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Further notes on the associated alpha-functions and related integrals in the theory of the light changes of eclipsing variables

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Abstract

With a view to furthering the theory of the light changes of eclipsing variables, developed before systematically by Z. Kopal, this paper presents a number of new (and computable) expressions for the associated alpha-function α n 0 ,λ(r1,r2,δ) (and also for its partial derivatives), where α n 0 ,λ(r1,r2,δ) represents the fractional loss of light suffered by an eclipse of a circular disc of fractional radiusr 1 (and darkened at the limb to thenth degree) by an opaque disc of radiusr 2, with their centres separated by a fractional (projected) distance δ, provided that the transparency of the occulting disc increases with the angle of foreshortening in the same manner as the limb-darkening of the eclipsed star (that is, when the transparency functiong(ϱ, ζ) of the second aperture is given by Equation (4) below). Many of the explicit expressions derived here are valid for any type of eclipse, occultation or transit, regardless of whetherr 1>r 2 orr 1<r 2. and for any degreen of the adopted law of limb-darkening. It is also pointed out how some of the results obtained in this paper are related to the various representations given earlier in the literature for the case λ=0.

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Srivastava, H.M., Kopal, Z. Further notes on the associated alpha-functions and related integrals in the theory of the light changes of eclipsing variables. Astrophys Space Sci 162, 181–204 (1989). https://doi.org/10.1007/BF00640737

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Keywords

  • Explicit Expression
  • Degreen
  • Related Integral
  • Light Change