Model history

7 Feb. 2008. v1.0. Presentation to Herschel calibration meeting and availability to community.

14 Feb. 2008. v1.1. Mars oblateness is taken into account. Brightness temperatures are unchanged, and fluxes are decreased by typically 0.6 %.

14 Feb. 2008. v1.2. Decimal hours are allowed.

April 2013. v1.3. A 100x100 grid is used.

Note: The parameters are calculated for Earth-based observations. For observations from any other vantange point,
the fluxes should be rescaled to the appropriate distance.
The brightness temperatures are not significantly different.

Model description

This model calculates the thermal emission of Mars for any date between 1990 and 2020
and any frequencies between 30 and 5000 GHz. Input parameters are:
- Date, including UT time
- The telescope beam (HPBW = FWHM) at a reference frequency of 300 GHz. This is used
to calculate the telescope beam at any frequency assuming it is exactly proportional
to wavelength.
- The surface roughness, expressed in rms degrees of the slopes
- The penetration length of the radiation, expressed in units of the wavelengths
(typically 12-15)
- The surface dielectric constant (typically 2.2-2.5)
- Four frequencies can be calculated at a time.

Output parameters are:

- a number of physical parameters of Mars for the considered date
- for each frequency:

* the beam HPBW

* the filling factor f of Mars in the beam, defined as
f = 1/ (1 - 2** [ -(R_app/HWHM)^2 ] , where R_app is the apparent radius of Mars
and HWHM = 1/2 HPBW is the beam half-width at half-maximum

* the total flux Phitot (Jy) emitted by Mars
If B(x,y) is the local radiance from Mars, Phitot is given by:
Phitot = int_disk B(x,y) dxdy (int_disk is the integral over the disk) and then expressed in Jy

* the associated mean Planck brightness temperature over the planet (Tb) , with no
beam convolution. Tb is given by:
Phitot = B_nu(Tb) * pi * R_app^2 , where B_nu is the Planck function

* the flux in the main beam Phi_mb(Jy), defined as:
Phi_mb = B_mb * pi/(4ln(2)) * HPBW^2 * 1/f

where B_mb is equal to the beam-weighted radiance, i.e.

B_mb = int_disk B(x,y) P(x,y) dxdy / int_disk P(x,y) dxdy
and P(x,y) is the beam pattern: P(x,y) = 2^ (-(x*x+y*y)/(HWHM*HWHM))

* the associated Planck brightness temperature Tb_beam , defined by: B_mb = B_nu(Tb_beam).
At low frequencies when the Planck temperature is identifical to the Rayleigh Jeans temperature, Tb_beam is
the mean (beam-weighted) brightness temperature over the regions of Mars encompassed in the beam.

* the main-beam Rayleigh-Jeans temperature (T_mb) . T_mb accounts for filling effects and is defined by T_mb = Trj(B_mb / f) .
T_mb is related to antenna temperature T_a* by Ta* = T_mb * eta_mb/eta_l , where eta_mb and eta_l are the main-beam and forward efficiencies.

Note: The last two temperatures are different because (i) T_mb accounts for the filling
factor, unlike Tb_beam (ii) T_mb is a Rayleigh-Jeans temperature, while Tb_beam
is a Planck temperature.

Examples:

- if the beam is much smaller than Mars, then Tb_beam will be the brightness
temperature at planet center. If in addition, the frequency is low enough that the
R-J approximation is valid, then Tb_beam = T_mb

- if the beam is much larger than Mars, then Tb_beam will be almost
equal to Tb. T_mb will be much smaller due to the filling factor.

T_mb is the most useful parameter for calibration purposes, since it can be
directly compared to observations.

All parameters relevant to the main beam were calculated by appropriate convolution
with an assumed gaussian beam. By clicking on the appropriate link, the user can
recover the individual local brightness temperatures at one frequency (the first one
he/entered) as a function of RA,DEC offset from disk center. This is useful for the
user to make his own convolution, e.g. for specific beam profiles or size, and/or
for beam profile studies. These temperatures are tabulated on a 100x100 point grid
suited to the size of Mars, and zero values correspond to points outside the disk.
A contour plot of these brightness temperatures can be viewed by clicking
on another link.

Methods

The code uses surface and subsurface temperatures taken from the European
Martian GCM (see http://www-mars.lmd.jussieu.fr, Forget et al. 1999, Millour et al. 2015)
and Martian ephemerides from IMCCE (www.imcce.fr). A standard dust scenario ("Climatology")
is used. For each user-provided date, the code computes the
aspect of Mars. The disk is split on a 100x100 grid, each of them having its
own latitude, longitude, and local time. On each point of the grid,
the usual radiative transfer equation (e.g. Eq. 5 of Rudy et al. Icarus 71, 159,
1987) is used. Radiative transfer in the surface/subsurface includes
a choice of an absorption coefficient, expressed as a radio absorption
length in unit of the wavelength. In addition, the thermal emission of the surface
includes an emissivity term, depending on the dielectric constant (a Fresnel reflection
coefficient description is used), and on the emission angle. An average
emission angle is calculated by averaging over a gaussian distribution of
angles. The width of the distribution is given by the surface roughness (in degrees;
a typical value for the roughness is 12°). Brightness temperatures and local
fluxes are thus calculated and then convolved with the gaussian beam, as explained
above.

Note: the smallest working frequency is 30 GHz. At high frequency, results
(at least the beam-averaged flux and temperatures) become inaccurate
when the beam becomes comparable to the grid mesh, i.e. 1/50 of a planetary diameter.

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