The analysis proceeds in a similar fashion to that for
, except that the magnitude of
is now written as [34,
18
]
where
is the projection of the system velocity onto the orbital plane.
The angle
, used in determining this projection in a manner similar to that
of Equation (18
), is now the angle between the line of sight to the pulsar and
the absolute velocity of the binary system.
The figure of merit for systems used to test
is
. As for the
test, the systems must be old, so that
, and
must be larger than the rate of Galactic rotation. Examples of
suitable systems are PSR J2317+1439 [27,
18] with a last published value of
in 1996 [28
], and PSR J1012+5307, with
[84
]. This latter system is especially valuable because observations
of its white-dwarf component yield a radial velocity
measurement [24], eliminating the need to find a lower limit on an unknown
quantity. The analysis of Wex [146
] yields a limit of
. This is comparable in magnitude to the weak-field results from
lunar laser ranging, but incorporates strong field effects as
well.
Thus, the self-acceleration depends strongly on the compactness of the object, as discussed in Section 3 above.
An ensemble of single (isolated) pulsars can be used to set a
limit on
in the following manner. For any given pulsar, it is likely that
some fraction of the self-acceleration will be directed along the
line of sight to the Earth. Such an acceleration will contribute
to the observed period derivative
via the Doppler effect, by an amount
where
is a unit vector in the direction from the pulsar to the Earth.
The analysis of Will [152
] assumes random orientations of both the pulsar spin axes and
velocities, and finds that, on average,
, independent of the pulse period. The
sign
of the
contribution to
, however, may be positive or negative for any individual pulsar;
thus, if there were a large contribution on average, one would
expect to observe pulsars with both positive and negative total
period derivatives. Young pulsars in the field of the Galaxy
(pulsars in globular clusters suffer from unknown accelerations
from the cluster gravitational potential and do not count toward
this analysis) all show positive period derivatives, typically
around
. Thus, the maximum possible contribution from
must also be considered to be of this size, and the limit is
given by
[152
].
Bell [16] applies this test to a set of millisecond pulsars; these have
much smaller period derivatives, on the order of
s/s. Here, it is also necessary to account for the ``Shklovskii
effect'' [119
] in which a similar Doppler-shift addition to the period
derivative results from the transverse motion of the pulsar on
the sky:
where
is the proper motion of the pulsar and
d
is the distance between the Earth and the pulsar. The distance
is usually poorly determined, with uncertainties of typically 30%
resulting from models of the dispersive free electron density in
the Galaxy [132
,
30
]. Nevertheless, once this correction (which is always positive)
is applied to the observed period derivatives for isolated
millisecond pulsars, a limit on
on the order of
results [16,
19
].
In the case of a binary-pulsar-white-dwarf system, both bodies
experience a self-acceleration. The combined accelerations affect
both the velocity of the centre of mass of the system (an effect
which may not be readily observable) and the relative motion of
the two bodies [19]. The relative-motion effects break down into a term involving
the coupling of the spins to the absolute motion of the centre of
mass, and a second term which couples the spins to the orbital
velocities of the stars. The second term induces only a very
small, unobservable correction to
and
[19]. The first term, however, can lead to a significant test of
. Both the compactness and the spin of the pulsar will completely
dominate those of the white dwarf, making the net acceleration of
the two bodies effectively
where
and
denote the compactness and spin angular frequency of the pulsar,
respectively, and
is the velocity of the system. For evolutionary reasons (see,
e.g., [21
]), the spin axis of the pulsar may be assumed to be aligned with
the orbital angular momentum of the system, hence the net effect
of the acceleration will be to induce a polarization of the
eccentricity vector within the orbital plane. The forced
eccentricity term may be written as
where
is the (unknown) angle between
and
, and
P
is, as usual, the spin period of the pulsar:
.
The figure of merit for systems used to test
is
. The additional requirements of
and
being larger than the rate of Galactic rotation also hold. The
95% confidence limit derived by Wex [146] for an ensemble of binary pulsars is
, much more stringent than for the single-pulsar case.
where
is a unit vector from the centre of mass to the periastron of
. This acceleration produces the same type of Doppler-effect
contribution to a binary pulsar's
as described in Section
3.2.2
. In a small-eccentricity system, this contribution would not be
separable from the
intrinsic to the pulsar. However, in a highly eccentric binary
such as PSR B1913+16, the longitude of periastron advances
significantly - for PSR B1913+16, it has advanced nearly 120
since the pulsar's discovery. In this case, the projection of
along the line of sight to the Earth will change considerably
over the long term, producing an effective
second
derivative of the pulse period. This
is given by [149
,
152
]
where
is the mass ratio of the two stars and an average value of
is chosen. As of 1992, the 95% confidence upper limit on
was
[133
,
149
]. This leads to an upper limit on
of
[149]. As
is orders of magnitude smaller than this (see Section
3.2.2), this can be interpreted as a limit on
alone. Although PSR B1913+16 is of course still observed,
the infrequent campaign nature of the observations makes it
difficult to set a much better limit on
(J. Taylor, private communication, as cited in [75
]). The other well-studied double-neutron-star binary,
PSR B1534+12, yields a weaker test due to its orbital
parameters and very similar component masses. A complication for
this test is that an observed
could also be interpreted as timing noise (sometimes seen in
recycled pulsars [73
]) or else a manifestation of profile changes due to geodetic
precession [79,
75].
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Testing General Relativity with Pulsar Timing
Ingrid H. Stairs http://www.livingreviews.org/lrr-2003-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |