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T. I. Larchenkova and O. V. Doroshenko
Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow 117810, RUSSIA
The distribution of matter in the Universe and its composition is one of the most fundamental problems in astrophysics. There is growing evidence for the presence of non-luminous matter in the Universe. As was shown by Oort (1960) and Bachall (1984), the distribution and motion of stars above and below the galactic plane imply a non-luminous mass in the plane of about , while the counts of all known stars, dust, and gas give a surface mass density of only one-half of this amount. Any detection of dark matter in our Galaxy can help to solve this problem.
Gravitational lensing and time delay of pulsar signal in the gravitational field of a mass are General Relativity effects which may be used as a tool to detect the observational parameters of dark matter in our Galaxy.
Gravitational lensing can be separated into: ``macro-lensing'' and ``micro-lensing''. Both kinds are due to the mass of a lensing galaxy cluster, galaxy, or compact object. Macro-lensing includes: (1) cases of multiply imaged sources when the distant source lensed by an intervening galaxy; (2) radio rings; and (3) giant luminous arcs and arclets when the distant source is lensed by a cluster of galaxies.
Micro-lensing is provided by an individual star or a compact object having a mass range from to , usually located in the Galaxy (Chang & Refsdal (1979)). It should be noted that there are two effects which are responsible for a time delay of photons travelling along the curved space-time of a gravitating mass. One effect is geometric and is due to the bending of the photon trajectory in the gravitational field. The second one is due to the relativistic time delay in the gravitational potential of this mass. Both effects were discussed by Krauss & Small (1991) who considered them as a source of magnification of luminosity and have suggested to use the observation of temporal variations of a pulsar pulse profile for finding of micro-lensing events in our Galaxy. In this work we propose to use observation of the extra time delay of pulses from pulsars to find gravitational delays in the field of non-luminous masses close to the line of sight. We discuss the possibility of finding such an event by measuring the time delay of pulses from pulsar, and apply it to data for PSR B0525+21. In the next section we recall some physical reasons for the time delay of pulses in a gravitational field. Section 3 deals with the method of analysis for reduced data. In the Section 4 we present and discuss the obtained results.
The time delay of the radio signals propagation in the gravitational field of a massive object (the Shapiro effect) can be used to evaluate the gravitating mass. The Shapiro delay causes an offset to the observed times of the signals travelling along the curved space-time is:
where vectors and are the unit length vectors pointed from the observer to the source and to the -th gravitating mass , is the speed of light in vacuum and is the Newtonian gravitation constant.
Pulsars are amazing galactic objects with a strict periodic radiation. The time delay of the arrival times of pulses in the gravitational field of the companion of a binary pulsar has been used as a tool for measuring the companion mass by Ryba & Taylor (1991). Several pulsars exhibit some unexplained distortions of the observed times-of-arrival (TOA) of their pulses. The differences between the observed TOAs and those calculated using the classical spin-down model of the pulsar rotation are the residuals of the TOA. Apart from those resulting from a companion, the residuals are believed to be caused by some instability in the pulsar rotation such as the fluctuations in the inertia tensor (glitches), torque variations, precession. They might also be caused by a mass moving close to the line of the sight. In general, any moving massive object located near the line of sight of the observer should cause a time delay which manifests itself as an additional noise in the observations of arrival times of the pulses.
There is no unambiguity in identifying the distortion caused by the gravitating mass because the Shapiro time delay causes a sharp growth of amplitude of TOA near conjunction of the mass with the line of sight of the observer. We have analyzed for several pulsars the timing data obtained at the Jet Propulsion Laboratory by Downs & Reichley (1983) and Downs & Krause-Polstroff (1986) from 1968 to 1983. The data on the arrival times were reduced using the standard fit of the pulsar astrometric and spin parameters based upon the data reduction algorithm developed by Doroshenko & Kopeikin (1990). From an analysis of the observed residuals (see also Cordes & Downs (1985)) we have concluded that amongst the studied objects PSR B0525+21 may present a case of micro-lensing. After subtracting the best fitting polynomial to the arrival times, we observe significant TOA residuals with a behaviour similar to that caused by a mass passing close to the line of sight. The observed residuals of the TOAs of the pulsar are shown as the dots on the Figure 1.
Figure 1: The TOA residuals for PSR B0525+21. The dots show the observed pre-fit-TOA residuals after the fit of the pulsar spin and astrometric parameters. The solid line is the Shapiro time delay for the fitted values of parameters of lensing object. The dashed line is the post-fit-TOA residuals after removing the Shapiro delay due to the lensing object.
We have used the estimation of the Shapiro delay effect (1) for this pulsar in terms of the relative velocity of the pulsar projected in the plane of the sky, with respect to a single gravitating mass , of the distance of the mass from the observer and of the time of conjunction of the mass with the observer's line of sight. We assumed that the variation of the angle between the vectors and can be approximated as:
Here denotes the distance of closest approach of the mass to the line of sight of the observer. Assuming that the parameters and are small enough, so that and , and introducing the parameters , and their ratio , one can obtain from equations 2.1 and 3.2 the modulation due to the Shapiro delay as:
where we have used the notation and left only the time-varying part of the delay omitting in the expression nearly constant contributions to the delay.
The procedure of fitting of the Shapiro delay parameters is based upon including the Shapiro delay (3) into the differential timing formula for the pulsar parameters and the further least-square fit of the pulsar spin and astrometric parameters, as well as the parameters , , , describing the time delay in gravitational field of mass (see details in Blanford & Teukolsky (1976)). For the modulation described by (3), one can see that the partial derivatives for the differential timing formula are:
One can see from these equations that if we assume the generally accepted model of slowing-down phase of the pulsar, describing the arrival time of th pulse from the pulsar as
then our parameters and will be nearly constants multiple of the pulsar spin frequency and its derivative respectively. And indeed, our calculation have shown that the constructed system of the normal equations for the global fitting of the pulsar spin, astrometric, and Shapiro delay parameters were almost degenerated, and that the parameters and are in strong covariance with and . For the total published data set for the pulsar B0525+21 the time delay parameters were obtained with a good level of confidence. We have used the model of slowing-down of the rotation of the pulsar to fit the pulsar period, period derivative, position, proper motion. In addition, the values of , , were also included in the fit. The initial values of , were evaluated from the observed residuals of the TOA and the data onto the proper motion of the pulsar.
Our fitted values for the mass and other parameters of the lensing object are:
With these values of the parameters the pre-fit and post-fit residuals are equal 15 ms and 3 ms respectively. The modulation of the residuals due to these parameters is shown as smooth line in the Figure 1. The post-fit residuals are plotted as dashed line in the same figure.
Because the pulsars are the fastest objects in Galaxy we can suppose that the pulsar velocity is larger than the lensing object velocity. The measured values of the proper motion of the pulsar and its distance (Taylor et al. (1993)) yield , so that our value for gives the distance of closest approach . One can suggest that the observed extra modulation is due to a time delay PSR B0525+21 caused by a mass passing near the line of the sight. The existence of stellar masses as high as is unlikely. So, we suggest that the gravitating mass may be a black hole.
It should be noted that the extra modulation of TOA residuals for this pulsar was supposed to be due to a glitch in 1974 ( JD) by Downs (1982), and that the behaviour of its residuals is very similar to the post-glitch recovery described by the two-component model of the pulsar interior, where the neutron star consists of a rigid crust and an viscous fluid (Baym et al. (1969),Lyne (1992)). However, our alternative explanation of the TOAs for this pulsar is also plausible. The fact that a lensing event manifests itself both by a relativistic time delay and by gravitation bending of the light can be used to assess the feasibility that the observed modulation is due entirely to a mass . Precise VLBI-measurements of the pulsar position near the time of conjunction should show a discrepancy between the corresponding positions and that before or after the event. The magnitude of the displacement in position (Misner et al. (1973)) for the pulsar B0525+21 would be , rather less than the present uncertainty in determination of the pulsar position by timing of . Simultaneous timing plus VLBI monitoring of the pulsars would give the first confirmation that some part of the significant TOA residuals is caused by a mass passing in front of the pulsar, even in the case when the characteristic time scale of the effect is too short compared to the interval between the observed sessions. Moreover, observation of modulation of the amplitude of the pulses can also be successfully explained in the way proposed by Krauss & Small (1991), if multi-path propagation of signal in the gravitational field of lensing mass causes the magnification of the observed amplitude. We believe that the described pulsar observations will help to detect non-luminous matter in our Galaxy.