The birth properties of Galactic millisecond radio pulsars

0.2 MB·English

by  Lilia Ferrario

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev1.4)
The birth properties of Galactic millisecond radio pulsars
Lilia Ferrario and Dayal Wickramasinghe
Department of Mathematics, The Australian National University,Canberra, ACT0200, Australia
Accepted. Received;inoriginalform
7
ABSTRACT
0
0
We model the population characteristics of the sample of millisecond pulsars
2
(MSPs)withinadistanceof1.5kpc.Wefindthatforabrakingindexn=3,thebirth

n magnetic field distribution of the neutron stars as they switch on as radio-emitting
a MSPs can be represented by a Gaussian in the logarithm with mean logB(G) = 8.1
J and σlogB =0.4 and their birth spin period by a Gaussian with mean P0 =4 ms and
16 σP0 =1.3ms.WeassumenofielddecayduringthelifetimeofMSPs.Ourstudy,which
takesintoconsiderationaccelerationeffectsontheobservedspin-downrate,showsthat

most MSPs are born with periods that are close to the currently observedvalues and
1
withaveragecharacteristicagesthataretypicallylargerbyafactor∼1.5comparedto
v
4 the true age. The Galactic birth rate of the MSPs is deduced to be >∼3.2×10−6 yr−1
4 near the upper end of previous estimates and larger than the semi-empirical birth
4 rate ∼ 10−7 yr−1 of the Low Mass X-ray Binaries (LMXBs), the currently favoured
1 progenitors.The meanbirth spinperiod deducedby us for the radioMSPs is a factor
0 ∼2 higher thanthe meanspin periodobservedforthe accretionandnuclear powered
7
X-ray pulsars, although this discrepancy can be resolved if we use a braking index
0 n = 5, the value appropriate to spin down caused by angular momentum losses by
/
h gravitationalradiationormagneticmultipolarradiation.Wediscusstheargumentsfor
p and against the hypothesis that accretion induced collapse (AIC) may constitute the
- main route to the formation of the MSPs, pointing out that on the AIC scenario the
o
low magnetic fields of the MSPs may simply reflect the field distribution in isolated
r
t magneticwhitedwarfswhichhasrecentlybeenshowntobebi-modalwithadominant
s
a component that is likely to peak at fields below 103 G which would scale to neutron
: star fields below 109 G, under magnetic flux conservation.
v
i Key words: pulsars:general,stars:neutron, stars:magnetic fields, X-rays:binaries.
X
r
a
1 INTRODUCTION thermore,proper motion studies haveshown that while the
average space velocity for normal pulsars is 400 km s−1
ThepropertiesoftheMSPsandthe“normal”radiopulsars ∼
(Hobbs et al. 2005), the MSPs form a low-velocity popula-
place them in two nearly disjoint regions in the spin pe- tion with typical transverse speeds of 85 km s−1 (Hobbs
riod (P) period derivative (P˙) diagram. In the normal pul- ∼
etal.2005;Toscanoetal.1999).Differencesintheincidence
sars, P and P˙ are distributed about mean values of 0.6 s
and 10−15 s s−1 respectively, with implied magnet∼ic field ofbinarity,eccentricityoforbits,andspacemotionsareusu-
stren∼gths in the range 1011 1013 G. In contrast, in allyattributedtodifferencesinthekickvelocityimpartedto
the MSPs, P and P˙ ar∼e distrib−uted about 5 ms and theneutronstarsatbirth(Shklovskii1970).Themagnitude
10−20 ss−1 respectively,withfieldstrengths∼intherange ofthekick,andits effecton thebinarysystem,will depend
∼ 108 109 G. This bi-modality in the field distributions on the nature of the system, and whether the neutron star
∼ − originates from the core collapse of a massive star with a
of the radio pulsars has been an enigma which has still to
supernovaexplosion,orfrom theaccretion inducedcollapse
befully resolved.
(AIC) of a white dwarf.
Therearealsomajordifferencesinthepopulationchar-
acteristics of these two groups of pulsars which provide im- In the standard model, the MSPs are considered to
portant clues on their origin. Most ( 85%) of the MSPs be the end product of the evolution of low-mass and
∼
are in binary systems (MSPs) on nearly circular orbits in intermediate-mass X-ray binaries where it is assumed that
contrast to the normal pulsars which tend to be isolated the neutron star was formed by core collapse (CC) of
and when they are not, exhibit more eccentric orbits. Fur- a massive (M > 8M⊙) star, and is subsequently spun
(cid:13)c 0000RAS
2 L. Ferrario & D. T. Wickramasinghe
up to millisecond periods during an accretion disc phase strength through the high-latitude survey of Burgay et al.
(Bhattacharya&vandenHeuvel1991;Bisnovatyi-Kogan& (2006) in the region of the sky limited by 220◦ < l < 260◦
Komberg1974).Weshallrefertothisclassofobjectsasthe and b <60◦ conductedwiththe20-cmmulti-beamreceiver
| |
“Core-Collapsed LMXBs and IMXBs” or, more briefly, the on the Parkes radio-telescope. If we restrict the pulsars in
LMXBs(CC)/IMXBs(CC). Accretion induced field decay is this survey to a distance of up to 1.5 Kpc, we find that
an integral part ofthis modelwhich appears plausible from all but one previously known radio-pulsars have been re-
atheoreticalview-point,particularlyifthefieldsinneutron detectedand4newobjects,outofatotalof16,discovered.
starsareofcrustalorigin (Konar&Bhattacharya1997,but We therefore apply a correction factor of 1.25 to obtain an
seeRuderman2006foranalternativemodel).Regardlessof estimateofthetotalnumberofMSPsinthe1.5kpcsample
theoriginofthelowfieldsintheMSPs,alongstandingprob- that we analyse, stressing the fact that this may only be a
lem with theLMXB/IMXB scenario hasbeen thedifficulty lowerlimitfortherealnumberofobjectsuptothisdistance.
in reconciling their semi-empirical birth rates with those We have restricted our analysis to binaries and iso-
of the radio MSPs (e.g. Lorimer 1995; Cordes & Chernoff lated millisecond pulsars with spin periods shorter than 30
1997).Theproblemwiththebirthrateshasbeenconfirmed ms. In the restricted sample to 1.5 kpc, the Australia Tele-
by recent population synthesis calculations which havealso scope National Facility (ATNF) catalogue (Manchester et
highlightedthedifficultiesinexplainingtheobservedorbital al. 2005) gives 24 MSPs in binaries and 11 isolated MSPs.
perioddistributionofMSPsontheLMXB(CC)/IMXB(CC) We do not distinguish between these two groups since the
scenario (Pfhal et al. 2003). isolated MSPs are likely to be an end product of binary
Another often discussed channel for the production of systems in which thecompanion has been tidally disrupted
MSPs involves the AIC of an ONeMG white dwarf (Michel or ablated (Radhakrishnan & Shukre 1986) with an other-
1987). Here, during the course of mass transfer, a white wise similar evolutionary history to the binary MSPs (see,
dwarf reaches the Chandrasekhar limit, and collapses to however, Michel 1987; Bailyn & Grindlay 1990 for alterna-
formaneutronstar(Bhattacharya&VandenHeuvel1991). tivepointsofview).Theyexhibitverysimilarobservational
In the AIC scenario, we may expect the magnetic field dis- properties, except, perhaps, for the radio-luminosity which
tribution of the MSPs to reflect in some way the magnetic seems to be slightly lower in the isolated MSPs (Bailes et
field distribution of theirprogenitor white dwarfs obviating al. 1997). Of the 24 binary MSPs, 14 have periods above
theneedforfielddecay.Alow-massorintermediate-massX- 10dandmosthavelowerlimitstothecompanionmassesin
raybinaryphasemayfollow thecollapseofthewhitedwarf therange 0.2 0.4M⊙ (Manchesteretal.2005)indicative
∼ −
and we shall refer to this class of objects as the “Accretion of the end state of evolution of binary systems that evolve
Induced Collapse LMXBs and IMXBs” or, more briefly, as to longer periods (beyond the bifurcation period, see Pod-
the LMXBs(AIC)/IMXBs(AIC). Population synthesis cal- siadlowski, Rappaport & Pfahl 2002) due to mass transfer
culations indicate that the expected birth rates from the from a low-mass giant (see sections 3.1 and 3.2) leading to
AIC channel may be significantly higher than those from Hewhitedwarfs. Theremaining shorterperiod systems ap-
theLMXB(CC)/IMXB(CC)route(Hurleyetal.2002;Tout peartohaveeitherHeorCOwhitedwarfsorverylow-mass
et al 2007; Hurley 2006, privatecommunication). companions.
In this paper, we present an analysis of the 1.5 kpc Theageofthemillisecondpulsariscalculatedaccording
sample of MSPs which is considered to be sufficiently sam- to
pled out (Lyne et al. 1998; Kramer et al. 1998) with the
aim of establishing the MSP birth properties and con- t= P 1 P0 n−1 (1)
straining the different models that have been proposed for (n 1)P˙ (cid:20) −(cid:16)P (cid:17) (cid:21)
−
their origin. Our estimate of the Galactic birth rate of
the MSPs is at the upper end of previous estimates (e.g. where n is the braking index, which is equal to 3 for the
Cordes & Chernoff 1997) and again brings into question dipolar spin-down model, and P and P0 are the observed
the LMXB(CC)/IMXB(CC) scenario as being the domi- and the initial period of the MSP respectively. If P0 <<P
nant route for the origin of the MSPs. The paper is ar- we obtain thecharacteristic age of the MSP:
ranged as follows. In section 2 we describe the data set
P
and our model. Our results are presented and discussed in τc = 2P˙. (2)
section 3 where we also present the case for and against
LMXB(CC)/IMXB(CC) progenitors and AIC progenitors In the next sections we will show that P0 is often too close
for the MSPs. Ourconclusions are presented in section 4. toP tobeabletorely onτc foran estimateof thetrueage
of a MSP.
The very low spin-down rates of the MSPs have so far
precluded any direct measurements of the braking index.
2 THE MODELLING OF THE RADIO
Anindexofn=3isindicated forold normal radiopulsars,
PROPERTIES OF THE MSPS
but values n 1.5 2.8 have been measured in younger
∼ −
In1998,Krameretal.conductedaverydetailedstudyaimed pulsars (Lyne 1996; Hobbs et al. 2004). In this work, we
atcomparingtheradioemissionpropertiesofMSPstothose have adopted n = 3, but it is conceivable that a different
of normal pulsars. In their work, they restricted their com- value of n may be appropriate for the MSPs. For instance,
parative studies to objects within 1.5 kpc, on the grounds if spin down is by multi-polar radiation, the braking index
that the population of all radio-pulsars is sufficiently sam- willbesomewhatlargerthan3,whileifangularmomentum
pled out up to this distance, as first pointed out by Lyne is lost mainly by gravitational radiation, we expect n = 5
et al. (1998). Recently, this assumption has gained further (Camilo, Thorsett & Kulkarni 1994). In our modelling we
(cid:13)c 0000RAS,MNRAS000,000–000
Birth properties of MSPs 3
adopt n=3 but we also discuss the implications of using a λ=b log L400 +a (6)
larger valueof n. (cid:18) L400 (cid:19)
h i
We synthesise the properties of the MSPs using essen-
andaandbareconstantstobedetermined(Hartmanetal.
tially the method described in Ferrario & Wickramasinghe
1997).
(2006, hereafter FW). However, in the present study there
Krameretal.(1998) findthatbyrestrictingtheircom-
aretwomain differences.Firstly,wedirectly assume an ini-
parisonanalysisofnormalradio-pulsarstoMSPstosources
tial magnetic field distribution for the MSPs without at-
up to 1.5 kpc, the mean spectral indices of normal radio-
temptingtorelate it backtothemagnetic propertiesof the
pulsars and MSPs are essentially the same, i.e., 1.6 0.2
(mainsequence)progenitors.Wethereforehaveasourbasic − ±
(MSPs)and 1.7 0.1(normalpulsars).Henceourdeduced
input the MSP birth magnetic field distribution, which we − ±
radio luminosity at 400 MHZ is scaled to 1400 MHZ using
describebyaGaussian in thelogarithm, and thebirth spin
a spectral index of 1.7 (as in FW).
distribution also described by a Gaussian. We stress that −
OncealltheintrinsicpropertiesofourmodelMSPsare
here with “birth” characteristics of MSPs we refer to those
determined,wecheckforpulsarsdetectabilityat1400MHZ
characteristics that theMSPs haveas theyswitch on asra-
by the Parkes multi-beam receiver (e.g. Manchester et al.
dioemitters,regardlessoftheirprevioushistory.Hence,the
2001; Vranesevic et al. 2004).
resultsofourcalculationsdonotdependinanywaysonthe
Furthermore,pulsarsradioemission isanisotropic with
specific route(s) leading to theformation of theMSPs.
pulsarsatshorterperiodsexhibitingwiderbeams,hencewe
Secondly, we take into consideration thethree Doppler
need to correct for this factor, since this will influence the
accelerationseffectscitedbyDamour&Taylor(1991)which
birthratesofMSPs.Forexample,largebeamswouldrequire
affect theobservedspin-downrateoftheMSPs,namely,(i)
smaller birth rates, since the MSPs would have a greater
the Galactic differential rotation, (ii) the vertical accelera-
chancetobedetected.However,thereisasyetnoagreement
tionKz intheGalacticpotentialand(iii)theintrinsictrans-
onthebeamingfraction-periodrelationship,particularlyfor
verse velocity of the pulsar. Thus, the observed spin-down
theMSPs.Rankin(1993),Giletal.(1993)andKrameretal.
rate is given by(e.g. Toscano et al. 1999)
(1994)pointedoutthatobservationalevidenceseemstosug-
P˙obs =P˙i+∆P˙ (3) gest that the opening angles of normal radio-pulsars (that
is,thelast opendipolar field line) isproportional to1/√P.
where P˙i is the “intrinsic” spin-down rate and ∆P˙ is the Intheabsenceofaconsensusonthisissue,weuseKramer’s
term dueto theaforementioned acceleration effects. Hence, (1994) model at a frequency of 1.4 GHz for the opening
when we compare our models to observations, we intro- half-angle θ (in degrees) of thepulsar beam:
duce these acceleration terms to our synthetic population 5.3◦
to mimic thebehaviour of the observed MSPs. θ= . (7)
P0.45
Wefollow themotionsofthestarswegeneratebyinte-
grating theequationsof motion in theGalactic potential of Thesevaluesofθ yielddutycyclesoflessthanunityforpe-
Kuijken & Gilmore (1989) assuming that the neutron stars riodsdowntoabout1ms.However,wewouldliketoremark
are born with a kick velocity given by a Gaussian distribu- that our results are quiteinsensitive to slight modifications
tion with velocity dispersion σv. totheaboveθ−P relationship.Then,byassumingthatthe
To fit the observations, we also model the radio lu- viewing angles of MSPs are randomly distributed, the frac-
tion f of the sky swept by the radiation beam is given by
minosity at 1400 MHZ and compare it to the members of
our list with a measured value at this frequency. The stud- (Emmering & Chevalier 1989)
ies of Kramer et al. (1998) and Kuz’min (2002) indicate π
that despite the large differences in periods and magnetic f =(1−cosθr)+ 2 −θr sinθr (8)
(cid:16) (cid:17)
fields, normal pulsars and MSPs exhibit the same fluxden-
where θr is the half-opening angle now in radians. We will
sity spectra, therefore pointing towards the same emission
usethisf tocompare ourMSPsyntheticpopulation to the
mechanism, although the MSPs tend to be weaker sources
data sample underconsideration.
on average (Kramer et al. 1998). Hence, similarly to many
previousinvestigators (e.g. FW;Narayan & Ostriker1990),
we haveassumed that theluminosity L400 at 400 MHZ can
bedescribed by a mean luminosity of theform 3 RESULTS AND DISCUSSION
1 P˙ Wehaveused as our observational constraints the1-D pro-
loghL400i= 3log(cid:18)P3(cid:19)+logL0 (4) jections of the data comprising the numberdistributions in
period P, magnetic field B, period derivative P˙, radio lu-
Here the luminosities are in units of mJy kpc2. We have minosity L1400, Z-distribution and characteristic age τc, as
determined from equation 3.
modelled the spread around L400 using the dithering func-
Similarly to FW, the best fit model to the observa-
tion of Narayan & Ostriker (1990) to take into account the
tions of the MSPs was determined “by eye” after conduct-
various intrinsic physical variations within the sources and
ing hundreds of trials. Our results are shown in Figure 1.
also variations caused by differentviewing geometries. This
function is given by The fit was obtained by setting σP0 = 1.3 ms about a
mean P0 =4 ms. The parameters for the luminosity model
ρL(λ)=0.5λ2exp(−λ) (λ≥0) (5) aMreSPhsL4a0r0eiim=p5a.r4t,edaw=ith1.5aaonnde-dbim=en3s.i0o.nFaulrntahtearlmkoircek,dtihse-
where persionvelocityof50kms−1,whichyieldsanaveragetrans-
(cid:13)c 0000RAS,MNRAS000,000–000
4 L. Ferrario & D. T. Wickramasinghe
Figure 1. Our model (solid line) overlapped to the MSPs data sample of up to 1.5 Kpc (shaded) taken from the ATNF catalogue
(http://www.atnf.csiro.au/research/pulsar/psrcat, Manchester etal.2005). TheerrorbarsindicatethePoissonstandarderrors.
verse velocity of 83 km s−1. Our model reproduces the ob-
served total number of MSPs with a local formation rate
of >∼4.5 10−9 yr−1 kpc−2 which translates into a Galac-
tic birth×rate of >∼3.2 10−6 yr−1. We emphasise that this
×
shouldbeseenasalowerlimitfortheMSPbirthrate,since
the1.5kpcsampleisstill likelytobesomewhat incomplete
even after the correction factor that we haveapplied.
OurcalculationsshowthattheobservedMSPmagnetic
field distribution can be modelled with a field which is ini- Figure 2. Intrinsic (solid histogram) and observed (dashed his-
tially(thatis,attheonsetoftheMSPradio-emissionphase) togram)spin-downratesofoursyntheticpopulationofMSPs.
Gaussian in the logarithm. We find that, similarly to our
modelling of normal isolated radio-pulsars (see FW), it is
notnecessarytoassumeanyspontaneousfielddecayduring
of these objects appear to be older than the Galactic disc
the lifetime of the radio-emitting MSPs. Thus, the “high”
(10 Gyr) if one uses equation 2 to assign them an age.
magnetic field tail of MSPs arises from the dependence of
Toscano et al. (1999) found that by correcting their spin-
the spin-down rate on the magnetic field, and not from a
down rates for the transverse velocity (Shklovskii) effect
complex birthfielddistribution assigned totheparentpop-
theyexacerbatedthisparadox,sincethecorrection resulted
ulation(e.g.asaresultofaccretion-inducedfielddecaydur- in a decrease in the spin-down rate P˙ (and thus of the de-
ing aprevious phaseof mass accretion). Inthis context,we
rivedmagneticfieldstrength)andanincreaseincharacteris-
notethat iftheMSPswereborn withhigherfieldstrengths
ticage.Asaconsequence,nearlyhalfoftheircorrectedsam-
thanpostulated byusand thendecayedduringtheirradio-
ple exhibited characteristic ages comparable to or greater
emissionlifetimetowardsweakermagneticfieldsvalues,then
than the age of the Galactic disc. Our theoretical results
wewouldexpectacontinuousfielddistributionfillingupthe
agreewiththeirfindingsandarepresentedinFigure2where
gap between the normal radio-pulsars and the MSPs. This wehaveplottedtheobservedP˙obsandintrinsicP˙ispin-down
was first noted by Camilo et al. (1994), who also point out
rates of our syntheticpopulation. This clearly supports the
that there is no indication that old globular cluster MSPs
view that transverse velocity effects do play an important
havemagneticfieldswhichare lowerthanthoseofMSPsin
role in theobservations of MSPs.
theGalactic disc.
Camiloetal.(1994)proposedthreepossibilitiestosolve
AnotherobservationalpeculiarityofMSPsisthatsome this paradox. The first is that the magnetic field structure
(cid:13)c 0000RAS,MNRAS000,000–000
Birth properties of MSPs 5
of MSPs is multipolar and thus the braking index that ap- period towards longer periods. For the lower mass donor
pears in equation 1 may be greater than 3. Alternatively, stars,masstransferphaseendswhentheheliumcoreofthe
due to gravitational radiation, n = 5. The second is that donorstarisexposedasalow-mass( 0.2 0.4M⊙)helium
∼ −
the magnetic field decays, so that high values of τc could whitedwarf. For themore massive donorstars, mass trans-
be attained bya certain choice for thedecay time-scale, al- fercanterminatewhenaCOoranONeMgwhitedwarfcore
though they discarded this possibility as outlined earlier. is exposed. Although the observed sample of binary MSPs
Finally, the third possibility that Camilo et al. (1994) pro- consistsofsystemswithalloftheabovecompanions,recent
posed isthat at least someMSPsmay havebeen bornwith populationsynthesiscalculationshavenotbeensuccessfulin
P0 P. modellingtheobservedorbitalperioddistributions(Pfahlet
∼
Ourmodellinghasshownthatifwestartwithaninitial al. 2003).
period distribution that isGaussian with mean at 4ms, we Thus,Pfahletal.(2003)findthattheLMXB(CC)route
obtainacurrentdayperioddistributionthatisclosetowhat leadstoasignificant population oflow period binaryMSPs
is observed. In particular, we can reproduce the sharp rise peaking at P 0.03 d, but this population is not repre-
orb
∼
in the observed period distribution near 3 ms. In our sentedintheobservedsampleofMSPs.Indeed,theshortest
∼
model,mostpulsarshaveinitialperiodsthatarequiteclose observed binary period for the MSPs is P = 0.1 d for
orb
totheobservedperiods.Thisyieldsanaveragecharacteristic PSRJ2051-0827 (Stapperset al. 2001). Ontheother hand,
age which is larger than the average true age (as given in the binary periods of the ultra-compact X-ray binaries are
equation 1) by nearly 50%. Further observational support generallysignificantlyshorterwithafewofthemexhibiting
in favour of birth periods being close to observed periods orbital periods of 0.03 days. Hence, this may be an indica-
in MSPs also comes from studiesof individualsystems. For tionthateither(i)astheirLXMB(CC)evolutioncontinues,
instance, for PSR J0437-4715, Johnston et al. (1993) and theywillendupablatingtheircompanionandthusappear-
Bell et al. (1995) find P = 5.757 ms, Porb = 5.741 d and ingasisolatedMSPs,or(ii)theultra-compactLMXBs(CC)
τc = 4.4 4.91. Sarna, Ergma, Ger˘skevit˘s-Antipova (2000) and theMSPs are not evolutionarly linked.
−
derive themass of the companion of PSR J0437-4715 to be In contrast, the IMXB(CC) route predicts a popula-
0.21 0.01 M⊙ with a cooling age of 1.26 2.25 Gyr. This tion of binaries that peaks roughly at the observed periods
± −
impliesthatPSRJ0437-4715 ismuchyoungerthaninferred P 6 30d,butwithawidththatfallsshortbyafactor
orb
∼ −
through its characteristic age and was born with a period of 10 100 (depending on assumptions on the common
∼ −
close to thecurrent period. envelope parameter) in comparison to the observations of
The lack of sub-millisecond pulsars is apparent in the binary MSPs. Indeed, the majority of the binary MSPs in
distribution of the MSPs and has also been noted in the theATNF sample (Manchester et al. 2005) do not haveor-
millisecond X-ray pulsars (Chakrabarty 2005). Our study bital periods that fall in the most probable region (Pfahl
indicates that this is consistent with our assumption of a et al. 2003) predicted for either LMXB(CC) or IMXB(CC)
Gaussian distribution of initial birth periods that peaks at evolution.
4 ms and rules out the possibility that most millisecond There is also the problem with the birth rates
pulsars are born at sub-millisecond periods. Here, selection mentioned in section 1. Attempts at reconciling the
effectsmay beplaying arole, however, Camilo et al. (2000) LMXB(CC)/IMXB(CC) rateswiththebirthratesofMSPs
estimated a loss of sensitivity of only 20% below about havenot beensuccessful(Pfahlet al.2003) andthepresent
2 ms, which is far too low to explain the sudden drop in resultsgointhedirectionofmakingthisdiscrepancylarger.
the number of MSPs below this period. This may suggest Ithasbeensuggestedthattheabovediscrepanciesmaydis-
that neutron stars can never achieve break-up spin peri- appear when more realistic models are constructed that al-
ods ( 0.4 0.7 ms depending on equation of state, Cook lowforlimit cyclesthatmayarisefromX-rayirradiation of
∼ −
et al. 1994). Hence, loss of angular momentum caused by thedonorstars, and for theintricacies in common envelope
gravitational radiation maylimit theneutron starspin rate evolution. This remains a possibility.
(Wagoner 1984). Current estimates to the lower limits for However, it should be noted that the semi-empirical
spin periods are about 1.4 ms (Levin & Ushomirsky 2001), birth rates are based almost entirely on the observed
which isclose tothespinoftherecentlydiscovered MSPin LMXBs, which, given the arguments above, cannot be the
theglobular cluster Terzan 5 (Hessels et al. 2006). dominant progenitors of the binary MSPs. The same com-
mentalsoappliestoourdiscussionoftheAICthatundergo
a phaseof mass transfer after collapse (see section 3.2).
In the evolution that leads up to LMXBs(CC) and
3.1 The LMXB(CC)/IMXB(CC) scenario and its
IMXBs(CC),oneofthestellarcomponents(usually,theini-
relation to birth properties
tiallymoremassive)evolvesintoaneutronstarthroughcore
IntheLMXB(CC)route,matteristransferedfromlow-mass collapse with a field distribution peaking near logB(G) =
main-sequence donors and binary evolution occurs towards 12.5 (as observed in the isolated radio-pulsars). The ob-
shorter periods driven by magnetic braking and gravita- served field distribution of the MSPs, on the other hand,
tional radiation. Mass transfer continues past the period peaksatlogB(G)=8.4.Thisdiscrepancyisoftenexplained
minimum over a Hubble time or until the companion is by accretion-induced field decay or evolution. The presence
evaporated.BinaryMSPsthatresultfromthisrouteareex- ofMSPsinold systemssuggeststhatifthelowvalueofthe
pected to have very low-mass companions with ultra short magneticfieldisduetofielddecay,itmustdosomainlydur-
orbital periods. In contrast, in the IMXB(CC) route the ing the accretion phase prior to the neutron star becoming
donor stars are of intermediate mass >∼2M⊙, and binary a radio MSP.
evolutionisdrivenbynuclearevolutionpastthebifurcation The manner in which the field is expected to decay in
(cid:13)c 0000RAS,MNRAS000,000–000
6 L. Ferrario & D. T. Wickramasinghe
accreting neutron stars depends on the origin of the mag- theMSPsastheybecomeradio-emitters,tobesimilartothe
neticfields,andherethereisnoconsensus.Thereisnoclear observedspinperioddistributionoftheLMXBs(CC).How-
evidence for field decay in ordinary pulsars on a time scale ever,observationsofaccretionandnuclearpoweredLMXBs
of107 108 yr.However,accretion canenhancefielddecay, show that their spin periods peak near 2 ms (Lamb & Yu
−
particularlyifthefieldsareofcrustalorigin.Twocompeting 2005). This could indicate either a different origin for the
effects have been considered. Accretion raises the temper- radio MSPs (see section 3.2), or that the braking index is
ature and reduces the conductivity in regions of the crust significantly larger than adopted by us. We have carried
thatcarrythecurrent,therebyenhancingfielddecay.Accre- outcalculationsfordifferentbrakingindicesandfindthata
tion also pushes the current forming region inward towards braking index of n=5 (appropriate to angular momentum
regions of higher density and conductivity where the field loss by gravitational radiation or magnetic multipolar radi-
can be frozen. Konar & Bhattacharya (1997) have shown ation)willbringthetwodistributionsintocloseragreement.
that these two effects, when taken together, could lead to
an asymptotic “frozen” field strength that is a factor of
10−1 10−4 below the initial field strength. The asymp-
−
toticvaluedependsontheaccretionrateandthetotalmass
3.2 The AIC scenario and its relation to birth
accreted.Ontheotherhand,Wijers(1997)presentedstrong
properties
evidenceagainst accretion-induced fielddecaywhichispro-
portionaltotheaccretedmassontotheneutronstar.Thus, According to current models, accretion induced collapse
thestandardmodeldoesnotexplaintheobservedcharacter- leads totheformation of arapidly spinning(a few millisec-
istics of the MSP birth field distribution as they switch-on onds) neutron star when an ONeMg white dwarf accretes
asradio-emitters.Forinstance,ifweconsidertheroutethat matter in a binary system and reaches the Chandrasekhar
contributes to the majority of binary MSPs, namely those limit. Recent calculations have re-affirmed that an AIC is
havingorbitalperiodsPorb 10 1000dwithlowmassHe theexpectedoutcomeofthermaltimescalemasstransferin
≥ −
WD companions arising from the evolution of intermediate suchsystemswithorbitalperiodsoftheorderofafewdays
massdonors,wemayexpecttheaccretionhistory,andthere- (Ivanova&Taam 2004). Themagnetic fluxesin thesecores
fore the birth field distribution, to depend on the orbital may thus reflect the magnetic fluxes seen in the isolated
period. It is therefore not immediately apparent why this white dwarfs.
fieldshouldhaveanearlyGaussiandistributionwithsucha Untilrecently,itwasbelievedthatthemagneticfieldsof
narrowwidth.Theproblembecomesevenmoreseverewhen theisolatedwhitedwarfscouldbedescribedbyasingledis-
morethanonechannelisconsidered(seediscussion inTout tribution.However,itisnowevidentthatthedistributionis
et al. 2007). bi-modal, comprising of a high and a low field component.
The detection of coherent X-ray pulsations with mil- The high field component (10 15% of all white dwarfs)
−
lisecond periodsin ahandfulofLMXBs(Lamb &Yu2005) has a distribution that peaks at logB(G) 7.5 with a half
∼
isoftenusedinsupportoftheideaofaccretioninducedfield widthσlogB=7.3(e.g.Wickramasinghe&Ferrario2005).
decay (Wijnands & van der Kliss 1998). However, whether Thisdistributiondeclinestowardslowerfieldswithveryfew
thisisevidencesimplyforfieldsubmersionandspinupdur- starsdetectedinthefieldrange105 106 G(magneticfield
−
ing an accretion disc phase, or for field decay and spin up, gap).Theincidenceof magnetism rises again towardslower
remains to be established. Cumming et al. (2001) have ar- magnetic fields with some 15 25% of white dwarfs being
−
guedthatthemajorityoftheLMXBsdonotshowcoherent magnetic at thekilo-Gauss level (Jordan et al. 2006). Since
pulsations because they may have fields significantly less the new detections of Jordan et al. (2006) are at the limit
than 108 G due to field submersion which, at face value, is of the sensitivity of current spectropolarimetric surveys, it
inconsistentwiththefieldsseenintheradio-MSPs,buttheir appearslikelythatall whitedwarfswillbefoundtobemag-
calculations also indicate that the field will re-emerge on a netic, with the majority ( 85%) belonging to the low field
timescaleof 1000yralthoughitisuncleartowhatvalue. group(<∼1,000G).Afield∼ofakilo-Gaussscalesundermag-
∼
Indeed, for the LXMB(CC)/IMXB(CC) standard scenario netic flux conservation to a neutron star field of 109 G. Al-
to be viable, the field would be required to re-emerge to though the peak of the low field distribution has still to be
valuesthataresimilartothoseobservedintheradioMSPs. established observationally, it is conceivable that the fields
If we adopt the contentious viewpoint that magnetic willbedistributedinaGaussian manneraboutapeakthat
fields do not decay due to accretion, but are simply tem- will map on to the observed field distribution of the radio
porarily submerged, and re-emerge to their original values MSPs.GiventhehighmasstransferratesrequiredforAIC,
ofafew 1012 GattheendoftheLMXB(CC)/IMXB(CC) theOhmicdiffusiontimescalewill bemuchlarger thanthe
×
phase,thenwemayexpectapopulationofhighfieldMSPs. accretiontimescale(Cumming2002),soweexpectthewhite
The objects in such a population would have a birth rate dwarffieldtobesubmergedduringthebuildupofthewhite
thatis10−4 timesthebirthrateofnormalradio-pulsarsand dwarf mass prior to collapse. For the above scenario to be
wouldthereforebeunlikelytoberepresentedinthecurrent viable, we need to postulate that the submerged field will
sampleofradio-pulsars.Furthermore,sincetheywouldspin re-emerge without decay to its flux conserved value at the
down very rapidly to much longer periods (with character- birthoftheneutronstar.Inthiscontext,wenotethatthere
istic time scales of only a few hundred years), they would is no evidence of accretion-induced field decay in the AM
have an even smaller chance to be detected as high field Herculis-type Cataclysmic Variables, where a highly mag-
radio-MSPs. neticwhitedwarfhasbeenaccretingmassoverbillionyears
Finally, we note that on the LMXB(CC)/IMXB(CC) from a companion. In fact, their well studied field configu-
hypothesis, we expect the birth spin period distribution of rations are very similar to those observed and modelled in
(cid:13)c 0000RAS,MNRAS000,000–000
Birth properties of MSPs 7
the isolated high field magnetic white dwarfs (e.g. Wickra- 4 CONCLUSIONS
masinghe & Ferrario 2000 and references therein).
WehavepresentedananalysisofthepropertiesoftheMSPs
We expect the white dwarf to be spun up to near
and placed constraints on the magnetic field and the spin
break up velocity prior to collapse (e.g. like the white
period distributions of the MSPs at the time they turn on
dwarfsobservedindwarfnovae).However,angularmomen-
as radio emitters. We find that if we assume a braking in-
tum (and mass) must necessarily be lost during the subse-
dex n = 3, the field distribution can be represented by a
quentcollapsetoaneutronstar(Bailyn&Grindlay1990)so
Gaussian in the logarithm with mean logB(G) = 8.1 and
that detailed models are required to establish the expected
birth spin and mass distributions of the resulting neutron σlogB = 0.4 and the birth spin period by a Gaussian with
star. Dessart et al. (2006) have conducted 2.5-dimensional mean P0 = 4 ms and σP = 1.3 ms. Our study, which al-
lowsforacceleration effectsontheobservedspin-downrate,
radiation-hydrodynamics simulations of the AIC of white
shows that (i) most MSPs are born with periods that are
dwarfs to neutron stars. Their calculations show that these
close to thecurrently observed values (ii) the characteristic
lead to the formation of neutron stars with rotational peri-
ages of MSPs aretypically muchlarger than theirtrueage,
odsofafew(2.2-6.3)milliseconds.Hence,evenifthebinary
and(iii)sub-millisecondpulsarsarerareordonotexist.We
systemweretobedisruptedfollowingakickduringtheAIC,
these“runaway”newly born neutronstarswould appearas yalrso−1fi.ndaGalactic birthratefortheMSPsof >∼3.2×10−6
isolated radio-MSPs of thetypecurrently observed.
We have used our results to discuss the relative merits
A proportion of the white dwarfs that could be sub-
oftheLMXB(CC)/IMXB(CC)andAICscenariosthathave
jected to AICwill inevitably belong to thehigh field group
been proposed for explaining the origin of the MSPs. Our
( 106 109 G), and will result in rapidly rotating (mil-
li∼second−) pulsars with fields in the range 1010 1014 G on main conclusions can be summarised as follows.
−
collapse, if we assume magnetic flux conservation (see Fig- (a) The birth rate that we deduce for the MSPs is signif-
ure 1 in FW). This proportion could be as high as 50% be- icantly larger (by a factor 100) than the semi-empirical
∼
causehighlymagneticwhitedwarfstendtobemoremassive birth rates quoted for the LMXBs/IMXBs and is more in
thantheirnon-magnetic(orweakly magnetic) counterparts accord with the expected birth rates from the AIC route
(mean mass of 0.92M⊙, Wickramasinghe & Ferrario 2005). from population synthesis calculations.
Assuming that the kicks are not field dependent and thus (b) The AIC scenario relates the MSP neutron star field
preferentially disrupt these systems, we may also expect a distribution totheir white dwarf ancestry without invoking
group of MSPs with high fields. However, given the much any kind of field decay, and finds some support from the
higher spin-downrates of theseobjects, and their low birth recently discovered bi-modality of the magnetic field distri-
rates as compared to normal radio-pulsars, we expect them butionofthewhitedwarfs.ThelowfieldsoftheMSPsmay
tomakeasmallcontributionwhich wouldbedominatedby simplyarisefromthelowfieldcomponentofthewhitedwarf
thelowestfieldobjectsinthedistributionwhichwouldhave magnetic field distribution that is expected to peak below
the longest lifetimes as radio-pulsars. A possible candidate 103 Gandtoscaletofieldsbelow109 Gundermagneticflux
could be the binary radio-pulsar PSR B0655+64 which has conservation. The nearly Gaussian distribution that we de-
a relatively high magnetic field (B = 1.1710 G), short or- ducefortheneutronstarbirthmagnetic fieldsin theMSPs
bitalperiod(Porb =1.03d)andisonanearlycircularorbit may thus have a more ready explanation on the AIC sce-
(Damashek, Taylor& Hulse1978; Edwards & Bailes 2001). nario.
The population synthesis calculations of Hurley et al. (c) On the standard LMXB(CC)/IMXB(CC) picture, the
(2002) yielded an AICrate that is two orders of magnitude majority of the MSPs which have intermediate or long or-
higher than theLMXB(CC) rate that results from theevo- bitalperiods,comefromtheIMXB(CC)route.However,the
lution of binary systems with primaries that are less mas- predictionsoftheexpectedorbitalperioddistributionsfrom
sive than 2M⊙. A more detailed investigation of the AIC populationsynthesiscalculationsarenotingoodagreement
and core collapse rates and orbital period distributions ex- with the observations of theMSPs. It remains to be seen if
pected from such calculations has been presented by Tout betteragreement with theobserved period distribution can
etal.(2007). Hereitisshownthataswiththecorecollapse be obtained with predictions from the AIC route which is
route, the AIC route also generates binary MSPs of all of expected to also produce systems with the same range of
theobserved types.Wenoteinparticularthataclassoflong orbital periods and companion masses at the end of mass
period(Porb 10d)binaryMSPswithHewhitedwarfcom- transfer prior to theradio MSP phase.
≥
panions is predicted, and this closely follows the observed (d) Populationsynthesiscalculationspredictthatthestan-
Porb MWD relationship (Van Kerkwijk et al. 2005). dard LMXB(CC) route results in a significant class of
−
We conclude this section by noting that neutron stars LMXBswith periods less than1 hr,and theseareobserved
thatareformedviaAICmayalsogothroughamasstransfer asultracompactLMXBs.However,thereisnotasyetasin-
phase prior to their switching-on as radio MSP. We there- gle MSP with such a low period. A likely possibility is that
fore expect that the known sample of LMXBs/IMXBs will the mass of the companion has been reduced to negligible
have a contribution from both neutron stars that have re- values by mass transfer, or that the companion has been
sulted from the core collapse of massive stars and from fully ablated by the time the pulsar turns on. These sys-
the AICs. According to current estimates of the AIC rates, temscouldresultinisolatedMSPs.However,theAICroute
the LMXBs(AIC)/IMXBs(AIC) may dominate over the also leads to similar systems and therefore end products.
LMXBs(CC)/IMXBs(CC). However, because of field sub- (e) The peak in the spin period distribution of accretion
mersion,itmaybedifficultatpresenttodistinguishbetween andnuclearpoweredX-raypulsarsoccursat 2ms,shorter
∼
these two possibilities. that the 4 ms that we have deduced for the birth spin
∼
(cid:13)c 0000RAS,MNRAS000,000–000
8 L. Ferrario & D. T. Wickramasinghe
period of MSPs assuming a braking index of n = 3. We HesselsJ.W.T.,Ransom, S.M.,Stairs,I.H.,FreireP.C.C.,Kaspi
find that the two distributions can be brought into closer V.M.,Camilo,F.,2006,Science, 311,1901
agreement if we assume a braking index n = 5 which may HobbsG.,LyneA.G.,KramerM.,MartinC.E.,JordanC.,2004,
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360,974
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SolankiS.K.,2006,astro-ph/0610875
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