which obeys the relation
If the source is emitting continuously and isotropically in all directions, then the total magnification factor is the sum of the two individual expressions in equation (7) and this forms the basis for the search for dark compact objects (MACHOS, black holes, etc.). The idea is that brightness enhancements of luminous sources will result from lensing by dark compact objects with gravitational fields large enough to produce magnification factors many times larger than unity.
However for pulsed, directional beams such as those emitted by pulsar sources, the finite time interval during which the beam can be seen by the observer must be taken into account. The magnification factors are then time dependent. If the arrival times for the two beams do not occur simultaneously then the total magnification is not simply the sum of the two factors. Delays in arrival times will occur due to path length differences, Shapiro delay differences and differences in the initial beam directions for each image. The pulse profiles are also time dependent and each of these effects need to be analyzed properly in order to compute the time dependent behaviour of each magnification factor and their contributions to the total overall magnification.
Such a situation requires a modification of the microlensing magnification equation that takes into account the pulse profile and the Shapiro delays suffered by each image. While the source is close to superior conjunction, situations can arise where the delay between the pulses is sufficient to prevent them from overlapping. In this case the pulse profile, the magnification of each beam and the delay between the pulses must be modeled accurately in order to obtain a more realistic prediction of the effect of the lens on the overall pulse profiles. In addition, as stated above, the directional property of the pulsar beam must also be taken into account. The rotational period of the pulsar will also contribute to the overall delay in arrival times since the orientation of the initial direction of the two beams does not occur simultaneously but requires some fraction of the pulsar period so that the two beams are positioned correctly in order for the pulses to reach the Earth bound observer.
In order to develop a method for the analysis of the gravitational lensing of pulsar beams, we need to introduce some new parameters (based upon the both orbital motions of the pulsar and the pulsar rotational period). These parameters then will be used to describe the the time dependent position of the pulsar in the source plane and the orientation of the beams associated with the creation of the two lensed images. By relating these parameters to the source angular position 3, one can then return to standard lensing theory to reconstruct image positions, time delays and magnification factors. It must be remembered that each image requires a specific initial alignment of the pulsar beam in order for it to reach the observer. Since two different alignments are required for the two lensed images, a "geometric time delay" will occur as the beam rotates from one initial angle to the other. This time delay required for the pulsar beam to change from one direction to the other will simply be a fraction of the total pulsar rotation period and will be added to the difference in the Shapiro and travel time delays associated with each beam.
The geometry associated with the motion of a pulsar in orbit around a compact lensing object is shown in Figure 2. The notation used in Figure 2 is the same as that introduced in Figure 1 but now the orbit of the pulsar (shown here as circular but higher eccentricity orbits can also be analyzed) is included along with the angles that will be used to locate the position of the pulsar with respect to the lens plane.
Assuming that the portion of the pulsar orbit of interest forms part of a circular arc with a radius R, an angle ^ is introduced and this will measure the pulsar position with respect the lens. Here ^ = 0 is defined to be the angular position of the source when it is on the optical axis OL. The angle ^ has a time dependence determined by the orbital period of the source. Since we are only interested in measurements made over short time scales, this approximation will be adequate over a few pulse periods. However if the orbit is significantly elliptic with a relativistic shift in periastron, then a more complicated mapping of the orbital radial and angular positions to the lensing source angle 3 and source position Ds will be required, especially if the lensing observations are to take place over many orbits of the pulsar.
The angles y1 and y2 which represent the specific directions of the primary and secondary pulsar beams (measured with respect to the radial vector pointing from the lens position to the position of the source) required for the beam to create the images Si and S2 as seen by the observer. These are shown in Figure 3 which represents a detailed picture of the region between the source located at the left vertex and the lens. The sum of these two angles y = y1 + y2 yields the angle between the two beams that produce the images S1 and S2. The time for the the pulsar to rotate through this angle gives the "geometric time delay" between the two beams.
The small angle deviation requirements can be translated into lower limits on the distances of closest approach:
£1,2 > 10Rs which is equivalent to a "weak-field" approximation where £ remains outside of the region defined by a radius of ten Schwarzschild horizon radii. This in turn places limits on the angular position ^ of the source:
10RsDs 1 Dds
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