Gravitational Lensing of Pulsar Sources

The assumptions that are made in order to develop a model of gravitational lensing in a compact binary system consisting of at least one observable pulsar are the following:

1. The light rays pass through empty (or nearly empty) space. This allows one to use the equations describing motion along null (zero-mass particle) geodesics of a vacuum spacetime to compute bending angles, time delays and magnification factors for the light beams as they traverse the region from source to observer.

2. The light beams pass by the lens at large distances compared to the size of the source and the deviations from straight line motion are small. This allows one to use weak lensing conditions with small angle approximations and a perturbation analysis where the geodesic equations are expanded in powers of ¿0/r ^ 1 where ¿0 is some appropriate scale length (mass of the black hole in geometric units, the radius of the star, or the impact parameter) and r is the instantaneous radial position of the photon.

3. The spin angular momentum of the lens is small, as is its translational motion. This means that v/c ^ 1 and a/r ^ 1 where v is the translational velocity of the lens, c is the speed of light, a is the spin angular momentum parameter of the star or black hole (again in geometric units).

4. Electromagnetic effects are small. That is, there are no residual charges on either the source or lens q « 0, magnetic fields are small B2r2 ^ M/r where B is the magnetic field strength and M is the mass of the lens (again as measured using geometric units).

5. The multipole moments associated with the distributions of mass in the lens are small i.e. the lens is nearly spherically symmetric. This and assumptions 1, 3, and 4 mean that one can use the vacuum Schwarzschild solution to approximate the motion of the photons.

6. The light source produces narrow, pulsed, directional beams of photons. This will require the modifications to the standard microlensing theory [9], where the sources emit isotropically and continuously in time.

In this section the lensing produced by a Schwarzschild-type object will first be reviewed which will then be followed by a introduction of the effects resulting from the addition of the last assumption.

To begin, the geometry of the Schwarzschild lens having a mass M is shown in Figure 1 along with the various angles and distances that will be used to compute the gravitational lensing effects.

The position of the source is labelled by S, the location of the observer is labelled by O, and the location of the lens is labelled L. It will be assumed that the optical axis of the system is defined by the straight line OL connecting the lens with the observer. Two planes perpendicular to the optical axis are then defined by the source position S (called the source plane) and the position of L (the lens plane). The following distance measurements are now introduced: Ds is the distance between the observer and source plane, Dd is the distance

Fig. 1. Geometry of a Schwarzschild Lens system

Fig. 1. Geometry of a Schwarzschild Lens system

between the observer and lens plane and Dds is the distance from the lens plane to the source plane. In the simple case of a Euclidean metric Ds = Dds + Dd.

One now defines two other distances that are measured orthogonal to the optical axis. The distance n measures the the position of the source with respect to the optical axis and ( measures the distance of closest approach of the bent light ray as it passes by the lens. Clearly from the definition of the source and lens planes, these measurements are restricted to lie within their respective planes.

In addition, the following angular measurements (assumed to be small) are introduced: 3 is the angular separation of the source position from the lens, 0 measures the angle between the incoming light ray and the line OL (this produces an image of the source located at S1 in the plane containing the source and perpendicular to OL). The angle a is the the deflection angle for the light ray while the angle a = 0 — 3 is the angular separation of the image from the source.

At conjunction, when n = 0 (3 = 0) there is no longer a preferred plane defined by OL and OS. This results in a characteristic angular deviation known as the "Einstein angle", a0, (thereby producing a ring - the "Einstein Ring" - of images) at an angular separation of:

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