5. Tropospheric scattering

  From earthbound measurements of the night sky brightness the contribution due to tropospheric scattering (see Eq. (1)) has to be subtracted in order to determine its uncontaminated extraterrestrial intensity and polarization. The strongest contributions to scattered light come from airglow, zodiacal light (ZL) and integrated starlight (ISL) - that is, the correction to be applied is in part determined by the brightness distribution of the sources under study themselves. The correction is of the order of , which corresponds to 15% or more of the Zodiacal light, and to typically of the ISL. Due to the limited accuracy to which the correction can be determined, it can be applied explicitly only to measurements aimed at the determination of ZL and ISL. The weaker components of the night sky brightness, DGL and EBL, must be determined by differential methods.

Detailed calculations on first order Rayleigh- and Mie-scattering (including linear and circular polarization) in the (spherical) Earth's atmosphere illuminated by a uniform, unpolarized source, by the Milky Way and by the Zodiacal light were performed by Staude (1975) for various values of the optical thickness of the Rayleigh and Mie components of the atmosphere, and assuming two different values for the refractive index m of atmospheric aerosols (m = 1.33, as for water vapour, and m = 1.5 - 0.1i, as for aerosols in dry air). The position and orientation of Milky Way and Zodiacal Light cone were varied independently over the whole range occurring in practice. Some results from this study are reported in the following.

5.1. A uniform unpolarized source of unit brightness

The brightness of tropospherically scattered airglow can be estimated using the results obtained for a uniform unpolarized source of unit brightness (extending over the entire visible sky) in the single scattering approximation, which are given in Figs. 13 and 14. They give the intensity of the scattered light and its polarization as a function of zenith distance of the observing direction z₀, for different values of the zenith extinction of the Rayleigh and of the Mie component.

 
Figure 13: Intensity and polarization of the atmospheric scattered light in a pure Rayleigh atmosphere, for a source of unit brightness and various values of the zenith extinction , as a function of zenith distance z

 
Figure 14: Same as Fig. 13 for two pure Mie atmospheres

The influence of multiple Rayleigh scattering was estimated using the work of Dave (1964) and of de Bary & Bullrich (1964), who determined the higher order contributions to the scattered light from a point source in a plane-parallel atmosphere. The derived correction factors for the intensity, and for the depolarization of scattered light are given in Table 7. All results for Rayleigh scattering given in the following are corrected for multiple scattering. For Mie scattering, de Bary (1964) concludes that higher order contributions are negligible for scattering angles . Therefore, since the main contribution by atmospheric aerosols to the scattered light comes from regions with , no corrections were applied to the first order results for Mie scattering.

 

Table 7: The correction factors for multiple scattering in a Rayleigh atmosphere for different values of the zenith extinction . See text for details

5.2. The integrated starlight

The integrated starlight scattered in the troposphere was calculated using an analytical model for the extraterrestrial brightness of the ISL: a two dimensional Gauss distribution was fitted to the blue isophotes given by Elsässer & Haug (1960). The constants were adjusted to give a model intensity I₁(l=0, b=0)=260 S₁₀, I₁(l=120,b=0)=I₁(l=240,b=0)=100 S₁₀, and S₁₀. At higher galactic latitudes an exponential decrease was assumed, with S₁₀, following the star counts of Roach & Megill (1961). The assumption of such a smooth brightness distribution is safe even for Mie scattering, since also in this case scattering angles up to contribute substantially to the integrated scattered light. Figure 15 shows the intensity of the scattered ISL as a function of zenith distance for the case that the galactic centre is at the zenith. In Table 8 (click here) the scattered intensity , and its degree and orientation of polarization (in percent) and are tabulated for this situation together with the assumed source brightness I₁ in the viewing direction and the transmitted brightness I₂ weakened by atmospheric extinction. In Table 9 (click here) the same values are given for the galactic anticentre at the zenith. The refractive index of the Mie particles is assumed to be m=1.33.

 
Figure 15: The intensity of the scattered integrated starlight as a function of zenith distance, for different azimuths and zenith extincion values of the Rayleigh resp. Mie components of the atmosphere. The galactic centre is assumed at the zenith, the galactic equator crosses the horizon at

   

l b A z I₁ I₂ I₂

0 0 0 0 260.0 235.3 4.0 10.4 90 213.0 8.2 8.7 90

330 0 90 30 220.8 196.8 4.5 12.9 90 175.4 9.0 11.4 90

333 14 120 30 112.5 100.3 4.4 11.6 69 89.4 8.9 10.2 69

344 26 150 30 57.3 51.1 4.4 8.3 41 45.5 8.8 6.9 43

360 30 180 30 52.3 46.6 4.3 6.0 0 41.6 8.7 4.4 0

300 0 90 60 151.9 124.5 7.0 18.3 90 102.0 13.4 17.4 90

304 26 120 60 54.1 44.3 6.8 15.9 82 36.3 13.1 15.4 82

319 49 150 60 36.3 29.7 6.5 9.9 74 24.4 12.5 10.1 77

360 60 180 60 29.4 24.1 6.4 4.7 90 19.7 12.2 6.1 90

360 0 0 0 260.0 247.4 5.9 0.6 90 223.9 15.8 0.5 90

330 0 90 30 220.8 208.4 5.8 0.7 90 185.7 15.4 0.6 90

333 14 120 30 112.5 106.2 4.9 0.7 67 94.7 13.0 0.7 68

344 26 150 30 57.3 54.1 3.9 0.6 40 48.2 10.2 0.6 42

360 30 180 30 52.3 49.4 3.5 0.5 0 44.0 9.4 0.4 0

300 0 90 60 151.9 137.5 7.2 1.0 90 112.7 17.4 1.1 90

304 26 120 60 54.1 48.9 5.1 1.1 86 40.1 12.3 1.3 87

319 49 150 60 36.3 32.8 3.5 0.9 86 26.9 8.6 1.1 88

360 60 180 60 29.4 26.6 3.0 0.7 90 21.8 7.4 1.0 90

Table 8: Intensity in S₁₀ and polarization of scattered integrated starlight for two pure Rayleigh and two pure Mie atmospheres with the given values of zenith extinction and . The galactic center is assumed at the zenith (z=0), the galactic equator crosses the horizon at A=90, 270; l and b are galactic coordinates

   

l b A z I₁ I₂ I₂

180 0 0 0 100.0 90.5 2.8 9.9 90 81.9 5.6 8.1 90

180 30 0 30 50.6 45.1 3.1 7.0 180 40.2 6.1 5.2 180

196 26 30 30 51.9 46.2 3.1 8.0 144 41.2 6.2 6.5 142

207 14 60 30 67.6 60.2 3.1 9.8 115 53.7 6.2 8.5 114

210 0 90 30 100.1 89.2 3.2 10.6 90 79.5 6.3 9.3 90

180 60 0 60 29.0 23.8 4.7 0.6 180 19.5 8.9 1.5 90

221 49 30 60 35.8 29.3 4.8 5.4 120 24.1 9.1 5.8 111

236 26 60 60 51.9 42.5 5.0 10.2 103 34.9 9.5 10.2 101

240 0 90 60 101.8 83.4 5.1 12.1 90 68.4 9.7 12.0 90

180 0 0 0 100.0 95.1 3.0 0.7 90 86.1 8.0 0.6 90

180 30 0 30 50.6 47.7 2.5 0.7 180 42.5 6.5 0.6 180

196 26 30 30 51.9 49.0 2.6 0.7 148 43.6 6.9 0.6 146

207 14 60 30 67.6 63.8 3.0 0.6 118 56.8 7.9 0.6 116

210 0 90 30 100.1 94.5 3.3 0.6 90 84.3 8.7 0.6 90

180 60 0 60 29.0 26.2 2.6 0.2 180 21.5 6.3 0 90

221 49 30 60 35.8 32.4 3.0 0.2 129 26.6 7.2 0.3 107

236 26 60 60 51.9 47.0 4.0 0.5 101 38.5 9.6 0.6 98

240 0 90 60 101.8 92.1 5.1 0.6 90 75.5 12.3 0.6 90

Table 9: Same as for Table 8, but with the Galactic center at A=0, z=180

5.3. The Zodiacal light

Intensity and polarization of Zodiacal light scattered in the troposphere were calculated assuming the brightness distribution given by Dumont (1965) at Å. For the linear polarization the values measured by Weinberg (1964) at the ecliptic were used, assuming that over the whole sky the polarization is a function of angular distance to the Sun (=elongation , see Sect. 3.5 (click here) alone (Dumont & Sanchez Martinez 1966). The polarization was assumed to be perpendicular to the direction of the Sun.
Figures 16 and 17 show the results for two cases, pure Rayleigh- and pure Mie-scattering (water vapor), respectively. In Tables 10, 11 and 12 the results are collected for three different positions of the Sun below the horizon. The ecliptic is assumed to be perpendicular to the horizon. All other quantities as in Tables 8 (click here) and 9 (click here).

 
Figure 16: The intensity of the scattered zodiacal light for a pure Rayleigh atmosphere with optical thickness . Position of the sun at azimuth , zenith distance , the ecliptic is perpendicular to the horizon

 
Figure 17: The intensity of the scattered zodiacal light for pure Mie scattering with optical thickness , for particles with refractive index m=1.33. Otherwise same as for Fig. 16

 

A z I₁ I₂ I₂

105 0 0 0 158.2 143.2 9.5 32.5 90 129.6 18.5 27.6 90

85 0 90 20 220.3 198.1 10.6 26.5 90 178.2 20.5 22.6 90

89 13 130 20 173.1 155.7 10.3 27.8 48 140.0 20.0 23.5 48

105 20 180 20 133.6 120.1 9.9 31.6 178 108.0 19.1 26.5 177

125 0 270 20 138.6 124.7 10.0 33.2 90 112.1 19.3 28.6 90

65 0 90 40 351.6 308.7 13.7 18.7 90 271.0 25.9 16.6 69

72 24 130 40 170.2 149.5 12.9 20.2 46 131.2 24.3 17.1 47

105 40 180 40 102.4 89.9 11.4 29.0 175 78.9 21.6 23.8 174

145 0 270 40 146.8 128.9 12.7 27.9 90 113.2 23.9 25.1 90

45 0 90 60 865.6 709.4 21.4 13.0 90 581.4 38.5 12.5 90

52 34 130 60 186.8 153.1 19.2 10.8 44 125.5 34.6 9.1 48

105 60 180 60 90.5 74.2 15.8 25.9 171 60.8 28.6 20.5 170

165 0 270 60 164.4 134.8 20.0 20.1 90 110.4 35.9 19.2 90

30 0 90 75 2200.0 1504.5 37.8 11.3 90 1028.9 61.6 11.7 90

34 38 130 75 201.4 137.7 33.1 3.0 45 94.2 54.1 3.1 66

105 75 180 75 78.8 53.9 26.4 23.9 169 36.9 43.3 18.2 166

180 0 270 75 180.0 123.1 36.4 15.2 90 84.2 59.3 15.4 90

105 0 0 0 158.2 150.5 6.5 17.6 90 136.2 17.3 17.3 90

85 0 90 20 220.3 208.9 8.9 18.9 90 187.9 23.4 18.8 90

89 13 130 20 173.1 164.2 7.9 18.8 52 147.6 20.8 18.7 52

105 20 180 20 133.6 126.7 6.3 17.4 4 113.9 16.7 17.2 4

125 0 270 20 138.6 131.5 5.9 12.8 90 118.2 15.6 12.7 90

65 0 90 40 351.6 329.4 16.0 17.5 90 289.2 40.4 17.6 90

72 24 130 40 170.2 159.5 10.9 17.9 52 140.0 27.6 18.0 52

105 40 180 40 102.4 95.9 6.5 17.1 7 84.2 16.6 16.9 7

145 0 270 40 146.8 137.6 6.8 6.7 90 120.8 17.6 6.7 90

45 0 90 60 865.6 783.6 38.4 14.8 90 642.2 88.6 15.1 90

52 34 130 60 186.8 169.1 17.5 15.3 50 138.6 40.7 15.6 51

105 60 180 60 90.5 81.9 7.9 16.6 10 67.1 18.8 16.3 10

165 0 270 60 164.4 148.9 9.9 9.0 90 122.0 23.6 3.1 90

30 0 90 75 2200.0 1819.3 92.3 12.7 90 1244.2 178.9 13.2 90

34 38 139 75 201.4 166.5 29.8 12.7 49 113.9 59.1 13.0 50

105 75 180 75 78.8 65.2 11.5 16.0 12 44.6 23.7 15.5 12

180 0 270 75 180.0 148.9 16.1 2.1 90 101.8 32.7 2.5 90

Table 10: Intensity in S₁₀ and polarization of tropospherically scattered Zodiacal light. The Sun is located at A=90, z=105, the ecliptic is perpendicular to the horizon

 

A z I₁ I₂ I₂

135 0 0 0 141.1 127.7 6.6 16.9 90 115.6 13.1 14.3 90

115 0 90 20 143.2 128.8 7.0 17.8 90 115.9 13.7 15.3 90

119 13 130 20 128.4 115.5 6.9 17.1 53 103.9 13.6 14.5 53

135 20 180 20 120.2 108.1 6.9 15.2 3 97.3 13.6 12.6 3

155 0 270 20 153.6 138.1 7.2 14.7 90 124.2 14.1 12.6 90

95 0 90 40 183.5 161.1 8.4 17.8 90 141.4 16.2 16.0 90

102 24 130 40 129.7 113.9 8.2 16.1 57 100.0 15.8 14.1 59

135 40 180 40 92.9 81.6 8.0 11.8 7 71.6 15.4 9.1 8

175 0 270 40 176.6 155.0 8.8 13.1 90 136.1 17.0 11.9 90

75 0 90 60 273.4 224.1 12.6 16.6 90 183.6 23.1 15.8 90

82 34 130 60 128.5 105.3 11.9 13.6 63 86.3 21.9 12.8 66

135 60 180 60 90.0 73.8 11.1 7.7 16 60.4 20.5 5.2 22

195 0 270 60 164.4 134.8 13.1 12.5 90 110.4 24.0 12.1 90

60 0 90 75 420.0 287.2 22.1 15.3 90 196.4 37.0 15.3 90

64 36 130 75 137.9 94.3 20.6 11.1 70 64.5 34.5 11.4 74

135 75 180 75 78.9 53.9 18.5 5.8 27 36.9 31.2 4.1 42

210 0 270 75 150.0 102.6 22.7 12.9 90 70.2 37.9 13.1 90

135 0 0 0 141.1 134.2 5.4 8.2 90 121.5 14.4 8.1 90

115 0 90 20 143.2 135.8 5.9 13.8 90 122.2 15.6 13.6 90

119 13 130 20 128.4 121.8 5.6 12.8 57 109.5 14.8 12.6 57

135 20 180 20 120.2 114.0 5.2 8.8 15 102.6 13.9 8.7 15

155 0 270 20 153.6 145.6 5.7 3.1 90 131.0 15.1 3.0 90

95 0 90 40 183.5 171.9 8.3 17.3 90 151.0 21.1 17.3 90

102 24 130 40 129.7 121.5 6.9 16.1 61 106.7 17.6 19.0 62

135 40 180 40 92.9 87.1 5.5 10.4 28 76.5 14.1 10.3 28

175 0 270 40 176.6 165.4 7.0 0.9 90 145.2 18.1 0.9 90

75 0 90 60 273.4 247.5 14.9 18.5 90 202.9 34.8 18.6 90

82 34 130 60 128.5 116.3 10.0 17.6 64 95.4 23.6 17.6 64

135 60 180 60 90.0 81.5 6.9 12.5 38 66.8 16.5 12.3 38

195 0 270 60 164.4 148.9 9.6 0.7 90 122.0 23.0 0.7 90

60 0 90 75 420.0 347.3 29.3 18.4 90 237.5 57.6 18.5 90

64 38 130 75 137.9 114.0 16.4 17.5 65 78.0 32.9 17.6 65

135 75 180 75 78.9 65.2 10.1 13.7 42 44.6 20.8 13.5 42

210 0 270 75 150.0 124.0 14.4 1.7 90 84.8 29.6 1.7 90

Table 11: Same as Table 10, with the Sun at A=90, z=135

 

A z I₁ I₂ I₂

180 0 0 0 180.0 162.9 6.0 5.9 90 147.4 12.0 4.8 90

160 0 90 20 158.0 142.1 6.3 7.0 90 127.8 12.5 6.0 90

164 13 130 20 144.5 129.9 6.3 6.0 57 116.9 12.5 5.0 58

180 20 180 20 130.0 116.9 6.2 4.2 0 105.1 12.4 3.2 0

140 0 90 40 144.0 126.4 7.4 10.1 90 111.0 14.4 9.2 90

147 24 130 40 117.4 103.1 7.3 7.7 71 90.5 14.2 7.1 74

180 40 180 40 90.0 79.0 7.2 0.3 87 69.4 14.0 1.3 89

120 0 90 60 140.0 114.7 10.5 13.8 90 94.0 19.6 13.2 90

127 34 130 60 101.9 83.5 10.3 11.2 82 68.4 19.2 11.1 83

180 60 180 60 90.0 73.8 9.9 6.2 90 60.4 18.6 7.1 90

105 0 90 75 158.2 108.2 17.9 16.0 90 74.0 30.4 15.8 90

109 36 130 75 102.3 70.0 17.4 13.7 86 47.9 29.6 13.9 87

180 75 180 75 78.9 53.9 16.7 9.9 90 36.9 28.4 10.9 90

180 0 0 0 180.0 171.2 5.7 0.7 90 155.0 15.2 0.6 90

160 0 90 20 158.0 149.8 5.7 1.6 90 134.7 15.1 1.5 90

164 13 130 20 144.5 137.0 5.5 1.2 72 123.2 14.7 1.2 73

180 20 180 20 130.0 123.3 5.3 0.1 88 110.9 14.1 0.2 88

140 0 90 40 144.0 134.9 6.5 5.5 90 118.5 16.7 5.3 90

147 24 130 40 117.4 110.0 5.9 4.8 85 96.6 15.2 4.7 85

180 40 180 40 90.0 84.3 5.3 3.8 90 74.0 13.7 3.8 90

120 0 90 60 140.0 126.7 8.9 10.7 90 103.9 21.3 10.5 90

127 34 130 60 101.9 92.2 7.6 10.0 88 75.6 18.2 9.9 88

180 60 180 60 90.0 81.5 6.6 9.2 90 66.8 15.9 9.1 90

105 0 90 75 158.2 130.8 14.4 13.9 90 89.5 29.2 13.7 90

109 38 130 75 102.3 84.6 11.7 13.2 89 57.9 23.9 13.0 89

180 75 180 75 78.9 65.2 9.7 12.4 90 44.6 20.0 12.3 90

Table 12: Same as Tables 10 and 11, with the Sun at A=90, z=180