Some remarks on the interpretation of apsidalmotion constants in close binary systems
Abstract
The aim of the present paper has been to establish explicit formulae for the computation of the constants kj of apsidal motions in close binary systems, invoked by the jth harmonic distortion of their components of arbitrary structure. If the degree of central condensation of such components is high, and their (equilibrium) density distribution p(a) remains analytic at the surface a=a1, it is shown that j+I  (a) = 2j + (ai) where the quantity 1(a1) is expressible by a series on the righthand side of equation (14) within the domain of its convergence (the existence of the latter defining the "stars of high condensation"). If the density distribution characterizing our stars is such that not all derivatives of p(a) are finite on the surface, it is shown that the values of kj may be effectively approximated by means of the equation (2j+ I)a125+'kj=3(j+2) f$ Da25 cia, where D p/p, p (a) denoting the mean density interior to a. The derivation of this formula reveals that the error of this representation of k5 should diminish with increasing value of j as well as with increasing degree of central conde'nsation. A numerical application to the polytropic family of models discloses that, if the ratio of the central to the mean density of our configuration exceeds (say) one hundred, the above approximate formula should furnish the values of k2 correctly within the limits of errors affecting their empirical determination in most practical cases, while the question of errors of our approximate values of k3 or k4 will scarcely ever arise.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 1953
 DOI:
 10.1093/mnras/113.6.769
 Bibcode:
 1953MNRAS.113..769K