Testing the rotating lighthouse model with the double pulsar system PSR J0737-3039A/B

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by  Zhu-Xing Liang

Mon.Not.R.Astron.Soc.000,1–??(2002) Printed5February2014 (MNLATEXstylefilev2.2)
Testing the rotating lighthouse model with the double
pulsar system PSR J0737-3039A/B
4 ⋆
Zhu-Xing Liang and Yi Liang
1
0 18-4-102 Shuixiehuadu, Zhufengdajie, Shijiazhuang, Hebei 050035,China
2
Joel M. Weisberg
n
a Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057, USA
J

1
3

]
R ABSTRACT
S
. EachofthetwopulsarsinthedoublepulsarPSRJ0737-3039A/Bsystemexhibits
h
not only the pulses emanating from itself, but also displays modulations near the
p
pulse period of the other. Freire et al. (2009, MNRAS, 396, 1764) have put forward
-
o a technique using the modulation of B by A to determine the sense of rotation of
r pulsar A relative to its orbital motion, among other quantities. In this paper, we
t
s present another technique with the same purpose. While the Freire et al. approach
a analyzes pulse arrival times, ours instead uses periods or frequencies (their inverses),
[
which can be experimentally determined via power spectral analysis similar to that

1 usedinpulsarsearches.Our technique is basedonthe apparentchangeinspinperiod
v of a body when it is measured from an orbiting platform (the other pulsar), and is
3 shown to be entirely analogous to the difference between the sidereal and solar spin
2 period of the Earth (i.e., the sidereal and solar day). Two benefits of this approach
5 are its conceptual and computational simplicity. The direct detection of spin with
0 this technique will observationally validate the rotating lighthouse model of pulsar
.
2 emission, while the detection of the relative directions of spin and orbital angular
0 momenta has important evolutionary implications. Our technique can be used on
4 other binary systems exhibiting mutually induced phenomena.
1
: Key words: binaries: general – stars: kinematics and dynamics – techniques: mis-
v
cellaneous – pulsars: individual (PSR J0737 - 3039A/B).
i
X
r
a
1 INTRODUCTION pulses and their mutual modulations which could yield the
sense of rotation of each pulsar with respect to its orbital
The doublepulsarsystem PSRJ0737-3039A/B was discov-
motion, among other quantities. In this paper, a comple-
ered by Burgay et al. (2003) and Lyneet al. (2004). This
mentarytechniqueispresentedwiththesameobjective,but
system consists of a 22-ms pulsar (hereafter A) and a 2.8-s
usingmeasuredperiodsratherthanarrivaltimes.Theprin-
pulsar(hereafterB)withanorbitalperiodof2.4hours.This
cipal benefit of our approach is that it is simpler and more
discovery has provided a laboratory for the study of rela-
intuitive.If thevalidity of either of these techniquesis con-
tivistic gravity and gravitational radiation (Kramer & Wex
firmed, not only will new insights be gained but also the
2009). The system has several strange features that chal-
correctness of the lighthouse model will be empirically as-
lenge the current understanding of pulsars and provide an
sessed beyond dispute.
uncommon opportunity to improve pulsar theories. One
of the most interesting properties is the observed modu-
lation of each pulsar’s signal by the energy flux from the
other, as evidenced by each pulsar’s modulation period be-
2 DETERMINING THE PRESENCE AND
ing approximately equal to the other pulsar’s pulse period
SENSE OF ROTATION OF A, AND
(McLaughlin et al. 2004a,b). Freire et al. (2009a) proposed
ANALOGIES TO THE EARTH-SUN SYSTEM
a technique for analysing the arrival times of the pulsars’
Our method is identical in principle to the determination
of the difference between two measurements of the Earth’s
⋆ [email protected](LZX);[email protected](LY) spin period – its sidereal and solar day. The difference be-
2 Z.-X. Liang, Y. Liang, & J. M. Weisberg
Table 1.Earth-Sunanddoublepulsaranalogies.Allquantitiesaredefinedasmagnitudes.
Rotatingobject: Earth(⊕) PulsarA
Orbitalperiod,Meanorbitalfrequency Porb(≡year),forb Porb,forb
Instantaneous orbitalfrequency forb(t) forb(t)
Rotatingobject’ssiderealspinperiod,frequency P⊕,0(≡siderealday), f⊕,0 PA,0,fA,0
Rotatingobject’sspinperiod,frequency(measuredatotherbody) P⊕at⊙(≡solarday),f⊕at⊙ PAatB,fAatB
≡Modulationperiod,frequencyatother bodyduetorotatingobject ≡Pmat⊙dueto⊕, ≡PmatBduetoA,
fmat⊙dueto⊕ fmatBduetoA
tween the two periods is caused by the kinematic effect of
Earth’s rotation and revolution. Because the earth rotates ω
A
and revolves in the same (“prograde”) sense, the solar day
is about 4 min longer than the sidereal day. If the earth θ ascending nodes
rotated in the opposite (“retrograde”) sense relative to its B
orbit, the solar day would be about 4 min shorter than the
sidereal day. A similar relationship should be seen in the r
AB
PSR J0737-3039A/B binary system if thelighthouse model
PSR
is correct, where the apparent period of one pulsar’s pulses r A
binary barycentre A
measured at theother pulsar– the otherpulsar’s “modula- apsidal line
tionperiod”–willbelongerorshorterthanthefirstpulsar’s r
ssiadmereAeoaltrlhpoopeurpgioohsdiotiuefrdtthiereecchfitnrioisqtnuieasscaaitn“ilsiignohrptbhriiotnuicnsigep.”lerboteautisnegditnotdhee- PSBR B line of nodes
termine whetheror not both A and B are spinningand the ω
rswneeeeanxlsftoedscaeouycfst,sioPounnclhsy,ri,ononitsoadratdeinoetanerlrotmwogoiiatnuvhasotriitdeoosncpAoeon’cfsftutshtsiodieoenstrph.eieaTnilrhoreoofnrtAbaEittaiiaonrlntthmhp’sioesrtsiiiaoodnnded-,, orbito rBbit A θAB (ourtB oBfl i npSSelaB onfe )sight
,0
PA,0. Simil⊕arly, Earth’s solar day, P at , is analogous to
themodulationperiodinB’s,signal,⊕P ⊙ .Table Figure1.OrbitsofPSRsAandBaboutthebinarybarycentre.
m at B dueto A
1 further illustrates the analogies between the Earth-Sun Thedirectsignalpathdiscussedinthetextisemittedfrompulsar
A antiparallel to the line of sight vector; while the “two-legged”
system and the double pulsar system, in terms of the vari-
signalpathalsostartsatAbutthentravelstoBbeforetravelling
ousperiodsandtheircorrespondingfrequenciesf =1/P.In
antiparalleltothelineofsightvector.
whatfollows, wewillgenerallyusespinandorbitalfrequen-
cies instead of periods, because the frequency calculations
are simpler.
To further understand the relationships among the ro- They will be intermingled at the radiotelescope since they
tation, revolution and modulation signals, consider the fol- are spatially indistinguishable at the solar system. In this
lowing suppositional extreme case. Assume that the spin section, we focus on the signals originating at A. (See Sec.
periods of A and B are both equal to the orbital period 4 for a discussion of signals originating at B.)
and that both stars rotate in the same sense as their or-
bital motion. Then theiremission signals will not modulate
eachotherbecauseeachstarcanreceiveonlysteadyenergy
3.1 Signal paths originating at A
fluxesfrom theotherstar. Inanyothercase,it isclear that
modulation of one pulsar’s signal by the other is possible, Signals are observed from from pulsar A via two different
and that theperiod of modulation will be affected not only paths: (1) the “direct beam” travelling directly from A to
bytherotationperiodofthepulsarcausingthemodulation thesolarsystem;and(2)the“two-legged”beamemittedby
but also its orbital period and rotation sense. A that first impinges on B, thereby modulating B’s beam
Fromtheanalysisabove,wehaveshownthataslongas every time that A shines on B; with the modulated signal
we can measure the modulation frequency of B’s signal, we then travelling from B to the solar system. Consequently,
can determinethe presence and sense of rotation of A. the modulated signal encodes information concerning both
B’s and A’spulses.
DetailsoftheorbitalgeometryareshowninFig.1.Note
thatitisthefirstlegof thetwo-legged pathwhichprovides
3 FREQUENCY DETERMINATIONS
thecapabilitytodistinguishthedirectionofA’sspin,ifany;
When a radiotelescope points to the system PSR J0737- because it enables us to measure A’s pulse period with re-
3039A/B, up to four periodicities may be received simulta- spect to a another vantage point (namely pulsar B’s), in
neously; namely A’s and B’s direct and modulated signals. addition toour own.
Testing the rotating lighthouse model with the double pulsar 3
3.2 Frequency calculations where ~vrel,A and r˙rel,A are, for the relative orbit of PSR A
aboutPSRB,thevectorvelocityanditsradial(withrespect
It will be adequate for our purposes to ignore kinematic
to B) part, respectively.
modifications to source frequencies beyond order v/c, leav-
ing only the first-order Doppler shift. The Doppler shift
modifies fX,0, the emitted frequency of source ‘X’, leading 3.2.2 Apparent frequencies at the solar system barycentre
toareceivedfrequencyfX,rcvaccordingtothefollowingpre-
scription: In this section, we discuss the transformations necessary to
find apparent frequencies of A’s two beams at a given time
fX,rcv =(cid:20)1+ ~vX·ˆrcX←rcv(cid:21)−1× fX,0 , (1) aDtotphpelesrolfaarctsoyrsst,eimntbrianrsyicceonrtbriet1a,lianntdersmpisnoffreeqmuiesnsicoinest.iTmaebs2le,
2 displays detailed expressions associated with these items.
where the dot-product projects the velocity of source ‘X’ The first (direct) beam from A: Theintrinsic frequency
with respect to receiver ‘rcv’ , ~vX, onto the receiver-to- of this beam, fA,0, will be Doppler-shifted upon receipt at
source-‘X’ line-of-sight unit vectorˆrX←rcv, thereby yielding thesolarsystembarycentreduetoA’sorbitalvelocityatthe
the “radial” (with respect to the receiver) velocity compo- timeofemission, yieldingf1,ssb =fA,ssb =D[A,ssb] fA,0.
×
nent. In what follows, source ‘X’ may be PSR A or PSR The second (“two-legged”) pulsed signal from A: This
B,and ‘rcv’maybelocated at thesolar system barycentre, signal travels along two segments in sequence: first from
ssb; or even at PSR B (see below). In each such case, the A to B and then from B to the solar system barycentre.
subscriptswillbereplacedbyappropriatesymbolstodenote Someinvestigators (e.g.,Freire et al.(2009a))suggest that
theparticular choices. this “signal” may consist of relativistic charged particles
Wedefinethe“Dopplerfactor”D[X,rcv]solelyforcom- ratherthan photons,but eitherwill behavesimilarly in our
pactness of notation: frequency-based analysis. (While the arrival-time model of
D[X,rcv] 1+ ~vX·ˆrX←rcv −1, (2) FbreetwireeeentAal’.sr(a2d00io9aa)ndalelloewctsrofomraganpetoiscsibbelaemlos,ntghitiusdceonosfftasnett
≡(cid:20) c (cid:21)
phase offset dropsout of our frequency-based analysis.)
such that Eq.1 becomes Thefirstleg’spulsedsignalwillbereceivedandreemit-
fX,rcv =D[X,rcv] fX,0. (3) ted by B as a “modulation” with an apparent frequency
× in B’s frame of f . This frequency is modified
m at B dueto A
(ThetwoargumentsofDdenotethesourceandthereceiver,
from its intrinsic value fA,0, primarily by A’s orbital mo-
respectively.)
tion about B in a fashion analogous to the modification of
Earth’sintrinsicspinfrequencytoitssolarone,asdescribed
3.2.1 Radial velocity calculations in Section 2. This requires that
Therearewell-developedtechniquesfordeterminingthera-
dialvelocityofanorbitingbodyX(thedot-productappear- fm at B dueto A = (6)
inginEqs.1and2)intermsofitsorbitalelements.Follow-
(fA,0 forb(t) ) prograde spin,
ing Freire et al. (2001, 2009b), the radial (with respect to −
tishegirveecneibvyer) velocity of object X, at any point in its orbit, D[A,B]×(ffAA,,00+forb(t) ) preutlrsoagtrioande(nspoin,
~vX·ˆrX←rcv = P2oπrb √aX1sinei2[cos(ωX+θ(t))+ecosωX], (4) as summarized inTable 2. spin);
−
whereaXisthesemimajoraxisoftheorbitof‘X’,inclination Itiseasytoshowthatthiskinematicconsequenceofthe
i is the angle between a vector normal to the orbital plane lighthousemodelleadstoexactly onefewer(progradecase)
andˆrX←obs,eistheorbitaleccentricity,ωX istheargument oroneextra(retrogradecase)pulsefromPSRAperorbital
ofperiastronofX’sorbit,andtrueanomalyθ(t)isthepolar period,comparedwiththeno-spinnumber.Theformercase
angleof‘X’atemissiontimet,measuredaroundtheprinci- is closely analogous to the (prograde) Earth - Sun system,
palfocusofitsorbit,startingatperiastron.Finally,inorder where there is one fewer solar day than sidereal days per
relateallofthesequantitiesexplicitlytotime,onemustde- Earth year.
termine the value of θ(t) at a given emission time via the Eq.6wouldhaveaparticularlysimpleformiftheorbit
iterative“Kepler’sEquation”technique,asgivenincelestial were circular, both because f could be treated as a con-
orb
mechanicstexts[e.g.,Roy(2005)].NotealsothatωX isalso stantandbecausetheDopplershiftfunctionD[A,B] 1in
≡
a(slowly andessentially linearly-varying)functionof t,due this case. The difference between the exact, elliptical orbit
toageneralrelativisticphenomenonthatiseasilyaccounted expression given in Eq. 6 and this fictitious, circular-orbit
for. Finally, for thespecial case of therelative orbit of PSR
A about PSR B, Eq. 2 becomes
1 It is standard procedure for pulsar observers to remove the
D[A,B]= 1+ ~vrel,A·ˆrA←B −1 = 1+ r˙rel,A −1 efrffeeqcutesnocfietsh,aenEdaarrtrhi’vsalptoimsiteisonbyarneddumciontgiotnhefmrotmotphuelisraerqupievraiolednst,
(cid:20) c (cid:21) (cid:20) c (cid:21)
valuesatthesolarsystembarycentre.
2 A desired emission time can be determined as a function of
= 1+ 2π arel e sinθ(t) −1, (5) ssb time, by calculating the propagation time between the two
(cid:20) Porb c√1 e2 (cid:21) locations.See§3.4fordetails.
−
4 Z.-X. Liang, Y. Liang, & J. M. Weisberg
Table 2.Magnitudes offrequenciesf(=1/P)ontwosignalpaths forA’semission.
Signal path 1: Direct beam from A to solar system barycentre (ssb)
IntrinsicspinfrequencyofA: fA,0
ApparentspinfrequencyofAatssb,
f1,ssb=fA,ssb: D[A,ssb]×fA,0
Signal path 2: Two-legged beam from A to B to solar system barycentre
lighthousemodel othermodel
A’srotationsenserelativetoorbit: prograde retrograde none
IntrinsicmodulationfrequencyofB,
fmatBduetoA: D[A,B]×(fA,0−forb(t))a D[A,B]×(fA,0+forb(t))a D[A,B]×fA,0
Apparentmodulationfrequencyof
Batssb,f2,ssb=fm,ssb: D[B,ssb]×D[A,B]×(fA,0−forb(t)) D[B,ssb]×D[A,B]×(fA,0+forb(t)) D[B,ssb]×D[A,B]×fA,0
a T˙heinstantaneousorbitalfrequencyforb(t)variesaroundtheellipticalorbit:forb(t)= 21πdθd(tt).Foracircularorbit,forb(t)→forb=1/Porb.
expression is intimately related to the difference between
the apparent and mean spin frequencies of the Earth with Two-legged path,
respect to the Sun which leads to a time-varying difference retrograde-spin A
between apparentand mean solar time, dubbedthe“Equa-
tion of Time” (Seidelmann 1992).
Thepulsetrainonthesecondportionofthistrajectory, Two-legged path,
leaving B with a frequency f of Eq. 6, will prograde-spin A
m at B dueto A
be Doppler shifted upon its reception at the solar system Two-legged path,
barycentre due B’s orbital motion, leading to modulation no-spin A
frequency
f2,ssb =fm,ssb =D[B,ssb] fm at B dueto A Direct path
×
B Dark B BrightB Dark B Bright B Dark phase
phase phase phase phase
(6a)
(fA,0 forb(t) ) prograde spin,
−
=D[B,ssb]×D[A,B]×(ffAA,,00+forb(t) ) preutlrsoagtrioande(nspoin, Foorfiiggtiuimnraeetifn2og.rAfMrpoJmpDarA5e,n41ta5tf5rt.ehqTeuhesoenlcafirreessqyousfetnedcmiyrebcotafratyhncedendttwirreoe,c-ltaesgpgaaetfdhunp(cAutliso→ens
 sp(6inb)). ossfbth),efm1,ossdbu,liastitohneipsuollsaetetdraciunrvfree,qwuhenilceyt,hfemt,hssrbee=pofs2s,isbsble, voafltuhees
“two-legged”path(A→B→ssb)aredepicted(witha40×exag-
gerationoftheir offsets fromthemiddle,no-spincurve) asthree
3.3 Spin-A-Induced Offsets in B’s modulated closely-spacedcurves.Verticallinesdelimitthepartsoftheorbit
whereBisvisible(labelled“Brightphase”)orinvisible(labelled
signal
“Darkphase”).Sincethetwo-leggedpathisactuallydetected as
Theessence of ourtest fortherotation ofA iscontained in a modulation of B’s signal, the two-legged path could never be
Eqs.6,sincethesense oreventheexistenceof A’sspinwill observed duringB’s“dark phases,”sothe two-legged curves are
shown as fainter lines at these phases. B’s modulation was ac-
select one of the three versions of Eqs. 6b for the value of
tually only detected in a small region of the first bright phase,
B’s drifting- subpulse-likemodulation frequency.Hence the
althoughwebelieveitwouldbeworthwhiletouseourtechniques
modulation has only three allowed values at a given orbital
tosearchformodulationsthroughout bothbrightphases.Asimi-
phase. Fig. 2 illustrates the direct-path and the three pos-
lar,butmirror-image,set ofcurves centered near 0.36Hzwould
sible two-legged-path pulse train frequencies of the beams becreatedifasimilarpairofbeamsoriginatesatPSRB(see§4).
fromAatthesolarsystembarycentreasafunctionoftime,
with the offset of the two extreme values from the central,
no-spin, curve for the two-legged beam exaggerated by a offset3 at the ssb between each of the three possible two-
factor of forty. legged paths, is
Indeed,B’spulsesareclearlymodulatedbyA’semission
(McLaughlin et al.2004a).Therefore,aslongasthedataare |∆f2,ssb|=|∆fm,ssb|=D[B,ssb]×forb, (7)
sufficiently ample, the modulation frequency fm,ssb should
be obtainable with enough precision to distinguish among 3 The instantaneous value, also derivable from Eqs. 6b, varies
thethree possibilities. aboutthemeanowingtotheellipticityoftheorbit,asdiscussed
AscanbeascertainedfromEqs.6b,themeanfrequency above.
Testing the rotating lighthouse model with the double pulsar 5
while the mean fractional offset δ is3: t[k] for the two-legged path from A to B to the ssb that is
responsible for the drifting-subpulse-like modulation. Then
δ= |∆f2,ssb| = |∆fm,ssb| = D[B,ssb]×forb we derive an expression for the phase of the modulation at
f2,ssb,no−spin fm,ssb,no−spin D[B,ssb]×fA,0 B, taking elliptical orbital motion into account. Finally, we
f
= orb. (8) show how todetermine thepresenceand natureof A’sspin
fA,0 bysearchingforperiodicitiesinthismodulationphasespace.
The value of δ is only 0.00026% [using f =
orb
1/(8834.535 s) and f0,A = 44.0540694 Hz (Kramer et al.
2006)]. While the three possible values of fm,ssb are thus 3.4.1 Freezing the frequency of B’s pulses
quite similar, we note that they are precisely calculable via
First, we calculate t [k], the time of the kth sample mea-
Eqs. (6b), as illustrated in Fig. 2. The next section details B
sured at B, by correcting for B’s propagation contributions
analgorithmfordeterminingthepresenceandsenseofrota-
along thesecond leg, thepath from B to the ssb:
tionofAfrom arealdataset,byfindingwhichofthethree
modulation frequencies best matches the data. L z k L z
tB[k]=t[k]− c − cB =t0+ fs − c − cB; (10)
3.4 The algorithm with L the distance between thessb and binary barycentre
bb4;andz ,theprojectionofthepositionofBwithrespect
The results of the above sections derive and illustrate the B
threepossiblevaluesofthessbmodulationfrequencyatany to thebb,~rB, onto theline of sight ˆrbb←ssb:
givenorbitalphase.However,thesefrequenciesvaryaround a sini (1 e2)sin(ω +θ)
theorbitduetoDopplershiftsandorbitalellipticity,thereby zB≡ˆrbb←ssb·~rB = B c (1−+ecosθ)B . (11)
rendering frequency searches difficult without special tech-
niques. Wediscuss such techniquesbelow. [SeeSec.3.2.1 foradditionaldefinitions,andRoy(2005)for
Consider the observed, sampled (though incompletely) a derivation.]
kthintensityI[k],measuredatsolarsystembarycentrictime Eq.10canbeusedtotransformI[k],theintensitydata
t[k]: sampled at the ssb, into a resampled space where the pre-
viously time-variable Doppler shift D[B,ssb] caused by B’s
k
t[k]=t + , (9) orbitalmotionisremoved.Inthisnewspace,thepulsesfrom
0 f
s Bwillyielda fixedpowerspectralpeakatfB,0.Indeed,itis
wheret isthessbtimeofsamplezeroandf isthesampling bythisverytechniquethatsurveydataareresampledinan
0 s
frequency. efforttosearchforabinarypulsarofgivenorbitalproperties
Full power spectral analyses could be employed on [e.g., Allen et al. (2013)] (in this case, properties matching
chunksofthistimeserieswhose durationsareshort enough B’s).
to avoid significant spectral smearing due to Doppler shifts
andorbitalellipticity.Thehighest-amplitudechannelofthe
power spectrum near the expected modulation frequency 3.4.2 Freezing the frequency of the modulation
could then be defined as the best estimate of the modu-
lation frequencyforagiven chunk;and itsvaluecould then In order to freeze frequency of the the modulation occur-
be compared with each of the three values of f predicted ingwhenA’spulsesinteract with B,wemust also calculate
for that chunk by Eqs. 6b, thereby determininmg the pres- tA[k],thetimeofthekth samplemeasuredatA,bycorrect-
ence and sense of rotation of A. However, since the three ingadditionallyforpropagationtime dAB/calongthepath
| |
expected values are so similar, it would be difficult to dis- from A to B:
tinguish among them in a relatively short and noisy chunk. d k L z d
Therefore, it wouldbehelpfultoeliminatetheDoppler tA[k]=tB[k]− | AcB| =t0+ fs − c − cB − | AcB|, (12)
and elliptical-orbit smearing of the modulation frequency
with [see also Freire et al. (2009a)]:
aroundtheorbitandbeyond,inordertoemploymuchlonger
data sets. Here we illustrate an algorithm to resample the
d r dr r dr
full extant data set in order to “freeze” the desired mod- | AcB| ≈ | AcB| + dAtB∆tAB = | AcB|(1+ dAtB), (13)
ulation at a constant frequency, whose value can be deter-
minedviaFourieranalysisofallthedataatonce.Thispro- where rAB istheinstantaneousseparationbetweenthetwo
| |
cedure will result in much greater spectral resolution and pulsars.
signal-to-noise. Theprocess issimilar toproceduresusedin Eq. 12 implicitly eliminates the Doppler factors
frequency-based searches of time series data for unknown D[B,ssb] and D[A,B] in Eq. 6b caused by both pulsars’
binary pulsars, wherein the Doppler shifts due to putative radial motions along thetwo-legged path.
orbital motion are eliminated via resampling of the data Nowletusmorecarefullyspecifythe(“unprime”)side-
(e.g., Allen et al. (2013)). However, while unknown binary realreferenceframecentredonAasonewhosex-axispoints
searchesrequireresamplingthedatawithmultipletrialsets fromAtoB’spositionatperiastron,andwhosey-axispoints
of parameters spanningan enormous search space, our pro- inthedirectionofB’smotionatperiastron.Aispulsing,or
cedurerequiresonlyoneparametersetforeachofthethree
possible rotational states, all of which are well-determined.
In orderto resample thedata, we first calculate times tB[k] 4 TheunknowndistanceLcanbeabsorbedintot0 byanappro-
and tA[k], the times at B and A corresponding to ssb time priateredefinition,withnolossofgenerality.
6 Z.-X. Liang, Y. Liang, & J. M. Weisberg
else spinning in a prograde or retrograde manner with re- Eq 16. We show below that this enables us to measure the
spect to B’s revolution, all at a rate whose magnitude is modulation frequency at B viaaperiodicity search,thereby
fA,05 (cf.Table 1), measured in this unprimeframe. determining A’strue spin state.
Therefore, the rotational or pulsational phase of A In order to search for modulation periodicities over a
in the unprime, sidereal frame centred on A would be rangeoffrequencies,wewillnowrepresentthemagnitudeof
essentially5 strictly proportional to tA : A’ssiderealrotationperiodbyageneralizedvariablefA,trial,
whosevaluewillbeallowedtovaryslightlyaboutitsknown
+2πfA,0 tA[k] prograde spin.
value:
ΦA,0[k]=−2πfA,0 tA[k] retrograde spin. (14)
2πfA,0 tA[k] pulsation (nospin).
WhilefA,0 isthesiderealfrequencyofA’spulsationsor fA,trial=zfA,0, (17)
rotations, it is not necessarily the frequency of A’s signal’s
interception and modulation of B (i.e., it is not necessarily wherethefrequencysearchfactorz 1.Thenforeachcho-
≈
equal to the modulation frequency). Specifically, if A emits senzands= +1, 1,0 ,themodulationphasemagnitude
a rotating lighthouse beam, a correction must be made due at B of thekth{ssb−samp}le is
to B’s orbital motion. (Conversely, if A is instead pulsat-
ing,nosuchcorrection isnecessary becausethepulsation is
presumed to be emitted simultaneously in all directions. Φm at B dueto A[k,z]s =2π(zfA,0)tA[k] sθ(tB[k]) (18)
| | −
In order to correct for orbital motion in the case of a
rotating lighthousebeam,consider asecond (“prime”) two-
We can now test for the presence of a modulation pe-
dimensionalcoordinatesystemlyingintheorbitalplaneand
riodicity due to one of the three possible spin states, by
centredonA,whoseaxes x’,y’ ,rotate(nonuniformly,due
{ } doing a periodicity search in the vicinity of each such state
to the elliptical orbit) such that its x’-axis points from A
inmodulation-phasespace.Weassociatetherelevantmodu-
toward B’s location at time t [k]. Thephase anglebetween
B lationphasefactorforagivenstates,witheachssb-sampled
thetwocoordinatesystems(e.g.,betweenthex-andx’-axes)
intensityI[k], and preparea Fourier powerspectrum of the
is just thetrue anomaly θ(t [k] (cf.Sec. 3.2.1).
The drifting-subpulse-liBke modulations, which are cre- product.TheresultingquantityPn(zfA,0)s,thepowerinthe
nth harmonic of the modulation corresponding to A’s trial
ated whenever A’s emission intercepts B, occur at inter-
vals separated by exactly one rotation or pulsation of A in spin frequency zfA,0 and spin state s, is given by
the prime frame. Therefore, we can write an expression for
the corresponding “modulation phase” Φ [k], 2
m at B dueto A
which isjust therotational orpulsational phaseof Ain the Pn(zfA,0)s=(cid:12) I[k] exp( i n Φm at B dueto A[k,z]s)(cid:12) ,
prime frame: (cid:12)(cid:12)Xk − | | (cid:12)(cid:12)
(cid:12) (1(cid:12)9)
(cid:12) (cid:12)
where k can extend over any subset of the full ssb-sampled
Φ [k]=
m at B dueto A dataset.
+2πfA,0 tA[k] θ(tB[k]) prograde spin, The true spin state will then be manifested as a sharp
−
−2πfA,0 tA[k]−θ(tB[k]) retrograde spin, (15) preesapkonindionngetooftzheth1reeanpdowitesrshpaercmtroaniacts,thwehfirleequtheencpyocwoerr-
+2πfA,0 tA[k] pulsation (nospin). ≡
spectrageneratedfortheothertwoputativespinstateswill
The modulations now occur at phase intervals of exhibitnosuchpeaksattheexpectedfrequencyanditshar-
∆Φm at B dueto A[k] = 2πj, with j any positive integer. In monics. If necessary, the sought-after signal can be further
this fashion, we have achieved our ultimate goal of “freez- enhanced by the process of harmonic summing, which is a
ing” the modulations at a fixed periodicity in this trans- well-established pulsar search technique.
formed space, even in the presence of ellipticity-induced, The theoretical framework applied here can be further
time-variable orbital frequencies. verifiedwith a fewadditional observational procedures.For
An observer can distinguish only the magnitude of the example, since the modulation of B’s pulses can occur only
modulation phase, Φm at B dueto A[k], with at those orbital phases where B’s pulses are present (de-
| |
Φm at B dueto A[k]s =2πfA,0 tA[k] s θ(tB[k]), (16) noted “bright phases” in Fig. 2), the sought-after modula-
| | − tion periodicity must also be visible only at those phases
where A’s spin state s = +1, 1, 0 for prograde spin,
{ − } { as well. [Indeed, the modulation phenomenon is only di-
retrograde spin, no spin but pulsation , respectively. (Note
} rectly visible near the beginning of the first bright phase
that the right side of Eq. 16 will always be positive, as de-
(McLaughlin et al. 2004a), although we believe it is worth
sired, since fA,0>>dθ/dt.) deploying our technique with multiple trials on data from
a wide range of bright phases.] Conversely, while thepulses
3.4.3 Determining A’s spin state s from observations arrivingdirectlyfrom Aareobservableatallorbitalphases,
theabove procedures should filter out their presence in the
We have expressed the modulation phase magnitude at B,
resampled data.This ability to filter for the direct signal is
ofthekth sampleforeachofthethreepossiblespinstatesin
key, especially if a pulsation (no spin) result appears. The
successofthefilteringoutofthedirectrayfromAcanitself
5 For a sufficiently long time series, fA,0 must be a function of be assessed by using our technique only on the (B-) dark
timetoaccount forpulsarA’sspindown. phases, where only thedirect A ray is emitted.
Testing the rotating lighthouse model with the double pulsar 7
4 AN ALTERNATIVE: DETERMINING THE that the second supernova, which created B, was relatively
PRESENCE AND SENSE OF ROTATION OF symmetric.Thereforethepresenceandsenseoftherotation,
B asrevealedbythisanalysis,willfurthertesttheirandothers’
(e.g.,Kramer & Stairs(2008);Farr et al.(2011))evolution-
An analysis of the apparent pulse train frequencies on the
ary scenarios.
two equivalent paths from PSR B to the ssb (rather than
While McLaughlin et al. (2004b) also found that A’s
theearlier case of paths originating at A) leads to identical
signal is modulated at B’s frequency near A’s eclipse, the
expressions as above, except with subscripts “A” and “B”
mechanism is different and not dependent on B’s rotation.
interchanged.
If, however, the modulation is observed in the future away
Indeed, McLaughlin et al. (2004b) and Breton et al.
from eclipse, it will provide an easier test for B’s rotation
(2012) show that A’s pulses are modulated by B. However,
than does theapproach delineated above for A.
the modulation has so far only been detected during the
Our analysis can also be used on other binary systems
short(partial!)eclipseofAbyB.Unfortunately,thismodu-
discoveredtopossessphenomenacausedbymutualinterac-
lationsignalcannotbeusedtodetermineB’srotationsense,
tions.
because it arises from a different mechanism and hence the
signal relationships differ from those derived above.
If A’s signal modulation by B can also be found in the
ACKNOWLEDGMENTS
non-eclipse phase, B’s rotation sense may be determined
more easily than A’s. In this case, with f0,B =0.360560355 JMW is supported by U.S. National Science Foundation
Hz(Kramer et al.2006),theequivalentmeanfractionalfre- Grants AST-0807556 and AST-1312843. ZXL and YL are
quency offset distinguishing the three possible spin states gratefultoXiang-PingLi,AliEsamdin,andJian-PingYuan
of B is a much larger 0.031%. The data segments corre- for very useful discussions.
sponding to the eclipse phases should be discarded in the
analysis in order to avoid contamination from modulation
originatingbyadifferentmechanism.WhileB’sradiosignal
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