitrarily set atmospheric properties at any radius from the
occulting body, giving it flexibility for applying models of vertical variation in
atmospheric conditions. The method is used to examine trends in lightcurve
characteristics resulting from changes in the atmosphere of the occulting body.
We find that for Pluto-like atmospheres, temperature and pressure variations
affect the characteristics of the lightcurve much more than the gas composition.
We also find that the half-light radius is more sensitive to atmospheric changes
than either the minimum normalized flux or the slope at half-light. Temperature is found to be the most easily-constrained atmospheric parameter, as the
gradients for changes in lightcurve characteristics are much more aligned with
the temperature axis of the atmospheric parameter space than any other axis.
Trends in lightcurve characteristics were examined in and around the parameter space occupied by the atmospheric conditions predicted for Pluto based
on the 2011 and 2013 occultation events. Our error analysis method produced
uncertainty values consistent with the reported uncertainties for half-light radius. This kind of lightcurve characterization is potentially useful for constraining the level of precision required in measuring given lightcurve characteristics
in order to provide certain uncertainty bounds on the atmospheric conditions
of the occulting body.
Thesis Supervisors: Richard P. Binzel, Amanda
5
S. Bosh, and Michael J. Person
6
Acknowledgments
I would like to acknowledge the support of my research supervisors, Amanda
Bosh and Michael Person. They have been an invaluable help throughout the course
of this work, from conception to final edits. They have given so generously of their
time and energy for this project, that I sincerely doubt that there are very many
other graduate students apart from myself at this institute or elsewhere who have
seen this kind of support for their work.
Thanks also go in particular to Amanda, who has had to put up with me as
a TA in three classes during the course of less than a year, sometimes while I am
in questionable degrees of consciousness, and often when it is far too cold for most
reasonable people to be standing outside staring up at night.
7
8
Part I
Introduction
1
The Stellar Occultation Method
A stellar occultation is an event in which light from a star is blocked by a body
coming between the star and an observer. When the positions and movements of
both the occulted star and the occulting body are known to high precision, such
events can be predicted ahead of time, allowing them to be recorded and analyzed.
As the body passes in front of the star, the flux received by an observer changes
when the star is partially or completely blocked by the body, and also when the light
is refracted or attenuated by any atmosphere that the body may have, resulting in a
non-flat lightcurve. The character of lightcurves recorded during occultation events
may be used to understand objects in the solar system, such as planets, moons,
asteroids, and rings.
The character of stellar occultation lightcurves differs depending on whether or
not the intervening body has an atmosphere.
For bodies without atmospheres,
lightcurves show sharp drops from full stellar flux to minimum flux, and sharp rises
back to full flux, as the solid mass of the intervening body occults the starlight. In
these cases, the angular size of the star and the edge diffraction phenomenon are controlling factors that determine the characteristics of the lightcurve drop, while the
size and velocity of the occulting body, and its distance from the observer, control
the length of the minimum-flux portion. However, when the occulting body has an
atmosphere, the structure of the lightcurve becomes more complex. In these cases,
atmospheric refraction, attenuation, and focusing will shape the lightcurve, lowering
the magnitude of the slope of the drop from full flux, and causing additional features
9
such as non-constant slope in the drop, and nonzero minimum flux for the entire
event, which obscures the surface of the body. Stellar occultations thus act as a
powerful way to study extraterrestrial atmospheres from Earth (Baum and Code,
1953, Elliot and Young, 1992).
Modeling of the lightcurve-generation process allows observed lightcurves to be
used as part of an inverse problem that returns constraints on the atmospheric properties of an occulting body. This procedure is made possible by the fact that atmospheric characteristics such as temperature, pressure, and composition determine the
amount of refraction that occurs, while the presence of aerosols or nontransparent
gases adds an additional attenuation effect (Eshleman, 1989, Elliot and Young, 1992).
Models of the behavior of light as it travels through the extraterrestrial atmosphere
use these refraction and attenuation effects in order to generate synthetic lightcurves
to be compared to collected data.
Previous atmospheric models used for occultation lightcurve modeling have generally relied on continuous nonlinear functions to describe the change in observed
flux over the course of an occultation event. Baum and Code (1953) were able to
relate the scale height and geometry of the occultation of a Arietis by Jupiter to calculate the mean molecular weight and temperature of the isothermal top section of
Jupiter's atmosphere, where the dominant cause of the flux decrease was differential
refraction. Elliot and Young (1989, 1992) extended this model to include possible
extinction effects, and relaxed previous assumptions tional force in the atmosphere -
such as a constant gravita-
in order to apply it to small rocky bodies like Pluto
(the Elliot and Young extension to the Baum and Code model is hereafter referred
to as the EY92 model). Under some assumptions for the structure of the occulting
atmosphere, these models allow the scale heights of various atmospheric parameters
to be found from the light curves. In the case of Pluto's atmosphere, modeling using these functional forms has been able to constrain the properties of the upper
10
atmosphere well.
2
Tenuous Planetary Atmospheres
Tenuous planetary atmospheres are a class of extremely low density atmospheres
that are found on some small solar system bodies.
These types of atmospheres
may be found on Mercury, Pluto, and several of the gas giant moons, and may be
formed in a variety of ways. For the cases of Mercury and the Moon, they arise
primarily from capture of solar wind particles, and from photosputtering, a process
by which energetic background radiation causes the release of chemical species from
the planetary surface (Sanchez-Lavega, 2011). The Galilean moons experience ion
bombardment due to their proximity to Jupiter's magnetic field, in addition to solar
effects, and, in the case of Io, volcanism from its own surface. Like Io, the Saturnian
moon Enceladus generates an atmosphere primarily from surface processes, though in
the latter case, it is from gas and water-ice plumes on the surface rather than from
volcanic activity, leading to very different atmospheric compositions.
Unlike the
previously-mentioned cases, Triton and Pluto present very similar atmospheres that
are dominated by sublimation and condensation processes caused by solar radiation
that affects their surface ices. The tenuous atmospheres of small planetary bodies
are somewhat similar to the extreme upper atmospheres of gas giants, which is why
the EY92 model for Pluto's atmosphere was a natural extension of the Baum and
Code model for Jupiter's upper atmosphere.
Here, we will focus our attention on the conditions present in atmospheres similar
to those of Triton and Pluto Pluto-like conditions -
with particular attention paid to atmospheres with
and the lightcurves that result from stellar occultation events
involving these bodies. Pluto and Triton are two small planetary bodies that are
widely regarded as being solar system "twins." They have similar masses and mean
11
densities, and for much of the time since their discovery, have been at about the
same distance from the Sun, at approximately 30 AU. Since our knowledge of Triton
is supported by the 1989 flyby of the moon by Voyager 2, Triton has been used as a
model for our study of Pluto, which has yet to be visited by a spacecraft.
The similarities between Pluto and Triton extend to their atmospheric conditions.
Not only are they both sublimation-dominated atmospheres, but the composition of
their atmospheres is also similar. Both being primarily nitrogen-based, with a small
amount of methane and carbon monoxide (Cruikshank et al., 1993, Owen et al.,
1993). The sublimation of surface ices to form these types of atmospheres lead to
a sublimation source location that follows the subsolar point, and generates a wind
that causes mass transport around the planet. This set of processes tends to create
isothermal conditions for much of the atmosphere (Sanchez-Lavega, 2011). Models
of the isothermal upper regions of these atmospheres have led to similar results
regarding their temperatures, both of which hover at around 110 K (Strobel et al.,
1996). In relating the thermal structure to the composition, Yelle and Lunine (1989)
found that a range of mixing ratios for N2 and CH4 could lead to such temperatures,
requiring only a small amount of CH4 to maintain them due to the heating effect of
solar absorption in the 3.3 pm band for CH4 (the "methane thermostat" model).
The first unambiguous detection of Pluto's atmosphere was made in 1988 using
the stellar occultation method (Elliot et al., 1989).
For bodies at the distance of
Pluto, this method allows atmospheric properties to be recovered at spatial resolutions down to a few kilometers, making it a particularly suitable tool for studying
the atmospheres of both Pluto and Triton. Occultations have revealed the existence
of two distinct sections of Pluto's atmosphere with different optical properties, since
the occultation lightcurve shows a break in the slope near the minimum, referred
to in the literature variously as the "break," the "bend," the "knee," and the "bite"
(Elliot et al., 1989, Eshleman, 1989, Hubbard et al., 1990). Such a change in slope is
12
consistent with a lower-atmosphere thermal gradient, where the temperature changes
from around 110 K in the isothermal region to less than 50 K near the surface (Eshleman, 1989, Hubbard et al., 1990). Alternatively, it may also be interpreted as a
lower-atmosphere haze layer, which has been observed on Triton, likely arising from
photolysis of hydrocarbons and nitriles (Elliot et al., 1989, Sanchez-Lavega, 2011).
Similarly dark plumes observed on Triton are believed to be composed of organic
polymers from methane reactions, or primordial hydrocarbons trapped during the
moon's accretion (Soderblom et al., 1990). If it is indeed a haze layer on Pluto that
is causing the bend in the lightcurve, then it is expected to be composed of tholins
-
nitrogen-rich organic polymers that are also found on Triton and Titan -
due to
the similar atmospheric compositions and pressures found in these atmospheres, as
well as the UV irradiation levels expected at Pluto distances (Khare et al., 1984).
The possibility of haze was explored by Stansberry et al. (1989), who modeled the
production, growth, sedimentation, and condensation of aerosol particles with varying optical properties on Pluto. The photochemical modeling showed that in order
for the kinds of optical depths that the Elliot et al. (1989) lightcurve fit requires, the
production of haze had to be twice the rate that was expected for Pluto, and the haze
had to be ten times as attenuating as other outer-solar-system hazes. In particular,
Stansberry et al. note that it is unclear whether their photochemical haze model
could be made to match the haze structure required by the Elliot et al. atmospheric
model.
It is thus beneficial to have an atmospheric model where the structure of
a haze layer does not have to follow the functional constraints of the EY92 model.
The EY92 model only works for a single exponential form for vertical variation of all
atmospheric parameters, including haze, temperature, and atmospheric composition.
In this work our atmosphere model may be arbitrarily specified for any part of the
atmosphere, so we do not have to follow any particular functional form for vertical
variation.
13
One way to resolve the ambiguity regarding the cause of the slope change in Pluto
occultation lightcurves is by comparing simultaneous lightcurves taken in different
wavelengths. The wavelength-dependent nature of haze-driven extinction makes it so
that if the bend was caused by haze, there would be a difference in the region of the
lightcurve under the bend when viewed in different wavelengths, since wavelength
must be shorter than the size of a particle for the particle to scatter the light.
However, if the bend was driven by a temperature gradient, the flux would not be
wavelength-dependent, and normalized lightcurves of different wavelengths would
not show a difference.
A set of such observations was conducted during Pluto's
occultation of the star P131.1 in August of 2002. Data taken during this event did
reveal a flux increase with wavelength around the lightcurve minimum, suggesting
that attenuation by haze, rather than by a thermal gradient, is more likely (Elliot
et al., 2003a). However, in this set of lightcurves, the bend seen in the 1988 event
was notably absent, complicating the conclusions that could be drawn from them.
Through the course of the study of Pluto's atmosphere in the last two and a half
decades, changes in the observed lightcurves have led to the conclusion that there
have been notable changes in the Plutonian atmosphere. In addition to erosion of
the bend in the lightcurves over time, changes in the overall size and pressure of the
atmosphere are also likely to have occurred (Elliot et al., 2007).
The overall size of the atmosphere is one of the factors that has been observed
to be changing.
A frost migration model developed by Hansen and Paige (1996)
predicted an increase in pressure (corresponding to an increase in the size of the
atmosphere, as measured by the half-light radius on the shadow plane) from the first
detection in 1988, to a maximum pressure occurring sometime between 2000 and
2010, followed by a drop. Occultation data collected since the model was published
have returned results that are generally consistent with it (Bosh et al., 2015).
A final interesting point to note about Pluto's occultation events is that the
14
lightcurves recovered from such events never show normalized flux values of zero
(normalized flux subtracts out the light of Pluto itself). The shows that its atmosphere is at least substantial enough so that the refraction effects from it cause some
light from an occulted star to be seen, even in the center of the shadow when observed from Earth. This effect, however, does make it so that the radius of the planet
cannot be precisely determined from occultation events, unlike the case for airless
bodies. Still, the timing of the start of ingress and the end of egress during the event
allows an upper bound to be calculated for Pluto's actual size, a radius found to be
1152
3
32 km (Elliot et al., 2003b, Zalucha and Gulbis, 2012).
Project Overview
In this project, a discrete numerical method for forward-modeling planetary atmospheres was developed and used for solving the inverse problem of retrieving atmospheric characteristics from occultation lightcurves. Instead of calculating observercollected flux from continuous analytical representations of the atmosphere, as previous models have done, this approach simulates occultation events through ray tracing
using discrete representations of layers of the atmosphere. Synthetic lightcurves from
occultations are generated by modeling the behavior of light as it passes through an
atmosphere with these discrete layers (referred to in this work as "shells"). These
shells surround the occulting body, and each one has associated physical properties that determine how much refraction and absorption occurs when beams of light
interact with it.
In contrast to the use of continuous analytical expressions for flux, this discrete
numerical approach allows the properties of different parts of the atmosphere to be
set independently, giving a more flexible definition of the vertical structure for fitting
purposes. Furthermore, discretizing both the atmosphere and the light subsumes the
15
process of joining multiple analytical expressions together to account for changes in
lightcurve slope, as each beam of light, all of the atmosphere shells it encounters, and
the resulting changes in direction and magnitude may be independently accounted
for. This feature also allows abrupt atmospheric changes of haze layers or temperature inversions -
such as the appearance
to be specified as part of the vertical
structure of the atmosphere. If very different atmosphere models result in similar
lightcurves, the simulation will be able to show that non-uniqueness, and more generally, be able to probe the sensitivity of lightcurve features to different changes in
the atmosphere. The atmospheric modeling method developed here may in theory be
applied to lightcurves of any occulting body, ranging from small bodies with tenuous
atmospheres, to gas giants.
Although the method developed here is generally applicable to any occulting
body, stellar occultation by small bodies with Pluto-like atmospheres was the focus
of the testing in this project.
Characterizing trends in the relationship between
atmospheric properties and the resulting occultation lightcurve properties of such
bodies is particularly appropriate now because of the imminent arrival of the New
Horizons spacecraft at the Pluto-Charon system in July of 2015. New Horizons will
be able to provide close-up sensing of the Pluto-Charon system that has never been
available before. In particular, the Alice and REX instruments on New Horizons
will be able to directly study the composition and structure of Pluto's atmosphere
through spectroscopic and radio methods respectively, providing extensive new data
that will significantly improve our understanding of the atmosphere (Young et al.,
2008).
A new method for modeling Pluto-like atmospheres is also well-timed for
ground-based observers, since Pluto is now passing through the galactic plane, which
makes stellar occultations much more frequent now than they were in the last two
decades.
Following this introduction, this thesis is divided into four additional parts. Part
16
II describes the forward-modeling component of the project, introducing the physical
basis for the model that allows synthetic lightcurves to be generated from arbitrarilyspecified occulting-body atmospheres. Part III goes over the testing process for the
model against the established EY92 analytical model. Parts IV and V detail the
application of this model to examining trends in the characteristics of occultation
lightcurves across a Pluto-like region of the atmospheric parameter space, and the
implications for calculating uncertainty requirements for stellar occultations.
Two appendices follow at the end.
The first is a reference for the coordinate
systems used throughout this thesis, and the other gives a more detailed description
of the software developed for this project.
17
Part II
Modeling Approach
1
Overview
The extraction of information about an atmosphere from the properties of occultation lightcurves is posed as an inverse problem. The approach used in this project
embeds a forward model of the atmosphere within the inverse problem by generating
synthetic lightcurve data from an atmosphere model using optical principles to calculate changes in light magnitude and direction as it passes through the atmosphere.
This synthetic lightcurve is then compared to the observed lightcurve, thus tying the
observed data to the parameters of the model atmosphere. The fit of the synthetic
and actual lightcurves is then used to inform successive iterations of atmosphere parameters that are used to generate new synthetic lightcurves, in order to find the best
model parameters to solve the inverse problem. A schematic of the entire process
may be seen in Figure 1. The discrete method used here may be contrasted with the
the continuous analytical atmosphere model developed by Elliot and Young (1992).
2
Model Assumptions
Models of planetary atmospheres used to fit stellar occultation data must make
certain assumptions about the planet, its atmosphere, and the light passing through
the atmosphere. Here, the assumptions inherent in this model are described, along
with some assumptions that are not made that should be noted in particular (in
contrast with previous models). The assumptions held by this model are as follows:
1. The occulted star may be regarded as a point source.
18
forward modeling to generate lightcurve
parameters
Figure 1: A forward-modeling component is embedded in the overall inverse problem in order to extract atmosphere parameters from observed lightcurve data. The
synthetic hightcurve resulting from the forward model is compared to the data and
used to adjust model parameters.
2. The atmosphere is spherically symmetric -
that is, atmospheric properties
only change as a function of altitude, rather than latitude or longitude.
3. The ideal gas law holds.
4. The atmosphere is in a state of hydrostatic equilibrium.
a. Gravitational lensing effects are negligible.
The point source assumption is made due to the relative angular sizes of the occulted star and the occulting body, where the star is typically much smaller than
the occulting body. Additionally, the integration time required for Pluto and Triton occultation events is long enough so that the binning of each flux data point
is generally unable to resolve the non-point-source nature of the star. If necessary,
any non-zero-size effect from the star or integration effect from the observer may
be accounted for in post-processing of the lightcurves. This can be done by using a
weighted moving average that treats the starlight not as individual beams of light
19
but as beam bundles, weighted by the brightness distribution of the star due to limb
darkening, as in the star brightness model given by Equation 8 of Elliot et al. (1984).
The spherical symmetry assumption is made in order to simplify the atmosphere
model, but is generally shown to be untrue due to the asymmetric nature of occultation lightcurves. However, there is enough symmetry in most lightcurves to use this
assumption as a starting point for forward modeling. Later extensions to the model
could dispense with this assumption by discretizing Pluto's atmosphere azimuthally
as well as radially, in order to account for asphericity in the atmosphere that leads
to asymmetric lightcurves.
The ideal gas assumption holds in the case of Pluto-like atmospheres despite
the low temperatures due to the extremely low pressures (typically on the order of
microbars or less), which extends the mean free path of gas particles to be much
larger than molecular sizes. Hydrostatic equilibrium is assumed, since changes in the
atmosphere due to any force imbalance would not be observable in the time span
of occultation events, which are typically on the order of minutes. Any significant
disturbances caused by a violation of the equilibrium would also likely be averaged
out across the data taken during an occultation and not resolvable in any case.
The assumption about gravitational lensing is made along the same lines as it
was in the EY92 model, since planetary masses are very small relative to the much
larger bodies that usually require such a consideration (generally black holes or entire
galaxies).
Our forward model accounts for refraction, absorption, and focusing.
Even
though refraction is generally the greatest atmospheric contributor to flux changes
in occultations for most of the duration of the lightcurve, absorption is specifically
included because attenuation by a lower-atmosphere haze layer is one of the candidate causes of the lightcurve bend in occultations by Pluto, and can strongly affect
features of the lightcurve near their minimum flux levels (Elliot et al., 1989, Stans-
20
berry et al., 1989). Additionally, in contrast to the method used by Baum and Code
(1953), local gravity is not assumed to be constant, making it possible to handle the
cases of small bodies, where significant changes in the local gravity value may be on
the same order as atmospheric scale heights.
3
Constructing an Atmosphere Model
3.1
Physical Basis for the Model
A model of a Pluto-like atmosphere was constructed by using discrete concentric
shells with associated physical properties. Spherical symmetry was assumed for the
atmosphere such that each shell had one set of constant properties that represented
the atmospheric properties at the altitude of the shell, which means that while radial
variation in the atmosphere is accounted for (giving vertical atmospheric structure),
azimuthal variation is not. Light ray interaction with shells resulted in two effects
-
refraction and absorption -
which were determined by the refractive index and
the attenuation coefficient, respectively. The model treats light as discrete vectors
coming from behind the occulting body towards the observer, and tracks the interaction of these vectors with the shells as they pass through different layers of the
atmosphere.
Refraction changes the direction of the rays (which also affect later
interactions with other shells), while absorption decreases the magnitude of the rays.
The index of refraction of each shell was determined by various atmospheric
properties assigned to the shell, as specified by the Lorentz-Lorenz equation. The
general form of the equation is given by
n2 - 1
47r
n2 +2
3
=
--
Na(1)
where n is the refractive index, N is the number density, and a is the mean
21
polarizability of the medium (the atmosphere shell). The number density N is found
by
N
NAP
RT'
(2)
where NA is Avagadro's Number, p is the pressure, R is the universal gas constant,
and T is the temperature. Taken together, Equations 1 and 2, give the desired form
of
n2 _ 1
n2+2
4r NAp
3 RT
(3)
This form leaves the composition, pressure, and temperature of any given shell as
free parameters (though pressure and temperature are not necessarily independent).
It should be noted that refraction may be expressed as n, the refractive index, or as
v, the refractivity, which is defined as v = 1 - n. Since there is no direct functional
solution for n in Equation 3, it must be solved for numerically when all other values
have been supplied.
Upon initial inspection, the simplified equation for refraction gives three free parameters per shell, which leaves the number of refraction parameters to be on the
order of the number of shells. In order to reduce the number of independent atmospheric parameters, the atmosphere may be broken down into particular regions
(referred to in this paper as "sections") where certain effects are dominant or are
absent, such as the presence or absence of temperature gradients, haze, or differentiated composition, or where variation of properties follows a certain functional form.
For example, an isothermal upper atmosphere might be assumed for Pluto-like atmospheres, using the result of Elliot et al. (1989) as a starting point. This assumption
removes the independence of the temperature parameter for all the shells in that region and reduces them to one variable. Applying a further assumption of hydrostatic
22
equilibrium makes the pressure a function of the constant reference pressure (defined
in this model as a radius of 1275 km for ease of comparison with existing literature,
though it may be adjusted as necessary), and local gravity, reducing the number
of independent pressure variables to one. In the case of small bodies, local gravity
cannot be assumed to be constant due to the relatively large size of the atmosphere
compared to the planet.
In the isothermal case, the pressure may be constrained by the following relation,
M~gh
p = po exp(- kT
(4)
'
where po is the pressure at the reference radius, Mp is the mass of the planet,
g is the height-variable local gravity force, h is the height above the surface (shell
location, r - rsurface), k is the Boltzmann constant, and T is the temperature.
However, since constant temperature is not a hard assumption for this model,
the pressure must be solved more generally. Thus, for each atmosphere shell, the
pressure at the shell radius r is found by numerically solving the boundary-value
problem specified by
p()=M(r)p(r) GM
p'(r) =
T(r)R
r
r2I
where p(ro)
=
Po,
(5)
where M(r), p(r), and T(r) are the radius-dependent mean molecular weight,
pressure, and temperature, and po is the pressure at a reference radius of ro.
In this way, separating the atmosphere into independently-specified sections of
shells significantly reduces the number of free parameters, which reduces the leastsquares minimization time for fitting atmosphere properties to the model correspondingly. A second effect of this division to continuous regions is that it decreases the
likelihood of overfitting specific shell parameters by smoothing out the relationships
between the properties of adjacent parts of the atmosphere.
23
In addition to the refraction, absorption is modeled by the Beer-Lambert Law,
which is expressed by
I(s) = Io exp(-
f
a(s')ds),
(6)
0
where Jo is the initial flux, a. is the attenuation coefficient, and s is the distance
traveled in the attenuating material. A simplification is introduced by assuming that
the attenuation coefficient is uniform across s in each shell (where s is the distance
in the direction of movement), which reduces the formula to
I(s) = Io exp(-ass).
(7)
We use a, to denote the attenuation coefficient, rather than just a, in order to
differentiate it from the a that we use above (Equations 1 and 3), for the mean
polarizability of the medium.
3.2
Atmosphere Construction by Section
As mentioned above, the specification of an atmosphere may be done in sections of
shells, rather than shell-by-shell. In order to implement this simplification, the code
that constructs atmosphere models was written in such a way so that the atmosphere
is specified by the boundary conditions of each section, and the kind of variation that
exists for different conditions within a given section. In this context, sections refer
to collections of one or more shells where temperature, attenuation coefficient, mean
molecular mass, and mean polarizability values are constant, or follow a linearlyvarying, or exponentially-varying form from the bottom to the top of the section.
The conditions for the top of one section are used as the boundary conditions
for the bottom of the section on top of it, ensuring continuity. Thus, an atmosphere
may be fully specified by giving the atmospheric parameters at the boundaries of
24
each section, as well as the locations of these sections, and the number of shells in
each section. See the Appendix and the corresponding Mathematica notebook for
full details on the software for atmosphere construction.
4
4.1
Generating Lightcurves
Ray Tracing and Interpolation for Lightcurves
Lightcurves are generated by simulating occultation events using ray tracing as beams
of light move from an infinite distance at the source star, through the atmosphere
of the occulting body, to the observer.
Light from the occulted star is assumed
to come in from a "source line" and move perpendicularly to the direction of the
relative star motion (along the source line) behind the occulting body, with all rays
starting parallel to each other, as shown in Figure 3. The problem is reduced to
a two-dimensional movement of light though what is essentially a cylindrical model
atmosphere, similar to what was assumed by Baum and Code (1953).
Each interaction with a shell (or lack thereof) is determined by the the intersection
of the parameterized equation of the ray path and the next circle representing a
shell. The ray then passes from shell to shell until it either reaches the other side
of the atmosphere or is entirely extinguished by an encounter with the surface of
the occulting body, which is simply defined to be the minimum-radius shell specified
in the model. Once all the beams have reached the shadow plane (the observer's
location), the focusing factor is calculated, which transforms the treatment of the
atmosphere from the cylindrical case to the full spherical case using the method
specified in the EY92 model (explained later).
As each light ray passes through shells, its direction is changed according to
Snell's law, which is given by
25
and changes.
Figure 2: Process diagram for calculating ray movement
sinG 1
sinG 2
_
T2
=
,
(8)
-i
of the shell, and the
where 0 is the entry/exit angle with respect to the normal
where the ray is exiting
two i values are the indices of refraction for the regions
the attenuation coefficient
from and entering into. Next, magnitude is changed by
associated with that shell according to Equation 7.
discrete disIn our model, shells are infinitesimally thin, and located at specific
to define the outer boundtances from the center of the planet, so shells are assumed
those of the shell for
aries of a layer of atmosphere with constant properties equal to
to be the surface of the
that space, except for the innermost shell, which is assumed
and does not reach
planet. Any beam that hits the innermost shell is extinguished,
movement and interaction
the shadow plane. The overall processing structure for ray
is shown in Figure 2.
model atmosphere,
To generate the lightcurve that would result from a particular
ray's ending position at the
the vector magnitudes and angles as a function of the
at the shadow plane. The
shadow plane are interpolated from the vectors that arrive
26
received light at the observer's position is then the flux that is incident to the shadow
plane, given by the magnitude of the ray multiplied by the cosine of the ray angle.
Unfortunately, this particular formulation fails in practice due to the extremely small
refraction angles, and the fact that -
to machine precision at least - the cosine of
these angles is simply equal to one. Thus, instead of using the angle, a measure of
beam density is used as a proxy instead, which is calculated by
dbeam = Sseurce
Sshadow
where Ssource and
shado,
(9)
are the distances between adjacent beams of light on the
source and shadow planes respectively. As beams are bent towards the center of the
planet, beam density on the shadow plane will generally be lower than the density
on the source plane during the occultation section by an amount proportional to how
occulted the starlight is. This makes the beam density an appropriate proxy for the
refraction term. The magnitude of individual beam vectors is set to 1.0 before any
atmosphere interactions, and is less than or equal to 1.0 upon reaching the shadow
plane. The beam density and the beam magnitude are interpolated independently
on the shadow plane to give continuous functions of both that would allow incident
flux to be found anywhere on the shadow plane.
Finally, the flux is modified by a focusing term.
The model so far has been
developed as a cylindrical scenario, where there is no curvature in the direction
perpendicular to initial ray direction (into and out of the diagram for Figure 3).
However, in the spherical case, the curvature in this perpendicular direction actually
enhances the received light in such a way that it is multiplied by a focusing term
expressed by
f
rsource
p
27
(10)
as given in Elliot et al. (1989). Here, rsorce is the distance from the center of
the parallel projection of the planet onto the source plane to the starting position
of a beam of light, and p is the analogous value for the same beam of light on the
shadow plane (the two values are not the same due to refraction, so p < rsorce for
any particular beam of light; see Figure 34 in the Appendix).
In combination, the three parameters give the incident flux on the shadow plane
as
<
=
dbeaml'f',
(11)
where I is the flux (beam magnitude of an individual beam, as in Equation 7),
and the primes denote the interpolated values for all parameters.
For generating a lightcurve, beams only need to be sent in to generate half the
curve, since symmetry allows the other half to be obtained by flipping the flux values
over the position/time axis. In the case of Pluto, refraction is significant enough such
that light from one limb crosses the center line from the planet to the shadow plane,
so the flipped flux values will overlap. In the region where the fluxes overlap, the light
from the two limbs is simply added. Refraction effects always bend light towards the
planet, so all beams starting at planet-plane impact parameters between the planet
surface radius and r = 0 will hit the planet and not reach the shadow plane, making
it unnecessary to generate these beams at all.
Since reflection is not considered in this model (reflection due to starlight is
negligible), beam generation ends when the first beam hits the planet surface. This
end condition significantly reduces the processing time required by the simulation,
since processing time is mostly taken up low-impact-parameter beams, as they are
the ones that have the most shell interactions.
28
0
(.1
I
77
I
,/
7, /~/
0
U)-
0
U)
'N
\
'N
<'N.
planet plane
Figure 3: Schematic ray diagram for forward modeling, where 3 is the total refracted
angle from the horizontal after all shell interactions. Interpolated beam densities and
magnitudes at the shadow plane may be flipped over the central axis and added to
the original results to obtain the full lightcurve. Note that although previous models
approximated all of the refraction as occurring on the planet plane, our model has
refraction occurring on a shell-by-shell basis. Because of this, refraction occurs before
a beam of light hits the planet plane, so the planet plane is only included as a reference
for previous models (Baum and Code, 1953, Elliot and Young, 1992).
4.2
Discretization and Model Resolution Considerations
The discrete nature of our model of the atmosphere presents a number of challenges that are absent in models that use continuous functional forms to describe the
lightcurve. Primary among these are two considerations: our model atmosphere has
a definite outer boundary, and the model interpolates values across a finite number
of sampling points (the beams of light). The discrete case stands in contrast to continuous atmospheric models like EY92, for which any given point on the lightcurve
may be calculated with infinite spatial resolution, and the precision at any point is
limited only by the input parameters. Examples of some of the effects we will discuss
may be seen in Figure 4.
The difficulty associated with the definite outer boundary of the atmosphere
is that beams with a planetary impact parameter greater than the radius of the
29
outermost shell in the model do not encounter any shells at all, and the first few
beams that encounters a shell do so nearly parallel to the tangent of the point
of interaction.
This is a problem if the refractivity of the outermost shell is not
sufficiently close enough to zero (for the vacuum of space beyond the outer edge of
the model), because there will be a jump in beam density at that point. The further
the refractivity of the outer shell is from zero, the more significant the jump will be.
Additionally, the first few beam to encounter a shell may experience more refraction
than subsequent beams, due to the fact that 01 in Snell's Law will be very high
for these beams. The effect is dampened for subsequent beams, since they not only
encounter shells at lower values of 01, but they also encounter further shells along
their path before they reach the shadow line. This effect on the lightcurve may be
seen on all of the lightcurves in Figure 4, where we note that the effect is shifted and
dampened in the thicker atmosphere (red) case, which presents a possible solution.
Another effect of discretization occurs due to the interpolation across finite test
points, which makes it so that the focusing term for the discrete model becomes
less and less similar to the continuous models the closer a point is to p = 0 (where
p is the shadow-plane analog of radius, so p = 0 is the center of the shadow of
the occulting body). For small values of p, focusing effects are the strongest, and
create a region called the "central flash," for which focused light causes flux values to
increase to a theoretically infinite value at exactly p = 0, a singularity in Equation
10. The continuous model is able to provide focusing solutions up to infinity due to
its infinite spatial resolution, but the interpolation for the discrete model is limited
by its sampling resolution for any particular calculation. Again, we can see how the
interpolation effects change depending on the model settings in Figure 4, where some
lightcurves clearly track the central flash effect better than others. Elliot and Young
(1992) note that the singularity at p = 0 does not cause a problem for data analysis,
since observations rarely ever probe low-p values. Even if they did, the singularity
30
10
s
Ihicer irtmosphere mvvas
further and
amm tiss magmitude
vcvwn jump ot
-
Jtaroctirs vary qwucktd whom bm
shell ratios arm too h i
- default settings
h-
0 0.4-C 0.4--N
* higher shell resolution
thicker atmosphere
beam to shell ratio too high
-
0.2
0.0
500
0
[H",l
1000
p (kmn)
1500
2000
shad resdlu*)ntracks
ttbecortraI Ibsh better
Figure 4: Examples of the effects of different model settings on a lightcurve produced
by the same atmospheric conditions.
could be removed by considering the effects of diffraction and by removing the pointsource and spherical symmetry assumptions for the star and the occulting body
respectively.
From the plot, we can see that there are a number of different effects that may
occur based on the model settings. We specifically point out that one way to alleviate
both the problem of upper-atmosphere transition to vacuum and interpolation of
the central flash is to increase the thickness of the atmosphere (the red line). The
problem of transitioning from the edge of the atmosphere to the vacuum of space can
be minimized when the outer edge of the atmosphere has a very low number density,
and a correspondingly low refractivity, making it closer to vacuum conditions. When
the transition to vacuum is smoother, the jump is less prominent. The central flash
region is likewise brought closer to the continuous model when a thicker atmosphere
is used, because a correspondingly higher number of beams of light are used to sample
the atmosphere, and more of these sample points are brought to shadow-line p values
31
near zero, improving the sampling for the interpolation there. Although increasing
the shell resolution is sometimes helpful, particularly in tracking the central flash
effect (the green line), it will sometimes cause numerical "jitter," a discretization
effect we will discuss shortly. Likewise, a higher beam to shell ratio (effectively a
sampling rate), quickly causes similar jitter effects, and does so across the entirety
of the lightcurve.
To demonstrate the effect of atmosphere thickness more systematically, results for
the EY92 model with a pure N 2 isothermal atmosphere at 109.56 K and 0.177641 Pa
at a reference planet radius of 1275 km were compared to discrete models for atmospheres with the same conditions, generated with different atmosphere thicknesses.
The effects of these changes are shown in Figure 5. The two points that appear in
the 800 km-thick atmosphere well off the zero residual line at approximately -1800
km show the effect of the transition to vacuum for a relatively thin atmosphere.
These points still exist for the other cases, but are much closer to zero, and moved
further from p = 0 (they are just barely visible at approximately -2200 km for the
1200 km-thick atmosphere). However, it should be noted that simulating thicker atmospheres can significantly extend the time required to model the occultation event,
since increasing the thickness directly increases the number of shell interactions that
must occur with many of the beams in the model, particularly if beam density is
kept constant (see Section 4.3 for the runtime analysis of our model). The figure
also shows the effect of changing interpolation order, which demonstrates a decrease
in agreement with the EY92 model with increasing interpolation order.
Another similar factor that may cause lightcurve anomalies is the occurrence of
non-smooth transitions within the atmosphere model.
This case is similar to the
atmosphere boundary condition described previously. Although the way the atmosphere model is constructed ensures continuity between atmospheric properties from
section to section, it does not ensure differentiability, so temperature, composition,
32
Residuals for a 800km-thick Atmosphere
001
000
In
0)
V
-001
-0 02 [
-0030
-6000
-5000
-4000
-3000
-2000
-1000
p (km)
Residuals for a 1200km-thick Atmosphere
0.01
-~>1
0,001(n
M
-0 011
01
m 2
.3
V
-0 021-003 [
-6000
Interpolation Order
-5000
-4000
-2000
-3000
-1
EL 4
00
p (Km)
0 01
Resi duals for a 1600km-thick Atmosphere
-/
0 001
U)
-001
ci)
-0.02
-003[
-6000
-5000
-4000
-3000
-2000
-1000
p (km)
Figure 5: Effects of interpolation order and atmosphere thickness on agreement
with the EY92 model for a given isothermal atmosphere. Note the general trend of
increasing residuals with interpolation order, and the trend of decreasing residuals
with increasing atmosphere thickness. In all cases, the points for interpolation order
2 are on top of those for order 1.
33
and haze transitions may not necessarily be smooth. Since these three parameters
are defined only as linear or exponential variations between the section boundary
conditions, it is possible to create sharp changes in the atmosphere that introduce
dramatic changes in the middle of the ingress and egress portions of the lightcurve.
These conditions are, of course, likely unphysical, since atmospheric properties tend
to vary more smoothly, but to avoid strictly imposing a particular smoothing function
in the model, the smoothness of the properties is left up to the user to determine.
One possible way to dampen these anomalies when it is necessary to transition
between very different gradients is to insert additional intermediate atmosphere sections that gradually change the gradient using one or more additional interpolations
of some smoother functional form between two primary sections. Again, the precise
conditions for smooth lightcurves is difficult to define, since it is dependent on the
atmosphere model and the desired level of smoothness, and so the implementation
is left up to the specific use case.
Apart from the overall thickness of the atmosphere model, the density of beams
being sent out from the source line and the density of shells in the atmosphere
model may be set arbitrarily.
However, it should be noted that there are certain
settings of the latter two parameters that cause numerical "jitter" in the resulting
lightcurve (one example is seen in the blue line of Figure 4).
The beam density
at the source plane controls the sampling rate of the atmosphere during lightcurve
generation, while the density of shells in the atmosphere controls the resolution of
the atmosphere model.
The jitter occurs when multiple beams of light cross the
same set of shells, such as in Figure 6. Intuitively, one would expect that any beam
with a smaller planet-plane impact parameter (b) would necessarily experience more
scattering than a beam with a larger b because the beam with smaller b goes through
the same regions of the atmosphere as the beam with larger b, as well as additional
portions of the atmosphere due to its ray path. However, in the case illustrated by
34
beami
beam 2
Figure 6: Schematic representation of the beam/shell density issue, demonstrating a
simplified combination of beam and shell densities that would likely cause jitter in the
resulting lightcurve. Here, two beams are crossing the same set of shells (two shells,
causing four shell interactions total for each beam). Beam 1 will experience more
refraction than beam 2, despite being further away from the center of the planet,
due to the higher 01 value it has with respect to the shells it is interacting with.
the figure, the two beams actually go through the same sets of discrete shells, but
the Snell's law 01 angles experienced by beam 1 are larger than those experienced
by beam 2 (up to the limit of 01 = 900 for the tangent interaction case). When the
01 angle is the only difference between the two beams' atmospheric interactions, it is
possible for the beam with larger b to experience more refraction than the beam with
smaller b, shifting its position on the shadow plane and affecting the calculation of
dbeam
(Equation 9). In extreme cases where beams enter discrete shells very close to
the normal (such that 01 ~ 900), refraction angles increase to the point where beams
that started out adjacent to each other on the source plane may cross and end up
on opposite sides on the shadow plane. These effects are not seen in occultation
data, since they are purely numerical artifacts arising from the discretization of the
atmosphere into shells.
Even when beams do riot cross, the discretization can cause jitter in the resulting
lightcurves that make them non-smooth, or in worse cases, may change the shape of
the lightcurve entirely, both of which affect the fit to real data. Due to the many
shell interactions that occur for each beam of light, and the different interactions and
results that occur from different atmosphere models, it is difficult to pinpoint exactly
what combinations of beam and shell density will cause these issues, except to say
35
that if the ratio of initial beam density to shell density is very high, then the problem
is more likely to occur. Jitter may be diagnosed visually as non-smooth "jumps" in
the lightcurve (e.g., the blue line of Figure 4), which indicate the model's attempt
to linearly interpolate some value (likely a shadow-plane beam density value) where
one or more beams has reached a shadow-plane location very far from where it would
have gone if the atmosphere had been continuous. This effect often occurs in several
places throughout the lightcurve, though in some rare instances, it may be more
localized. Typically, such effects cause the lightcurve to look very different from one
jitter case to another even when just one of the atmospheric parameters is changed
slightly, giving us another way to confirm the effect.
In general, this problem was avoided by initializing the y-locations of each beam
on the source plane to be halfway between adjacent shells to stay away from tangent
interactions as much as possible (though the y-locations of these beams will have
changed somewhat by the time they reach the planet plane due to interactions with
the first half of the atmosphere). A beam-to-shell density ratio of 0.5 (meaning 1
beam was generated for every two shells per distance in y) and a shell density of 5
km/shell were used as default values for Pluto-like atmosphere scenarios after some
experimentation to avoid the numerical jitter. Appropriate values for these ratios
will vary depending on the size of the occulting body and the optical conditions of
the model atmosphere.
4.3
Runtime Analysis
Previously, we briefly mentioned a difficulty with extending the size of the atmosphere
to arbitrarily large numbers of shells in order to alleviate issues with discretization.
In order to show why increasing the number of shells to an arbitrarily large number
may be an issue, we perform a runtime analysis based on the number of beam/shell
interactions.
36
First, we note that for nearly all of the tests we have performed, the vast majority
of the runtime is taken up in the ray tracing portion of the program, and this is also
the only part that is significantly affected by increasing the number of shells in
our model.
With this in mind, we shall consider how the number of beam/shell
interactions (each of which requires a fixed number of calculations) scales with the
number of shells, neglecting all beams that do not hit the atmosphere, as well as all
initialization and interpolation portions of the software. We consider the nominal
case where the beam to shell ratio is 1, such that each shell has a corresponding
beam starting at a source-plane y-location halfway between the radius of the shell in
question and the radius of the shell immediately beneath it (neglecting the innermost
shell). We will now make the assumption that refraction effects are not significant
enough to cause a beam starting at some initial yo to interact with a shell of radius
rshell
< yo. This assumption is, of course, broken in some cases, particularly when
the change in y within the atmosphere for a given beam of light is on the order of
the distance between shells, but we present here a lower bound for the nominal case.
With these assumptions in mind, we can easily tell that for the outermost beam
that encounters the atmosphere, the number of interactions will be two -
one for
entering the shell, and one for leaving it. The next beam will have these two shells
interactions, as well as two more for the shell beneath the outermost one, for a total
of four interactions. Subsequent beams would have 6, 8, 10... and so on number of
interactions, giving us
n
2i,
total interactions =
(12)
i=1
where n is the number of shells. Simplifying the summation, we obtain,
Z2i = n(n + 1).
37
(13)
Thus, our runtime T(n) (measured by the proxy of beam/shell interactions) grows
quadratically with the number of shells, and for a given beam density, we have the
lower-bound runtime condition of
e(T(n))
=
n2 + n.
(14)
We can see then why there are practical limits to how much we ought to increase
the extent of the atmosphere model in order to counteract discretization effects. For
reference, we have found from our tests that for Pluto-like atmospheric conditions
(refractive angles in the atmosphere are generally small compared to inter-shell distances, ensuring that the assumptions above generally hold), where the occulting
body has a radius of 1000 km, and using a shell-to-beam ratio of 2 shells/beam, and
a shell resolution of 5 km/shell, has runtimes on the order of a minute for 800 km
thick atmospheres, and runtimes of approximately 40 minutes for 2000 km thick atmospheres on a quad-core Intel i7 machine with 8GB RAM, running Windows 8 and
Mathematica 10.
38
Part III
Model Testing
Our model was tested against the EY92 analytical model that was designed for
the case of Pluto. Model lightcurves for given atmospheres were generated by our
forward model and compared to lightcurves of the same atmospheres generated by
EY92. Test cases involved varying the temperature and haze layer structure of the
atmosphere, and covered three transit conditions with different shadow-plane impact
parameters.
1
Clear Atmospheres
The analytical model used by Elliot and Young (1992) has historically performed well
in fitting to the upper half of observed occultation lightcurves for Pluto, and returns
the gross properties and scale heights of the Plutonian atmosphere as a result of
these fits. As such, a comparison was done between the lightcurves generated by the
forward model and ones generated by the EY92 model under the same conditions.
For this comparison, six occultation lightcurves were generated using each model,
following two different atmospheric temperature profiles and three different transit
conditions. No fitting was done here, as each pair of lightcurves were simply generated
from identical given conditions for comparison. In each case, the atmosphere was
composed of a single section without haze; three cases had isothermal atmospheres,
and three had exponentially-varying temperature profiles with a b value (thermal
gradient power index, not impact parameter) of -2, following the form in Elliot and
Young (1992), Equation 3.3, reproduced below,
39
T(r)
=
T
(15)
7' 0
where ro and To are the reference radius (usually taken to be 1275 km) and
the temperature at the reference radius, respectively. The parameters used for these
atmospheres are shown in Table 1. For the three transit conditions, we used a central
transit, a close transit, and a far transit, which have shadow-plane impact parameters
of 0, 250, and 1000 km, respectively. The Earth-Pluto distance was assumed to be
30 AU, and the mass of Pluto was assumed to be 1.305 x 1022 kg.
Table 1: Test cases for clear atmosphere lightcurves. Temperature and pressure are
defined at a reference radius of 1275 km (at the planet).
Isothermal Case
Temperature Gradient Case
Temperature (K)
109.56
88.49
Thermal Gradient Parameter (b)
Pressure (Pa)
0
0.177641
-2
0.120633
For the purposes of this test, our model used the same exponentially-varying
temperature profile for generating shell properties, and in all cases, our model used
a surface radius of 1000 km, and an upper atmosphere boundary radius of 3000 km
(atmosphere thickness of 2000 km), with one shell every 5 km. The surface radius
was chosen so that the presence of the surface would not cause the lightcurve to drop
to zero flux, since minimum flux for Pluto lightcurves are always greater than zero.
In both the isothermal and gradient cases, our model showed an approximately
1% difference in the residuals in the region from p = 400 km to p = 1200 km, which
covers much of the region where the lightcurve probes the atmosphere (p values less
than 400 encounter the central flash, where there is a known discrepancy due to the
discretization in our model). These residuals show that there is a notable difference
between our model and the EY92 model. Unfortunately, the residuals are on the
order of differences that would be considered significant errors in Pluto occultation
40
iso
Is therma Atmosphere and Centiat Transit
isothermal Atmosphere and Far Transit
tidosit
rrra~ A rtsph a tid 1)
'0
00
212F
110
0 1-
061
214
04
021
012
002
-022
-101
(f
-
0
1000
2000
Gradient
Gradent Atmosphere and Cointral Transit
II
-101
-2010O
10(11
2000
?000
0
-1000
1000
200
Gradient Atmosphere and rar Transit
Almoshere and GCase iTrasit
*1 0
08
io
01
'K!
02?
08
02
06
11A
-0
200 1100
lori
21 000
207
00
110
-0
-012
-02
-
-20V
100
100
2000
-2000
0
-1000
1000
?OM0
0
-1000
-;0M0
1low
Figure 7: Comparison between the our model (red) and EY92 model (black)
lightcurves for clear atmospheres. Axes are p distances in km (x)., and normalized
flux (y). Residuals are shown in blue and shifted down by 0.2 to avoid clutter.
Gradient AImosphere Lightcurve Residuals
Isothermal Atmosphere Lightcurve Residuals
0 04
004
x
002
0 02
0 00
0 00
-002
-0 02
-
)K
-004
-2000
-1000
0
p (kl)
1000
_-0
01
2000
-2000
V.
-v, I
.j
m Central
m Close
a Far
-1000
0
1000
2000
p (Km)
Figure 8: Magnified plot of the residuals for the isothermal reference atmospheres.
The center-most pairs of dips and spikes are from the focusing factor, while the
outermost dips occur at approximately p = 1300 km.
41
2000
data. Since these differences cannot be resolved in the time allotted, actual lightcurve
data cannot be properly analyzed using our forward modeling method. However, we
find that our model tracks the trends in the changes of lightcurve characteristics
induced by changing the atmospheric characteristics, which allows us to conduct
trend analysis instead.
2
Atmospheres with Haze
An additional set of tests was conducted to compare the two models in cases where
a haze layer was introduced.
Here, an additional complication is involved due to
the differences in the way haze is specified in the EY92 model and in this model.
In the former, haze is defined by the tangential optical depth that is calculated by
a line integral of the linear absorption coefficient along the line of sight (Elliot and
Young 1992, Equation 2.6). The line integral ignores the actual path of a beam of
light going through the haze, but rather approximates the path as being straight
up until the point of closest approach to the planet, and then straight after an
instantaneous angle change, where all of the refraction is approximated to happen
at the point of closest-approach to the planet. This approximation gives haze purely
as an optical depth that varies with the tangential planet-close-approach distance
of a beam. In contrast, our model represents haze not by optical depth, but rather
by levels atmospheric attenuation coefficient a, that are associated with individual
atmosphere shells, such that flux changes as light moves from shell to shell according
to Equation 7.
Due to the differences in the way haze is handled, it is not possible to simply
generate lightcurves with identical conditions from the two models as was done for
the clear atmospheres. Instead, with all other atmospheric parameters remaining the
same as the clear atmosphere case, lightcurves were generated using the EY92 model,
42
toothernaol
ilo,erral Atmosphere aund Central Transit
Isothermnal Atmouspreie and far Transa
a
t~nsd Gklose 7fno0
02
00
-200
12
0 10
'
-00
-r
-02 --0
-
00
040
0
'OK
Gradrenti Adr
Grwerlt Atmosphiere ando Central Transit
p
e
8)
d (Okore Ratin
/ 4
02
-1000
6
1nonOoo
00
0
SM
-1000
Gradient Atmorphere and Far Transi
1000
0
2000
-2000
-02
1010M00 .?
-1000
O-1000 0 1000
200
Figure 9: Comparisons between the forward (red) and analytical (black) model
10-5
lightcurves for atmospheres with haze, using an attenuation coefficient of 1.745 x
distances
p
at 1000 km, and 0 at 1200 km with linear variation between. Axes are
in km (x) and normalized flux (y). Residuals are shown in blue and shifted down by
0.2 to avoid clutter.
where we held optical depth at 1 for a radius of 1175 km, set the optical depth scale
height to 30 km, and ended the haze layer at 1200 km (clear atmosphere above this
radius). To approximately match these conditions, a two-section atmosphere was
used for the discrete forward model, where the lower section had a linearly-varying
attenuation coefficient that ended at 1200 km, and the attenuation coefficient at
the base was set by fitting just that parameter with the isothermal central transit
lightcurve generated by the EY92 model. The fit returned an attenuation coefficient
of 1.745 x 10-
5
at a reference radius of 1000 km, which decreased linearly to a value
of 0 at 1200 km.
From this value of the attenuation coefficient, shown in Figure 9, we can see that
in both cases, the addition of a haze layer decreases the minimum flux levels of the
lightcurve down to zero, with the forward model roughly following the shape of the
EY92 lightcurves at the bottom, particularly with the change in slope in the curve at
just under p = 1000 km, which indicates the start of the haze layer. However, from
43
Hazy Gradient
Hazy Isothermal Atmosphere Lightcurve Residuals
Atmosphere Lightcurve Residuals
0 04
004
002o,?
-0
04
-2000
m Close
-002
L-002
-0
1000
-1000
2000
-2000
Far
p (Kin)
04-1000
0
10
2000
p (kmn)
S(kM)
Figure 10: Residuals for both the isothermal and gradient versions of the hazy atmosphere models. Here, the differences between our model and the EY92 haze model
are clear in that there is significantly more difference throughout the lightcurve than
in the clear atmospheres.
the residuals plot in Figure 10, we can see that the residuals for both the isothermal
and the gradient cases are larger than in the clear atmospheres. This larger difference
was expected and is likely due to the differing ways that the EY92 and our forward
model handle attenuation, constraining the ability of our model to match EY92's
lightcurves without treating the match as its own inverse problem.
An interesting note about haze layer, though, is the fact that its structure in
Pluto-like atmospheres is poorly constrained. EY92 uses an exponential model for
the linear absorption coefficient in regions where the haze is present, but such a
model was chosen primarily for ease of analysis, rather than on an actual model for
haze (Elliot and Young, 1992). The lack of a constraining model makes the choice
of variation structure fairly arbitrary in both EY92 and in our model, so the only
important difference in the choice of the haze characteristics is that our model offers
more flexibility in setting it, a feature that has effects on the bottom of occultation
lightcurves that are explored later in Section 7.
44
3
Potential Solutions to the Model Discrepancy
Since our model should return the same solution as the EY92 continuous case when
brought up to sufficient resolution and when we are using the same model of the
atmosphere (assuming we can avoid the previously-mentioned discretization effects),
there should exist some solution for the seemingly-inherent 1% difference we found
in the clear atmosphere examples such that we could get arbitrarily low residuals
when comparing to the EY92 model. A number of potential solutions have already
been explored. For example, in Part II, we considered the effects of increasing the
thickness of the atmosphere model, as well as increasing the interpolation order
between points. While the former change certainly caused an improvement in the
smoothness of the lightcurve, the latter was shown to increase the residuals (Figure
5).
We had also previously demonstrated the effect of increasing shell resolution
(Figure 4, green line), which improved our model's ability to track the central flash.
However, we can see from the plots of residuals for the clear atmosphere cases in
Figure 8 that the p values for which the high residuals occur begin at around 1500
km -
approximately the same p as where the lightcurve starts to come down from
its full flux value -
which means that the differences between the models are not
limited to the central flash.
One potential avenue to explore is to run a simulation for an extremely thick
atmosphere, on the order of 10000 km or more, which is about 10 times the thickness
of the atmospheres we typically simulate (generally ranging from 1000 to 1800 km
thick). Of course, as we described in the runtime analysis in Section 4.3, this kind
of testing can quickly take prohibitively long periods of time to complete, since the
model runtime scales quadratically with the number of shells. Other approaches may
be to attempt a spline interpolation between sampled points on the shadow plane,
which may smooth out differences between the discrete and continuous models that
may be caused by the linear interpolation, without necessarily forcing higher-order
45
interpolations between all points, as we did in the interpolation order tests in Part II.
Finally, if it is a problem with the specification of the atmosphere, it may be useful
to reduce the number of factors we are considering and track, for example, a single
beam as it moves from the source plane to the shadow plane, and compare each of the
angle changes it experiences when the integral form of the same ray-tracing process
done in the EY92 continuous case. If our discrete model follows the continuous model
well, then increasing the number of interactions for the single beam (which avoids
the quadratic scaling issue of simulating an entire lightcurve) should get arbitrarily
close to the single-point prediction of the continuous model. The latter test would
need to be compared to just the calculated angle changes in the EY92 model, and not
the actual shadow-plane flux values, since those also involve a shadow-plane beam
density measurement, as well as a focusing effect (Elliot et al., 1989, 1992).
4
Current Model Capabilities
Despite the discrepancy with the EY92 model, we still conducted a small set of tests
in order to integrate our forward model with a least-squares fitting routine and see
how well it performs in obtaining atmospheric parameters. Occultation data taken at
the 2.3 m Australian National University telescope at Siding Spring, Australia during
the 2006 June 12 occultation of P384.2 by Pluto were used as our test set. First,
we plot these data along with the model lightcurve generated by giving our discrete
model the adopted atmospheric parameters from Elliot et al. (2007) in Figure 12,
showing only the data above 0.4 normalized flux (following the procedure in Elliot
et al., since the fit was only for data with flux > 0.4). It is important to note here that
the data are only for Siding Spring site, but the fitted parameters are from a global
fit of five lightcurves taken for that event (Figure 11), so the model parameters are
also driven by data that are not present in the figure. We see that for the occultation
46
1.5
d
z-
0.
-
'T 0.5
0
1000
4000
3000
2000
Data Point Nunter
5000
Figure 11: EY92 model lightcurves for the five lightcurves obtained during the 2006
occultation event, reproduced from Elliot et al. 2007. Siding Spring is the fourth
lightcurve from the left. Red indicates the fitted global model for this event using
the EY92 method, which only used points above 0.4 normalized flux. Green indicates
an extension of the fitted model to points below 0.4 normalized flux.
region, the residuals (in blue) are generally shifted slightly down from zero, and there
is also a trend in the residuals that goes further downward the closer the points are
to p = 0. Given the high signal-to-noise ratio of the Siding Spring data, it is likely
that the primary cause of the shift in the residuals is from the difference between our
discrete model and the EY92 model, though some of the difference may also come
from the fact that the EY92 fit used additional data from that event.
The temperature gradient exponent b, the reference temperature, and the reference pressure were then fitted in succession using the discrete forward model and
least-squares minimization. The results of the fit may be seen in Figure 13, where
we note that the distinct downward shift of the residuals previously seen is largely
missing, and the residuals are generally centered well around the zero line.
Table 2 shows the fitted parameter comparisons between the fit done by the EY92
47
10
x
06
04
E
02
0
M data
ll
00
'~
02
-2000
-4000
2000
0
.
.*m
residuals
a model
4000
p (km)
Figure 12: Lightcurve model generated using discrete model, following the adopted
atmospheric parameters in Elliot et al. 2007. The parameters are from a fit of five
lightcurves, but the data here are only from Siding Spring. Black lines near the
residuals show 5% for reference. Note the general downward curve of the residuals
in the occultation region.
08
04
E
o
02
U data
U model
U residuals
C
00-0 2.__________________________________
-4000
-2000
2000
0
4000
p (km)
Figure 13: Fitted model lightcurve for Siding Springs data. Black lines near the residuals show 5% for reference. Note the reduced downward trend of the occultationregion residuals compared to the previous model.
48
Table 2: Fitted parameter comparison between Elliot et al. (2007) adopted fit and fit
obtained by our discrete model. Temperature and pressure are given at a reference
radius of 1275 km.
Parameter
EY92 Fit
b
-2.2 t 0.7
temperature (K)
pressure (Pa)
97
5
0.158 t 0.014
Discrete Model Fit
-2.7
99
0.7
2
0.183 t 0.008
model and our discrete model. Here, we can see that the values for b and reference
temperature obtained from our model overlap with the values obtained by the EY92
model, but the pressure does not. Since the pressure generally has the greatest effect
on where the occultation region begins to be noticeable, and how quickly the flux
drops in that region, it is understandable that it would be the value that is different
from the adopted parameter.
This difference in pressure is likely responsible for
most of the shift in the residuals seen in the previous two figures. Finally, we should
emphasize again that the different fit results are likely attributable to the fact that
there is still a small difference between the results we obtain from our model and
those from the EY92 model.
49
Part IV
Lightcurve Characterization
1
Goals and Testing Approach
The forward model developed here allows for an exploration of the effects of varying
atmospheric parameters on the characteristics of the resulting lightcurves. Although
testing has shown that our model does not precisely match with the values generated by the EY92 model for specific atmosphere cases, the overall trends in lightcurve
characteristics do occur appropriately when the atmosphere is changed. Thus, we
are able to use our model to examine the connections between lightcurve characteristics and the atmospheres that generate them. By studying these trends, we aim
to understand how the appearance of lightcurves change when the atmospheres of
the occulting bodies are varied, which can also help in measuring the uncertainty
in claims of atmospheric characteristics based on uncertainties in measurements of
lightcurves.
As a point of clarification, we will be referring often to two different sets of
characteristics: those of lightcurves, and those of atmospheres.
Lightcurve characteristics refer to features such as the half-light radius at the
shadow plane, the minimum normalized flux, and the slope at half light, which can
be determined from the lightcurve data with little to no model-fitting involved. We
chose to examine these characteristics because when taken together, they give a good
first approximation of the overall shape of a lightcurve, and also because each one
of these characteristics is indicative of a different aspect of the atmosphere.
The
half-light radius at the shadow plane can be considered a proxy for the extent of
the atmosphere (Bosh et al., 2015), while the slope at half-light is indicative of the
50
1.0
0.8
0.6
-E
0
-htaH4JW*
rah icm
0.4
0.2
minimum normalized flux
0
5
6
i
a
I
shadow-plane distance p (km)
of the
Figure 14: Schematic representation of a lightcurve showing the definitions
characteristics being examined.
general extinctive effects of the atmosphere, and the minimum flux level measures
not
the conditions in the lower portion of the atmosphere. These characteristics are
entirely independent, though they are different enough to be able to give a more
complete picture of the state of the atmosphere than any one of them alone. They
of
give the advantage of letting us extract key features that give us a good deal
information without requiring us to do full comparisons of entire lightcurves, which
may introduce further confounding factors (though the latter is a process that we
will consider in a different way in a later section of this work).
Atmosphere characteristics refer to the temperature, pressure, and composition
of the occulting-body atmosphere.
Note that since temperature and pressure are
to
often expressed as their values at the half-light radius, this half-light radius refers
the radius at the planet, expressed in r, which is different from the half-light radius
and
when we refer to it as a lightcurve characteristic, which is at the shadow plane,
due to
is expressed in p; both are measured in kilometers. The two are not the same
51
the refraction of light in the atmosphere.
For this analysis, we use a clear isothermal atmosphere near Plutonian atmosphere conditions as the test case. Isothermality is assumed in order to reduce the
dimensionality of the parameter space, and also because the half-light radius is affected primarily by the upper atmosphere, which is the main driver for flux from the
full-flux baseline down to shadow-plane radii slightly below half-light (Elliot et al.,
2007). Haze is not considered in the atmosphere parameter space because we do not
have a good model for its structure, leaving it fairly unconstrained in the types of
additional effects that it can introduce to the lightcurve. However, haze is examined
separately for certain cases in order to illustrate potential difficulties in disambiguating lightcurves when models allow for haze layers.
The set of model atmospheres used for examining the parameter space all share
a number of common characteristics. Each one extends from 1000 km to 2000 km
from the center of the planet, with a shell density of 5 km/shell, and a beam density
of 2 shells/beam.
A single parameter X, the N2 mole fraction, is used to control
the compositional variation, which determines the
N2
to CH4 ratio of the entire
atmosphere. The use of X subsumes the determination of both the average molecular
weight of the atmosphere Matm and the mean polarizability acatm (not the a, for the
attenuation coefficient, which remains at zero for our clear cases), so that the two
were determined via an additive variation given by
Matm= XMN 2 + (1
aatm
XN 2
t (1
x)MCH4 ,
(16)
X)aCH4
(17)
For the figures in the following sections, Matm is shown along one axis, while
aatm is not shown, because any particular value of Matm assumes a corresponding
52
value of
&aot
for that mixture of CH4 and N2 . As we shall see, the lightcurve
characteristics are not very sensitive to changes in composition, except in a few rare
cases. Pressure is specified at a reference radius of 1275 km for ease of comparison
with existing literature and varied from 0.1 Pa to 1.0 Pa (1 to 10 pbar). All results
are shown under central transit conditions, and half-light radius is considered the
first occurrence of 0.5 normalized flux during ingress or the last occurrence of 0.5
normalized flux during egress to avoid confusion with the half-light flux that occurs
at lower p values due to the central flash.
2
Half-Light Radius
The main results from the exploration of half-light radius variability are shown in
Figures 15 and 16. Figure 15 shows the 3D contour plot of constant half-light radius
across the three atmospheric parameters of pressure, temperature, and molecular
weight. Half-light radius has a variable range of nearly 400 km in p across the entire
parameter space we examined. For temperatures below approximately 150 K, the
isosurfaces are nearly parallel, with the direction of the gradient closely aligned with
the temperature axis, which means that at these temperatures, half-light radius is
relatively insensitive to changes in pressure and mean molecular weight. For temperatures greater than 150K, the direction of the gradient changes somewhat, and
deviates more from the temperature axis to point slightly towards higher pressure
and higher molecular weight.
From Figure 16, which is the pure-N2 slice of the Figure 15, we find that the
slope of the contour lines for these types of atmospheres are largely aligned with the
pressure axis, though they show a definite positive slope in the selected portion of
temperature/pressure space. We can see that the lines are less vertical and more
diagonal in the lower-pressure regions than in the higher-pressure regions.
53
This
difference in slope means that although half-light radius is generally insensitive to
pressure changes, it will be somewhat more sensitive when pressures are lower. The
location of the higher-sensitivity region is fortunate, since Pluto-like atmospheres
exhibit pressures in the range of approximately 0.1 to 0.3 Pa. It would have been
much more difficult, for example, to constrain the uncertainties on atmospheres closer
to 1.0 Pa, since the contour lines there are not only more vertical, but are also spread
further apart, so sensitivity of the half-light radius to both temperature and pressure
would be lower in that region.
These results also demonstrate cases where atmospheric conditions can give nonunique half-light radius values.
For clarity, Figure 17 illustrates the variation in
half-light radius with pressure for the 50 K and 100 K atmosphere cases alone. Here,
we can more easily see how low-pressure 50 K atmospheres can have half-light radii
equivalent to high-pressure 100 K atmospheres by following the horizontal line on
the plot. As the general trend is for the values of half-light radius to shift downward
with increasing temperature (increasing their slopes only slightly in comparison to
the 50 and 100 K cases), it becomes easier to distinguish between atmospheres of
different temperatures with just the half-light radius.
54
Figure 15: Two views of the same 3D contour plot of half-light radius across temperature, pressure, and mean molecular weight for parameter values near Pluto conditions. Isosurfaces occur every 30 km in p. Note the much stronger variation with
temperature compared to the other parameters and the large region of near-constant
half-light radius in the high-temperature, and low-molecular-weight portion of the
plot.
The overall trends shown in the parameter space (Figures 15-17) have implications
for the similarity of lightcurves for widely-varying atmospheric conditions. For Plutolike conditions where pressures are around 0.2
are around 100
0.1 Pa, and isothermal temperatures
10 K (Bosh et al., 2015, Person et al., 2008, 2013, Elliot et al.,
2007), different atmospheres that yield equivalent half-light radii likely exist in the
parameter space for temperatures up to 160 K, when pressures are increased at an
approximate rate of 0.02 Pa/K, following the slope of the half-light radius contour
lines of Figure 16. This phenomenon of different atmospheric conditions producing
similar lightcurves is consistent with previous modeling work done by Zalucha and
Gulbis (2012).
Of course, an equivalent half-light radius does not necessarily mean similarity for
an entire lightcurve, since this is but one lightcurve characteristic that we are looking
entire
at so far. Quantifying similarity for other lightcurve characteristics, and for
lightcurves is addressed in the following sections.
55
200
150
110I
200
?0
1
'2"l
Figure 16: A slice of the contour plot shown in Figure 15 taken at a constant molecular weight of 28.013 g/mol, showing half-light radius variability for a pure N2 atmosphere. The gradient magnitude at a particular point is used as a measure of
the sensitivity in the plot on the right. Note that at low temperatures, the gradient
is nearly parallel with the temperature axis, meaning that pressure is poorly constrained by measurements of the half-light radius for low temperatures. We discuss
the portion of the temperature/pressure space relevant to Pluto later in Section 5.1.
Half-Light Radius Variation with Pressure
1350
1300
sothermal Temperatures (Pa)
0
0
*50K
m 100K
$1250
1200
0.2
0.4
0.8
0.6
1.0
Reference Pressure (Pa)
Figure 17: Half-light radius values for selected temperatures. This figure is included
in addition to Figure 16 to facilitate numerical comparison of temperatures that show
lower sensitivity. Higher temperatures retain the same shape, but appear lower on
the y-axis and cross wider ranges of half-light radii for the same range of pressures.
The horizontal line is one example where atmospheres of different temperatures and
pressure can give the same half-light radius. Such cases can occur with these two
temperatures for half-light radius values ranging from 1280 km to slightly over 1300
km.
56
3
Minimum Normalized Flux
The trends and sensitivity of the minimum normalized flux were examined in the
same way as the half-light radius. We see from Figure 18 that variation in composition plays a relatively minor role for atmospheres under 100 K. For temperatures
higher than 100 K, there is a region of high-pressure parameter space where the
minimum flux is elevated to 30% or even more (the reddish region in the plot on
the right), though that region is small compared to the larger diagonally-cut portion
of the space trending towards higher temperatures and lower pressures where the
minimum flux tends to be less than 10% of full flux. From both Figures 18 and 19,
we can see that the gradient across temperature actually changes direction so that
the extreme ends of temperature have more similar minimum flux values than the
middle, where the concentrated high-minimum-flux portion exists.
Furthermore, in comparing the two plots in Figure 19, we see that although the
contours "wrap around" the high-flux space, the gradient is much larger on one side
of the space than the other, where we see pure white regions (off the scale) in the
lower-right-hand corner of the sensitivity plot, in contrast to the lower-sensitivity
regions on the left side of the same plot.
57
Mir murn Nu wiai
FrIuA
020
/ 15~
0
/
0650T
10
Fb
T
100
W
"1/200
f.IZA
50
normalized flux across
Figure 18: Two views of the same 3D contour plot of minimum
values near Pluto
temperature, pressure, and mean molecular weight for parameter
radius, though
half-light
for
conditions. Here, the variation is more complex than
the minimum flux is also most sensitive to temperature variations.
riit
Mirimun Normaizred Fluo
Serr!AAy
105
,060614
I 0k)
150
Tw1
200
56-
250
20
at a constant molecFigure 19: A slice of the contour plot shown in Figure 18 taken
for a pure
variability
flux
ular weight of 28.013 g/mol, showing minimum normalized
point is used as a measure
N2 atmosphere. The gradient magnitude at a particular
right shows that higher
the
on
plot
of the sensitivity in the plot on the right. The
of the minimum flux
isothermal temperatures dramatically increase the sensitivity
higher than the colorbar
value (pure white regions on the lower right are significantly
by a preciselyconstrained
least
maximum). Like half-light radius, pressure is also
measured minimum flux value.
to changes in temperThe minimum normalized flux is found to be most sensitive
and mean molecular weight for
ature, and relatively insensitive to changes in pressure
58
conditions below ~ 150K. Past 150K, we find a region of rapid changes to minimum
normalized flux, where the gradient is pointed diagonally towards one corner of the
parameter space, meaning all three atmospheric parameters we are examining play
a role in shaping this value. At Pluto-like conditions, the minimum normalized flux
is slightly sensitive to changes in temperature, and relatively insensitive to changes
in both pressure and mean molecular weight.
4
Slope at Half-Light
A third lightcurve characteristic of interest is the absolute value of the slope at halflight. The symmetry of the two limbs of a lightcurve make the slope of the model
at ingress equal to the negative of the slope at egress, so a single absolute value is
just taken here to be the slope at half-light. This slope is taken as a representative
value for the lightcurve, since the actual change in flux over p varies across the extent
of the lightcurve. In cases where the slope is extremely high, the lightcurve would
be indicative of a very thin atmosphere, approaching the limit of infinite slope (a
vertical line) for an airless body (neglecting edge diffraction effects).
In Figures 20 and 21, we can see that the variation of the slope at half-light at low
temperatures is again similar to the variation in the half-light radius and the minimum normalized flux in that the gradient is roughly aligned with the temperature
axis. A notable characteristic of the trends here is that apart from a small portion
of the space in the high-temperature, low-pressure, low-molecular-weight corner, the
quarter of the parameter space defined by the lowest temperatures exhibit the most
sensitivity, which is markedly different from the relatively low sensitivities of halflight radius and minimum flux in the low-temperature regions. With slope, we also
see that the shapes of the contours in the higher-temperature spaces are different
than those of the plots for half-light radius and minimum flux.
59
3
1oom0
OU2
Figure 20: Two views of the same 3D contour plot of slope at half-light across
The
temperature, pressure, and mean molecular weight for Pluto-like conditions.
parameter
most rapid changes in slope occur at a rough plane that cuts across the
space diagonally in the low-pressure, high-temperature, and low-molecular-weight
region.
For N2 atmospheres, Figure 21 also indicates that the gradient for slope becomes
much weaker past 125 K. In the higher-temperature regions here, we can see from
the scale that the slopes are decreasing, giving rise to more gradual decreases in flux
that also take more parameter variation to alter.
60
ope at
013
Half-Liqht
15C
2030
91D
250
25
molecFigure 21: A slice of the contour plot shown in Figure 20 taken at a constant
pure N 2
ular weight of 28.013 g/mol, showing slope at half-light variability for a
in p
change
over
flux
normalized
%
in
atmosphere. Slopes are measured in change
than
less
temperatures
(%/km). The lines of constant slope are mostly vertical for
100 K, but change rapidly with temperature in that region.
in
On the whole, we find that the slope at half-light is most sensitive to changes
weight. The
temperature, and relatively insensitive to changes in mean molecular
characteristic is also largely insensitive to pressure, except at the high-temperature/lowatmopressure corner of the parameter space. These trends hold true for Pluto-like
parameter
spheres, though sensitivity to temperature changes in that region of the
from
space is higher than it is in most other parts of the space, a notable difference
the sensitivities of the half-light radius and the minimum normalized flux.
5
Application to Lightcurve Analysis
that it
From the previous analysis on specific lightcurve characteristics, we can see
and
is possible to retrieve scalar values to represent specific features of lightcurves
more holistic
examine the trends across them. However, we ultimately want to take a
to examine
view of the model lightcurves when comparing them, which requires us
these values in a more unified fashion.
We wish to combine the results from the
61
last three sections into a single kind of plot that will aid in determining atmospheric
parameters from measured lightcurve characteristics. Conversely, we may also use
the combined results to understand what signal-to-noise ratio an observer must have
in order to constrain a given atmospheric parameter to a certain precision. Here, we
will use the term "lightcurve characteristic contours" to refer to plots of the variation
of one or more of the aforementioned lightcurve characteristics in the parameter space
(Figures 22-26, and Figure 28), where the axes of the parameter space are defined
by the characteristics of the occulting body's atmosphere.
In the ideal case for data analysis, all lightcurve characteristics would show high
sensitivity to all atmospheric changes, and these changes would have completely
independent and different effects on each lightcurve parameter, such that any particular measure of half-light radius, minimum flux, and slope at half-light would result
in a highly-constrained set of atmospheric parameters. The gradient of all of these
characteristics would point towards different corners of the atmospheric parameter
axes. Of course, as we have seen, this is not really the case with lightcurve trends.
The different ways that the lightcurve characteristics change and the extent to which
they change within any given region of the parameter space is thus useful to analyze
with all three characteristics together, in order to help discern where uncertainties
are likely to be large, and where they are likely to be smaller.
When looking at the three-dimensional characteristic contours, we note that overall, the variation in lightcurve parameters with composition within the confines of a
nitrogen-methane atmosphere tends to be much weaker than variation effects caused
by temperature and pressure (particularly for nitrogen-dominated atmospheres). As
such, we will focus the analysis on a pure-N2 atmosphere, which is close to the
expected composition of the Plutonian atmosphere.
62
5.1
Combined Lightcurve Characteristic Contour Plots
Figure 22 combines the contour plots of the three lightcurve characteristics (Figures
16, 19, and 21) into a single plot for a nitrogen atmosphere. A notable feature of
this combined plot is that all of the contours appear nearly parallel with the pressure
axis for temperatures less than 150 K. Fortunately, slightly different slopes (and in
both positive and negative directions) aid in constraining pressure and temperature
conditions based on the lightcurve characteristics. From these overlapping contours,
one may map out the approximate intrinsic uncertainties in the measurement of pressure and temperature based on the uncertainties from measuring the three lightcurve
characteristics.
It should be noted, however, that due to slight difference between
this model and the EY92 model at present, the mapping for any specific value will
not be exact for lightcurve data fitted with the EY92 model, though the contours are
instructive in showing where particular lightcurve characteristics may provide more
or less constraint on the atmospheric conditions.
The black box on the figure denotes the approximate portion of the parameter
space occupied by accepted values for model fits to the upper portion of Pluto occultation lightcurves. Most of these fits do, however, involve a gradient atmosphere
with a thermal gradient parameter of -2.2, which is not something that is reflected
here since this strictly an isothermal model.
Since many of the model fits for Pluto's atmosphere have a temperature gradient
of b = -2.2 (Bosh et al., 2015, Person et al., 2013, 2008, Elliot et al., 2007), a new
set of pure-N2 atmospheres was generated that had this temperature gradient. For
these plots, the values on the temperature axis represents the temperature at the
reference radius of 1275 km. These results are shown in Figure 23.
We find that there is strong similarity between the b
mal plot (b
=
=
-2.2 plot and the isother-
0) in the structure of the contours for the left half of the parameter
space (T < 150 K). The only noticeable differences in the region occupied by Pluto's
63
Lightcurve Characteristics for Pure-N2 Isothermal Atmospheres
(
I-
/
1,0
K
0,8
0,6
0~
(1
0~
04
0.2
50
100
150
T (K)
200
250
* Half-Light Radius
" Minimum Normalized Flux
* Slope at Half-Light
Figure 22: Combined contour plots of three lightcurve characteristics for pure- N 2
isothermal atmospheres. Contours occur at every 30 km in p for half-light radius,
4
5% flux for minimum normalized flux, and 5 x 10- % flux per km in p for slope at
half light. Most of the contours remain fairly vertical (near-constant with pressure)
for temperatures less than 150 K. The black box denotes the approximate range of
accepted values for fits to the upper portion of Pluto's atmosphere (Elliot et al.,
1989, 2003a, 2007, Person et al., 2013, Bosh et al., 2015).
64
atmosphere is that the minimum normalized flux contours (blue) for the gradient
case are spread further apart, and contours for slope at half-light (green) are closer
together than the isothermal case (Figure 22). That being said, due to the nature
of Pluto's lower atmosphere, the minimum normalized flux is likely not a good characteristic to use from this set models used in our exploration of the parameter space
in any case, since we do not include the effects of haze or a strong near-surface
temperature gradient.
The slopes of the characteristic contours for Pluto-like atmospheres show that all
of the lightcurve characteristics are somewhat diagonal in that region (the black box
in Figure 23), meaning that they all show some sensitivity to both temperature and
pressure. The half-light radius (blue) is the most sensitive to variation in temperature and pressure, since its contour lines are furthest from vertical across the space.
The lines for the slope at half-light (green) are also somewhat diagonal, and have a
nearly opposite slope compared to the half-light radius contours, which is a fortunate
occurrence, since it allows these two lightcurve characteristics to more easily act in
concert in constraining the uncertainty on temperature and pressure, something that
we discuss further in the next two sections. The behavior of the minimum normalized
flux (red) lines is interesting in the Pluto atmosphere region, because the contours
bend inward, so depending on what pressure value is returned by a fit (and how large
the uncertainty on the pressure is), there will be some variability in how sensitive
the minimum normalized flux is to changes in the atmospheric conditions.
Further away from Pluto-like conditions, for the right half of the parameter space
in Figure 23 (T > 150 K), the triangular region of rapidly-changing minimum normalized flux in the low-pressure portion of the space is preserved from the isothermal
plot, as is the transition back to more slowly-changing minimum flux values further
down to the corner. However, both features occur in lower pressures for the gradient case than in the isothermal case. The minimum normalized flux contours on the
65
upper-right also differ slightly in that they do not curve back up to higher pressures at
the high-temperature end quite as much as the lines for the isothermal atmospheres
do. Notably, the slope at half-light undergoes a dramatic change to much higher values at the mid-temperature, low-pressure region, where the lightcurves drop much
more rapidly from the baseline flux, a behavior not seen in the isothermal cases.
5.2
Uncertainty Analysis Procedure
Earlier, we alluded to the fact that that the lightcurve characteristic plots would allow
us to calculate uncertainties on atmospheric conditions directly from the uncertainties
on the lightcurve characteristics themselves. Here, we will demonstrate a case where
that may be done for a nominal set of data in order to show how the procedure would
be carried out.
First, we assume that the atmosphere we are looking at is a clear, pure-nitrogen
atmosphere with a temperature gradient of b
be referring to the contours of Figure 23.
=
-2.2,
which means that we will
Now, given a set of p/flux data from
observation, we may do the following:
1. Measure lightcurve characteristics from the data, along with the uncertainties
associated with the measurements.
2. Fit the lightcurve and get the resulting temperature and pressure at 1275 km,
as well as the uncertainties from the fit.
3. Place a point on the lightcurve characteristic contour plots where the fitted
value occurs, and adjust the contours so that lines for all three characteristics
cross the point.
4. Adjust the resolution of the contours so that the distance between contours is
equal to the degree of the uncertainty for the corresponding lightcurve characteristic.
66
Lightcurve Characteristics for Pure-N2 b=-2 2 Atmospheres
10
08
0,6
0,4
0,2
50
100
150
200
250
T (K)
* Half-Light Radius
m Minimum Normalized Flux
* Slope at Half-Light
Figure 23: Lightcurve characteristics for pure-N2 atmospheres with a temperature
gradient of -2.2, following the fitted temperature structure of several previous occulfor
tation events. Contours occur at every 30 km in p for half-light radius, 5% flux
minimum normalized flux, and 5 x 10-% flux per km in p for slope at half light.
Note the similarity in the contour structure to the isothermal case in the region where
Pluto's atmosphere falls. The black box denotes the approximate range of accepted
values for fits to the upper portion of Pluto's atmosphere.
67
5. Use the lightcurve characteristic uncertainties with the characteristic contour
plots to give bounds for the answer in temperature/pressure space.
6. Compare the uncertainties from the characteristic measurements to the uncertainties from the fit.
For the purposes of this demonstration, we will assume a case where we find from
step 1 that the uncertainty on the half-light radius measurement is +5 km, the
uncertainty on the minimum flux measurement is +0.5%, and the uncertainty on
the slope at half-light is +0.01%/km.
Next, we will assume that from step 2, we
find that the fitted temperature and pressure at 1275 km was 97.5
0.2
7.5 K, and
0.05 Pa, respectively. Using these data, we proceed to steps 3 and 4. We first
plot the fitted point in the temperature/pressure space and then center a contour line
for each of the lightcurve characteristics on that point. Then, we find the contour
line resolution that makes the contour lines immediately next to the fitted point give
us the uncertainty on the respective characteristics (e.g., 5 km/line for the half-light
radius). The results of these steps are shown in Figure 24.
After we have this set of bounded lightcurve characteristic plots, we may overlay
them, as seen in Figure 25. With this combined plot, we may carry out step 5 by
seeing which uncertainties are most constraining. In our case, we can see that the
uncertainties for the half-light radius and the minimum flux completely constrain the
result. The contours thus allow us to show the bounded region of the pressure and
temperature space that is given by the measurement uncertainty on the lightcurve
characteristics, as seen in Figure 26.
We are thus able to characterize uncertainties in the temperature/pressure result
directly from the uncertainties in the measurement of the lightcurve characteristics.
From this result, we see a few key features about the uncertainty bounds now displayed in the temperature and pressure space. First, the bounded region is neither
rectangular nor rhomboidal, nor is it aligned with either axis, since its boundaries fol-
68
km
Half Light Radius Uncertamty.
/
024
022
018
016
04
02
D0
9
W9 100
(a)
102 10
T !K)
S%
Minimum Flux Uncertainty
024-
022
i.
020
018
016
90
92
94
96
(b)
98
100
102
104
T(K)
Slope at Half-Light Uncertainty
:t 9(1%/Kff
024
i 020
018
0.16,
90
92
94
90
91
100
102
104
T(K)
(c)
flux,
Figure 24: Lightcurve characteristic contours for half-light radius, minimum
by
separated
is
line
contour
Each
point.
and slope at half-light around the fitted
on
uncertainties
the
5 km, 0.5%, and 0.01%/km for each plot, respectively, following
the measurement of each of those characteristics.
69
024
C
0.20
.
022
018
016
90
92
94
T (K)
their correspondFigure 25: Lightcurve characteristic plots overlaid and shown with
ing uncertainty bounds.
Combined Lightcurve Characteristic Uncertainties
024
022
06
020
018
016
90
92
94
98
96
100
102
104
T (K)
given the
Figure 26: Uncertainty constraint space for pressure and temperature
measurement uncertainty on the lightcurve characteristics.
70
low the curves on the characteristic contours. Second, given different circumstances,
the space could have been constrained by all three characteristics, two of them (as
in this case), or even in some rare instances, just one (e.g., if it were on one of the
contour "islands" in the low-pressure region of Figure 23). Third, we can see that
this space is smaller than the temperature/pressure uncertainties that we got as a
result of the fit, which in the figures is simply the boundaries of the plots.
With this kind of analysis, the uncertainty regions returned by the fit may not
be the same as the ones we get from the characteristic uncertainty, and either one
may be larger, though they should not result in extremely different spaces. Note
that we do not specifically claim that the uncertainty regions of one kind of analysis
will necessarily be larger than the other, since that is something that will require
more testing to show. Also note that we avoid making claims on the values of the
lightcurve characteristics (i.e., in step 3, we did not place a dot in the plot where
the measured value of the lightcurve characteristics actually occurred), due to the
fact that our model returns lightcurves that are slightly different from the EY92
model. Despite the difference, the trends in how the characteristics vary still hold,
allowing our model to be used to characterize uncertainties in this way. That being
said, once our discrete model is brought into agreement with the continuous model,
the placement of the fitted point in temperature/pressure space should match its
location in the lightcurve characteristic (contour) space.
5.3
Application to 2011 and 2013 Pluto Lightcurves
Having shown how uncertainty analysis may be done directly on measurements of
the lightcurve characteristics, we will now attempt to apply it to the fits of two actual lightcurves. Unfortunately, of the three lightcurve characteristics we have been
examining, only the half-light radius at the shadow plane is typically reported. However, we know that the uncertainties on the reported fits of pressure and temperature
71
come from the uncertainties on the model fits to the lightcurves, so instead, we will
examine the expected uncertainty in the lightcurve characteristics given the uncertainty on the temperature and pressure, and compare it to the reported values for
the half-light radius. If our lightcurve characteristic trends are correct, the values for
the uncertainties on the characteristics should be consistent with what is reported,
given the temperature and pressures of the fit. This procedure is, of course, different
from the one shown in the previous section, which is the ideal use case of our characteristic contours, though this test with the actual lightcurves is a good consistency
check for our general procedure.
Bosh et al. (2015) illustrates an example where observations from the June 2011
and May 2013 Pluto occultation events produced lightcurves that were nearly indistinguishable above the half-light level (Figure 27), which was the portion that
was used to perform the fits in each case, even though the fitted solutions returned
half-light radius pressures with non-overlapping error bars. Table 3 shows the temperature and pressure values of the two events at a reference radius of 1275 km,
using the reported fits that held the thermal gradient constant at b = -2.2.
We note
that the uncertainties for both the pressure and the temperature do not overlap, and
thus the fits return distinctly different values for each parameter. These results will
be used to test our uncertainty analysis procedure. We will be examining these two
lightcurves through the fitted parameters for the b = -2.2 cases for each, as well as
the reported half-light radius values to check for consistency with our characteristic
contour plots.
The lightcurve characteristic contour plots allow us to examine the approximate
range of allowable uncertainty in the measurement of lightcurve characteristics in
order to obtain the range of atmospheric condition uncertainties reported by the
fits. To do this, we look at the smaller region of the contour plots bounded by the
uncertainties on temperature and pressure for the two fits, as shown in Figure 28
72
Table 3: Temperature and pressure at 1275 km from the fixed-b (-2.2) model fits of
the 2011 June 23 and 2013 May 4 Pluto occultation events.
Event Date (UT) P at 1275 km (Pa) T at 1275 km (K)
Reference
2011 June 23
0.206 0.004
95.5 0.1
Person et al. 2013
2013 May 4
0.294 0.020
101.4 + 0.9
Bosh et al. 2015
(a subset of the contour plot for b = -2.2 atmospheres shown in Figure 23). We
emphasize that given the differences in the sizes of the error bars for each result, it is
necessary to plot the zoomed-in contours separately, and to use different resolutions
for the contour lines, which are given in Table 4.
We calculate the measurement uncertainty for each lightcurve characteristic by
counting the number of contour lines that occurs in the region, and multiplying one
less than that number (for the number of spaces between lines) by the resolution
of the characteristic for that plot. From the plot, we also know that our counting
only gives a lower bound for the uncertainty, since there are two spaces on either
end of the plot for which we cannot count a contour line. Thus, we add one more
resolution unit to our count, and give the uncertainty on the measurement (a kind
of second-order uncertainty) as
one resolution unit.
For example, for the 2011 plot in Figure 28a, we count seven blue lines for the
half-light radius. This gives us six counted resolution units (spaces between contour
lines), each with a value of 0.25 km, according to column 2 of Table 4. However, we
see that there are two partial spaces beyond the spaces between the counted contour
lines, which when accounted for, gives us a total range on half-light radius variation
of (7 x 0.25)
(1 x 0.25) = 1.75
0.25. Since this is the range of values across
the extent of the selected temperature/pressure space, the actual uncertainty on the
measurement is half of that, or 0.875 i 0.25.
Table 5 gives the measurement uncertainty for the three lightcurve characteristics
that would allow for the reported error bars on the b = -2.2 cases of the fits.
From the results presented in Table 5, we find that the half-light radius uncertain73
1.0
x
0.8
0.6
half-light level
0.4
0.2
0.0
10
0.5
0.0
-0.5
-1.0
Distance from shadow center (units d6 hl-IgrW
-1.5
15
i*us)
and
Figure 27: Lightcurves from the Pluto occultations on 2011 June 23 (black),
2013 May 4 (red), reproduced from Bosh et al. (2015).
2013 May 4 Oc union
2011 June 23 Occufttion
0'M)
A
n 20')
0 2X)
ri
4
0 20 t
0 2W)
0 275
0
(a)
202
AO 95A
Or,0
9550i-
~ 95
10(
60
k- 57
I5
101
0
101
1
102
0
(b)
T (K)
* Half-Light Radius
* Minimum Normalized Flux
s Slope at Half-Light
two Pluto occulFigure 28: Region of parameter space occupied by light curves for
uncertainties for
tation events, where the limits of each plot are the limits of the
and Figure
each axis. Figure (a) shows the region of uncertainty for the 2011 event,
limit of
likely
the
show
lines
(b) shows the region for the 2013 event. The contour
to get
the allowable uncertainties on measurements of the lightcurve characteristics
clarity, the
the reported atmospheric parameters. Note that in order to maintain
sizes of
different
the
to
due
contours are drawn at different resolutions for each plot
the uncertainties (see Table 4).
74
Table 4: Contour line resolution for the plots of the two occultation events.
Event
Half-Light Radius (km)
Minimum Flux (%)
Half-light Slope (% flux/km)
2011 June 24
2013 May 4
0.25
1 x 10-2
5 x 10-
1.00
5 x 10-2
5 x 10-4
4
Table 5: Allowable uncertainty (and second-order uncertainty) in lightcurve characteristic measurements required to obtain the reported temperature/pressure uncertainties, according to the characteristic contours. See text for definitions and calculation procedure. The reported half-light radius uncertainties are from the b = -2.2
fits in Bosh et al. (2015) and Person et al. (2013). The uncertainties we calculate for
the half-light radius are consistent with the reported uncertainties
Event
Half-Light
Reported Half-Light
Radius (km)
Radius Uncertainty
Minimum Flux (% flux)
Half-light Slope (%/km)
(km)
2011 June 24
0.875
2013 May 4
4.5
0.25
1.1
0.03
0.01
1.25 x 10-
1.0
4.3
0.15
0.05
1.5 x
3
10-3
5 x 10-4
5X
10-4
ties from the contours (second column) are consistent with the reported uncertainties
(third column). We cannot do the same check for the minimum flux or for slope at
half-light, since these values are not reported. In any case, our values for the minimum flux likely do not correspond to the observed values, since our model does not
include the strong thermal gradient or haze effects that are believed to affect the
region of the lightcurve near minimum flux for occultation by Pluto. Still, we show
that our procedure for calculating the uncertainties on temperature and pressure
return values that are consistent with the uncertainties reported by the model fit.
The allowable uncertainties presented here aid in estimating the signal-to-noise
ratios of observations that would be required to make certain claims on the precision
of the atmospheric characteristics from the fitted data. Specifically, the range of
half-light radii calculated to be within the pressure/temperature error bars of the
2011 fit show that in order to define the values of pressure and temperature to be
0.206
0.004 Pa, and 95.5
0.1 K respectively (as the fitted models do), the value
for the half-light radius must be measured to within an uncertainty of
0.875 km,
in addition to the other characteristics being measured according to their respective
75
allowable uncertainty ranges.
These ranges provide a different way to define the
uncertainties on the atmospheric conditions, because they are based on uncertainties
in the measurement of the lightcurve characteristics themselves, rather than on those
of the model fits to the lightcurves. This method is potentially a way to resolve the
problem of near-indistinguishable lightcurves producing distinctly different fits to
atmospheric conditions that Bosh et al. (2015) notes.
5.4
Implications for Half-Light Radius as a Proxy for Pressure
Half-light radius on the the shadow plane (p) has historically been used as a proxy
for the pressure, or, more generally, the "extent" of the atmosphere of Pluto (Elliot
et al., 2007, Person et al., 2013, Bosh et al., 2015). Now, with the ability to test the
changes in lightcurve characteristics over the parameter space, we may check how
valid this proxy is. When treated as a proxy, it has been noted that there is a general
trend of increasing pressure in Pluto's atmosphere between 1988 and 2013 implied
due to the increasing half-light radius, which has been compared to the atmospheric
pressure models reported by Hansen and Paige (1996). The comparison shows that
the shadow-plane half-light radius values generally track the relative positions of the
corresponding events as far as fitted pressure, but their uncertainties do not allow
for any particular atmospheric model (of the three proposed by Hansen and Paige)
to be favored.
To see how useful the shadow-plane half-light radius is as a measure of pressure,
we plot the extent of the fitted uncertainties in temperature and pressure on top
of the contour plots of shadow-plane half-light radius, generated in the same way
as the previous contour plots, but with a higher-resolution sampling of lightcurves
around the Pluto-like atmosphere region of the parameter space. Table 6 shows the
temperature, pressure, and shadow-plane half-light radius of occultation events from
1988 to 2013, mostly fitted with a temperature gradient exponent of -2.2, and Figure
76
30 shows the uncertainty regions in temperature and pressure for these events on the
half-light radius contours for a pure-N2 , b = -2.2
gradient atmosphere. We have
excluded the 2012 event due to the poor pressure constraint that it provides. Note
that because the 1988 and 2012 fits were for isothermal atmospheres, the contours do
not actually apply directly to them, so their uncertainty regions are simply provided
for reference and are not discussed in comparison to the other uncertainty regions.
Table 6: Temperature and pressure at 1275 km, and shadow-plane half-light radius
from the fixed-b model fits for Pluto occultation events from 1988 to 2013.
Event Date (UT)
T (K)
1988 June 9*
106 ] 9
Half-Light Radius (km)
P (Pa)
0.10
0.07
1168
Reference
10
Elliot et al. 2007
2002 August 21*
109
9
0.18 i 0.1
1218 i 6
Elliot et al. 2007
2006 June 12
2007 March 18
97
98
5
0.16
1208 t 4
Elliot et al. 2007
2011 June 23
95.5
1
0.1
0.06
0.20: 0.03
0.206 + 0.004
1207
4
Person et al. 2008
1205
2
Person et al. 2013
Bosh et al. 2015
1200 11
Bosh et al. 2015
1213 t 4
101.4 0.9 0.294 t 0.020
2013 May 4
* 1988 and 2002 fit results used an isothermal model atmosphere (b = 0). All other results in the
table came from atmospheres with a b = -2.2 temperature gradient.
2012 September 9
97.5
0.1
0.2
0.2
It is possible to understand how well half-light radius acts as a proxy for measuring
pressure changes by looking at how the contour lines flow in Figure 30. We see, for
example, that for the occultations from 2006 onward (those for which we have b =
-2.2 fits), the uncertainty regions in temperature/pressure space are aligned in such
a way that the contour line extend from a set of overlapping regions (the 2006, 2007,
and 2011 events) up and to the right, generally in the direction of the uncertainty
region for the 2013 event. Taking a look at which lines actually cross which regions,
we see that while the 2007 and 2013 error regions are actually distinguishable by their
half-light radii -
since they do not have contour lines that cross them both -
the
2006 and 2011 events are not necessarily distinguishable from the 2013 event by their
half-light radii, since they occupy overlapping regions in half-light radius. The region
of temperature/pressure space occupied by Pluto's atmosphere experiences half-light
77
radius changes for which the gradient is not strictly aligned with the pressure axis,
but rather is diagonal. This feature means that a change in half-light radius is driven
not only by a pressure change, but also by a temperature change, and both must be
considered when using the half-light radius as a proxy.
In addition to the differences we can see in the fits by the half-light radius contours, we may also use the reported uncertainties from the fits to get a sense of how
this particular value constrains the results in pressure/temperature space by examining how the temperature/pressure uncertainties span contour lines. Here, we are
limited to only using the half-light radius uncertainties, since we do not have the
uncertainties for the other two lightcurve characteristics. Still, we note that for the
2006 and 2007 events, the extent of the uncertainties likely makes the constraint on
pressure slightly stronger than what was provided by the corresponding fits, since
the region spanned by their reported half-light uncertainties of
4 km is thinner
than what is spanned by the diagonal of the temperature/pressure uncertainty box.
On the other hand, the 2011 and 2013 events have half-light uncertainties that are
consistent with the contours spanned by the limits of the fitted uncertainties in temperature and pressure. However, given the fact that there are only four instances
where we can make this comparison, it is probably too little information to make
a claim as to whether or not the half-light radius is generally more constraining
or not, particularly since the contour slices actually extend well past the extent of
the uncertainty boxes when we cannot constrain them with additional lightcurve
characteristics (as is the case here).
We can conclude then, that although the half-light radius is perhaps not the best
proxy for pressure changes in general due to the weak variability in its value with
changes in pressure, there is enough of a slope in the contour lines in the region of
the parameter space occupied by fits to Pluto's atmosphere to suggest a difference
in pressure in the history of these occultation events purely from their shadow-plane
~ 78
Med Prswe Over Time
Haff-LUgt Radius Over Time
~-
0.4-
1220
0.3
Dt
1209
a-
-j
* 11W
S
I
0.2
a-
-
------------
*; visa
0
.
1IM0
19M5
2100
200$
9.. 0I010
law
2010
99
2D00
2005
2010
Year
Figure 29: Shadow-plane half-light radius and atmospheric pressure at 1275 km from
fitted results over time, using either b = 0 or b = -2.2 fits, following Table 6.
half-light radii, as long as where the associated fits place them on the temperature
axis is also considered.
The contour lines indicate that the 2011 and 2013 events
are distinguishable purely from their half-light radii, though other pairs are not
necessarily distinguishable (excluding the 1988 and 2002 events), which is consistent
with the uncertainties seen in Figure 29.
79
Half-Light Radius
0.3
.
. .
2007
,-
2011
---
-.
0.2
CL0.20
0.15
0.10
0.5 90
-100
- - ----110
120
T (K)
Figure 30: Extent of the fitted temperature and pressure results for three occultation
events, overlaid on contour lines of shadow-plane half-light radius for a b = -2.2
gradient atmosphere. Contour lines are spaced out at a rate of 10 km per line. The
dashed boxes represent isothermal fits, rather than b = -2.2 fits.
6
Full Pairwise Comparison of Lightcurves
In the previous sections, we performed a dimensionality reduction of model lightcurve
features from the total number of sampled p/flux points that specify a lightcurve
down to three scalar values for three representative characteristics. Though we have
examined the trends across these three characteristics, this kind of dimensionality
reduction is not the only way to look at lightcurve characteristic trends.
In addition to measuring characteristics individually, another approach to quantifying lightcurve similarity is to do a full comparison between pairs of lightcurves.
Overall similarity apart from the characteristics we have been discussing, can be an
issue when fitting lightcurve models to data. As suggested by the contour plots in
the previous sections, it is quite possible for lightcurves that come from very different
atmospheres to have some very similar characteristics when we are only measuring
80
Lrptrtcrves for Isothermal Atmospheres at 100 K
LUghtcurves for Isothermal Atmospheres at 50 K,
1210
Reference Pressure (Pa)
00,2
30-4
a 06
* 0.8
03
0
a
0
36
2, '
iS
11
04
10
0
02
-000
-10
0
1000
'0
_ 1000
0
100
2000
rangFigure 31: Lightcurves for five isothermal atmospheres at 50 K, with pressures
the
between
similarity
ing from 0.2 to 1.0 Pa at 1275 km. Note in particular the
flash
lightcurves for 50 K atmospheres for nearly the entire curve up until the central
region.
them at certain points, so here we take this second approach by examining "families" of lightcurves (sets that hold one or more atmospheric parameters constant), to
determine how much they differ from each other on the whole.
To illustrate the effect of the pressure variation across families of isothermal
of
atmospheres, a more detailed atmosphere model (with an atmosphere thickness
test
1800 kin, maintaining a shell density of 5 km/shell) was run for a number of
cases. Figure 31 shows two pure-N 2 isothermal families of model lightcurves, where
one was held at 50 K, and the other was held at 100 K, with pressures ranging from
0.1 to 1.0 Pa at a reference planet radius of 1275 km. These lightcurves are notable
breaks
in that apart from the differences in the central flash region (where the model
down due to its inability to handle an infinite focusing factor), the lightcurves can be
fairly close to each other in the 50 K case, but are less so at 100 K. The lightcurves
follow the characteristic trends seen in the previous sections, but now we may take
a step further in examining their overall similarity to each other.
We calculate the area between pairs of lightcurves in each family as a measureand
ment for similarity between the curves. Lower-area pairs are closer to each other,
for pairs
are thus less distinguishable than higher-area pairs. Table 7 shows the areas
error
of atmospheres under 50 K and 100 K isothermal assumptions. Since numerical
81
is known to error in the central flash region due to the linear interpolation required
of our discrete model, we only compare the area between pairs of lightcurves from
p
=
800 km to p = 2000 km, allowing us to avoid the central flash region.
From the tables, we can see that there is a trend of increasing lightcurve dif-
ferences between pairs that have greater differences in pressure. This trend holds
for both the 50 K and the 100 K atmospheres, and shows that on the whole, there
are increasing differences between the lightcurves when the entire curve is examined,
despite the very slight changes in the three lightcurve characteristic parameters for
isothermal atmospheres of those temperatures. An interesting feature that is revealed
here is in the difference between the area values for the two sets of atmospheres. On
the whole, the 100 K atmospheres show higher areas than the 50 K atmospheres, up
to nearly twice the corresponding 50 K values in some cases. This means that in
general, 100 K atmospheres are easier to differentiate than their 50 K counterparts,
confirming the difference in Figure 31 that may be seen visually between the two
plots.
82
Table 7: Area between for pairs of model lightcurves (one limb, p = 800 km to
p = 2000 km) for atmospheres of different reference pressures (Pa, along rows and
columns). Lower values (red) indicate stronger similarity, and higher values (green)
indicate more difference. Values of 0 indicate complete agreement of models (occurring along the diagonal, since a lightcurve is being compared to itself). Tables are
diagonally symmetric.
Subtable 7a: Area between lightcurves for 50 K isothermal atmospheres at different pressures
1.0
0.9
0.8
0.7
0.6
0S
OA
03
0.2
0.1
3892M1
0.1
0.2
0.3
0.4
17.2U445'II'5
13.31526 17-51162 2094221
14.3145 17.65069 :;M
U3
0.5
0.6
3
0
14.88094 17.52786
13.17165
0.8
0.91
1.0
Subtable 7b: Area between lightcurves for 100 K isothermal atmospheres at different pressures
1.0
0.9
0.8
0.7
0.6
0S
OA
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
5
2&.735"
2A9427,515%
334Bi7
22.57333 28.J0162
33,10311
7,6
24,19721 28,55Z22
21.10831
0.7
0.8
0.9
1.01
7
Haze Effects on Near-Minimum-Flux Lightcurve
Features
Now that we have looked at the clear atmosphere cases, we turn our attention
to atmospheres with haze. One of the concerns when analyzing Pluto occultation
lightcurves is the cause of a change in slope towards the bottom of the lightcurve
(the "bend"), for which one of the candidate causes is a strong thermal gradient
(Eshleman, 1989, Hubbard et al., 1990) where a layer of colder air lies below the
isothermal region, while the other is a lower-atmosphere haze layer akin to the one
seen on Triton (Elliot et al., 1989, 1992). Since the haze models that are used in
83
EY92 simply follow an exponential function of tangential optical depth, and do not
come from a physical model of haze, the specific effects of the haze on a lightcurve
could be difficult to constrain when given our model's flexibility to arbitrarily assign
attenuation values to any part of the atmosphere.
We can qualitatively check the differences between a clear isothermal atmosphere
lightcurve, an atmosphere with a thermal gradient, and an atmosphere with haze.
To do this, a nominal clear, isothermal, pure-N2 atmosphere was generated for a
planet with a surface radius of 1000 km, that extended from the surface to a radius
of 2000 km.
In the nominal case, the temperature was set to be 100 K, and the
pressure at 1275 km was set at 0.15 Pa.
Three variants were created from the nominal case. A thermal gradient atmosphere was generated by an exponential temperature gradient in the lower 200 km of
the atmosphere, where the temperature varied from 20 K at 1000 km (the surface),
to the isothermal temperature of 100 K at 1250 km. We tested two models of haze:
one with a linearly-varying attenuation coefficient (a.), with a value of 2 x 10-
5
at
1000 km that reaches zero at 1250 km, and one with a constant attenuation coefficient, where the value was held at 6 x 10-7 for all shells from 1000 km onward until
it abruptly drops to zero at 1250 km. The specifications for these may be seen in
Table 8, and Figure 32.
Table 8: Parameters used in the lightcurve comparisons. All changes to the isothermal atmospheres started at 1000 km and ended at 1250 km.
Onset Value End Value
Thermal Gradient
20 K
Constant Haze
Linearly-Varying Haze
6 x 10- 7
2 x 10-5
100 K
6 x 10-
7
0
In Figure 33, we see that the thermal gradient produces a lightcurve that drops
down lower than the clear isothermal case, as expected, and that the two haze models
match the drop to different degrees. The "bend" in the lightcurve occurs at approxi-
84
Attenuation Coefficient as a Function of Radius
E
2000
1800
0 1600
m constant haze
C
m linearly-varying haze
'O 1400
E
0
1000
''-
0
0.000015
0.00001
5. x 10-6
attenuation coefficient
0.00002
Figure 32: Attenuation coefficient in the two models of haze. Note that attenuation
coefficient is zero in both cases above 1250 km.
mately p = 1200 km, where the variant lightcurves deviate from the clear isothermal
case.
The constant haze model generally follows the thermal gradient lightcurve down
to approximately p = 1000km (though it continues to track the shape of the gradient
lightcurve down to approximately 800 km), while the linearly-varying model follows
it slightly less, down to approximately p = 1100 km. We can also see that any of
the changes from the clear isothermal case will cause a drop in the magnitude of the
central flash.
However, the linearly-varying haze case suppresses the central flash
entirely, while the others do not, which is understandable, given that the lowest part
of the atmosphere where the focusing effect is most prominent is attenuated the most,
given the higher attenuation coefficient used in this model compared to the constant
haze model (Table 8).
The haze models presented here follow a pattern of variation that allows them
to be fully specified by one or two parameters. However, with our discrete forwardmodeling approach, it is possible to individually set the attenuation coefficient of
each shell in the lower atmosphere, allowing for much closer matches to the thermal
85
LUnearty-Varying Haze Case
Constant Haze Case
04
0 4!
02
02
0
o00
1000
1500
200
0 clear Isothermal
k Themnal Gradient
M Hae
0
500
1000
1500
2000
p (km)
p (km)
Figure 33: Lightcurves for clear isothermal, thermal gradient, and haze atmospheres.
Note the differences between the clear isothermal case and the others, and the continued similarity between the gradient and haze cases beyond where they split from
the clear isothermal case.
gradient case. Of course, this is likely to produce haze patterns that are non-physical,
a modeling problem with the haze that we discuss later. Despite the flexibility in
specifying the haze, it appears that the central region of the lightcurve would be
difficult to reproduce with haze of any kind of variation if the thermal gradient
was not severe enough for rather extreme suppression of the central flash, since the
haze directly decreases the magnitude of the light beams with for every amount of
distance that the beam travels through it. At first, it may appear that this difference
would allow the central flash region to play an important role in distinguishing haze
from thermal effects, but in practice, this is not as useful as it seems. Observations
of occultation events rarely have small enough shadow-plane impact parameters to
allow for very deep probing of the atmosphere, where the central flash has a noticeable
effect. Central flashes for occultations by Pluto, for example, do not appear until p
values of approximately 75 km or lower are probed, which has only been done once,
by one site during the 2007 occultation (Olkin et al., 2014).
Thus, simultaneous
observation of lightcurves in different wavelengths remains the best way to distinguish
between haze and thermal effects, since the former will have a wavelength dependence
while the latter will not (Elliot et al., 2007).
86
Part V
Discussion and Future Work
1
Implications for Lightcurve and Uncertainty Characterization
The modeling method presented in this work provides a way to characterize trends in
occultation lightcurve features when the atmosphere of the occulting body is varied
across a parameter space. The flexibility of this method also allows for direct comparison between arbitrarily-specified atmospheres. These features of the model give us
the ability to estimate the uncertainty on the atmospheric characterization directly
from uncertainty in the observation, rather than through the uncertainty in the fitted
model. It also allows us to pinpoint areas in the atmosphere characteristic parameter
space where similar lightcurves might appear, even if these areas are well-separated
from the region where the fitted parameters lie. The latter provides a pathway to
tackle the problem of non-uniqueness in lightcurves produced by distinctly different
atmospheres, something we touch upon in our qualitative examination of problem
with fitting models to the lower portion of Pluto-like lightcurves given the possibility
of haze and/or a strong thermal gradient.
We find that for Pluto-like atmospheres, characteristics of occultation lightcurves
are relatively insensitive to changes in composition and pressure, but are highly
sensitive to changes in temperature. That being said, depending on the specific fitted
values for temperature and pressure, there can be enough of a slope difference in the
constant-value contours of the lightcurve characteristics to define a small uncertainty
region in temperature and pressure space, due to the fact that the contour lines have
slopes of opposite signs at Pluto-like conditions.
87
We also find that in applying
our uncertainty constraint procedure on reported fits 2011 and 2013 lightcurves, we
obtain uncertainty bounds that are consistent with the reported uncertainties for the
fits, lending some credibility to the reliability of our method.
The trends revealed by the pairwise comparisons follow the general trends seen
in the lightcurve characteristic contours, indicating that the latter are a suitable
method for measuring differences between atmosphere models.
The two methods
offer distinct advantages and disadvantages when exploring the parameter space.
The contours are useful in that they act as a tool to reduce the dimensionality of
the data, making each lightcurve a vector of as many characteristics as we wish to
examine (3-element vectors in our case), and allow us to easily compare features
across groups of lightcurves. In contrast, the pairwise comparison does not reduce
the dimensionality of the data at all, but compares lightcurves more directly, offering
a comparison advantage at the cost of obscuring trends in specific features.
2
Future Work
Although the forward model presented here has the potential for significant advantages over an analytical model as far as flexibility in specifying atmospheric structure
goes, it is clear that there are a number of outstanding issues with the model. From
testing the model it was shown that there are inherent differences between our model
and the established EY92 model in the non-baseline lightcurve region, generally giving residual values of 1-2% in situations where they should agree -
a variation that
makes it unsuitable for characterizing actual lightcurve data. We acknowledge that
though the trends in general lightcurve characteristics when traversing the parameter
space hold, the remaining issues with the model need to be addressed before it may
be used for fitting.
We have shown that increasing the thickness of the atmosphere improves our
88
model's ability to simulate the central flash, but neither it nor increasing the interpolation order seems to bring us closer than about 1% difference from the EY92
model. A number of potential avenues of exploration are proposed in Part III, Section 3. Such methods include simulating an extremely thick atmosphere model, on
the order of ten times the usual thickness we use for Pluto-like occulting bodies,
using spline (or some other kind of interpolation) rather than simple polynomial interpolation on the shadow plane when calculating values between sampled points,
and examining the path of a single beam through our model in comparison to the
same beam going through the EY92 model, where multi-beam effects such as beam
density changes and focusing have been removed.
Apart from that correction, there are a number of modifications that may be
made to further generalize the model and remove certain assumptions that it must
currently follow. First, the forward model only allows a single dimension (r) for
atmospheric parameters to vary along. Clearly, the fact that the atmosphere is only
discretized along the radial direction means that the model is strictly adherent to
the spherical symmetry assumption for the atmosphere. An extension to allow for
azimuthal discretization would make it possible to model effects such as the suspected
Rossby waves detected in Pluto's upper atmosphere by Person et al. (2008).
A further, more powerful generalization of the model would be to make it truly
three-dimensional. Right now, the forward model still draws heavily from its heritage in the Baum and Code and Elliot and Young occultation models, and retains
the property that it remains a cylindrical model that is turned into a spherical one
by a focusing factor approximation. Being a cylindrical model inherently limits atmospheric specification to at most two dimensions (if the aforementioned azimuthal
discretization was implemented), a limitation that would be removed by this generalization. In a full three-dimensional instance with latitudinal and longitudinal
discretization, general circulation models of the atmosphere could be implemented
89
on the occulting body, allowing for specific atmospheric patterns to be implemented
and tested. However, three-dimensional specification of the atmosphere is not the
only advantage here. The extension would expand the source and shadow lines used
in the current model to be full source and shadow planes. The focusing factor approximation could then be removed and replaced with the flux interpolation across the
entire shadow plane that contributes all appropriate sources of light coming from all
sides of the occulting body. Additionally, data could be matched to moving observers
(e.g., airborne observatories) by moving the observer in the shadow plane in the same
pattern as the actual data was collected, rather than projecting all the coordinates
onto p, a consideration which comes into play for asymmetric atmospheres.
Apart from the model itself, the parameter-space exploration that we have done
here hint at at least two additional avenues of future research. The flexibility of our
model makes it particularly suitable for tackling the lightcurve uniqueness problem,
though the issues of haze and the way that lightcurves models are currently fit to
data need additional work in order for uniqueness to be addressed in a better way.
A model of haze layers that might be found in tenuous planetary atmospheres would
help to greatly constrain the possible search space for fitting lightcurve models when
haze is invoked as an effect in the atmosphere. Of course, since our model currently
assumes spherical symmetry, and it is quite likely that haze sources are unevenly
distributed on planetary bodies, the three-dimensional extension to the model may
be a prerequisite to the proper application of haze constraints.
Secondly, we have
noted that given the proper atmospheric parameter space, it seems possible for similar lightcurves to appear in separated regions of the space.
Currently, lightcurve
model fitting is done using a linear least-squares method, which is very sensitive to
local minima. If multiple minima exist for a particular dataset case for non-unique lightcurves -
as would be the
linear least-squares would not necessarily find the
global minima, nor would it reveal the existence of multiple minima. More extensive
90
characterization of the lightcurve contours across different forms of the atmospheric
parameter space would be necessary to show whether or not this would actually be
a problem for any particular planetary atmosphere. If it were, a randomized search
approach, such as particle swarm optimization or rapidly exploring random tree may
be tried as possible alternative fitting methods.
91
Appendix: Coordinate Conventions
locations of various parts
Three reference planes are generally used to communicate
plane, and the shadow
of a stellar occultation event: the source plane, the planet
together, and give their
plane. Here, we tie the coordinates in the three planes
way that is consistent with
relations to various physical locations and distances in a
used in this project.
existing literature, and that is consistent with the conventions
possible to avoid confusion.
Various approximations are also explicitly stated where
planes are parallel to
The reference planes are pictured in Figure 34. The three
origin of the planes occurs
each other and orthogonal to the atmosphere cutout. The
that is perpendicular to
at the line joining them and the center of the occulting body
and Code cylindrical
all three planes. The circular cutout is analogous to the Baum
atmosphere model when extended in the z direction.
source line
circular atmosphere cutout
shadow line
rIp
source
plane
shadow
plane
planet plane
z
in stellar occultation
Figure 34: The three reference planes used to describe locations
reducing the problem to two
events. The source and shadow lines are a result of
the occulting body.
dimensions. The shadow plane cuts across the center of
92
A
The Source Plane
The source plane is the origin location for the light coming from the occulted star (the
aforementioned "source").' Due to the distances of stars from the Earth compared to
all other relevant distances in an occultation event, for the purposes of occultations
the occulted star (and thus, the source plane) is located at an infinite distance from
the shadow plane. For calculating events in time, the source plane may be thought of
as entirely lit (sending light towards the shadow plane from all points simultaneously)
as the observer moves through the shadow plane collecting different amounts of flux.
Alternatively, the occulted star may be thought of as moving through the source
plane -
approximated as either a point source or a ray bundle, depending on the
angular size of the star -
while the observer remains stationary on the shadow plane.
The modeling approach used in this project takes the former view as far as the code
goes, along with the point-source approximation for the star.
The distance r, which is the projection of a radial point on the occulting body to
the source plane, is the primary unit of concern. Since it is a straight-line projection,
the source-plane r is the same distance as the planet-plane r, which allows us to
directly use the formulation for the focusing factor given by Elliot, Dunham, Bosh,
Slivan, Young, Wasserman, and Millis (1989). The source plane may alternatively be
defined by right-handed Cartesian coordinates, with
+
pointing towards the center
of the occulting body, and the y coordinate being taken as the direction along which
the occultation event occurs, in order to simplify coordinate considerations for the
event. The source line is the slice of the source plane that is considered when the
occultation is reduced to the two-dimensional case, in the same way that Baum and
Code (1953) constructed their occultation model.
Since our approach models all
occultation events as central transits, the origin of the Cartesian system is simply
at the center of the projection of the occulting body to the source plane. Thus, for
this model, the source line is always on the y-axis, a condition that also means that
93
distances in y are equal to distances in r.
B
The Planet Plane
The planet plane is parallel to the source and shadow planes, cutting across the center
of the spherical occulting body. In previous work, all refraction was assumed to occur
at this plane, which is not the case in this atmosphere model (Baum and Code,
1953, Elliot and Young, 1992). The atmosphere cutout is the slice of the occulting
body taken by the ray path of the occulted star during an occultation event, and is
perpendicular to the three reference planes. The atmosphere, as well as the movement
of light rays through the atmosphere, are both specified on the atmosphere cutout.
For specifications of atmosphere shells, r is typically used, representing the vector
from the center of the occulting body to the location of the interaction, with shell
locations being given at various values of
I1r
. However, for beam-shell interactions,
Euclidean x and y are typically used on an "atmosphere plane" where the cutout
lies. An extension of the cutout to infinity would be equivalent to the cylindrical
treatment of the atmosphere used by Baum and Code.
C
The Shadow Plane
The shadow plane (also called the observer plane in some literature) is the righthanded Cartesian plane at Earth. In the Pluto case, a plane is assumed because
both the curvature and the radius of the Earth is small compared to the EarthPluto distance.
In the two-dimensional case, the slice of the shadow plane under
consideration is the shadow line, which is on the y-axis of this plane.
For stationary observers, the shadow line is simply a straight line, but for moving
observers (e.g., airborne observatories), the line curves depending on the path of
the observer as the distance from the observer to the center of the occulting body's
94
shadow changes.
Matching the observations of moving observers is still possible,
since here, p is the primary value of concern, and is the distance from the center
of the shadow to any point on the shadow plane.
Because it is a scalar, it does
not necessarily have to lie on a straight shadow line, so for moving observers, the
coordinates of the data gathered along their path are simply given the corresponding
p values. When p is used for plotting lightcurves, negative values denote the ingress
portion, while positive values denote the egress limb.
95
Appendix: Software Documentation
A
Introduction
This appendix contains the detailed specifications of the software used for forward
modeling of tenuous planetary atmospheres and resulting stellar occultation lightcurves.
The code for this project was written entirely in Mathematica 10, and does not have
any other platform dependencies. Atmosphere model and synthetic lightcurve generation code is entirely contained in the Mathematica notebook occultationForwardModeling.nb, which can be used by itself. Please note that while this documentation
covers all major functions of the code, some minor functions (that serve as computational aids, transforms, or are expressions of physical laws, for example) are
not described here. All functions are, however, commented with at least a brief
description of their role in the code itself.
By default, the code uses linear variation in atmosphere parameters, though for
some instances of tests done for this project, a second version of the code was used
that allowed for exponentially-varying parameters. Here, we will refer solely to the
linearly-varying cases for simplicity.
B
Quick Start Guide for Fitting
The forward-modeling code may be used in conjunction with the Planetary Astronomy Lab's Least Squares Fitting routines to fit atmosphere models to lightcurve
data. The follow steps describe how to use the two pieces of software together.
1. Run initialization cells in occultationForwardModeling.nb.
2. Run initialization cells in Least Squares Fitting notebook.
96
3. In the Least Squares Fitting notebook, import the test data into the testLC
variable as a n x 2 table with n data points comprising of the shadow line
y location in the first column, and the normalized flux value in the second
column.
4. Import the fluxes from testLC to the variable lsData.
5. Define the occultation conditions: distToObserver (the distance from the occulting body to the observer, km), and MPlanet, the mass of the occulting body (otherwise, they are defined using the default values stored in the
constants.xlsx file when occultationForwardModeling.nbis initialized..
6. Set lsUseModel to use the appropriate forward model. Models already exist for
1-, 2-, and 3-section atmospheres (fcnlSection, f cn2Section, and fcr3Section,
respectively). Additional types of atmosphere models may be constructed as
necessary, and relationships between atmospheric section boundary conditions
defined as needed.
7. Set lsNames and initialValues to the appropriate initial parameters for the
chosen model, and set lsChangeFracand lsStep accordingly.
8. Run fitting according to jleGroup package methods, but ensure that the correct
atmosphere model is used in the end when plotting the resulting atmosphere.
C
Atmosphere Generation
The atmosphere generation portion of the software takes in a set of boundary locations in the atmosphere, along with the associated boundary conditions for those
locations, and generates an atmosphere that matches those conditions.
Multiple
atmospheres may be generated by specifying a variable parameter and the range
97
and resolution of the variation. In general, makeAtmosphere[] is called with boundary conditions supplied as arguments, which then uses makeAtmosphereSections[] to
generate specific sections, which are joined by joinAtmosphereSections[].
C.1
Specifying an Atmosphere
An atmosphere is constructed using a set of the following conditions:
" radii (boundaryR) - a list of boundary locations where atmospheric properties
are set; units in meters
- radii are specified from least to greatest, with the least assumed to be the
surface of the planet, and the greatest assumed to be the outer edge of
the atmosphere
" temperatures (boundaryT) - a list of temperatures at the boundary locations;
units in Kelvin
" mean molecular weights (boundaryM) - a list of mean molecular weights at the
boundary conditions, used as a proxy for atmospheric composition; units in
kilograms per mole
" number of shells per section (sectionShells) - a list of the number of atmosphere
shells to be generated between the boundary locations; dimensionless integers
" reference pressure (P0) - a single number specifying the pressure at a reference
radius given by the global variable rPRef, which is the only place where pressure is specifically set (all other pressures are calculated assuming hydrostatic
equilibrium); units in Pascals
o attenuation coefficient (boundaryaz) - a list of coefficients that act as a proxy
for the amount of haze in the atmosphere at the boundary locations; dimensionless numbers
98
"
mean polarizability (boundarya) - a list of mean polarizabilities of the atmosphere at the boundary locations, which are a function of the atmospheric
composition; units in m 3
"
planet mass (MPlanet) - mass of the underlying planet; units in kilograms
The number of radii, temperatures, mean molecular weights, and attenuation coefficients must be the same (integer values > 1), while the number of values for
number of shells per section must be one fewer. If there are n distinct sections of the
atmosphere, there will be 6n+7 total parameters that specify the full atmosphere.
A full atmosphere is given by lists of radii, temperatures, mean molecular weights,
pressures, number densities, attenuation coefficients, and refractivities.
Tempera-
ture, mean molecular weight, and attenuation coefficient are either constant or vary
linearly between boundaries, according to specifications. Pressure and number density are determined numerically from the other state variables according to the ideal
gas and hydrostatic equilibrium assumptions, refractivity is determined from the
Lorentz-Lorenz equation, and attenuation is specified directly.
C.2
Function Descriptions
function makeAtmosphere[boundaryR, boundaryT, boundaryM, sectionShells, PO, boundarya z, boundarya, MPlanet, exportFolderPath=Null,
atmosphereNumber=Null]
Constructs and optionally exports an atmosphere according to a set of boundary conditions. Returns a list of lists representing the constructed atmosphere's properties
at arbitrary resolution.
* boundaryR, boundaryT, boundaryM, sectionShells, PO, boundary az, boundarya,
and MPlanet are explained in Subsection C.1
99
"
exportFolderPath - a string pointing to the folder where generated atmospheres
are to be saved
- generated atmospheres are saved in .tsv format
- the default value for this is null, and if a path is not supplied, the result
will be returned, but not exported
" atmosphereNumber - an integer which represents the number to append to the
file name of the generated atmosphere if it is exported
function makeAtmosphere[params, exportFolderPath=Null, atmosphereNum-
ber=Null]
Alternate form of the previous function, where params is the list {boundaryR, boundaryT, boundaryM, sectionShells, PO, boundaryaz, boundarya, MPlanet}. Returns
and exports the same way as the previous version does.
function makeAtmosphereSection[rO, ri, MPlanet, sectionShells, TO, T1,
MO, Ml, PO, azO, azl, aO, al]
Constructs a section of an atmosphere based on supplied boundary conditions. All
boundary conditions that are given for both the start and the end of the section (radius, temperature, mean molecular weight, and attenuation coefficient) are assumed
to either remain constant or vary linearly. Pressure is determined by the boundary
condition at the base of the section, which is numerically integrated upwards for each
subsequent point (shell) in the atmosphere, following the ideal gas and hydrostatic
equilibrium assumptions and using the temperature provided at any given point in
the section. Returns a list of lists representing the constructed atmosphere section's
properties at a resolution specified by sectionShells.
* rO, ri - start and end radii of the atmosphere section; units in meters
100
*
MPlanet - mass of the underlying planet; units in kilograms
" sectionShells - the number of shells (points where atmospheric properties are
to be calculated) that this section has; dimensionless integer
" TO, T1 - temperatures at the start and end radii; units in Kelvin
" MO, M1 - mean molecular weights at the start and end radii; units in kilograms
per mole
" P0 - pressure at the start radius; units in Pascals
" &zO, azl - attenuation coefficients at the start and end radii; dimensionless
numbers
3
" &o , al - a list of mean polarizabilities at the start and end radii; units in m
function joinAtmosphereSections[sections]
Given a list of atmosphere sections (the outputs of makeAtmosphereSection[]), this
function returns a list of lists that joins the corresponding lists of properties together.
function makeAtmosphereSet[baseAtm, exportFolderPath, variableProperty, initNum=1I
Constructs and exports a set of atmospheres that have one linearly-varying boundary
parameter. Returns null.
o baseAtm - a list of atmospheric parameters {boundaryR, boundaryT, boundaryM, sectionShells, PO, boundaryaz, MPlanet}
- this set of parameters acts as the base from which all the other atmospheres will be generated, since only one parameter value will be varied
at a time
101
"
exportFolderPath - a string pointing to the folder where generated atmospheres are to be saved
- generated atmospheres are saved in .tsv form
" variableProperty - a list containing the following {var, arrayLocation, min,
max, stepSize}
- var is a string that specifies which atmospheric property is to be varied,
and may be any of the following: r, t, m, s, p0, az, or MPlanet
- var is case-insensitive
- arrayPosition is positive integer that specifies the array position of the
boundary for which which var is the one to be changed
- min, max, and stepSize represent the range and resolution over which to
generate atmospheres
*
if the variable property is radius, then min and max will be constrained by the values of the radii in the array positions immediately
before and after it, which must be less than and greater than its value,
respectively
" initNum - an integer which represents the first number to append to the
file names of the generated atmosphere, to be incremented by one for each
subsequent atmosphere
D
Lightcurve Generation
The lightcurve generation portion of the software takes the description of an atmosphere from the Atmosphere Generation portion and uses it to create an atmosphere
model consisting of discretized shells, each representing atmospheric conditions at a
102
given location. Discretized beams of light (represented by vectors) are then passed
through the atmosphere, where interaction with each shell alters their direction (via
refraction) and magnitude (via absorption). The pattern of vectors emerging from
the other side of the atmosphere is then collected and used to create a synthetic
lightcurve at the shadow plane for the specified atmosphere.
D.1
Atmosphere Model Function Descriptions
function shell[r, n, az]
A shell is the basic description of atmospheric properties at a given radius.
The
radius of the innermost shell is assumed to be the surface radius of the planet for
the purposes of occultation. Returns a list containing the following: a Mathematica
Circle object of radius r centered at (0,0), a mathematical description of the shell
geometry in the form of x 2 + y 2
=
r 2 , the radius of the shell, the index of refraction
of the shell, and the attenuation coefficient of the shell. An "atmosphere" object is
composed of an ordered list of shells, sorted from least to greatest radius.
" r - a number describing the radius of the shell
" n - the index of refraction at that shell
" az - the attenuation coefficient of the shell
function makeAtmosphereFromOpticalParams[params, rArrayPos =
1,
AfArrayPos = 8, azArrayPos= 6]
Given a table of atmospheric properties, returns an atmosphere model made using
the specified properties.
* params - a table of atmospheric parameters generated by the Atmosphere
Generation portion of the software
103
- if the Import function is being used, the first line of the file should be
excluded (using "HeaderLines" -+ 1, because it is a description of the
atmosphere parameters)
* rArrayPos, fiArrayPos, azArrayPos are the columns in the params file
that correspond to radius, refractivity, and attenuation coefficient
D.2
Graphics Functions
function makeAtmosphereGraphic[atm]
Given an atmosphere object (a sorted list of Shells), returns a Graphics object consisting of a set of concentric circles that are of proportionally appropriate sizes for
each shell.
function makelntersectionPointsGraphic[intersections]
Given a list of points where light beams originated or intersected with an atmosphere
shell, returns a Graphics object consisting of all these points, connected appropriately
to represent the movement of vectors from shell to shell.
D.3
Ray Tracing Functions
function moveLightRay[rayPath, rayAngle, atmi
Given the past and present state of a light ray, and the atmosphere it is passing
through, updates the position of the light ray, which is either the next shell that it
intersects, or a position on the observer plane, the angle of the vector (by calling
refract[]), and the magnitude of the vector (calculated via the Beer-Lambert law).
Returns null.
104
"
rayPath - a list of {x,y} points of previous intersection locations of the light
beam
" rayAngle - the current angle of the light beam in radians, where zero is defined
to be normal to the light source plane towards the observer plane (left on all
ray tracing graphics)
" atm - an atmosphere object (list of shells) that the light is interacting with
function refract[rayPath, rayAngle, nI, n2]
Given the past and present state of a light ray, and the index of refraction of the
shell it is currently interacting with, returns the new angle of the light ray.
" rayPath - a list of {x,y} points of previous intersection locations of the light
beam
" rayAngle - the current angle of the light beam in radians, where zero is defined
to be normal to the light source plane towards the observer plane (left on all
ray tracing graphics)
" n2 - index of refraction for the atmosphere shell that the ray is entering
function rayTrace[beamToShellRatio=0.5, maxLightY = 3, minLightY
01
Creates a set of unit light vectors on a light source line behind the planet (representing
the occulted star at infinite distance) and calls moveLightRays[] and refract[] to
model light movement through the atmosphere of the planet.
Sets a number of
global variables that represent the eventual positions, magnitudes, and directions of
the light once it reaches the observer plane. Returns null.
105
* beamToShellRatio - the ratio of beams generated to atmospheres shells per
unit distance in the atmosphere (in the vertical direction)
- this ratio defines the sampling rate of the atmosphere, and is initially set
to 1 beam for every two shells
- depending on the atmosphere, some ratios may cause numerical jitter, as
explained in Part III, Section 4.2; 0.5 was found to be a good ratio for
Pluto-like atmospheric conditions
" maxLightY - the initial factor that is multiplied by the largest atmosphere
shell to define the y position of the first light ray in the set
- rays starting at y positions greater than the largest shell (maxLightY >
1) are required to define a baseline value for the lightcurve, since they do
not interact with the atmosphere
- rays are generated from greatest y position to least
" minLightY - the minimum multiplication factor to define the y position of
the set of light rays
- this number may not actually be reached, depending on what it is set to
be, since the function terminates after the first light ray hits the planet
surface (the innermost atmosphere shell)
D.4
Lightcurve Generation Functions
function interpolateLC[interpolationOrder = 11
Given the global variables set by rayTrace[], four InterpolatingFunctions are returned,
which represent pairs of light beam density and light beam magnitudes as a function
of y on the observer plane. These functions exist as pairs because each function in
106
each pair represents the beams from one limb of the occultation (from y position
parameterized by maxLightY down to the planet surface).
* interpolationOrder - sets the interpolation order for the beam magnitude,
beam density, and focusing factor for lightcurve generation on the shadow
plane; see Section 4.2 for discussion on interpolation order effects
function twoLimbLC[var, varIsRho
=
True]
Given a position on the observer plane, returns a normalized flux value received at
that position. The position var may be given in distance from the shadow center p
(default), or in y, the distance along the shadow line. If it is given in y, the default
argument of varIsRho must be changed to False.
function makeLightCurveGraphic[
Determines the domains of the interpolation functions, the appropriate way to combine the interpolations (based on whether or not they are overlapping) and calls
twoLimbLC[] with appropriate bounds on y to generate the full lightcurve.
getLCProperties[spikeLimit = 0, innerLimit = 0]
Returns a list containing the half light radius, the minimum normalized flux value,
and the slope at half-light for a lightcurve.
The spikeLimit and innerLimit are p
values that may be set to avoid the atmosphere-edge boundary spike and the focusing
factor, which in some cases results in an incorrect reading of the half-light radius, if
they cross 0.5 flux.
107
References
WA Baum and AD Code. A photometric observation of the occultation of sigma
arietis by jupiter. The Astronomical Journal, 58:108-112, 1953.
Amanda S Bosh, MJ Person, SE Levine, CA Zuluaga, AM Zangari, AAS Gulbis,
GH Schaefer, EW Dunham, BA Babcock, AB Davis, et al. The state of pluto's
atmosphere in 2012-2013. Icarus, 246:237-246, 2015.
Dale P Cruikshank, Ted L Roush, Tobias C Owen, Thomas R Geballe, Catherine
De Bergh, Bernard Schmitt, Robert H Brown, and Mary Jane Bartholomew. Ices
on the surface of triton. Science, 261(5122):742-745, 1993.
JL Elliot and LA Young. Analysis of stellar occultation data for planetary atmospheres. i-model fitting, with application to pluto. The Astronomical Journal, 103:
991-1015, 1992.
JL Elliot, RG French, KJ Meech, and JH Elias. Structure of the uranian rings.
i-square-well model and particle-size constraints. The Astronomical Journal, 89:
1587-1603, 1984.
JL Elliot, EW Dunham, AS Bosh, SM Slivan, LA Young, LH Wasserman, and
RL Millis. Pluto's atmosphere. Icarus, 77(1):148-170, 1989.
JL Elliot, A Ates, BA Babcock, AS Bosh, MW Buie, KB Clancy, EW Dunham,
SS Eikenberry, DT Hall, SD Kern, et al. The recent expansion of pluto's atmosphere. Nature, 424(6945):165-168, 2003a.
JL Elliot, MJ Person, and S Qu. Analysis of stellar occultation data. ii. inversion,
with application to pluto and triton.
The Astronomical Journal, 126(2):1041,
2003b.
108
John L Elliot, MJ Person, AAS Gulbis, SP Souza, ER Adams, BA Babcock,
JW Gangestad, AE Jaskot, EA Kramer, JM Pasachoff, et al. Changes in pluto's
atmosphere: 1988-2006. The Astronomical Journal, 134(1):1, 2007.
VR Eshleman. Pluto's atmosphere: Models based on refraction, inversion, and vaporpressure equilibrium. Icarus, 80(2):439-443, 1989.
Candice J Hansen and David A Paige. Seasonal nitrogen cycles on pluto. Icarus,
120(2):247-265, 1996.
WB Hubbard, RV Yelle, and JI Lunine. Nonisothermal pluto atmosphere models.
Icarus, 84(1):1-11, 1990.
Bishun N Khare, Carl Sagan, ET Arakawa, F Suits, TA Callcott, and MW Williams.
Optical constants of organic tholins produced in a simulated titanian atmosphere:
From soft x-ray to microwave frequencies. Icarus, 60(1):127-137, 1984.
Catherine B Olkin, Leslie A Young, Richard G French, Eliot F Young, Marc W
Buie, Robert R Howell, Jeffrey Regester, Catherine R Ruhland, Tim Natusch,
and David J Ramm.
Pluto's atmospheric structure from the july 2007 stellar
occultation. Icarus, 239:15-22, 2014.
Tobias C Owen, Ted L Roush, Dale P Cruikshank, James L Elliot, Leslie A Young,
Catherine De Bergh, Bernard Schmitt, Thomas R Geballe, Robert H Brown, and
Mary Jane Bartholomew. Surface ices and the atmospheric composition of pluto.
Science, 261(5122):745-748, 1993.
MJ Person, JL Elliot, AAS Gulbis, CA Zuluaga, BA Babcock, AJ McKay, JM Pasachoff, SP Souza, WB Hubbard, CA Kulesa, et al. Waves in pluto's upper atmosphere. The Astronomical Journal, 136(4):1510, 2008.
109
MJ Person, EW Dunham, AS Bosh, SE Levine, AAS Gulbis, AM Zangari, CA Zuluaga, JM Pasachoff, BA Babcock, S Pandey, et al. The 2011 june 23 stellar
occultation by pluto: Airborne and ground observations. The Astronomical Journal, 146(4):83, 2013.
Agustin Sanchez-Lavega.
An introduction to planetary atmospheres. CRC Press,
2011.
LA Soderblom, SW Kieffer, TL Becker, RH Brown, AF Cook, CJ Hansen, TV Johnson, RL Kirk, and EM Shoemaker. Triton's geyser-like plumes: Discovery and
basic characterization. Science, 250(4979):410-415, 1990.
John A Stansberry, Jonathan I Lunine, and Martin G Tomasko. Upper limits on
possible photochemical hazes on pluto. Geophysical research letters, 16(11):12211224, 1989.
Darrell F Strobel, Xun Zhu, Michael E Summers, and Michael H Stevens. On the
vertical thermal structure of pluto's atmosphere. Icarus, 120(2):266-289, 1996.
RV Yelle and JI Lunine.
Evidence for a molecule heavier than methane in the
atmosphere of pluto. Nature, 339(6222):288-290, 1989.
LA Young, SA Stern, HA Weaver, F Bagenal, RP Binzel, B Buratti, AF Cheng,
D Cruikshank, GR Gladstone, WM Grundy, et al. New horizons: Anticipated
scientific investigations at the pluto system. Space Science Reviews, 140(1-4):93127, 2008.
AM Zalucha and AAS Gulbis. Comparison of a simple 2-d pluto general circulation model with stellar occultation light curves and implications for atmospheric
circulation. Journal of Geophysical Research: Planets (1991-2012), 117(E5), 2012.
110