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 0n ,λ(r1,r2,δ) (and also for its partial derivatives), where α 0n ,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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Appell, P.: 1880,Compt. Rend. Acad. Sci. Paris 90, 296.
Appell, P.: 1884,J. Math. Pures Appl. Sér. 3,10, 407.
Appell, P.et Kampé de Fériet, J.: 1926,Fonctions Hypergéométriques et Hypersphériques; Polynômes d'Hermite, Gauthier-Villars, Paris.
Bailey, W. N.: 1933,Quart. J. Math. Oxford Ser. 4, 305.
Bailey, W. N.: 1935a,Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Cambridge University Press, Cambridge; reprinted, by Stechert-Hafner, New York, 1964.
Bailey, W. N.: 1935b,Quart. J. Math. Oxford Ser. 6, 233.
Bailey, W. N.: 1936,Proc. London Math. Ser. 2 40, 37.
Bailey, W. N.: 1938,J. London Math. Soc. 13, 8.
Bateman, H.: 1905,Proc. London Math. Soc. Ser. 2 3, 111.
Burchnall, J. L.: 1942,Quart. J. Math. Oxford Ser. 13, 90.
Demircan, O.: 1978,Astrophys. Space Sci. 59, 313.
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.: 1953a,Higher Transcendental Functions, Vol. I, McGraw-Hill, New York.
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.: 1953b,Higher Transcendental Functions, Vol. II, McGraw-Hill, New York.
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.: 1954,Tables of Integral Transforms, Vol. II, McGraw-Hill, New York.
Kopal, Z.: 1977,Astrophys. Space Sci. 50, 225.
Kopal, Z.: 1979,Language of the Stars; A Discourse on the Theory of the Light Changes of Eclipsing Variables, D. Reidel Publ. Co., Dordrecht, Holland.
Kopal, Z.: 1983,Astrophys. Space Sci. 90, 445.
Lauricella, G.: 1893,Rend. Circ. Mat. Palermo 7, 111.
Magnus, W., Oberhettinger, F., and Soni, R. P.: 1966,Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged ed., Springer-Verlag, New York.
Rice, S. O.: 1935,Quart. J. Math. Oxford Ser. 6, 52.
Slater, L. J.: 1966,Generalized Hypergeometric Functions, Cambridge University Press, Cambridge.
Srivastava, H. M.: 1966,Proc. Nat. Acad Sci. India Sect. A 36, 145.
Srivastava, H. M.: 1987,J. Phys. A: Math. Gen. 20, 847.
Srivastava, H. M. and Chang, C.-C.: 1987,Chinese J. Math. 15, 153.
Srivastava, H. M. and Exton, H.: 1979,J. Reine Angew. Math. 309, 1.
Srivastava, H. M. and Karlsson, P. W.: 1985,Multiple Gaussian Hypergeometric Series, Halstead Press (Ellis Horwood Limited, Chichester), John Wiley, New York.
Srivastava, H. M. and Manocha, H. L.: 1984,A Treatise on Generating Functions, Halstead Press (Ellis Horwood Limited, Chichester), John Wiley, New York.
Szegö, G.: 1975,Orthogonal Polynomials, Fourth ed., American Mathematical Society Colloquium Publications, Vol. XXIII, Amer. Math. Soc., Providence, Rhode Island.
Watson, G. N.: 1922,Proc. London Math. Soc. Ser. 2 20, 189.
Watson, G. N.: 1944,A Treatise on the Theory of Bessel Functions, Second ed., Cambridge University Press, Cambridge.
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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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DOI: https://doi.org/10.1007/BF00640737