PHYSICAL REVIEW D 71, 123002 (2005)

Gravitational wave astrometry for rapidly rotating neutron stars and estimation of their distances

Naoki SetoTheoretical Astrophysics, MC 130-33, California Institute of Technology, Pasadena, California 91125, USA

(Received 10 November 2004; published 17 June 2005)

1550-7998=20

I discuss an astrometric timing effect on data analysis of continuous gravitational waves from rapidlyrotating isolated neutron stars. Special attention is directed to the possibility of determining their distancesby measuring the curvature of the wave fronts. I predict that if continuous gravitational waves from anunknown neutron star with a stable rotation are detected around 1 kHz within 1=3 yr by initial LIGOdetectors and the ellipticity parameter � is smaller than 10�6, the distance r to the source can be estimatedwith relative error �r=r of �10% by using the broadband configuration of advanced LIGO detectors over3 years. By combining the observed amplitude of the waves with the estimated distance, information onthe parameter � can be obtained purely through gravitational wave measurements.

DOI: 10.1103/PhysRevD.71.123002 PACS numbers: 95.85.Sz, 04.80.Nn, 97.10.Vm

I. INTRODUCTION

Over the last 5 years, several large-scale interferometricgravitational wave detectors, US LIGO [1], JapaneseTAMA [2], and British-German GEO [3], were constructedand have given us scientific data. Italian-French VIRGO[4] will be soon in operation. In the next decade, secondgeneration detectors (e.g. advanced LIGO [5], LCGT [6])are expected to be available. More ambitious third genera-tion detectors (e.g. EURO [7]) will also be realized in thefuture. While the current central objective of scientific runsis the direct detection of gravitational waves in 10 Hz to afew kHz band, gravitational wave astronomy will be rap-idly established with the advent of new and more sensitivedetectors [8,9]. Once gravitational waves from a popula-tion of sources are detected with relatively small signal-to-noise ratios (SNRs), follow-on detectors would reveal newastronomical information at high SNRs. This kind of con-sideration is important for the design and operational strat-egy in planing next generation detectors.

In this paper, I study an astrometric effect relevant to thedata analysis of gravitational waves from rapidly rotatingisolated neutron stars and discuss the possibility of esti-mating their distances. If gravitational waves from a rotat-ing neutron star are detected, the amplitude containsinformation about the distance r and the ellipticity parame-ter � through the combination �=r. The parameter � char-acterizes the nonaxisymmetry of the star. However, it is noteasy to separate these two quantities individually fromgravitational waves. Here, I show that by a long-termobservation of gravitational waves from a source, we canmeasure a small phase shift induced by an astrometriceffect. This means that rapidly rotating neutron stars arethe ideal targets to apply the long-term astrometric studieswith planned ground based detectors. Gravitational wavesfrom merging neutron stars or supernovae would last atmost order of seconds [8,9] and are not suitable for detect-ing the shift. It is also interesting to understand how thequality of the astronomical information depends on theobservational duration. For a very short phenomenon, the

05=71(12)=123002(6)$23.00 123002

only information improved by a long data span is the eventrate.

To confirm gravitational waves from a rotating neutronstar, we have to integrate the signal for a long time to get asufficient signal-to-noise ratio [10,11]. The required com-putational cost for the data analysis depends strongly onwhether the neutron star is known or unknown. If itslocation is unknown, we need a large number of templates,and the data analysis is a very challenging task [12]. Theastrometric effect is larger for a closer object, and weimplicitly assume a nearby unknown neutron star as ourmain target. Once gravitational waves from an unknownneutron star are detected, we might identify anelectromagnetic-wave counterpart using information ob-tained from the gravitational wave measurement, and esti-mate its distance with various traditional methods (see e.g.[13]). But a search for an electromagnetic-wave counter-part would be very difficult for a cold neutron star with alarge peculiar velocity and a very weak magnetic field (e.g.less than �107 gauss critical for millisecond pulsars [14]).

This paper is organized as follows: In the next section Iformulate the phase shift induced by the astrometric effectfor a rapidly rotating neutron star, and explain a way toestimate its distance though the phase shift. Then, using theFisher matrix approach, I obtain an expression for themagnitude of error associated with this estimation. Thisexpression is our central result. In section III I apply it toobservational situations with specifications for futureprojects, such as, advanced LIGO. I examine requiredconditions for the ellipticity � and the source distance rin order to detect gravitational waves from rotating neutronstars and to determine their distances with the proposedmethod. Then I statistically study whether there are nearbyneutron stars to apply our method. Section IV is devoted toa discussion including a brief summary.

II. FORMULATION

I first study how well we can extract information on thedistance from the phase modulation of gravitational waves.

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NAOKI SETO PHYSICAL REVIEW D 71, 123002 (2005)

Similar arguments have been made for radio pulsar timinganalyses [16,17], but this is the first attempt at applying itto gravitational wave data. Our basic aim is to develop amodel relevant for evaluating parameter estimation errors.I do not try to make a detailed model required for actualsignal analysis, e.g., including general relativistic timedelays in the solar system that are small and introduce noadditional source parameters [11]. In addition, the effectsof diurnal rotation are not included here, because we aredealing with the case in which the signal integration is oforder a few years. Following Ref. [11], I use the solarsystem barycenter (SSB) rest frame and investigate theeffects induced by the motions of the detector as well asthe source. I denote the time dependent position of thedetector rd (rd � jrdj � 1AU), and assume that the sourceis moving with a constant velocity vns as rn0 � vns�t�r=c� in the SSB frame. Here rn0 is the position of thesource (n0; source direction) from which the emitted wavefront reaches the SSB at t � 0. I define n0 ��sin cos; sin sin; cos� in the spherical SSB coordi-nate with being the angle between the source direction n0and the normal vector of the orbital plane (ecliptic) of theEarth. I write the gravitational wave phase �SSB observedat the SSB in the form �SSB�t� � �0 �2�

P3k�0 fkt

k�1=�k� 1�! and perturbatively include thephase modulation due to the detector’s and source’s mo-tions with two small expansion parameters; � � jrd=rj and� � jvnstj=r. After some algebra, the phase �d observedby the detector is given to O��m�n� �n�m � 2� as

�d � �0 � 2�X3k�0

fktk�1

�k� 1�!�2�rc

�n0

rdr�

vns?tr

rdr�

1

2r2�1� �n0 rd�

2�

�X3k�0

fktk

k!

� higher order terms, (2.1)

where vns? is the velocity component of vns perpendicularto the direction n0 [16]. There are no terms of order O���and O��2� [11]. In the large parenthesis in the aboveequation, the O��� term is nothing, but the plane waveeffect that has been analyzed in the literature and can beused to determine the direction of the source. This term isperiodic with a 1 yr�1 frequency. The O���� term wasdiscussed to some extent for gravitational wave data analy-sis in Ref. [11,18] and provides information on the propermotion of the source on the sky. The distance r to thesource is estimated through the O��2� term. This termprobes the curvature of the wave fronts as they deviatefrom plane wave propagation. In the O��2� term, theconstant contribution �1=�2r2� is effectively absorbed inthe phase constant�0, while the time dependent oscillatingterm (with a 2 yr�1 frequency) is proportional to sin2.The order of magnitude of the time shift induced by theplane wave O��� term in 1 yr is written as �rd cos�=c�

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500 sec with � being the angle between the two vectors n0and rd. This angle changes by��� rd=r in 1 yr due to theparallax. Therefore the O��2� term �r2d=cr�10�4�r=30pc��1 sec can be regarded as a time shift inducedby the parallax. Even at f� 1 kHz, this time shift pro-duces a phase shift much smaller than the 1=4 cycle that iscritical for detection of gravitational waves, unless thesource is very close to the solar system. Therefore thedistance r cannot be estimated without a high SNR obser-vation well beyond the required level for gravitationalwave detection.

Using a model h � hc sin��d� (hc: constant) forthe gravitational wave signal, I calculated the Fishermatrix for the following 10 fitting parametersff0; f1; f2; f3;�0;vns;?; r; ; g and examined the relativeestimation error�r=rwith various sets of input parametersas well as the total observational time Tobs * 1 yr. I foundthat the result�r=r is independent of the parameters,�0

and inputs ff1; f2; f3; f4;vns?g, as long as these inputs areclose to reasonable values for real neutron stars. For aintegration time Tobs longer than �2 yr, the resolution�r=r is approximated well by the following expression

�rr

� 0:11�

r100 pc

��SNR

500

��1�sinsin�=3

��2�f

1 kHz

��1:

(2.2)

I take � �=3 for a reference value for the angle , as itbisects the area of a half sphere 0 � � �=2. With thespecific choice of parameters � �=3, f � 1 kHz, r �100 pc and SNR � 500, the prefactor 0.11 in Eq. (2.2)becomes 9.14, 0.13, 0.109, and 0.107 for Tobs � 1; 2; 3and 4 yr, respectively. This indicates that for Tobs * 2 yrthe correlation between r and other parameters decreasessignificantly and becomes nearly stationary. The coeffi-cient becomes 0.107 when I remove f3 or vns? from thefitting parameters with a fixed observation period Tobs �3 yr. In our model I have not directly dealt with the effectsof the acceleration of the source. However, these numericalexperiments imply that, even if I include acceleration in arelatively simple manner, it would not change the resolu-tion �r=r significantly. When the source is a binary (e.g.with its orbital period �1 yr), the situation could changeconsiderably.

III. PROSPECTS WITH PROPOSED DETECTORS

Next, I discuss how well we can estimate the distance toa rotating neutron star with the planned gravitational wavedetectors, such as, advanced LIGO. I first describe thestandard detection criteria of its gravitational wave signalbased on SNR for a matched filtering method [10,12]. ThenI use a expression for SNR to evaluate the distance esti-mation error given by Eq. (2.2). For these studies, I pay aclose attention to dependence of the ellipticity parameter �and the distance r to the source.

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FIG. 1. Relation between the distance r and the gravitationalellipticity � for given observational conditions of an unknownneutron star with gravitational wave frequency f � 1 kHz. Thethin dashed lines represent the maximum distance for detectionof gravitational waves with SNR � 20 using initial LIGO andadvanced LIGO for 1=3 yr integrations. The thick solid lines arethe maximum distance for resolution �r=r � 0:1 through thetiming parallax method, shown for advanced LIGO andxylophone-type EURO for 3 yr integrations. The dashed line � �10�6 is an upper limit of theoretical predictions for the ellipticityparameter.

GRAVITATIONAL WAVE ASTROMETRY FOR RAPIDLY . . . PHYSICAL REVIEW D 71, 123002 (2005)

If the rotation axis of the neutron star is identical to oneof the principal axes of the inertial moment tensor Iij, theorientation averaged amplitude of gravitational waves be-comes hc � 8�2G

�����������2=15

pIf2�=c4r, where gravitational el-

lipticity is defined as � � �I22 � I11�=I� 1, and I � I33(x3: the rotation axis) [8]. With typical numerical values, Iobtain

hc � 7:7� 10�26��

10�8

��I

1045 g cm2

��100 pc

r

�

�

�f

1 kHz

�2: (3.1)

Hereafter I fix the inertial moment at I � 1045 g cm2 andthe gravitational wave frequency at 1kHz. When the rota-tion axis is different from a principal axis, the star pre-cesses and emits gravitational waves at multiplefrequencies [8,19]. In this paper I use the above Eq. (3.1)as a reference to relate the characteristic amplitude hc withtwo parameters � and r. For the sensitivity of initial andadvanced LIGO at 1kHz, I adopt the numerical valuesgiven in Figure 1 in Ref. [5] (see also [15]). I take hd �1:5� 10�22 Hz�1=2 for a single 4km detector of initialLIGO and hd � 4:2� 10�24 Hz�1=2 for advanced LIGOwith a broad band configuration. After averaging withrespect to the direction of the source, the effective sensi-

tivity for a monochromatic signal becomes heff �

hd�������������������2�15T�1

obs

qfor an observational period Tobs. The factor

2�1=2 comes from the number of 4km detectors and anotherfactor 51=2 is due to the angular average with respect to thesource direction. I also study the case for the proposedxylophone-type EURO detector [7] with hd � 3:6�10�26 Hz�1=2 at 1 kHz. With these concrete specifications,the signal-to-noise ratio is calculated as

SNR �hcheff

; (3.2)

and the resolution �r=r is now given as a function of r and� from Eqs. (2.2), (3.1), and (3.2).

The characteristic SNR threshold required for detectingan unknown rotating neutron star is discussed in [10] and Itake SNR � 20 as a reference value for a 1=3 yr integra-tion. In Fig. 1 I plot the r� � relation for four observa-tional conditions; SNR � 20 with (i) initial LIGO and (ii)advanced LIGO for 1=3yr integrations, and �r=r � 0:1with (iii) advanced LIGO and (iv) xylophone-type EUROfor 3 yr integrations. We have r / � for a given SNR, butr / �1=2 for a given resolution �r=r. If the parameter � isfixed, we have SNR / r�1 and �r=r / r2.

To detect gravitational waves from neutron stars, alonger integration time is statistically advantageous. Thisis because of the fact that the amplitude of the detectablesignal decreases and the effective survey volume (or equiv-alently the event rate) increases. However, the number of

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templates required for the matched filtering method de-pends strongly on the integration time, and a manageabledata length is limited by computational resources. In real-ity, it might be difficult to deal with data of 1=3 years,especially for a young neutron star with large coefficientsfk (k � 1) (see Eq. (2.1)). In this case, we can detectneutron stars that are closer than the distances shown bythe dotted lines in Fig. 1. However, with a total �10Teraflops computational power and efficient searching al-gorithm, the differences between the dotted lines and themaximum detectable distances for initial and advanceLIGO would be less than a factor of 2 for most neutronstars (see figure 5 in [12]). Once gravitational wave from aneutron star is confirmed, it would be a relatively easy taskto analyze longer data streams with a finer spacing oftemplate parameters. But the available data span wouldnot be much larger than 3 years, considering the time scaleof major upgrades to next generation detectors. This is thereason why I studied the case with a 3 yr integration for afollow-up analysis of the distance measurement.

In Fig. 1, the solid line for advanced LIGO and thedotted line for initial LIGO intersect around the interestingvalue � � 10�6, close to the upper limit from theoreticalpredictions [10]. Almost all the triangle region in��; r�-plane bounded by � � 10�6 (dashed line) and thedotted line for initial LIGO is below the sold line for

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NAOKI SETO PHYSICAL REVIEW D 71, 123002 (2005)

advanced LIGO. From viewpoints of future prospects ofthe gravitational wave astronomy, this fact can be re-phrased as follows: if an unknown neutron star is detectedaround 1 kHz with initial LIGO by �3�1 yr signal inte-gration and the ellipticity has a reasonable magnitude � &

10�6, advanced LIGO can determine its distance with errorof �10% level within 3 years. This is quite encouragingbecause blind searches for neutron stars are very hard tasksrequiring a huge computational cost, as mentioned earlier[12]. Once it is detected with initial LIGO, advanced LIGOwould provide information about its distance, one of thefundamental astronomical parameters for a source. Theellipticity parameter � may then also be found throughknowledge of the amplitude hc. Both parameters are thusdetermined purely through gravitational wave measure-ments. Note that advanced LIGO has the potential tomeasure the distance to an unknown neutron star thatcannot be detected by initial LIGO. The effective sensitiv-ity of the single EURO xylophone-type detector is 82 timesbetter than advanced LIGO detector at 1kHz, and thedistance r for a given resolution �r=r is 9.1 times larger.

I have discussed the case in which the gravitational wavedata would be integrated without a break (100% dutycycle) for 3 years or so. But our result for �r=r would beapproximately valid for a coherent analysis of intermittentdata streams that are sampled with appropriate intervals( � 1 yr) and in a sufficient span (e.g. 3 years). If thenarrow band configuration is available at the target fre-quency of a source, we can reach the same resolution �r=rwith a shorter total integration time. For example, thesensitivity hd could be improved by a factor �3 at 1kHzwith the narrow band configuration of advanced LIGO, andthe total integration time could be a factor �10 timessmaller than for its broad band configuration.

From Fig. 1, we can estimate the distance r for therequired resolution �r=r� 0:1 as a function of the pa-rameter �. Here let us statistically study whether there is atleast one rapidly rotating pulsar within this distance. Forthis purpose I use the argument originally made byBlandford assuming a population of neutron stars whosespin evolutions are primarily determined by energy lossthrough gravitational radiation [8,10]. I define ��1B as theformation rate of the population in our Galaxy with aninitial gravitational wave frequency higher than 1 kHz. Themagnitude of the time �B is highly uncertain but should bemuch larger than �B � 100 yr corresponding to theGalactic supernova rate. I first evaluate the minimumdistance rmin to a pulsar of this population, and thencompare rmin with the distance allowed for the resolution�r=r � 0:1. If these two distances coincide, we can expectto have at least one pulsar with resolution �r=r better than0.1. Our goal here is to calculate the required formationrate �B for this condition as a function of the ellipticity �.

In Blandford’s model, the amplitude hc from a neutronstar with a fixed distance becomes larger for a larger �. On

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the other hand, the evolution time scale �GW becomessmaller for a larger �, and the smallest distance rmin wouldbe larger. Here the evolution time scale �GW is defined as

�GW �f_f�

5c5

32�4GI�2f4

� 1:8� 108��

10�8

��2�f

1 kHz

��4yr: (3.3)

The gravitational frequency becomes smaller by �30%within a single �GW . The time scale �GW becomes theage of the universe �1010 yr at �� 10�9�f=1 kHz��2.For the spatial distribution of the population in ourGalaxy, I assume a uniform density in a cylinder withradius Rns � 10 kpc and height Hns � 1 kpc. If the pa-rameter � is in the range 10�9�f=1 kHz�2 < �<10�6��B=200 yr�

�1=2�f=1 kHz��2, we have

rmin � �3V�B��1GW=4�

1=3

� 44��B

200 yr

�1=3

��

10�8

�2=3

�f

1 KHz

�4=3

pc (3.4)

for the smallest distance. The distance rmin becomes largerthan the height Hns for � > 10�6��B=200 yr��1=2��f=1 kHz��2. In this case we have rmin � Rns��B��1GW� /�1=2. With these expressions and results presented in Fig. 1,we can now estimate the required formation rate �B forobserving at least one pulsar with resolution �r=r � 0:1.For 3 yr advanced LIGO observation at 1 kHz, we have�B < 84��=10�8��1=2 yr in the range 10�9 < �< 10�5.Therefore, a very small �B (or a large formation rate ��1B )is required from this rough statistical argument. In the caseof 3 yr EURO-xylophone-type data, we obtain �B < 6:3�104��=10�8��1=2 yr for 10�9 < �< 10�7.

So far, I have studied the case with �r=r� 1. Here, Idiscuss the opposite case in which the distance to a sourceis large, and we cannot detect the signature of the parallaxtiming shift. In the case of usual angular parallax measure-ment, the amplitude of the apparent annual motion of asource on the sky is inversely proportional to its distance r.Therefore, it is straightforward to fit distance with a formr�1 in the parameter estimation. We can understand thisfrom the original meaning of the distance unit ‘‘parsec’’that represents 1=arcsec. The situation is similar to thefitting for the parallax timing. Note also that the error��r�1� � ��r�=r2 for r�1 does not depend on the distancer itself, as shown in Eq. (2.2). In the low resolution limitwith a fitting result j�r�1�fitj ���r�1�, we can set a con-straint for the value r�1 in the form r�1 <maxf�r�1�fit; 0g � N��r�1� with N � % statistical signifi-cance. At the end we get a lower limit of the distance r as

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GRAVITATIONAL WAVE ASTROMETRY FOR RAPIDLY . . . PHYSICAL REVIEW D 71, 123002 (2005)

r > �N � x��1f��r�1�g�1 � 220�SNR

500

��sin

sin��=3�

�2

�

�f

1 kHz

��N � x4

��1pc

(3.5)

for Tobs * 2 yr with x � maxf�r�1�fit=��r�1�; 0g. We canalso set a lower limit to the parameter � by using thegravitational wave amplitude hc.

IV. DISCUSSIONS

In this analysis, the phase�SSB�t� at the SSB is assumedto be simple and be described by a low-order Taylorexpansion. Even though many pulsars are known to beextremely accurate clocks, the phase �SSB could have adeviation from this simple expression. The origin of thesetiming deviations can be due to the intrinsic evolution ofthe pulsars or other effects (e.g. the low frequency gravi-tational wave background). If the deviation is large, espe-cially around the frequency �2 yr�1, it would hamper theestimation of the distance through parallax timing evenwith a high SNR measurement. Here, I briefly describe theactual radio observation of certain pulsars following [20].PSR B1855 � 09 is a binary millisecond pulsar with spinperiod 5.36 msec and orbital period 12.3 d, and PSRB1937 � 21 is a single millisecond pulsar with spin period1.56 msec. They are considered to be recycled pulsars.Timing analysis for both pulsars have been performedwith data taken at Arecibo Observatory in a �8 yearspan. A systematic timing deviation of order a few & secis indicated for PSR B1937 � 21, but not for PSRB1855 � 09. The timing residual of the former is expected

123002

to be dominated by the intrinsic evolution of the pulsar andnot by other effects, such as, the low frequency gravita-tional wave background. As for the estimated distancesthrough the timing parallax, PSR B1855 � 09 has a fittedvalue � � �1 kpc=r� mas � 1:1� 0:3 mas correspondingto r � 0:9�0:4�0:2 kpc (1%), and PSR B1937 � 21 has a bound�< 0:28 mas (2%) corresponding to r > 3:6 kpc. Theirdistances are also estimated from dispersion measurestogether with a model of the Galactic distribution of inter-stellar free electrons. They are 0.7 kpc for PSR B1855 �09 and 3.6 kpc for PSR B1937 � 21 with uncertainties of�25%, and are consistent with the estimation through thetiming parallax. Therefore, the effects of the timing devia-tion would be moderate in these cases.

To summarize, I have discussed an astrometric effect forthe analysis of gravitational waves that would allow us todetermine the distance to a rapidly rotating isolated neu-tron star around f� 1 kHz. The resolution �r=r improvessignificantly, if the integration period of the signal is longerthan �2 years. By using advanced LIGO detectors for 3years at its designed sensitivity in the broad band configu-ration, we can estimate its distance with an error of�r=r &

0:1 up to r� 150��=10�7�1=2 pc, if its spin evolution isrelatively simple.

ACKNOWLEDGMENTS

The author is grateful to Teviet Creighton for valuablediscussions. He also thanks Sterl Phinney, Kip Thorne andYi Pan for their help and useful conversations, and AsanthaCooray and Steve Furlanetto for carefully reading themanuscript. This work was supported by NASA GrantNo. NNG04GK98G and the Japan Society for thePromotion of Science.

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