15-digit accuracy calculations of Chandrasekhar’s $H$-function for isotropic scattering by means of the double exponential formula

Abstract

This work shows that it is possible to calculate numerical values of the Chandrasekhar $H$-function for isotropic scattering at least with 15-digit accuracy by making use of the double exponential formula (DE-formula) of Takahashi and Mori (Publ. RIMS, Kyoto Univ. 9:721, 1974) instead of the Gauss-Legendre quadrature employed in the numerical scheme of Kawabata and Limaye (Astrophys. Space Sci. 332:365, 2011) and simultaneously taking a precautionary measure to minimize the effects due to loss of significant digits particularly in the cases of near-conservative scattering and/or errors involved in returned values of library functions supplied by compilers in use. The results of our calculations are presented for 18 selected values of single scattering albedo $\varpi_{0}$ and 22 values of an angular variable $\mu$, the cosine of zenith angle $\theta$ specifying the direction of radiation incident on or emergent from semi-infinite media.

Expansion coefficients for the Hopf constant

The Hopf constant $q_{\infty}$ can be calculated using the formula by Placzek and Seidel (1947):

$$ q_{\infty}=\frac{6}{\pi^{2}}+ \frac{1}{\pi} \int_{0}^{\pi/2} \biggl(\frac{3}{x^{2}} - \frac{1}{1-x\cot x} \biggr)dx. $$

(16)

Following Viik (1986), we expand the integrand of Eq. (16) in a power series of $x$ up to the term of $x^{n}$, and carry out the integration analytically, to get the following result:

$$ q_{\infty}\simeq\frac{6}{\pi^{2}}+\frac{1}{\pi}\sum _{n=2}^{\infty}\frac{(-b_{n})}{2n-3} \biggl( \frac{\pi}{2} \biggr)^{2n-3}. $$

(17)

It is found that terminating the series at $n=250$, the numerical value given by Viik (1986) can be reproduced to the 59-th decimal place by rounding off the 60-th decimal figure 8 of our result:

$$\begin{aligned} &0.7104460895987630727325241416991536719932 \\ &\quad{}01333958785239092798 \end{aligned}$$

(18)

(see also Loyalka and Naz 2006, which gives this value to the 20-th decimal place).

We present in Table 7 the coefficients $(-b_{n})$ up to $n=17$ in fraction form, ignoring all the terms beyond $(\pi/2)^{33}$, and in Table 8 in decimal form. We have confirmed that the value of $q_{\infty}$ can be reproduced to the 15-th decimal figure or 0.7104460895987631 using a FORTRAN code in double-precision arithmetic by incorporating the coefficients given in Table 8. It must also be noted that the values for $b_{n}$ ($n=2, 3, 4, 5$) shown here are in full agreement with those cited in Viik (1986).

Table 7 Expansion coefficients for the Hopf constant $q_{\infty}$ in fraction form

Table 8 Expansion coefficients for the Hopf constant $q_{\infty}$ in decimal form