Fig. 4. Magnification of pulses during superior conjunction

When ^ = 0 the difference between the path lengths and Shapiro delays vanish due to an exact cancellation and the total delay is due to the time required for the beams to move through the angle 7. For black holes of intermediate mass (on the order of twenty to thirty solar masses) the Shapiro delay and the geometric delays are of comparable length.

In both cases it can be seen that when the magnification factors are large, the time delays are short and a doubling of the magnification is possible if the peak of the pulses is of sufficient duration.

In order to visualize what this effect might be on the pulse profiles, a time series graph (Figure 4) of the pulses observed at the Earth are presented where the black hole lens mass is 10 Mq. The pulses shown are those that occur as the pulsar passes through superior conjunction. The pulses themselves are modeled by simple gaussian pulses with a half-width of 0.01 sec without any noise. The total magnitude and the differences in magnitude are plotted next to each other. The most noticeable effect is the intensity enhancement that occurs close to superior conjunction, which if not obscured by intervening material would act as an obvious signature of gravitational lensing. However unlike the standard microlensing scenario where the total magnitude of the unresolved images is simply the sum of the magnitudes of the two images, if the time delay between the two images is significant enough there will be no overlap of the pulse profiles and the total intensity enhancement is decreased.

Such a situation is shown in Figure 5 where the time delay between the two pulses passing on either side of a 50 Mq black hole is large enough that the pulse profile is more complicated than that expected from the sum of two gaussian pulses where the peaks occur at nearly the same time. Thus if the


Fig. 5. Two magnified pulses with significant time delay pulses are of short duration compared to the total delay, the magnifications do not simply add and more complicated situations arise.

Of course pulsar pulses are noisy and one can model the pulse profiles by adding a background of noise to the signal. However the intensity enhancement is such that it should provide a significant signal appearing well above the noise associated with the individual pulses. For example with a 50 Mq black hole lens, adding a 15% random noise to the individual pulses will not significantly affect the intensity enhancement if the time delays between the two pulses is small as is shown in Figure 6.

3.2 Application to the Double Pulsar: PSR J0737-3039

The analysis developed above assumed that the inclination angle of the binary system was i « n/2 to within one or two degrees at the most. If this is not the case then the magnification of the secondary image which undergoes the most light bending is too small to be observed. If the magnification enhancements can be observed during a number of eclipses, then the intensity measurements could be used to determine the inclination angle since the maximum possible intensity should be a measurable quantity. The recent discovery of the double pulsar PSR J0737-3039 with an inclination angle close to 90° has stirred much interest in the relativity community since it should be able to act as a test of general relativity (and alternative gravity theories) with an unprecedented accuracy. The system consists of two massive compact stars (Ma = 1.4Mq,

Black Hole Mass = 50Mo psi0 = -.006


Gravitational Lensing in Compact Binary Systems 79 Pulses with Noise

Gravitational Lensing in Compact Binary Systems 79 Pulses with Noise

time (sec)

Fig. 6. Magnification of noisy pulses Table 3. Physical Parameters for the Pulsar System: PSR J0737-3039A/B


Orbital Period (hrs)

Distance to PSR J0737 (pc)

Distance between lens and source (km)

Inclination angle

Orbital Eccentricity

22.70 2773 1.338 1.249 2.453 500-600 4000±2000 90.29° ±0.14° 0.08778

Mb = 1.3Mq ) in a nearly circular orbit (e = 0.088). Both stars are pulsars that can be seen from Earth. Some of the important physical characteristics of this system are shown in Table 3.

Model of Lensing in J0737-3039

The parameters given in Table 3 along with a model of the integrated pulse shapes for the pulsar sources provides the necessary information required to describe the light curve one might expect to observe during the eclipse of Pulsar A by Pulsar B. The averaged pulse profile of Pulsar A consists of two unequal pulses occurring during its 0.0227 second spin period. The averaged pulses are computed from the true light curves measured in the frequency range 427 to 2200 MHz. [8] In this analysis the light curve is approximated by two skewed gaussian pulses as shown in Figure 7. While the true light curves


Fig. 7. Model for the Pulsar A pulse profile


Fig. 7. Model for the Pulsar A pulse profile will be more complicated functions of frequency, it is the relative amplitudes and the duration of the sub-pulses that will be important in modeling the time delays and the magnification factors that occur during superior conjunction.

During the eclipse of Pulsar A by Pulsar B, the observations indicate that there is an attenuation of the light curve by the atmosphere of Pulsar B. [2,5] The attenuation is not symmetric with respect to the phase. The eclipse ingress is longer in duration than egress. The depth of the attenuation is also deeper after conjunction than it is before. In addition there is also a frequency dependent behaviour and these observations have provided a challenge to theoreticians to create a model of the atmosphere of Pulsar B along with its magnetic field structure. However in order to create a model for the gravitational lensing effects we follow the method used by Kaspi, et al [2] who fit the orbit-averaged light curve attenuation with two functions of the form:

where $ is the orbital phase defined such that at conjunction $ = 0. The two free parameters $0 and width a have been computed for both ingress and egress and for our model we use:

The functional form for the attenuation factor is plotted in Figure 8.

Using the physical parameters presented in Table 3 and applying the method discussed in the analysis of black hole-pulsar binaries one can compute the expected pulse profiles and magnifications subject to the light curve attenuation. The resulting pulse profiles are shown in Figure 9 where it can be seen that the lensed pulses from Pulsar A can be magnified significantly even in the presence of attenuation if the inclination angle is very close to 90°. If on the other hand the inclination angle is less than 90° the enhancement for a lens

Gravitational Lensing in Compact Binary Systems Light Curve of Pulsar A at 820MHz

Gravitational Lensing in Compact Binary Systems Light Curve of Pulsar A at 820MHz

0.002 0.004 0.006

Phase (degrees)

Fig. 8. The Attenuation curve for the eclipse of A by atmosphere of B

0.002 0.004 0.006

Phase (degrees)

Fig. 8. The Attenuation curve for the eclipse of A by atmosphere of B

PSR A eclipsed by PSR B with attenuation by B's atmosphere

Time (seconds)

Fig. 9. Gravitational lensing of Pulsar A by Pulsar B where sin i = 1

Time (seconds)

Fig. 9. Gravitational lensing of Pulsar A by Pulsar B where sin i = 1

of just larger than one solar mass is not very significant beyond Ai « 0.15°. In this case the magnification factor of the primary beam is reduced to unity and any significant attenuation of the signals close to superior conjunction would mask the gravitational lensing effects. Since Pulsar A has such a short rotational period, one can expect a very large enhancement of the magnification during conjunction since the pulsar source is now very close to being located at ^ = 0 when the emission of the beam occurs. The maximum value of A^ away from superior conjunction can easily be estimated by:

Fig. 10. Pulse profiles for the Eclipse of Pulsar B by Pulsar A Wmax = 2^

where t is the orbital period of the system. Using Tp « 22 ms and t « 2.5 hr leads to A^max = 1.58 x 10~5.

While the eclipse of Pulsar B by Pulsar A has not been observed, the methods presented here can predict what one might expect from gravitational lensing effects. Therefore the gravitational lensing of Pulsar B by Pulsar A has also been computed. Since Pulsar B has a period of about two seconds, the phase of the pulses will not always allow it to be as close to the superior conjunction position as is the case for pulsar A. The enhancements in intensity can be expected to be significantly less. The observed pulse profile for Pulsar B is much simpler than that for Pulsar A [1]. These pulses can be modeled by very sharp gaussian profiles with one pulse per pulsar period. One can compute the effects of Pulsar A's gravitational field on the light pulses emitted from Pulsar B and plot them as shown in Figure 10. As expected the intensity enhancements are found to be smaller than those for the observed eclipse. Here however both noise and/or attenuation, if present, have been ignored. As in the case of the eclipse of Pulsar A by Pulsar B the increase in magnification may be sufficient so as to render the signal observable during conjunction, even if the remainder of the light curve is of low intensity.

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