A grid of polarization models for Rayleigh scattering planetary atmospheres

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Volume 504 / No 1 (September II 2009)

A&A, 504 1 (2009) 259-276

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Table 1:

Reflectivity $I(\alpha )$ , polarization fraction $q(\alpha )$ and polarized intensity $Q(\alpha )$ phase curves for a very deep ( $\tau =30$ ) conservative $(\omega =1)$ Rayleigh scattering atmosphere above a perfectly reflecting Lambert surface (surface albedo $A_{\rm S}=1$ ).
$\alpha$ [ $^{\circ }$ ]	$I(\alpha )$	$q(\alpha )$ [%]	$Q(\alpha )$	$a(\alpha)$
2.5	0.795	0.0	0.0000
7.5	0.785	0.4	0.0031
12.5	0.766	1.1	0.0084
17.5	0.740	2.1	0.0155
22.5	0.708	3.4	0.0241	1.85
27.5	0.671	5.1	0.0342	1.86
32.5	0.630	6.9	0.0435	1.87
37.5	0.587	9.1	0.0534	1.89
42.5	0.542	11.4	0.0618	1.91
47.5	0.497	13.9	0.0691	1.94
52.5	0.453	16.6	0.0752	1.98
57.5	0.410	19.3	0.0791	2.03
62.5	0.368	22.0	0.0810	2.08
67.5	0.329	24.6	0.0809	2.14
72.5	0.292	27.0	0.0788	2.21
77.5	0.259	29.1	0.0754	2.29
82.5	0.228	30.7	0.0700	2.39
87.5	0.199	31.9	0.0635	2.49
92.5	0.174	32.5	0.0566	2.62
97.5	0.150	32.5	0.0488	2.77
102.5	0.130	31.8	0.0413	2.95
107.5	0.111	30.5	0.0339	3.16
112.5	0.094	28.6	0.0269	3.42
117.5	0.079	26.2	0.0207	3.76
122.5	0.066	23.4	0.0154
127.5	0.054	20.3	0.0110
132.5	0.043	17.0	0.0073
137.5	0.033	13.7	0.0045
142.5	0.025	10.4	0.0026
147.5	0.018	7.3	0.0013
152.5	0.013	4.4	0.0006
157.5	0.008	2.0	0.0002
162.5	0.005	0.0	0.0000
167.5	0.002	-1.4	0.0000
172.5	0.001	-1.9	0.0000
177.5	0.000

This model approximates well a conservative, semi-infinite Rayleigh scattering atmosphere. Additionally the fit parameter $a(\alpha)$ for the parametrization of the polarized intensity $Q(\alpha )$ (Eq. (5)) is given for relevant phase angles.
The statistical error of the Monte Carlo calculation for $I(\alpha )$ is smaller than 0.001 for all $\alpha$ . The uncertainty of the polarization fraction is less than 0.1% for phase angles between 5 and 165 degrees. Extrapolating the intensity I towards ${\alpha =0^\circ }$ with a quadratic least-squares fit to the first four points ( $\alpha = 2.5^\circ,\ldots, 17.5^\circ$ ) yields a value $I(0^\circ) = 0.7970$ . This agrees with the exact solution $I(0^\circ)=0.7975$ from Prather (1974) to the third digit.

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