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:**

- for each frequency:

* the beam HPBW

* the filling factor

and

* the total flux

If B(x,y) is the local radiance from Mars,

* the associated mean Planck brightness temperature over the planet

beam convolution.

* the flux in the main beam

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

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 is related to antenna temperature T_a* by

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: **

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.

(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.