Gravitational lensing by an extended mass distribution

Abstract

We continue our investigations of the optical properties of the solar gravitational lens (SGL). We treat the Sun as an extended axisymmetric body and model its gravitational field using zonal harmonics. We consider a point source that is positioned at a large but finite distance from the Sun and, using our new angular eikonal method, we established the electro-magnetic (EM) field on the image plane in the focal region behind the SGL and derive the SGL's impulse response in the form of its point-spread function (PSF). The expression that we derive describes the extended Sun in all regions of interest, including the regions of strong and weak interference and the region of geometric optics. The result is in the form of a single integral with respect to the azimuthal angle of the impact parameter, covering all lensing regimes of the SGL. The same expression can be used to describe gravitational lensing by a compact axisymmetric mass distribution, characterized by small deviations from spherical symmetry. It is valid in all lensing regimes. We also derive results that describe the intensity of light observed by an imaging telescope in the focal region. We present results of numerical simulations showing the view by a telescope that moves in the image plane toward the optical axis. We consider imaging of both point and extended sources. We show that while point sources yield a number of distinct images consistent with the caustics due to zonal harmonics of a particular order (e.g., Einstein cross), extended sources always result in forming an Einstein ring. These results represent the most comprehensive wave-theoretical treatment of gravitational lensing in the weak gravitational field of a compact axisymmetric gravitational lens.

Multipole decomposition of gravitational lensing

“…In Ref. [2,5], while studying the Maxwell equations on the background space-time (1), we developed a solution to the Mie problem for the diffraction of the EM waves on a large gravitating body (λ/R ≪ 1, see discussion in [1,11]) and found the EM field at an image plane located in any of the optical regions behind the lens (see Fig. 1).…”

“…To explore the solution ( 9)- (10), in Refs. [2,5], we considered the case of a lens with arbitrary multipole structure. We used a form of U representing an axisymmetric gravitational field of a body (such as the Sun) with mass M and equatorial radius R, with its external gravitational potential reduced to k = 0 spherical harmonics, C ℓ0 , and expressed [14,15] in terms of the usual dimensionless zonal harmonic coefficients J ℓ = −C ℓ0 :…”

“…These considerations allow us to evaluate the radial integral in (10) by the method of stationary phase. As a result, in the case of an axisymmetric lens, the complex amplitude of the EM field (10) takes the form (see [2,5]):…”

“…where σ 0 = −kr g ln kr g /e − π 4 is constant, discussed in [16,17], and B al (x) is the remaining angular integral of complex amplitude of the EM field for an axisymmetric lens [2,5]:…”

“…where a new shorthand notation ∂ a ≡ ∂/∂b a has been used and τ is defined by (4). With this representation (25), we can compute the relevant integral (where from (5) we have…”