There are numerous available radiative transfer models, yet these packages typically require complex installation and compilation procedures
for them to operate, and they are particularly restrictive in operational scope (e.g., planet, type of atmosphere/surface) and wavelength
(e.g., spectral database, file formats). For instance at the internet encyclopedia (Wikipedia), there is an entry for "Atmospheric radiative transfer codes",
listing dozens of packages and their capabilities (wavelength range, geometry, scattering, polarization, accessibility/licensing, etc.).
There have been several attempts to quantify the differences between different packages (e.g., Alvarado et al. 2013),
and a commercial internet facility (www.spectralcalc.com) implements a small subset relevant to Earth/Mars science.
With this online tool, we pursue to provide to the community an accurate and comprehensive (UV/Vis/IR/mm) planetary tool,
allowing to synthesize realistic spectra for a broad range of objects (rocky planets, gas giants, icy bodies, TNOs, comets, exoplanets, etc.),
and assisting with the planning and execution of current and future NASA missions. This is achieved by integrating two core radiative-transfer models:
the accurate and versatile PUMAS atmospheric/scattering model, and the cometary emission model (CEM).
Components of the PSG radiative transfer:
the diagram shows the different components considered by the radiative transfer modules. By performing a layer-by-layer analysis, PUMAS
intrinsically integrates and calculates the different flux contributions across the wavelength grid. For comets,
the molecular calculation is performed separately by CEM from the surface fluxes, and later added to compute integrated fluxes.
Atmospheric radiative transfer models
In order to compute atmospheric transmittances and radiances, the molecular modules ingest the parameters defined in the "atmosphere" section and compute observable fluxes:
Linelist
Capabilities
Reference
PUMAS Planetary and Universal Model of Atmospheric Scattering
PUMAS integrates the latest radiative-transfer methods and spectroscopic parameterizations, in order to compute high resolution
spectra via line-by-line calculations, and utilizes the efficient correlated-k method at moderate resolutions.
The scattering analysis is based on a Martian scattering model (Smith et al. 2009), while the line-by-line calculations
have been validated and benchmarked with the accurate GENLN2 model (Edwards 1992).
Villanueva, G. L. et al., Science, Volume 348, Issue 6231, pp. 218-221 (2015). Smith, M. D. et al., JGR, Vol. 114, E00D03, (2009). Edwards, D. P., "GENLN2: A general line-by-line atmospheric transmittance and radiance model, Version 3.0 description and users guide", NCAR/TN-367-STR, National Center for Atmospheric Research, Boulder, Co. (1992).
CEM Cometary Emission Model
CEM incorporates excitation processes via non-LTE line-by-line fluorescence model at short wavelengths (employing GSFC databases), and ingests
several spectral databases to compute line-by-line LTE fluxes. It operates with expanding coma atmospheres and temperatures lower than 300K, and accurately
computes photodissociation processes for parent and daugther species released in the coma.
Villanueva, G. L. et al., The molecular composition of Comet C/2007 W1 (Boattini): Evidence of a peculiar outgassing and a rich chemistry. Icarus, Volume 216, Issue 1, p. 227-240 (2011) Villanueva, G. L., The High Resolution Spectrometer for SOFIA-GREAT: Instrumentation, Atmospheric Modeling and Observations. PhD Thesis, Albert-Ludwigs-Universitaet zu Freiburg, ISBN 3-936586-34-9, Copernicus GmbH Verlag (2004)
Sampling of the planetary disk
General Circulation Model (GCM) sampling (heterogeneous atmospheric/surface properties and geometries)
Realistic modeling of full disk and transit planetary fluxes requires to capture the heterogeneous properties of
the atmosphere and the surface across the observable disk and the terminator. PSG allows to ingest General
Circulation Model (GCM) 3D data of temperature, pressure and abundance profiles, together with surface properties,
such as albedo. The user can upload atmospheric data from any GCM, after converting the 3D fields (typically in netCDF format)
into a PSG GCM binary file. The 3D and temporal atmospheric data are then aggregated and mapped into a 2D projected
observationally grid that is fed to the radiative transfer modules. PSG currently provides templates and
conversion scripts for several GCM models, including Rocke3D,
Laboratoire de Météorologie Dynamique (LMD),
and the Community Atmosphere Model (CAM). In order to ingest GCM 3D data, these are the steps required:
1) Run the GCM model in your personal machine, and store the resulting 3D fields and parameters into
a GCM formatted file (typically a netCDF file).
2) By employing a script (e.g., python, IDL) convert the GCM netCDF file into a PSG GCM binary file.
Several example conversion scripts are available at the GlobES application site, permitting to
convert netCDF ouput files from Rocke3D, LMD and CAM. This script should be run at the user's local machine.
3) In the GlobES site, upload the PSG GCM file by clicking on "Load GCM data". If the upload process was
successful, the site should display graphically the GCM outputs for the different variables.
4) Select the observing geometry (e.g., transit, direct imaging) directly on the GlobES site, or at the "Target and Geometry" section of PSG, and
verify and update the molecules and aerosols to include in the simulation by visiting the "Atmosphere and Surface" section of PSG.
5) In the GlobES site, click on "Generate 3D spectra". For each position on the planet, the algorithm
will update the atmospheric profiles, surface properties and geometry parameters and will run the radiative transfer.
These results will be integrated employing the appropiate weights. Results will be displayed in the "Home" site of PSG.
Simulation of realistic spectra employing 3D GCM data:
ingestion of 3D atmospheric data is possible in PSG, and this allows to accurately
capture the heterogenity of the atmospheric and the surface across the observable disk and terminator.
For transit observations, the GlobES algorithm will perform radiative transfer simulations
across the terminator, and will integrate the different spectra employing equal weights.
For direct imaging and secondary eclipse simulations, the algorithm performs radiative transfer
simulations across the whole observable disk, and the individual spectra are integrated
considering the projected area of each bin.
Radiative transfer sampling (homogeneous atmospheric/surface properties, heterogeneous geometries)
Simulations that encompass the whole observable disk would require sub-sampling in order to properly
capture the diversity in incidence and emission angles. Specifically, when the FOV is much smaller than
the object disk (e.g., nadir, limb, occultation and looking-up observations),
a single set of geometry parameters is typically sufficient when performing the radiative transfer calculation.
The issue is when the FOV samples a broad range of illuminations and surface properties
(e.g., the FOV is comparable and/or bigger than the object disk); in this case, one would need to compute
radiative transfer simulations over different geometries, which would be then integrated to produce
a single total planetary flux. By default, PSG currently defines one set of geometry parameters
and one radiative-transfer calculation per simulation, yet we integrated into PSG an algorithm
that allows to sub-sample the observable disk into sub-sections of similar incidence / emission angles.
The algorithm takes as input N, or the disk sub-sampling parameter (entered in the target section), which defines the number of angle bins for the
incidence and emission angles. Since these angles range from 0 to 90, a N=5 would lead to a sub-division with
bins of 18° and leading to 22 distinct radiative-transfer regions as described in the figure below.
The computational requirement scales quadractic with N, so a conscious choice has to be made between accuracy
and performance. The default is N=1, leading to typical errors less than 5% in flux, while N=3 reaches typical errors lower than 1%, and N=10 reaches flux accuracy better than 0.1%.
Disk sampling methodology:
when the beam samples a broad range of incidence and emission angles, PSG allows to divide the
observable disk into sub-sections of similar incidence / emission angles. The sampling parameter (N, disk sub-sampling) defines
the number of angle bins for the incidence and emission angles. Typically, N=3 reaches errors lower than 1%, and N=10 reaches a flux accuracy better than 0.1%.
Atmospheres in disequilibrium (non-LTE)
In situations when collissions are infrequent enough to equilibrate the radiative populations of the molecules, an atmosphere is considered to be
in non local thermodynamical equilibrium (non-LTE). When this occurs, the classical equations of Boltzmann distribution for populations
and Planck's function for the source radiation terms are no longer valid. This primarily occurs in the tenous regions of the atmosphere,
where collissions are infrequent and solar radiation is strong and unattenuated, leading to strong emissions as the molecules cascade back to their ground state.
The radiative equilibrium equation for a two-level system can be written as:
where n are the populations of the upper (u) and lower (l) levels, g the statistical weights, T the local kinetic temperature, C_{lu} and C_{ul}
the collisional excitation and relaxation rates respectively, B_{lu} and A_{ul} being Einstein coefficients for photo-absorption and
spontaneuous emission, and ρ the impinging excitation flux. The Einstein coefficients are radiative properties of the levels
in question, while the collision rates are dependent on the local temperature (α √T) and density.
At high pressures, collisions dominate the radiative processes (C ≫ A_{ul} or C ≫ B_{lu}), and the relative
populations of the levels can be simply described by Maxwell–Boltzmann statistics (LTE) dependent on the local kinetic temperature.
In regions with strong external radiative fields (ρ), and more infrequent collisions (C ≪ A_{ul} or C ≪ B_{lu}), the radiative parameters start to define the energy balance of the molecule and its emitting spectrum (non-LTE).
As nicely summarized by Appleby (1990), non-LTE starts to become relevant for methane in the atmosphere of giant planets
(e.g., Jupiter, Saturn, Uranus, Neptune) in the mesosphere at pressures below 0.1 mbar.
In CO_{2} rich atmospheres (e.g., Mars, Venus), the efficient collisional rates for this molecule,
keeps LTE up to much lower densities (> 1 ubar, Lopez-Puertas & López-Valverde 1995).
In the atmospheres of hot-giants, stronger radiation fields will bring this limit deeper into the atmosphere,
while higher kinetic temperatures will keep LTE further up in the atmosphere. Chemical reactions also lead molecules into highly excited non-equilibrium states, from which diagnostic photons are released,
as it is the case for the dayglow O_{2}(^{1}Δ) emission tracking the photodestruction of O_{3} in terrestrial atmospheres (Novak et al. 2002).
Implementation of non-LTE in PSG: the radiative transfer module of PSG, PUMAS, allows to ingest non-equilibrated
populations for the energy levels, and it will use these to compute the opacity and source functions for every ro-vibrational level
at each layer (Kutepov et al. 1998). Specifically, PSG permits to model the most common case of non-LTE, in which the vibrational levels
are assumed to be in disequilibrium while the rotational populations are assumed to be equilibrated. For that purpose, the user must provide vertical profiles of vibrational temperatures [K] (computed relative to the ground-state)
for every isotopologue and vibrational band, in the form of "Tvib[MOL:ISO:IDvib]" (MOL:HITRAN molecular ID, ISO:Isotopologue ID, IDVib: vibrational band ID as
defined for the isotopologue [see vibrational bands in the linelist help]). For instance "Tvib[6:1:7]" would correspond to the v3=1 level of the main isotope of CH_{4}.
Modeling of non-LTE radiation:
As the atmospheres are bombarded by high-energy photons from their partner stars, the upper-layers emit efficiently via non-LTE processes.
This disequilibrium has been observed across many planets in our solar system (shown is the detection of non-LTE methane in Jupiter; Encrenaz et al. 1996; Drossart et al. 1999), and is certainly active in many exoplanets’ atmospheres.
Instrument parameters
In this section, the user defines the desired characteristics of the synthetic spectra (wavelength range, resolution, desired radiance flux, noise performance, etc.).
When observing with ground-based observatories, PSG allows to affect the synthetic spectra by telluric absorption. The tool has access to
a database of telluric transmittances pre-computed for 5 altitudes and 4 columns of water for each case (20 cases in total). The tool can also perform
noise (and signal-to-noise ratio) calculation by providing details about the detector and the telescope performance.
PSG allows to define several type of telescope/instrument modes, while in all cases, the integration of the fluxes is done over bounded
and finite field-of-views and spectral ranges, with no convolutions applied to the fields.
Single telescope: this mode is the classical observatory / instrument optical setup, in which the etendue is defined
by the effective collecting area of the main mirror A_{Tele} and its corresponding solid angle Ω.
Interferometer: this mode is characterized by an etendue that is proportional to the number of antennas (e.g., ALMA), in which
A_{Tele} = n_{Tele}⋅π⋅[D_{Tele}/2]^{2}.
Coronagraph: the model for the coronagraph is relatively simple, and it is mainly intended for identifying regimes of operation.
This mode assumes that the throughput is minimum (1/contrast) within half the inner-working-angle (IWA),
it reaches 50% at the IWA, and the throughput is maximum (100%) at 1.5 times the IWA. Relevant to this mode is also the "Exozodi" level,
which indicates the level of zodiacal dust in the exoplanetary system w.r.t to Earth's zodiacal fluxes.
AOTF+grating: this mode characterizes telescope/instruments in which an Acousto-Optical-Tunable-Filter (AOTF) is located
between the entrance optics and the spectroscopic grating (e.g., ExoMars/TGO). The AOTF can be described with 6-parameters (only the center is required): center [cm^{-1}],
width [cm^{-1}], sidelobes factor, base, gauss width [cm^{-1}], factor. The grating is described with 2 parameters:
center [cm^{-1}, required] and width [cm^{-1}].
LIDAR system: in this mode, the "laser" source is injected in the FOV as an additional radiation flux, with the beam [FWHM]
describing the laser divergence of the source, and D_{Tele} the diameter of the receiver. Two parameters describe
the laser intensity, its peak power [W] and the duty cycle [%] (ON/ON+OFF). The integration time is considered to be for the complete
spectral scan, so an integration time of 1 sec for 100 spectral points, would correspond to an integration time of 10 ms per spectral point.
The standard case assumes that the receiver is colocated with the laser-source for nadir/observatory geometries (with the surface behaving
as a Lambertian), while for limb geometries, PSG assumes that the receiver is located at the other side of the occultation.
When entering a second value in the duty-cycle field, one defines that the receiver/source are also colocated for limb geometries,
with this additional value defining the reflectivity of the reflector at the other side of the occultation.
We have compiled a database of instrument models for a diverse range of telescope/instrument combinations, assisting the user when
defining the basic parameters of the instrument. Additional instrument models are being developed.
Instrument model:
We have identified 14 key parametes that are sufficient to describe the overall capabilities and performance of a particular telescope/instrument combination,
and we are now in the process of compiling a database of instrument models, that will assist the user when
defining the basic parameters of the instrument.
We have developed an echelle simulator for iSHELL (Immersion Grating Echelle Spectrograph) at the NASA Infrared Telescope Facility (IRTF).
It employs a set of ab-initio grating equations and simplified parameters that are accurate enough to describe
the overall behavior of the cross-dispersors and the gratings.
Radiance and wavelength units
PSG allows to compute synthetic fluxes in a broad range of possible units.
The constants used for the conversion are: λ is the wavelength in microns (μm), c is the speed of light (299792458 m/s), ASR is arcseconds^{2} per steradian (4.2545166E+10), h is Planck's constant (6.6260693E-34 W s^{2}), k is Boltzmann's constant (1.380658E-23 J/K),
A_{Tele} is the total collecting area of the observatory (m^{2}, n_{Tele}⋅π⋅[D_{Tele}/2]^{2}), Ω is the field-of-view of the observations (steradian).
F = L ⋅ t_{exp} ⋅ n_{exp} ⋅ A_{Tele} ⋅ Ω ⋅ dλ ⋅ λ ⋅ 1E-6 / hc
pt
Photons measured
F = L ⋅ n_{eff} ⋅ t_{exp} ⋅ n_{exp} ⋅ A_{Tele} ⋅ Ω ⋅ dλ ⋅ λ ⋅ 1E-6 / hc
pm
Irradiance
W / m^{2}
E = L ⋅ dλ ⋅ Ω
Wm2
erg/s / cm^{2}
E = L ⋅ dλ ⋅ Ω ⋅ 1E3
erg
Spectral irradiance
W / m^{2} / μm
E = L ⋅ Ω
Wm2um
W / m^{2} / cm^{-1}
E = L ⋅ Ω ⋅ λ^{2} / 1E4 ⋅
Wm2cm
Jy
E = L ⋅ Ω ⋅ λ^{2} / c ⋅ 1E20
Jy
mJy
E = L ⋅ Ω ⋅ λ^{2} / c ⋅ 1E23
mJy
Relative
Contrast
T = L / L_{stellar}
rel
Transmittance (I/F)
T = L / L_{cont}
rif
Magnitude
m = -2.5 ⋅ log(L ⋅ Ω / L_{Vega})
V
Online unit conversion tool
Input
Output
Radiation value:
From radiation unit:
To radiation unit:
Frequency/wavelength:
Resolving power:
Field-of-view:
Telescope diameter [m, effective]:
1.4498685715765913e-11
Frequency / wavelength value:
From unit:
To unit:
1.0
Telluric absorption
When observing with ground-based observatories, PSG allows to affect the synthetic spectra (or simply show) by telluric absorption. The tool has access to
a database of telluric transmittances pre-computed for 5 altitudes and 4 columns of water for each case (20 cases in total).
The altitudes include that of Mauna-Kea/Hawaii (4200 m), Paranal/Chile (2600 m), SOFIA (14,000 m) and balloon observatories (35,000 m), while the water vapor column was established by scaling the tropical water profile
by a factor of 0.1, 0.3 and 0.7 and 1.
Opacities at 225 GHz, a typical metric to quantify water at radio wavelengths, can be estimated from
the reported water column as τ225_{PSG} = 0.0642 x PWV, where PWV is the ammount of water in precipitable millimeters.
Noise simulator
PSG currently includes a noise calculator for quantum and thermal detectors, with the primary goal of providing users with quick look simulations
for planning observations, and to assist with the development of new instrument/telescope concepts. The user can provide a constant value
across all wavelengths (RMS), or a constant value with background noise (BKG), or can choose from several detector noise simulators.
At short wavelengths (e.g., optical or near IR), the background photon counts follow a Poisson distribution,
and the fluctuations are given by √N where N is the mean number of photons received (see review in Zmuidzinas et al. 2003).
This Poisson distribution holds only in the case that the mean photon mode occupation number is small, n<<1.
For a thermal background, the occupation number is given by the Bose-Einstein formula,
n_{th}(v,T) = [exp(hv/kT)-1]^{-1}, so the opposite classical limit
n>>1 is the usual situation at longer wavelengths for which hv<<kT.
When n>>1, the photons do not arrive independently according to a Poisson process but instead are strongly bunched,
and the fluctuations are of order N, instead of √N.
This is why the Dicke equation is used to calculate sensitivities for the receiver temperature mode (TRX),
which states that the noise is proportional to the background power rather than its square root. The formalism
employed for the TRX module is based on the ALMA sensitivity calculator (Yatagai et al. 2011).
PSG assumes that the instrument has a defined spatial resolution (beam [FWHM]), defined by the user
for the center wavelength. This spatial resolution will change across wavelength (proportional to λ),
and therefore the solid-angle (Ω) will be proportional to λ^{2}. Since Ω is changing with wavelength,
and the instrument has a fixed spatial resolution (defined by the optical design and detector properites),
the number of pixels encompassing Ω will be also proportional to λ^{2}.
Type of noise
Parameters
Detector specific noise formulas
TRX Receiver temperature (radio)
T_{RX} [K]: receiver temperature
g: sideband factor (0:SSB, 1:DSB)
n_{pol}: 1 (number of polarizations)
f_{N} = 1 (number of baselines)
For interferometric systems (e.g., ALMA):
n_{pol}: 2 (assuming dual / full configuration)
f_{N} = n_{tele}(n_{tele}-1)
NETD [mK] at T=300K, f=50 Hz and Δλ=1 μm
S [μm]: pixel size
This value is measured by the detector manufacturer by performing a defined measurement on a source of temperature T (e.g., 300K),
with a repetition f (e.g., 50 Hz). The noise will be dependent on the operating spectral coverage of the detector (e.g., Δλ=1 μm for a 12-13 μm response).
To convert from a NETD obtained with another source temperature (T [K]), sampling frequency (f [Hz]) or detector bandwidth (Δλ [μm]):
In an idealized optical sytem, the incidence etendue is equal to the sampled etendue by the detector.
The tool below allows to perform simple conversions applying this principle:
Inputs
The noise components with Poisson statistics (i.e., UV, optical, IR) are calculated as:
Assuming these parameters and units:
L [W / sr / m^{2} / μm]: spectral radiance of the source
L_{back} [W / sr / m^{2} / μm]: spectral radiance of the background sources
t_{exp} [s]: time per exposure
n_{exp}: total number of exposures
n_{pixels}: total number of pixels for Ω and dλ.
n_{ezo}: Exozodiacal dust scaler relative to Solar System zodiacal dust
T_{optics} [K]: temperature of the optics
ε_{optics}: emissivity of the optics
η_{eff}: total throughput of the system (including quantum efficiencies)
Ω [steradian]: is the solid angle of the observations. It is wavelength dependent.
A_{Tele} [m^{2}]: is the total collecting area of the observatory (n_{Tele}⋅π⋅[D_{Tele}/2]^{2})
λ [μm]: is the wavelength in microns
trn_{ground}: terrestrial transmittance
T_{ground} [K]: temperature of the terrestrial atmosphere - 280
h [W s^{2}]: is Planck's constant - 6.6260693E-34
c [m / s]: is the speed of light - 299792458
k [J / K]: is Boltzmann's constant - 1.380658E-23
Background noise sources:
When observing faint astronomical sources, the sensitivity is affected by the shot noise introduced
by background and diffuse sources (Leiner et al. 1998, A&A, v.127, p.1-99). From space, the background is dominated by the faint
and diffuse emission (thermal and scattered sunlight) from zodiacal dust, while
airglow (a mixture of photoionization emissions, chemiluminescence and scattered sunlight) dominates
the background for ground-based observations. Zodiacal dust fluxes depend greatly on the ecliptic longitude/latitude - in PSG,
the noise simulator considers a scaling of 2 with respect to the minimum ecliptic pole values. PSG also employs a rudimentary (as shown), yet relatively effective, approximation for atmospheric airglow.