Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/yerac/larchenkova/larchenkova.ps Äàòà èçìåíåíèÿ: Wed Feb 22 22:51:13 1995 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 06:13:48 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: massive stars

Possible manifestation of the gravitational
lensing effect in single pulsar timing
By T. I. L a r c henk o va AND O. V. Do r o s h e nko
Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow 117810,
RUSSIA
We propose to use observations of the time delay of pulses from pulsars to detect lensing objects
located close to the line of the sight, in order to study the distribution of dark matter in our
Galaxy.
1. Introduction
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
non­luminous 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 0:185M fi pc \Gamma3 , while the counts of all known stars,
dust, and gas give a surface mass density of only one­half 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: ``macro­lensing'' and ``micro­lensing''.
Both kinds are due to the mass of a lensing galaxy cluster, galaxy, or compact object.
Macro­lensing 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.
Micro­lensing is provided by an individual star or a compact object having a mass
range from 10 \Gamma4 M fi to 10 6 M fi , 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 space­time 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 micro­lensing 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 non­luminous 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.
2. Time delay
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
1

2 Larchenkov & Doroshenko: Gravitational lensing in pulsar timing
delay causes an offset to the observed times of the signals travelling along the curved
space­time is:
\Deltat = \Gamma
X
B
2GMB
c 3
\Delta ln(1 \Gamma ~
R \Delta ~x B ); (2.1)
where vectors ~
R and ~x B are the unit length vectors pointed from the observer to the
source and to the B­th gravitating mass MB , c is the speed of light in vacuum and G 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 times­of­
arrival (TOA) of their pulses. The differences between the observed TOAs and those
calculated using the classical spin­down 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.
3. Application to the pulsar B0525+21
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 con­
junction 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 & Krause­Polstroff (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.
We have used the estimation of the Shapiro delay effect (1) for this pulsar in terms of
the relative velocity VP of the pulsar projected in the plane of the sky, with respect to
a single gravitating mass M , of the distance of the mass from the observer x and of the
time of conjunction T 0 of the mass with the observer's line of sight. We assumed that
the variation of the angle ` between the vectors ~
R and ~x can be approximated as:
cos ` =
\Theta 1 \Gamma
\Gamma (t \Gamma T 0
) 2 \Delta V 2
P =R 2 + d 2 =R 2
\Delta\Lambda 1=2
: (3.2)
Here d denotes the distance of closest approach of the mass to the line of sight of the
observer. Assuming that the parameters d and ` are small enough, so that d Ü R and
` Ü 1, and introducing the parameters ff = d=R, —
ff = VP =R and their ratio fi = —
ff=ff =
VP =d, one can obtain from equations 2.1 and 3.2 the modulation due to the Shapiro
delay as:
\Deltat = \Gamma2r \Delta ln(1 + fi 2 \Delta (t \Gamma T 0 ) 2 ); (3.3)

Larchenkov & Doroshenko: Gravitational lensing in pulsar timing 3
­20
­10
0
10
20
30
40
1970 1972 1974 1976 1978 1980 1982
Residuals,
ms
yrs
Figure 1. The TOA residuals for PSR B0525+21. The dots show the observed pre­fit­TOA
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
post­fit­TOA residuals after removing the Shapiro delay due to the lensing object.
where we have used the notation r j GMB =c 3 and left only the time­varying 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 least­square fit of the pulsar spin and astrometric parameters, as well as the
parameters r, fi 2 , T 0
, 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:
@ \Deltat
@r = \Gamma2 ln(1 + fi 2 (t \Gamma T 0 ) 2 ) (3.4)
@ \Deltat
@fi 2
= \Gamma 2r(t \Gamma T 0
) 2
1 + fi 2 (t \Gamma T 0 ) 2
(3.5)
@ \Deltat
@T 0
= \Gamma 4rfi 2 (t \Gamma T 0 )
1 + fi 2 (t \Gamma T 0 ) 2
: (3.6)

4 Larchenkov & Doroshenko: Gravitational lensing in pulsar timing
One can see from these equations that if we assume the generally accepted model of
slowing­down phase of the pulsar, describing the arrival time of Nth pulse from the
pulsar as
N = N 0 + f(t \Gamma t 0 ) + 1
2
—
f (t \Gamma t 0 ) 2 + : : : ; (3.7)
then our parameters @ \Deltat=@T 0 and @ \Deltat=@fi 2 will be nearly constants multiple of the
pulsar spin frequency f and its derivative —
f 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 T 0 and fi 2 are in strong covariance with f and —
f . 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 slowing­down of the rotation
of the pulsar to fit the pulsar period, period derivative, position, proper motion. In
addition, the values of r, fi 2 , T 0 were also included in the fit. The initial values of T 0 ,
fi 2 were evaluated from the observed residuals of the TOA and the data onto the proper
motion of the pulsar.
4. Results and Discussion
Our fitted values for the mass and other parameters of the lensing object are:
M j r \Delta c 3 =G = 330 \Sigma 50M fi ; (4.8)
VP =d = (1:0 \Sigma 0:7) \Theta 10 \Gamma7 s \Gamma1 ; (4.9)
T 0 = 2442040:0 JD: (4.10)
With these values of the parameters the pre­fit and post­fit 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 post­fit 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 ¯ = 21 mas yr \Gamma1 and its distance R = 2:3 kpc (Taylor et al. (1993))
yield VP ¸ ¯R ú 200 km s \Gamma1 , so that our value for VP =d gives the distance of closest
approach d ú 13 AU. One can suggest that the observed extra modulation is due to a
time delay PSR B0525+21 caused by a mass M = 330M fi passing near the line of the
sight. The existence of stellar masses as high as 330M fi 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 (T 0 = 2442064 JD) by Downs (1982), and that
the behaviour of its residuals is very similar to the post­glitch recovery described by the
two­component 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 M . Precise VLBI­measurements 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 ' = 4GM=(c 2 d) (Misner et
al. (1973)) for the pulsar B0525+21 would be ' ¸ 0 00 :2, rather less than the present
uncertainty in determination of the pulsar position by timing of ¸ 5 00 :0. Simultaneous

Larchenkov & Doroshenko: Gravitational lensing in pulsar timing 5
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 VP =d 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 multi­path 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 non­luminous matter in our Galaxy.
The authors are very grateful to N.S.Kardashev, B.V.Komberg, V.N.Lukash and the
staff of the Theoretical Astrophysics Department of the Astro Space Center for discus­
sions. Also we thank S.Kopeikin, T.Fukushima, K.Ohnishi, M.Hosokava.It is a pleasure
to thank J.Lequeux for his comments on manuscript. TIL was supported in part by
Grant MEZ000 from International Science Foundation.
REFERENCES
Bachall J.N., 1984, ApJ, 276, 169.
Baym G., Pethick C., Pines D., Ruderman M., 1969, Nature, 224, 872.
Blanford R. & Teukolsky S., 1976, ApJ, 205, 580.
Chang & Refsdal, 1979, Nature, 282, 561.
Cordes J.M., Downs G.S., 1985, ApJS, 59, 343.
Doroshenko O.V. & Kopeikin S.M., 1990 SvA, 34, 496.
Downs G.S., 1982, ApJ, 257, L67.
Downs G.S., Reichley P.E., 1983, ApJS, 53, 169.
Downs G.S., Krause­Polstroff J., 1986, ApJS, 62, 81.
Krauss L.M., Small T.A., 1991, ApJ, 378, 22.
Lyne A.G., 1992, Phil.Trans.R.Soc.Lond. A, 341, 39.
Misner C.W., Thorne K.S.& Wheeler J.A., 1973, Gravitation, (W.H.Freedman, San Francisco).
Oort J.H., 1960, Bull.Ast.Inst.Netherlands, 15, 45.
Ryba M.F.& Taylor J.H., 1991, ApJ, 371, 739.
Taylor J.H., Manchester R.N. & Lyne A.G., 1993, ApJS, 88, 529.