Spectral functions librator approximations

Using the so-called planar libration-regular precession (PL-RP) approximation, it is possible to reduce the double integral for the spectral function to a simple integral. The interval of integration is divided in the latter by two intervals, and in each one the integrands are substantially simplified. This simplification is shown to hold, if a qualitative absorption frequency dependence should be obtained. Useful simple formulas are derived for a few statistical parameters of the model expressed in terms of the cone angle (5 and of the lifetime x. A small (3 approximation is also considered, which presents a basis for the hybrid model. The latter is employed in Sections IV and VIII, as well as in other publications (VIG). 

Using this result, we may simplify calculation of the spectral function Liz) by neglecting the precessional contribution to L. We shall estimate also in this approximation the peak frequencies X ib and xrot of the absorption bands determined by the librational and the rotational subensembles. 

In accord with the planar libration approximation, we first come from representation of the spectral function for motion of a dipole in a plane, where integration over l is lacking by definition, so only integration over energy h is employed. We shall find in this way the function (203). As a next step we carry out integration over l, so that a rather simple expression (171) for the spectral function L(z) will be obtained. 

Assumptions 1 and 2 constitute the planar libration-regular precession approximation. In Gaiduk et al. [56] and in GT2 the corresponding spectral functions L(z) and L(z) are found in analytical form, as well as the SF L(z) for the rotators. These SFs are expressed as simple integrals from elementary functions over the full energy of a dipole. The total spectral function is thus represented as... 

---