3.2 Preferred-frame effects and non-conservation of momentum

3.2.1 Limits on αˆ1

A non-zero ˆα1 implies that the velocity w of a binary pulsar system (relative to a “universal” background reference frame given by the Cosmic Microwave Background, or CMB) will affect its orbital evolution. In a manner similar to the effects of a non-zero Δnet, the time evolution of the eccentricity will depend on both ω˙ and a term that tries to force the semi-major axis of the orbit to align with the projection of the system velocity onto the orbital plane.

The analysis proceeds in a similar fashion to that for Δ net, except that the magnitude of e F is now written as [3719Jump To The Next Citation Point]

1 ||m − m || |w | |eF| = ---ˆα1||--1-----2||------------⊥--------1∕3, (19 ) 12 m1 + m2 [G(m1 + m2 )(2π∕Pb )]
where w ⊥ is the projection of the system velocity onto the orbital plane. The angle λ, used in determining this projection in a manner similar to that of Equation (18View Equation), 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 ˆα1 is P 1b∕3∕e. As for the Δnet test, the systems must be old, so that τc ≫ 2π∕ω˙, and ˙ω must be larger than the rate of Galactic rotation. Examples of suitable systems are PSR J2317+1439 [2819] with a last published value of −6 e < 1.2 × 10 in 1996 [29Jump To The Next Citation Point], and PSR J1012+5307, with − 7 e < 8 × 10 [85Jump To The Next Citation Point]. This latter system is especially valuable because observations of its white-dwarf component yield a radial velocity measurement [25], eliminating the need to find a lower limit on an unknown quantity. The analysis of Wex [146Jump To The Next Citation Point] yields a limit of αˆ1 < 1.4 × 10 −4. This is comparable in magnitude to the weak-field results from lunar laser ranging, but incorporates strong field effects as well.

3.2.2 Limits onˆα3

Tests of ˆα3 can be derived from both binary and single pulsars, using slightly different techniques. A non-zero αˆ3, which implies both a violation of local Lorentz invariance and non-conservation of momentum, will cause a rotating body to experience a self-acceleration aself in a direction orthogonal to both its spin Ω S and its absolute velocity w [105]:

1- -Egrav aself = − 3 ˆα3(mc2 )w × ΩS. (20 )
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 αˆ 3 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 P˙ via the Doppler effect, by an amount

P˙ = P-ˆn ⋅ a , (21 ) ˆα3 c self
where ˆn is a unit vector in the direction from the pulsar to the Earth. The analysis of Will [149Jump To The Next Citation Point] assumes random orientations of both the pulsar spin axes and velocities, and finds that, on average, |P˙ˆα | ≃ 5 × 10−5|ˆα3| 3, independent of the pulse period. The sign of the ˆα3 contribution to P˙, 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 10–14 s/s. Thus, the maximum possible contribution from αˆ3 must also be considered to be of this size, and the limit is given by −10 |ˆα3| < 2 × 10 [149Jump To The Next Citation Point].

Bell [17Jump To The Next Citation Point] applies this test to a set of millisecond pulsars; these have much smaller period derivatives, on the order of 10–20 s/s. Here, it is also necessary to account for the “Shklovskii effect” [117Jump To The Next Citation Point] in which a similar Doppler-shift addition to the period derivative results from the transverse motion of the pulsar on the sky:

˙ 2d- Ppm = P μ c , (22 )
where μ 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 [131Jump To The Next Citation Point32Jump To The Next Citation Point]. Nevertheless, once this correction (which is always positive) is applied to the observed period derivatives for isolated millisecond pulsars, a limit on |αˆ3 | on the order of 10–15 results [1720Jump To The Next Citation Point].

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 [20Jump To The Next Citation Point]. 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 Pb and ω˙ [20]. The first term, however, can lead to a significant test of ˆα3. 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

1 aself = -ˆα3cp w × ΩSp, (23 ) 6
where cp and ΩSp denote the compactness and spin angular frequency of the pulsar, respectively, and w is the velocity of the system. For evolutionary reasons (see, e.g., [22Jump To The Next Citation Point]), 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
cp|w-|Pb2-----c2----- |eF| = ˆα3 24π P G (m1 + m2 ) sin β, (24 )
where β is the (unknown) angle between w and ΩSp, and P is, as usual, the spin period of the pulsar: P = 2π ∕ΩSp.

The figure of merit for systems used to test ˆα3 is P2∕(eP ) b. The additional requirements of τc ≫ 2π∕ω˙ and ˙ω 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 − 19 ˆα3 < 1.5 × 10, much more stringent than for the single-pulsar case.

3.2.3 Limits on ζ2

Another PPN parameter that predicts the non-conservation of momentum is ζ2. It will contribute, along with α3, to an acceleration of the centre of mass of a binary system [148Jump To The Next Citation Point149Jump To The Next Citation Point]

acm = (α3 + ζ2) ---πm1m2---(m1--−-m2-)----enp, (25 ) Pb[(m1 + m2 )a(1 − e2)]3∕2
where n p is a unit vector from the centre of mass to the periastron of m 1. This acceleration produces the same type of Doppler-effect contribution to a binary pulsar’s ˙ P as described in Section 3.2.2. In a small-eccentricity system, this contribution would not be separable from the P˙ 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 a cm 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 ¨ P is given by [148Jump To The Next Citation Point149Jump To The Next Citation Point]
( )2 P¨= P-(α3 + ζ2)m2 sin i 2π- X--(1-−-X-) e-˙ω-cos-ω-, (26 ) 2 Pb (1 + X )2 (1 − e2)3∕2
where X = m1 ∕m2 is the mass ratio of the two stars and an average value of cosω is chosen. As of 1992, the 95% confidence upper limit on ¨ P was 4 × 10–30 s–1 [132Jump To The Next Citation Point148Jump To The Next Citation Point]. This leads to an upper limit on (α3 + ζ2) of 4 × 10–5 [148]. As α3 is orders of magnitude smaller than this (see Section 3.2.2), this can be interpreted as a limit on ζ2 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 ¨ P (J. Taylor, private communication, as cited in [76Jump To The Next Citation Point]). 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 P¨ could also be interpreted as timing noise (sometimes seen in recycled pulsars [74Jump To The Next Citation Point]) or else a manifestation of profile changes due to geodetic precession [8076].
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