The strong-field formalism instead uses the parameter , which for object “i” may be written as. 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  write the time-dependent eccentricity vector as [43, 145]
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 107 days2 . However, as pointed out by Damour and Schäfer  and Wex , 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 , 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  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
Next comes the task of estimating the projection of onto the orbital plane. The projection can be written as. The values of and can be determined from models of the Galactic potential (see, e.g., [84, 2]). The inclination angle can be estimated if even crude estimates of the neutron star and companion masses are available, from statistics of NS masses (see, e.g., ) and/or a relation between the size of the orbit and the WD companion mass (see, e.g., ). However, the angle is also usually unknown and also must be assumed to be uniformly distributed between 0 and .
Damour and Schäfer  use the PSR B1953+29 system and integrate over the angles and to determine a 90% confidence upper limit of . Wex  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  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 ) 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 , its comparatively large observed eccentricity of 2 × 10–5 and short orbital period mean it will not significantly affect the current limits.
© Max Planck Society and the author(s)