Approximate analytical description of the high latitude extinction

Abstract

The distribution of visual interstellar extinction $A_V$ has been mapped in selected areas over the Northern sky, using available LAMOST DR5 and Gaia DR2/EDR3 data. $A_V$ was modelled as a barometric function of galactic latitude and distance. The function parameters were then approximated by spherical harmonics. The resulting analytical tridimensional model of the interstellar extinction can be used to predict $A_V$ values for stars with known parallaxes, as well as the total Galactic extinction in a given location in the sky.

Appendices

Appendix A. Spherical harmonics

We use orthonormalized spherical harmonics $Y_n^m$ to calculate 2D approximation of extinction parameters. Only real parts of harmonics are used, hence we use harmonics of non-negative order $0 \le m \le n$. Therefore spherical harmonics are calculated by the following formulas:

$$\begin{aligned} Y_0^0 (l, b)&= \frac{1}{2} \sqrt{\frac{1}{\pi }}, \quad Y_1^0 (l, b) = \frac{1}{2} \sqrt{\frac{3}{\pi }} \cos b,\\ Y_1^1 (l, b)&= \frac{1}{2} \sqrt{\frac{3}{\pi }} \cos l \sin b, \quad Y_2^0 (l, b) = \frac{1}{4} \sqrt{\frac{5}{ \pi }} \left( 3 \cos ^2 b - 1\right) ,\\ Y_2^1 (l, b)&= \frac{1}{2} \sqrt{\frac{15}{\pi }} \cos l \sin b \cos b, \quad Y_2^2 (l, b) = \frac{1}{4} \sqrt{\frac{15}{\pi }} \cos 2 l \sin ^2 b. \end{aligned}$$

Fig. 12

$\chi ^2$ scan for 17777, 28666, 29945

Fig. 13

Best-fit for 34455, 35483, 36249

Fig. 14

$\chi ^2$ scan for 34455, 35483, 36249

Fig. 15

Best-fit for 47350, 49084, 62853

Fig. 16

$\chi ^2$ scan for 47350, 49084, 62853

Fig. 17

Best-fit for 64053, 79217, 82277

Fig. 18

$\chi ^2$ scan for 64053, 79217, 82277

Fig. 19

Best-fit for 83604, 84386, 84622

Fig. 20

$\chi ^2$ scan for 83604, 84386, 84622

Fig. 21

Best-fit for 96855, 97942, 98650

Fig. 22

$\chi ^2$ scan for 96855, 97942, 98650

Fig. 23

Best-fit for 99901, 100961, 112048

Fig. 24

$\chi ^2$ scan for 99901, 100961, 112048

Fig. 25

Best-fit for 112305, 113239, 114025

Fig. 26

$\chi ^2$ scan for 112305, 113239, 114025

Fig. 27

Best-fit for 136719, 137487, 138857

Fig. 28

$\chi ^2$ scan for 136719, 137487, 138857

Fig. 29

Best-fit for 149151, 151503, 152806

Fig. 30

$\chi ^2$ scan for 149151, 151503, 152806

Fig. 31

Best-fit for 160342, 161428, 162401

Fig. 32

$\chi ^2$ scan for 160342, 161428, 162401

Fig. 33

Best-fit and $\chi ^2$ scan for 178840

Appendix B. Figures 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33. Best-fit $\chi ^2$ minimization and $\chi ^2$ scan solutions for the areas used for determination of the interstellar extinction.