Two new analytic approximations of the Chandrasekhar's H function for isotropic scattering

Rational approximation formula for Chandrasekhar’s H-function for isotropic scattering

“…Note that similar numerical tables are also given by Bosma and de Rooij (1983), but for the combinations of ̟ 0 = 0.5, 0.7, 0.9, 0.99, 0.999, and 1 and for µ from 0 to 1 with a step of 0.1. We found that N = 100 is sufficient to bring our results for H(̟ 0 , µ) into complete agreement with those of Davidović et al (2008) and Bosma and de Rooij (1983) down to the 10th decimal place (11 figures altogether). On the other hand, even if the 300th-degree Gauss-Legendre quadrature is employed for the single interval [0, π/2], the accuracy of the resulting value of the H-function for the conservative scattering is much lower: we obtain, e.g., H(1, 1) = 2.9077901976 instead of the reference value 2.9078105291.…”

“…We varied the degree N of the Gauss-Legendre quadrature to calculate the values of H(̟ 0 , µ) for all the combinations of 48 values of ̟ 0 (0.01 plus 47 values employed by Davidović et al (2008) for their Table 1) and 21 values of µ (0 plus 20 values employed by Davidović et al (2008) for their Table 1). Note that similar numerical tables are also given by Bosma and de Rooij (1983), but for the combinations of ̟ 0 = 0.5, 0.7, 0.9, 0.99, 0.999, and 1 and for µ from 0 to 1 with a step of 0.1.…”

Solution of a Riemann–Hilbert problem for a new expression of Chandrasekhar’s H-function in radiative transfer

“…We have used Simpson's one-third rule to evaluate the integrals in (76) and (78) for 0 < w < 1 and w = 1, respectively, for U(u) = w/2 and it is seen that our tabulated values of H (x) for different albedo w are accurate to the ninth places of decimals and better than those values tabled by Chandrasekhar and Breen (1947), Placzek and Seidel (1947), Placzek (1947), Stibbs andWeir (1959), Hovenier et al (1988), Steinfelds et al (1997), and Davidonic et al (2008).…”

“…The mathematical forms of H-functions presented by different authors namely Chandrasekhar (1950), Kourganoff (1952), Busbridge (1960Busbridge ( , 1962, Fox (1961), Dasagupta (1974Dasagupta ( , 1977, Das (1979Das ( , 2008, Zelazny (1961), Mullikin (1964), Zweifel (1964), Ivanov (1973), Siewert (1980), Garcia and Siewert (1987), Sulties and Hill (1976), Siewert (1998, 1999), Rutily (1996, 1998), Dasgupta (1962), Abhyankar and Fymat (1969), Steinfelds et al (1997), Wegert and Von Wolfersdorf (2006), and Davidonic et al (2008) from different standpoints are therefore considered to be useful from theoretical as well as a numerical point of view. Those forms are not so easy for numerical integration because H-function appears within the integral for iteration.…”

The Chandrasekhar function revisited

“…Thus, much effort has been devoted to create simple and reasonably accurate formulas for the Chandrasekhar function. Davidović et al [8] derived general expressions for an approximate analytical representations of the H function. These authors derived a general expression of the form H(µ, ω) = ξ (µ, ω) + µ ξ (µ, ω) + µ(1 − ω) 1/2 (3) where the function ξ (µ, ω) is suitably approximated to ensure reasonable accuracy of Eq.…”