The photospheric parameters such as polar temperature, bolometric albedo, gravity brightening exponent, or limb-darkening coefficient of the model enter only in the function I (cos y; g, T, X). They do not change the other terms in the integrand of (4.5.16), and they do not change the range of integration or the characteristic function Xs(0). Therefore the derivatives of Ij(0) with respect to this set P1 of parameters can be calculated by taking the derivatives of the integrand, i.e., for all p e P1 we have di , r r

Analytic formulas for the Linnell model are given by Linnell (1989, Appendix). The advantage of this procedure is that the derivative is found by a single integration, rather than by taking the difference of two integrals. This helps to improve accuracy. As an example we consider the albedo A in the Wilson-Devinney model and its monochromatic flux integrand I in formula (6.3.5). Only the reflection factor R depends on the albedo A, and the dependence is described by (3.2.46). The bolometric albedo is by far the easiest parameter because it leads to an analytical expression for dlj/d A. Therefore, we get the simple term di,

Fs cos y

where the ratio of bolometric fluxes Fs /Ft is already available from the computation of R itself (see Sect. 3.2.5). In the case of other parameters p e P1, the partial derivative dlj/dp needs to be computed numerically.

The geometric parameters (p e P2) are more difficult to treat. Linnell (1989) suggests applying Leibniz's rule. In order to do so, (4.5.16) is rewritten as r 6u r <Pn (6 )

Telescopes Mastery

Telescopes Mastery

Through this ebook, you are going to learn what you will need to know all about the telescopes that can provide a fun and rewarding hobby for you and your family!

Get My Free Ebook

Post a comment