This online article is a supplement to our manuscript in preparation about studying the Solar Gravitational Lens (SGL) at arbitrary distances from the optical axis.
In this paper, we obtained an integral expression that can be used to either model the light field produced by the SGL in its focal region, or the view of an imaging telescope looking back at the SGL from that location.
Combined with the efficient, closed form quartic solution of the SGL with its J2 zonal harmonic, we have a formalism that can efficiently model the SGL in all regions. To demonstrate the power of this formalism, we modeled the view by a spacecraft equipped with a 1-m imaging telescope as it approaches the optical axis while viewing a distant point source.
In all these animations, the telescope (image plane) is located at 650 AU from the Sun, observing near infrared light at λ = 1 μm.
Preapproach animations
The animation shows early approach, as the spacecraft travels from 1,000,000 km (a distance comparable in magnitude to the solar radius) from the optical axis to 4,000 km. The Sun is indicated by a thin yellow circle. The image that we see at first (or rather, don't see because it would be obscured by the solar disk initially) is the "secondary" image; the "primary" image (which begins its existence as the unobstructed, unamplified view of the distant source) floats into the viewing area from the right. Both images become brighter as a result of amplification by the SGL, such that by the time we reach 4,000 km, the ring-like Airy pattern, produced by the diffraction-limited optical telescope, becomes dominant.
Final approach animations
Between 4000 km and the extreme vicinity of the optical axis, very little else happens, except that both images of the distant source brighten significantly. For this reason, this stage of the approach is not animated.
Instead, we pick up the trail when the telescope is only 10 meters from the optical axis. The contribution of the J2 quadrupole is set to be small. We begin with a simulation with the telescope positioned at the angle βs = 5.74° from the solar axis of rotation (corresponds to sin βs = 0.1). As we can see, as the telescope approaches the optical axis, the Einstein cross forms as expected.
At half that angle, βs = 2.87°, however, no Einstein cross forms. The quadrupole contribution is too small: the monopole gravitational lens reasserts itself at the selected wavelength, yielding a full Einstein ring.
Picking an intermediate angle, βs = 3.44°, we see an Einstein cross in the form of widened arcs that have not quite become large enough to merge into a full Einstein ring.
All these approaches were made in the direction of one of the principal directions, astroid cusps of the quadrupole pattern. The next animation shows when the telescope approaches from a 45° angle instead, coming through the astroid fold:
Finally, we also simulated an intermediate angle of 30°:
