The strong-field formalism instead uses the parameter
[41
], which for object ``
i
'' may be written as
Pulsar-white-dwarf systems then constrain
[41
]. If the SEP is violated, the equations of motion for such a
system will contain an extra acceleration
, where
is the gravitational field of the Galaxy. As the pulsar and the
white dwarf fall differently in this field, this
term will influence the evolution of the orbit of the system.
For low-eccentricity orbits, by far the largest effect will be a
long-term forcing of the eccentricity toward alignment with the
projection of
onto the orbital plane of the system. Thus, the time evolution
of the eccentricity vector will not only depend on the usual
GR-predicted relativistic advance of periastron (
), but will also include a constant term. Damour and
Schäfer [41
] write the time-dependent eccentricity vector as
where
is the
-induced rotating eccentricity vector, and
is the forced component. In terms of
, the magnitude of
may be written as [41
,
145
]
where
is the projection of the gravitational field onto the orbital
plane, and
is the semi-major axis of the orbit. For small-eccentricity
systems, this reduces to
where M is the total mass of the system, and, in GR, F = 1 and G is Newton's constant.
Clearly, the primary criterion for selecting pulsars to test
the SEP is for the orbital system to have a large value of
, greater than or equal to
[145
]. However, as pointed out by Damour and Schäfer [41
] and Wex [145
], two age-related restrictions are also needed. First of all,
the pulsar must be sufficiently old that the
-induced rotation of
has completed many turns and
can be assumed to be randomly oriented. This requires that the
characteristic age
be
, and thus young pulsars cannot be used. Secondly,
itself must be larger than the rate of Galactic rotation, so
that the projection of
onto the orbit can be assumed to be constant. According to
Wex [145
], this holds true for pulsars with orbital periods of less than
about 1000 days.
Converting Equation (16) to a limit on
requires some statistical arguments to deal with the unknowns in
the problem. First is the actual component of the observed
eccentricity vector (or upper limit) along a given direction.
Damour and Schäfer [41
] assume the worst case of possible cancellation between the two
components of
, namely that
. With an angle
between
and
(see Figure
4), they write
. Wex [145
,
146
] corrects this slightly and uses the inequality
where
. In both cases,
is assumed to have a uniform probability distribution between 0
and
.
Next comes the task of estimating the projection of
onto the orbital plane. The projection can be written as
where
i
is the inclination angle of the orbital plane relative to the
line of sight,
is the line of nodes, and
is the angle between the line of sight to the pulsar and
[41
]. The values of
and
can be determined from models of the Galactic potential (see,
e.g., [83
,
1
]). The inclination angle
i
can be estimated if even crude estimates of the neutron star and
companion masses are available, from statistics of NS masses
(see,
e.g., [136
]) and/or a relation between the size of the orbit and the WD
companion mass (see,
e.g., [114]). However, the angle
is also usually unknown and also must be assumed to be uniformly
distributed between 0 and
.
Damour and Schäfer [41] use the PSR B1953+29 system and integrate over the angles
and
to determine a 90% confidence upper limit of
. Wex [145] uses an ensemble of pulsars, calculating for each system the
probability (fractional area in
-
space) that
is less than a given value, and then deriving a cumulative
probability for each value of
. In this way he derives
at 95% confidence. However, this method may be vulnerable to
selection effects; perhaps the observed systems are not
representative of the true population. Wex [146
] later overcomes this problem by inverting the question. Given a
value of
, an upper limit on
is obtained from Equation (17
). A Monte Carlo simulation of the expected pulsar population
(assuming a range of masses based on evolutionary models and a
random orientation of
) then yields a certain fraction of the population that agree
with this limit on
. The collection of pulsars ultimately gives a limit of
at 95% confidence. This is slightly weaker than Wex's previous
limit but derived in a more rigorous manner.
Prospects for improving the limits come from the discovery of
new suitable pulsars, and from better limits on eccentricity from
long-term timing of the current set of pulsars. In principle,
measurement of the full orbital orientation (i.e.,
and
i) for certain systems could reduce the dependence on statistical
arguments. However, the possibility of cancellation between
and
will always remain. Thus, even though the required angles have
in fact been measured for the millisecond pulsar
J0437-4715 [139
], its comparatively large observed eccentricity of
and short orbital period mean it will not significantly affect
the current limits.
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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 |