Transcendental Harmonic Mappings and Gravitational Lensing by Isothermal Galaxies

Abstract

Using the Schwarz function of an ellipse, it was recently shown that galaxies with density constant on confocal ellipses can produce at most four "bright" images of a single source. The more physically interesting example of an isothermal galaxy has density that is constant on homothetic ellipses. In that case bright images can be seen to correspond to zeros of a certain transcendental harmonic mapping. We use complex dynamics to give an upper bound on the total number of such zeros.

On the Number of Solutions of a Transcendental Equation Arising in the Theory of Gravitational Lensing

“…which occurs as a model of gravitational lensing of a point source w by an elliptic object whose density equals c/r on the homothetic ellipses rE where E is a fixed ellipse, cf. [7,9,10]. The branch of arcsin in (2) is the principal branch which is defined in C\[−1 , 1].…”

“…The branch of arcsin in (2) is the principal branch which is defined in C\[−1 , 1]. As in [7,10] we assume that the density is zero outside of E and thus is equal to c/r on rE only for 0 < r ≤ 1. We note, however, that in the astronomy literature (cf., e. g., [9]) it is usually assumed that this formula for the density holds for 0 < r < ∞; see [10] for a discussion of the different models.…”

“…As in [7,10] we assume that the density is zero outside of E and thus is equal to c/r on rE only for 0 < r ≤ 1. We note, however, that in the astronomy literature (cf., e. g., [9]) it is usually assumed that this formula for the density holds for 0 < r < ∞; see [10] for a discussion of the different models. We also refer to the above papers for the derivation of equations (1) and (2).…”

“…The solutions of (1) that lie outside E correspond to the so-called bright images of the source. Khavinson and Lundberg [10] proved that the number of solutions of (1) in | Re z| < π/2 is finite and does not exceed 8. Up to 4 bright images by a single lensing galaxy have been observed by astronomers [10,3].…”

“…The upper estimate in [10] is based on a miraculous trick: an application of Fatou's theorem from holomorphic dynamics. This method originated in [13], and it was applied by Khavinson and Neumann [11] to obtain the estimate 5d − 5 for the number of solutions of the equation…”