The influence of the low-frequency timing noise on the precision of
measurements of the Keplerian and post-Keplerian orbital parameters in binary
pulsars is studied. Fundamental limits on the accuracy of tests of alternative
theories of gravity in the strong-field regime are established. The
gravitational low-frequency timing noise formed by an ensemble of binary stars
is briefly discussed.Comment: 4 pages, contributed paper to the proceedings of the IAU167
  colloquium on pulsars, Bonn, August-September 199

Kopeikin, Sergei M.

Potapov, Vladimir A.

English

arXiv

The influence of the low-frequency timing noise on the precision of measurements of the Keplerian and post-Keplerian orbital parameters in binary pulsars is studied. Fundamental limits on the accuracy of tests of alternative theories of gravity in the strong-field regime are established. The gravitational low-frequency timing noise formed by an ensemble of binary stars is briefly discussed

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arXiv:astro-ph/9911408v1  22 Nov 1999Pulsar Astronomy — 2000 and BeyondASP Conference Series, Vol. 3× 108, 1999M. Kramer, N. Wex, and R. Wielebinski, eds.Binary Pulsar Tests of General Relativity in the Presenceof Low-Frequency NoiseSergei M. Kopeikin1,2 and Vladimir A. Potapov3Abstract. The influence of the low-frequency timing noise on the preci-sion of measurements of the Keplerian and post-Keplerian orbital param-eters in binary pulsars is studied. Fundamental limits on the accuracyof tests of alternative theories of gravity in the strong-field regime areestablished. The gravitational low-frequency timing noise formed by anensemble of binary stars is briefly discussed.1 Dept. of Physics & Astronomy,University of Missouri-Columbia,Columbia, MO65211, USA2 On leave: ASC FIAN, Leninskii Prospect 53, Moscow, Russia3 PRAO, ASC FIAN, Leninskii Prospect 53, Moscow, Russia1. BackgroundThe discovery of the first binary pulsar in 1974 by R. Hulse and J.H. Taylor hadopened fascinating new opportunities for testing alternative theories of gravi-tation in the radiative and in the strong-gravitational-field regimes outside theboundaries of the Solar system. Presently, two distinguished binary pulsars -PSR B1913+16 and PSR B1534+12 - serve as primary extra-solar astrophys-ical laboratories for gravitational physisists. The radiative ω˙ − γ − P˙b test ofGeneral Relativity (GR) in the case of PSR B1913+16 has been performed witha precision about 0.4% (Taylor & Weisberg 1989, Damour & Taylor 1991). Inaddition, the binary pulsars PSR B1534+12 strong-field ω˙ − γ − s test of GRhas been done with the precision 1% (Stairs et al. 1998). What is worth notingis that the binary pulsar tests of GR have been carried out with an accuracybeing comparable to, or in some aspects, higher than the GR tests in the Solarsystem which have been always performed in the near, weak-field zone of thesolar system gravitating bodies.Reflecting about the future of gravitational experiments in binary pulsarsone can ask about whether the accuracy of GR tests can be further improvedand, if not, what physical influences can limit this accuracy. It is well knownthere are two main obstacles for improving precision of any experiment - system-atic and random errors1. Systematic errors cause the bias in the mean values1There exist a third reason which restricts the precision of GR tests in binary pulsars. It relatesto the fact that some additional (”hidden”) parameters appear in the form of additive linearcorrections to measured values of the main relativistic effects. These parameters can not beeffectively separated from the parameteres making the biggest contribution to the effect so that12 S.M.Kopeikin & V.A.Potapovof estimated parameters and can lead to inconsistent conclusions on the validityof the theory of gravitation under investigation. Fortunately, continuous im-provements being made in observational techniques and growing knowledge ondynamics of our galaxy can siginificantly reduce or even eliminate the influenceof systematic errors (Damour & Taylor 1991). On the other hand, random er-rors fully determine the magnitude of the numerical value of the dispersion ofmeasured parameters which itself crucially depends on the nature of noise inthe timing measurements. Usually, on rather short time spans the white noiseof measurement errors dominates. However, to improve the accuracy of the GRtests one has to observe binary pulsars over time intervals of about 10 years andlonger when the white noise is strongly supressed and low-frequency (or ”red”)noise dominates in observed times-of-arrivals (TOA) of pulsar’ pulses. The rednoise worsens estimates of measured parameters and increases the errors in theirmean values. What kind of problems will one meet in processing such timingobservations contaminated with the presence of the red noise ? Before tryingto answer this question let us briefly descibe the timing model of binary pulsarsand the properties of red noise.2. Timing ModelThe conventional timing model is based on the Damour-Deruelle analytic param-eterization of the two-body problem (Damour & Deruelle 1985, see also Klioner& Kopeikin 1994). Schematically it reads as followsN(T ) = N0 + νT +12ν˙T 2 + ǫint(T ) , (1)where N is the number of observed pulse, N0 is a constant, ν and ν˙ are the pul-sar’s rotational frequency and its time derivative, ǫint(T ) is the pulsur’s intrinsicnoise, T is the pulsar’s proper time, which is related to the observer’s propertime t by the equationT = t+∆(T ) + ∆⊙(t) + ǫpr(t) . (2)Herein, ∆(T ) and ∆⊙(t) describe the classic and relativistic time delay correc-tions for the binary system and for the Solar system respectively, ǫpr(t) are noisesproduced by a number of diverse perturbations during the time of propagationof pulses from the pulsar to observer, inaccuracies in ephemerides of the solarsystem bodies, imperfectness of atomic clocks used for the time metrology, andelectronic equipment used for timing observations. The equipment noise is per-formed to be white irrespective of the longivity of the observations. On the otherhand, any other kind of noise present has low-frequency components which showup when the measurements are carried out over a sufficiently long time span.It is worth noting that standard procedures of data processing were workedout only for white noise which has a gaussian distribution of timing residuals. Iftiming residuals are dominated by red noise, the standard statistical estimationsgive unrealistic (overestimated) numerical values of the measured parameters.one can not improve its numerical value (for more details see, for example, Damour & Taylor1991, Kopeikin 1994, 1996, Wex & Kopeikin 1999).Binary Pulsar Tests of General Relativity in the Presence of Low-Frequency Noise 3Table 1. Dependence of variances of the pulsar’s rotational fre-quency ν and some of the Keplerian and post-Keplerian parametersfrom the total span of observations τ in case of presence of red noisewith the spectrum S(f) = hn/fn, where h0, h1,...,h6 are constantscharacterizing the intensity of the noiseS(f) ν T0 x Pb x˙ P˙bh0 τ−3 τ−1 τ−1 τ−3 τ−3 τ−5h1/f τ−2 τ−1 τ−1 τ−3 τ−3 τ−5h2/f2 τ−1 τ−1 τ−1 τ−3 τ−3 τ−5h3/f3 const. const. const. τ−2 τ−2 τ−4h4/f4 τ τ τ τ−1 τ−1 τ−3h5/f5 τ2 τ2 τ2 const. const. τ−2h6/f6 τ3 τ3 τ3 τ τ τ−13. Red Noise and Limits on Estimates of ParametersRecently, we have begun to study the problem of pulsar data processing in thepresence of red noise (Kopeikin 1997, Kopeikin 1999a, Kopeikin & Potapov1998). To model the red noise we have used a shot-noise approximation witha specific choice of step function in such a way that any rational spectrum ofthe red noise could be restored, including flicker noise of phase ( 1/f), randomwalk of phase ( 1/f2), and so on. The noise model includes both stationary andnon-stationary parts of the autocovariance function which are well separatedalgebraically. In addition, an exhaustive treatment of the polynomial drift ofthe noise was worked out in full detail. Applying the model for processing fakedata of a binary pulsar in a circular orbit we set upper limits on the numericalvalues of the parameter’s variances. These upper limits depend on the totalspan of observations τ and can either decrease or grow as τ increases. The timedependence some of the limits are shown in Table 1.One can see that in the case when the spectral index n = 0, 1, 2, ... of thenoise is big enough variances of some, or even all, parameters grow such that onecan not get improvements in testing GR, which prevents better determination ofmasses of neutron stars and other physical characteristics of the binary system.Special methods of observations and/or data processing should be suggested toovercome this difficulty.This problem can also be considered from a different point of view. Thefact is that some of the red noises have a specific astrophysical origin and theirstudy would deliver extremely valuable information about physical processes4 S.M.Kopeikin & V.A.Potapovgenerating such noises2. One particular example represents a low-frequencytiming noise produced by the variable gravitational fields of binary stars.4. Gravitational timing noise from binary starsPrecise calculations of the autocovariance function of the gravitational timingnoise from binary stars requires having a mathematically complete solution tothe problem of propagation of electromagnetic waves in variable gravitationalfields of localized self-gravitating sources. Significant progress in solving thisproblem has been achieved recently by Kopeikin et al. (1999) and Kopeikin& Scha¨fer (1999). The relativistic time delay in an arbitrary time-dependentgravitational field has been presented as a function of the relative distancesbetween observer, source of light, and localized source of the non-stationarygravitational field as well as the intrinsic characteristics of the source. Using theexprerssion for the time delay, the gravitational timing noise from an ensembleof binaries in our galaxy has been evaluated for the case of PSR B1937+21 undersome simplifying assumptions (Kopeikin 1999b).Using the same approach the gravitational timing noise from an ensembleof binaries in a globular cluster can also be parameterized and predicted. Long-term precise timing monitoring of the bunch of millisecond pulsars in 47 TUC (orother globular cluster) will be required to test the presence of the noise and itsproperties. It will help to better understand spatial distribution, mass function,and other statistical properties of binaries in the globular cluster.5. AcknowledgementWe thank D. Moran for careful reading of the manuscript and useful comments.ReferencesDamour, T. & Taylor, J. H., 1991, ApJ, 366, 501Damour, T. & Deruelle, N., 1985, Ann. Inst. H. Poincare (Phisique Theorique),43, 107Ilyasov, Yu.P., Kopeikin, S.M., & Rodin, A. E., Astron. Lett., 24, 228Klioner, S. A. & Kopeikin, S. M., 1994, ApJ, 427, 951Kopeikin, S. M., 1994, ApJL, 434, 67Kopeikin, S. M., 1996, ApJL, 467, 93Kopeikin, S. M., 1997, MNRAS, 288, P. 129Kopeikin, S. M. & Potapov, V. A., 1998, ”Millisecond and Binary Pulsars as Na-ture’s Frequency Standards. III. Fourier Analysis and Spectral Sensitivity2A striking example which comes to mind is the case of electromagnetic cosmic backgroundradiation (CMB) which was discovered by Pensias &Wilson as an excess noise in the equipmentthey used for radio survey of the sky.Binary Pulsar Tests of General Relativity in the Presence of Low-Frequency Noise 5of Timing Observations to Low-Frequency Noise”, e-print physics/9811014,to be submitted to MNRASKopeikin, S. M., 1999a, MNRAS, 305, 563Kopeikin, S. M., 1999b, e-print gr-qc 9903070, to be published in the proceed-ings of the XXXIVth Rencontres de Moriond Meeting on ”GravitationalWaves and Experimental Gravity”, Les Arcs, France, 23-30 January 1999Kopeikin, S. M., Scha¨fer, G., Gwinn, C. R. & Eubanks, T. M., Phys. Rev. D,59, 084023Kopeikin, S. M. & Scha¨fer, G., Phys. Rev. D, 1999, Nov 15Stairs, I. H., Arzoumanian, Z., Camilo, F., Lyne, A. G., Nice, D. J., Taylor, J.H., Thorsett, S. E., Wolszczan, A., 1998, ApJ, 505, 352Taylor, J. H. & Weisberg, J. M. , 1989, ApJ, 345, 434Wex, N. & Kopeikin, S. M., 1999, ApJ, 514, 388

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