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Pulsar Highlights - Best Limits on Dipolar Gravitational Wave Emission

Best Limits Ever on Dipolar Gravitational Wave Emission

PSR J1738+0333 is a 5.85-ms pulsar in a binary system with an orbital period of 8.5 hours and a companion white dwarf (WD) with a mass of about 0.2 solar masses. This millisecond pulsar (MSP) was found with the Parkes 64-m Radio Telescope in a 20-cm Multi-Beam search for pulsars in intermediate Galactic latitudes ( 15° < | b | < 30°), lead by Bryan Jacoby (then at Caltech) and Matthew Bailes (Swinburne University).

Paulo Freire has been timing this pulsar with the Arecibo 305-m telescope for the last 3 years (project P1684) using the Wide-band Arecibo Pulsar Processors (WAPPs). He has obtained good timing precision, in fact this is now one of the most precisely timed pulsars ever, the 1-hour averages of the timing residuals have a root mean square of 200 ns per WAPP per hour. There is a small apparent eccentricity of about 0.0000011, which indicates that the orbit, even with a semi-major axis of 102,000 km / sin i, does not deviate from a circle by more than 80 µm / sin i! (and, yes, this number is correct). However, there seem to be small dispersive delays near superior conjunction. Taking these into account, we get an eccentricity that is less than 0.0000005. That would imply that the orbit does not deviate from a circle by more than 14 µm / sin i.

In a few double neutron star (DNS) systems, like PSR B1913+16 (the original Hulse-Taylor binary pulsar), PSR B1534+12 (see related highlight), PSR J0737−3039 (the double pulsar) and more recently PSR J1906+0746 (see related highlight), the observation of several relativistic effects in the timing of the neutron star (NS) observable as a pulsar (normally the object that has been recycled, but not always) has allowed precise estimates of the mass of the pulsar and of its NS companion. These effects, the precession of periastron and the varying relativistic time delay at different orbital phases (Einstein delay), can only be measured for pulsars with eccentric orbits, that is always the case among DNSs. Knowing the component masses, we can predict, assuming the validity of general relativity (GR), a few other relativistic effects, like the Shapiro delay or the rate of orbital decay due to the emission of gravitational waves. We can test GR if we are able to measure the latter parameters and check if their numerical values are as predicted.

While the recycled pulsars in DNS systems spin tens of times per second, MSPs like PSR J1738+0333 spin hundreds of times per second. They had much longer accretion episodes, which means that their companions were much longer lived stars of relatively low masses. These MSPs are normally found in circular orbits with WD stars. Because of these circular orbits, we can't measure the rate of advance of periastron nor the Einstein delay. This makes it difficult to measure the masses of these objects, and almost impossible to use them to test GR. This is a pity for many reasons:

MSPs have very high timing precision, so the aforementioned relativistic effects could be measured with extreme precision if such pulsars were in eccentric orbits.
MSPs have accreted matter from their companions for a very long time, so they could be substantially more massive than the NSs found in DNS binaries. Finding how massive a neutron star can be is important for determining how matter behaves at very high densities, an yet unsolved problem.
Theoretically MSP-WD systems could be the best probes for testing alternative theories of gravitation. The reason for this is the following: all alternative theories of gravitation (but not GR), predict violation of the Strong Equivalence Principle (SEP). The observables consequences of SEP violation are generally proportional to the difference of gravitational binding energies of the two components of the binary, or a theory-specific function of them. Therefore, DNSs that pass very precise GR tests (for the latest and best example, see Kramer et al. 2006) do not necessarily lead to tight constraints on SEP violation and alternative theories of gravitation: the binding energies of the neutron stars could be very similar, causing SEP violation to go undetected. In a MSP-WD binary, on the other hand, the binding energy of the WD is negligible compared to that of the MSP, so if SEP is violated, there will be observable consequences. These would be:
- If the orbital period is long, SEP violation would cause a secular increase in the eccentricity of the orbit. This would be due to the fact that the two components would be accelerating at slightly different rates in the gravitational field of the Galaxy. The Arecibo observation of several pulsar-WD systems in wide orbits and low eccentricities has lead to tighter limits on SEP violation (Stairs et al. 2005).
- If the orbital period is short, SEP violation could in principle lead to observable emission of dipolar gravitational waves. This would appear as a contribution to the orbital decay of the system in addition to that caused by the emission of quadrupolar gravitational waves predicted by general relativity. Unfortunately, for most MSP-WD systems, the masses of both components are not known, therefore we can't calculate the quadrupolar emission predicted by GR and compare it with the observational values.

Initial timing of PSR J1738+0333 sought to determine the companion and pulsar masses from a measurement of the Shapiro delay - this small propagational effect allows an estimate of the pulsar and companion masses even in the absence of any eccentricity. Generally, high timing precision is required for a measurement, but having an inclination close to 90° also makes the effect much easier to measure. For 1738+0333 this measurement was not possible because of the low orbital inclination of the system. This binary system might therefore be completely useless.

Fortunately, it was possible to determine the masses of the components independently. This comes from recent optical work of Marten van Kerkwijk and Bryan Jacoby. Using the Magellan telescope on Las Campanas, Chile, they detected the companion star and measured its spectrum accurately (see Fig. 1).

Figure 1: Spectrum of the white dwarf companion of PSR J1738+0333. Note the sharp absorption features, these were used to measure accurate radial velocities for this object. Image provided by M. van Kerkwijk.
The spectrum is very similar to that of the companion of PSR J1909−3744; which has as mass of 0.203 solar masses, measured by Shapiro delay (Jacoby et al. 2005). The companion of PSR J1738+0333 must therefore a very similar mass. More recently, the radial-velocity curve was measured using Gemini South on Cerro Pachón (see Fig. 2), from this we can derive the mass ratio of the system, 8.1 ± 0.3. The pulsar mass is therefore about 1.6 ± 0.2 solar masses (we have assumed here a 10% uncertainty in the mass of the companion, this still needs to be estimated more precisely). This is an interesting value per se, if measured more precisely it could exclude some models for the behavior of matter at densities higher than that of the atomic nucleus. This was the original motivation for the timing of 1738+0333.

Spectrum of the white dwarf companion of PSR J1738+0333

Figure 2: Radial velocity measurements for the companion of PSR J1738+0333 as a function of orbital phase, with best fit in blue. The red curve represents the variation of the pulsar's orbital velocity along the line of sight. For this binary pulsar, we have now information on the absolute radial velocity of the center of mass, something that is not normally available for binary pulsars. Image provided by M. van Kerkwijk.

This is also important because it allows a calculation of the expected rate of orbital decay due to the emission of quadrupolar gravitational waves: −(3.4 ± 0.6) × 10⁻¹⁴s/s. This period derivative is about 60 times smaller than what was measured for the Hulse-Taylor binary pulsar; the 8.5-hour orbital period should become approximately 1 microsecond shorter every year!

Fortunately, the timing precision for PSR J1738+0333 is so high that we can already measure this value after only three years of timing, although not with much significance: it is −(4.4 ± 2.9) × 10⁻¹⁴s/s. What is more important is the difference between predicted and observed values is the smallest ever measured. This introduces the tightest constraints ever on dipolar gravitational wave emission. If we interpret the limit on the emission of gravitational waves as a constant "omega" in Brans-Dicke gravity, we obtain ω > 2300 (s/0.2)² (a), the previous pulsar limit is ω > 1300 (s/0.2)², derived from Arecibo timing of PSR J0751+1807 (Nice et al. 2005, see related highlight). For GR, this value is infinite. This is not as good as the result from the Cassini spacecraft (ω > 40,000, Bertotti et al. 2003), but it is obtained in the strong-field regime, the only that can constrain all alternative theories of gravitation.

This, however, is not the main result. For 1738+0333, there is great potential for further improvement in the test, given that the component masses in this case are relatively well known from the optical studies. Continued timing of PSR J1738+0333 over the next 5(10) years will increase the precision of our measurement of the orbital period derivative by a factor of 10(40). By then, if we assume that the measured value conforms to the prediction, the uncertainty of the prediction itself (6 × 10⁻¹⁵ s/s) will be the limiting factor in the precision of this test. This will be equivalent to ω > 15,000 (s/0.2)², an order of magnitude improvement on all previous pulsar tests. If the optical measurements can be improved, this limit could be substantially larger.

One of the advantages of the high timing precision of PSR J1738+0333 has been a precise measurement of the proper motion (6.814 ± 0.017 mas/yr in RA and 4.90 ± 0.06 mas/yr in Dec) and the parallax (1.43 ± 0.10 mas). This allows a very precise correction of the kinetic contribution to the orbital period derivative.

Furthermore, improving the mass ratio (definitely possible by averaging more measurements) and using a precise measurement of the orbital decay will be used to determine the mass of the pulsar and the companion very accurately, assuming that general relativity applies. These values might be important for the study of the equation of state for dense matter. PSR J1738+0333 might therefore be a great physics laboratory, relevant for the study of gravitation and the study of the equation of state.

Figure 3: Mass constraints for the PSR J1738+0333 binary system. The system has to be in the intersection of the companion mass (m₂) and mass ratio (R) regions. The dashed lines indicate 1-sigma limits for the measurement of the orbital decay. The solid lines indicate constant orbital inclinations, we can see that the orbital inclination of this binary system is slightly larger than 30 degrees. The gray bar indicates the range of precisely measured neutron star masses.

(a) In Brans-Dicke theory, the variable "s" is the change of the binding energy of the neutron star as a function of the gravitational constant G, a parameter that is not fixed for that theory. The numerical value of s depends on the equation of state, it is predicted to range from 0.1 to 0.3.