*Polarization from Illuminated Atmospheres*

As a "toy" application, we consider a planet with an optically thick, Rayleigh-scattering
atmosphere, illuminated by a star (assumed far enough from the planet that we may consider
all the incident rays to be paralel).

The radiation incident upon the atmosphere will span the full range of mu_i = cos(theta_i) values,
where theta_i is the angle between the incoming radiation and the normal to the surface of
the atmosphere. We have thus computed an array of solutions for 22 values of mu_i :
mu_i = 0.0174 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.965 0.9986 1. This allows interpolation to any desired mu_i. The total optical depth of the slab was set to
2T=9 (mid-plane depth T=4.5) and the albedo set to unity. At each scattering, the "photon" is split,
with the fraction that escapes recorded, while the fraction that does not reach the boundary is just
rescattered. Since we keep the emerging Stokes parameters for each of the scatterings, we
can obtain the solution for any single-scattering albedo by simply multiplying the nth set
of Stokes parameters by albedo^n and summing the results.

The solutions were computed with the code Pol_beam.ijs. We ran
groups of 600,000 photons, each scattered 60 times; this was repeated 800 times (i.e., a total of 0.48 billion photons
for each of the 22 mu_i cases.) The outgoing photons were binned into 30 values
of mu and 36 values of phi (over the range of 0-180 deg only due to symmetry).

Here are results PB_tau_4.5 in a form easily read by a J routine:
read_beams45.ijs.

Here is the J code to evaluate the scattered radiation at about 1500 points on
the illuminated disk:
PLANET.ijs.

It should be pointed out that while the radiation which
emerges after one scattering must be polarized perpendicular to the plane defined
by the star and the observer, the multiply scattered radiation will have its
plane of polarization influenced by the local tilt of the surface. Since the
overall tilt of the polarization angle is generally small, the angle has been
multiplied by a scaling factor for these illustrations. The length of the blue
segment indicates the magnitude of the polarization (scale at upper right). The
orientation of the segment indicates the local plane of polarization (but the
deviation from 0 deg - vertical - has been multiplied by the "Pol. tilt scale" factor).

Polarization at 30 deg phase angle.

Polarization at 45 deg phase angle.

Polarization at 60 deg phase angle.

Polarization at 75 deg phase angle.

Polarization at 90 deg phase angle.

Polarization at 110 deg phase angle.

Polarization at 125 deg phase angle.

Polarization at 135 deg phase angle.

(The J code to plot these figures is pol_pict.ijs.)

Since the polarization tilts of the upper and lower hemispheres are opposite,
and thus cancel, the integrated
polarization will be exactly perpendicular to the direction of the
illumination. Here are the resultant values of intensity and polarization integrated
over the visible disk:
Flux and Polarization vs. Phase.

We can average the Stokes parameters over just the upper hemisphere, to see
how the tilt varies with phase:
Tilt vs. Phase.

The tilt of the lower hemisphere is just opposite, so the net polarization
will be exactly vertical. (But when the phase exceeds ~165 deg, the tilt passes 45 deg
and the net polarization from both hemispheres switches from vertical to horizontal.)

Finally, here is a comparison of the brightness of the planet as a function of phase
compared to the "Lambert sphere": Flux vs Lambert
Sphere. (A Lambert sphere is a sphere with a surface such that any incoming beam
of light is reflected with equal intensity in all directions.)