Time-Variable Emission from Transiently Accreting Neutron Stars In Quiescence due to Deep Crustal Heating

12 Pages·0.41 MB·English

by  Greg Ushomirsky

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev1.4)
Time-Variable Emission from Transiently Accreting
Neutron Stars In Quiescence due to Deep Crustal Heating
Greg Ushomirsky1 and Robert E. Rutledge2
1Theoretical Astrophysics, California Institute of Technology, MS130-33, Pasadena, CA 91125; [email protected];
2Space Radiation Laboratory, California Institute of Technology, MS 220-47, Pasadena, CA 91125; [email protected]
9January2001
1
0
0
2 ABSTRACT
Transiently accreting neutron stars in quiescence (LX∼<1034 ergs−1) have been ob-
n
served to vary in intensity by factors of few, over timescales of days to years. If the
a
J quiescent luminosity is powered by a hot NS core, the core cooling timescale is much
longerthantherecurrencetime,andcannotexplaintheobserved,morerapidvariabil-
9
ity. However,the non-equilibrium reactionswhichoccur in the crust during outbursts

1 deposit energy in iso-density shells,from which the thermal diffusion timescale to the
v photosphere is days to years. The predicted magnitude of variability is too low to
1 explain the observed variability unless – as is widely believed – the neutrons beyond
4 the neutron-drip density are superfluid. Even then, variability due to this mechanism
1 inmodels withstandardcoreneutrino coolingprocessesis lessthan50per cent–still
1 too low to explain the reported variability. However,models with rapid core neutrino
0 cooling can produce variability by a factor as great as 20, on timescales of days to
1 years following an outburst. Thus, the factors of ∼few intensity variability observed
0
fromtransientlyaccretingneutronstarscanbe accountedfor by this mechanismonly
/
h if rapid core cooling processes are active.
p
- Key words: stars: neutron – X-rays: binaries – nuclear reactions, nucleosynthesis,
o abundances
r
t
s
a
:
v
1 INTRODUCTION accretion in quiescence (Van Paradijs et al. 1987), possibly
i
X through an advection-dominated accretion flow (Narayan
Many neutron star (NS) X-ray binaries go through accre- et al. 1997; Menou et al. 1999); accretion onto the NS
r
a tion outbursts (LX∼1037 ergs−1) followed by long periods magnetosphere (Campana et al. 1997) following a “pro-
(months–decades)ofrelativequiescence(LX∼<1034 ergs−1). peller phase”, in which the NS magnetosphere is larger
The origin of these outbursts remains under debate, al- thanthekeplerian orbit with an orbital period equalto the
though many agree that in the wider binaries an accretion spin period of the NS (Illarionov & Sunyaev 1975; Stella
disk instability is the cause (Van Paradijs 1996). During et al. 1986); and to thermal emission from a hot NS core,
these outbursts, on the order of ∼1023 g may be accreted, heated by non-equilibrium reactions deep in the NS crust
over a period of ∼few-30 days (for a review of these tran- (Brownetal.1998,BBR98hereafter).Ofthesepossibilities,
sients see Tanaka & Shibazaki 1996; Campana et al. 1997; only deep-crustal heating predicts the similar luminosities
Chen et al. 1997). observed in the thermal spectral component of these sys-
Following an outburst, the NSs return to quiescence, tems.Theobservedtemperaturesofthethermalcomponent
and have typically been detected with luminosities in the (kT=0.08–0.20 keV)areasexpectedinthisscenario. Inad-
1032–1033 ergs−1 range (Van Paradijs et al. 1987; Verbunt dition,theemissionarearadiiofthethermalcomponentare
et al. 1994; Asai et al. 1996a; Asai et al. 1996b; Campana consistent with theoretically predicted NS radii (∼10 km;
et al. 1998a; Garcia & Callanan 1999; Rutledge et al. 1999; Rutledge et al. 1999; Rutledge et al. 2000); this supports
Campanaetal.2000;Rutledgeetal.2000).ThequiescentX- the interpretation of this spectral component as a thermal
ray spectrum hasbeen described as beingcomprised of two NSphotosphere.Itispossiblethatthequiescentluminosity
components,asoft(black-bodykT ∼0.2keV)thermalcom- isduetoacombinationofthesethreemechanisms,inwhich
ponent, and a power-law component dominating the emis- case they are only separable by their spectral and intensity
sion above2keV (Asaiet al. 1996b; Campana et al. 1998b; variability properties.
Campana et al. 2000).
Thedetectedfluxduringquiescencefromsomeofthese
The basal luminosity has been attributed to continued transientshasbeenobservedtovary,byafactorofasmuch
(cid:13)c 0000RAS
2 Greg Ushomirsky and Robert E. Rutledge
as ∼3 or more, over timescales of days to years. No clear ∼1 year, the core heats to an equilibrium temperature
cause of this variability has yet emerged. There are three in ∼104 yrs, after which the core temperature varies by
means discussed in theliterature: ∼0.5% in response totheperiodic input of energy from the
non-equilibrium reactions during individual accretion out-
(i) Variable accretion onto the NS surface. If accretion
bursts. They confirmed the steady-state assumption made
ontotheNSsurfacedominatestheluminosityinquiescence,
by BBR98. Our study is complimentary to that of Colpi
changes in the accretion rate could account for variations
et al., as we examine the time dependence of the tempera-
in the thermal (kTeff∼0.2 keV) part of the spectrum, since ture in the crust and the thermal luminosity from the NS
thephotosphericspectrumatthelowimpliedaccretionrate
surface between outbursts. We follow in detail the thermal
is thermal in the absence of a shock (Zampieri et al. 1995),
relaxation of the crust and find that, depending on the mi-
assumingquasi-steady-stateaccretionattheappropriateac-
crophysics and the accretion history, the magnitude of the
cretion rates.
variabilitycanbeassmallas<1%,oraslargeasafactorof
(ii) Variable accretion onto the NS magnetosphere. The
20. Only the reactions that deposit energy at depths where
power-law spectral “tail” which can dominate emission
above 2 keV may be due to accretion onto a NS magneto- τth∼<τrec lead to variable thermal emission. The luminosity
duetoindividualreactions islargely blendedtogether,sim-
sphere(Campanaet al. 1998a). Variations in accretion rate
plybecause thedifferencebetween theτth of adjacent reac-
ontothemagnetospherewould causeintensityvariabilityin
tionsistypicallymuchshorterthanτth.However,theemis-
this power-law component.
sion from thereactions intheoutercrust asawhole is typ-
(iii) Variableabsorption column density (NH).Outbursts ically well-separated from that due to the reactions in the
maybeaccompanied byoutflows,increasinglocally thecol-
innercrust,resultingincharacteristic“double-hump”lumi-
umn density of absorbing material. This would cause vari-
nosityevolution.Inaddition,theamplitudeofthevariability
ations in the absorption column density (NH), which may isnotsimplyrelatedtotheenergydepositedintheindivid-
accountforthefactorsof2-3variationinintensityobserved
ualreactions.Wequantifythedependenceofthisvariability
(Rutledgeet al. 1999).
ontheconductivityofthecrustandthephysicsofneutrino
Moreover,intensityvariabilityduetoaccretion–eitherpho- emission in the core.
tospheric or magnetospheric – would likely be stochastic; In §2 we briefly recount the observational evidence for
there have been no theoretical estimations of its magni- quiescent intensity variability. In §3, we describe detailed
tude or variability timescale. In the variable-NH scenario, calculations oftime evolution inthethermalNSflux,inre-
theamountofabsorptionshoulddecrease(andintensity,in- sponsetoa“delta-function”(1-daylong)accretionevent,for
crease) with time after outburst. different time-averaged accretion rates (and thus core tem-
Inthispaper,weinvestigatethetimedependenceofthe peratures), different dominant conductivies and core neu-
thermalemissionmechanismproposedbyBBR98.Compres- trinocoolingprescriptionsWealsopresenttheresultinglight
sion of the NS crust by accretion during outbursts induces curves in this section. Finally, in §4, we compare these re-
electron captures, neutron emissions, and pycnonuclear re- sults with observations, and in §5, we conclude.
actions deep in the crust (at densities > 109 g cm−3), and
these reactions deposit Qnuc ≈ 1 MeV per accreted baryon
ofheatintothecrust(Haensel&Zdunik1990b).BBR98ar-
2 OBSERVATIONS OF VARIABILITY IN
guedthatthesereactionsheattheNScoretoanequilibrium
QUIESCENCE
temperature ≈108 K, and, during quiescence, the hot core
shineswithatypicalluminosityLq ≈hM˙i(Qnuc/mb),where Intensityvariabilityinquiescenttransientsis,atpresent,not
hM˙i is the mean accretion rate, averaged over the thermal well-studiedobservationally,largelyduetothelowsignal-to-
time of the core, i.e., many recurrence intervals τrec. Con- noiseofthedata,fromasmallnumberofobservations(only
sequently,the core luminosity is not expected to change on a few of which are made with the same instrumentation),
timescales shorter than its thermal time, i.e., ∼105 yrs. anddifferentassumedintrinsicenergyspectra.Multipleob-
Transient X-ray emission from the non-equilibrium re- servations in quiescence (at L <1034 ergs−1) are found in
q∼
actions has not been previously considered. Similar work the literature only for three NSs, Cen X−4, 4U 2129+37,
regarding transient energy deposition in NSs has either fo- and AqlX−1.
cussed on the atmosphere, well above the crust (Eichler & ForCenX-4,VanParadijsetal.(1987)foundthelumi-
Cheng 1989), or investigated the thermal response to pul- nosityincreasedbyafactorof∼2−5over∼5.5yr(Jul1980–
sar glitches (van Riper 1991; Chong & Cheng 1994; Hirano Feb 1986), for the same assumed thermal bremmstrahlung
et al. 1997; Cheng et al. 1998). However, the crucial differ- spectrum.Therewasnointerveningoutburstobserved,and
ence is that the depths and amounts of energy deposited theobservations weremade usingtwodifferent instruments
by non-equilibrium reactions can, at least in principle, be (EXOSAT/LE and Einstein/IPC). Campana et al. (1997)
obtained from an ab initio calculation (and have been, for reanalysed these data, and contrarily concluded that they
thecaseofiron,byHaensel&Zdunik1990b),while, onthe are consistent with the same luminosity. Also, Campana
other hand, the total amount of energy deposited by the found that in observations with ROSAT/HRI of Cen X−4
crust-breakingglitchesisnotpreciselyknown(itisbounded over a 4-8 day period, the source countrate varied by ∼3,
by 2IΩδΩ), and neither the depth nor the distribution of with an average luminosity of ∼7×1031 ergs−1; Campana
energy deposition is well constrained. did not discuss whether the 1995 ROSAT/HRI observa-
Colpi et al. (2001) examined the effect of non- tions were consistent with theluminosities of the EXOSAT
equilibrium reactions on the temperature of the core. They and Einstein observations; a comparison between their val-
found that in transient NSs with recurrence timescales of ues and passbands indicates that the luminosities of Van
(cid:13)c 0000RAS,MNRAS000,000–000
Time-Variable Emission from Transiently Accreting Neutron Stars 3
Paradijs et al. were more than a factor 3 greater, although surface flux from the crust and core to a “delta-function”
part of this may be due to a different assumed spectrum. accretion profile, it is unimportant that the implied 1-day
Finally,Rutledgeetal.(2001)foundthatCenX-4variedby accretion rates may be super-Eddington, and we do not in-
<18% (0.2-2 keV) during a 10 ksec observation, while the cludeaccretion luminosity in our results.
0.5-10.0 keV luminosity had decreased by 40±8% in com-
parisonwithanASCAobservationtaken5yearspreviously.
Aql X-1 was also found to have a variable flux in qui-
3.1 Microphysics of the Crust and Core
escence (Rutledge et al. 1999) between three observations
with ASCA (Oct 1992, Mar 1993, and Oct 1996). If the The hydrostatic and steady-state thermal models of the
same intrinsic spectrum is assumed, the three observations crust used in this paper are substantially identical to the
are inconsistent with having the same intensity with high ones used by Ushomirsky, Cutler, & Bildsten (2000). We
confidence (probability=10−6). The difference in observed briefly recount the major ingredients of these models, and
fluxis a factor of ∼2-3. then describe our time-evolution code.
4U 2129+37 decreased in flux by a factor of 3.4±0.6 Sincethethermaltimescaleforthecrust(daystoyears,
between Nov-Dec 1992 (ROSAT/HRI) and March 1994 see below) is so much longer than the sound crossing time
(ROSAT/PSPC) (Garcia & Callanan 1999; Rutledge (milliseconds), and the equation of state is nearly indepen-
et al. 2000) in the unabsorbed luminosity, for the same as- dentoftemperature,thehydrostaticstructureofthecrustis
sumed spectrum. effectively decoupled from its thermal evolution. Therefore,
The observations of Cen X-4, with ROSAT/HRI, and onecansolvethethermalevolutionequationsassumingthat
of Aql X-1, with ASCA are the only two instances of re- thedensityandpressureasafunctionoftheposition inthe
peatedobservationswiththesameinstrumentation.CenX- stararefixed.This approach tomodeling thethermalstate
4 hasnot been observed in outburst since May,1979 (Chen of a neutron star is fully described by Brown (2000) and is
et al. 1997), while Aql X-1 goes into outburst every ∼220 followed here.
days,and itthereforeseems likelythattherewereinterven- The composition of the crust (i.e., the mass A and
ing outburstsbetween thethree ASCAobservations. charge Z of the nuclei, as well as the neutron fraction X )
n
Thus, observations of transiently accreting NSs in qui- is taken from the tabulation of Haensel & Zdunik (1990b;
escence, taken at face value, indicate that their quiescent 1990a). The pressure is the sum of contributions from de-
luminosities vary by a factor of up to 3−5 on timescales generate, relativistic electrons and free neutrons (Negele &
of days to years. However, it is not clear what fraction of Vautherin1973) at densities exceeding neutron drip (ρnd ≈
this variability is intrinsic, and what can be attributed to (4−6)×1011 g cm−3, Haensel & Zdunik 1990a). With the
systematic differences in instrumentation. equationofstateasdescribedabove,wesolvetheNewtonian
equations of hydrostatic balance (which, for a thin crust,
are exactly equivalent to the fully relativistic equations).
3 THERMAL EMISSION FROM THE CRUST We take M = 1.4M⊙, and R = 10 km. Our crust has a
thicknessof 1.1 km and a mass of 0.06M⊙.
We now describe our simulations of the thermal relaxation With the hydrostatic structure specified, we solve the
ofthecrustfollowing an accretion outburst,andtheresult- heat equation,
ing time dependence of the quiescent thermal emission. As
∂T 1 ∂ ∂T
argued byBBR98 andconfirmedbyColpi etal. (2001), the ρcv ∂t = r2∂r r2K∂r +ρ(ǫnuc−ǫν), (1)
nuclear energy release in the crust heats the NS core to a (cid:16) (cid:17)
temperaturecorrespondingtosteadyaccretionatthecorre- where c is the heat capacity per gram (which has contri-
v
sponding time-averaged accretion rate. We performed sev- butionsfromtheioniclattice,degenerateelectrons,andfree
eralsimulationsstartingwithaverycoldNSandsubjecting neutrons in the inner crust), ǫnuc is the nuclear energy re-
it to a series of accretion outbursts and found that indeed, lease, ǫ is the neutrino emissivity, and K is the thermal
ν
thecoreisheatedtotheappropriatetemperatureona104yr conductivity.Weneglecttheoveralldownwardmotionofthe
timescale, in agreement with Colpi et al. (2001). For the material duetoaccretion, aswell as theheat release dueto
remainder of our analysis we first construct a steady-state compression (these effects are not important in the crust).
thermalmodelcorrespondingtothetime-averagedhM˙i,and Therefore, the only energy source in the crust is the heat
then subject it to outbursts. After several outbursts, the release due to non-equilibrium electron captures, neutron
modelreachesalimit cycle. Inthispaper,wereport there- emissions, and pycnonuclear reactions. The detailed treat-
sults of only these latter simulations. Our accretion events mentoftheenergyreleaseisdescribed inUshomirskyetal.
(“outbursts”) are 1-day long “delta-functions”, with no ac- (2000); wenoteherethatǫnuc isproportionaltotheinstan-
cretion outside of these events. More realistic time-evolved taneousmassaccretionrateM˙.Thetotalenergyreleasedin
light-curves may in principle be found by convolving our the crust is taken to be Qnuc = 4πr2ρǫnucdr =1.45 MeV
“delta-function” thermal response light curves with a more peraccretedbaryon(Haensel&ZRdunik1990b).Atthetem-
realistic outburst accretion time-dependent profile for the peratures of interest, neutrino losses in the crust are negli-
small perturbation light curves (δL/L<1); lightcurves in gible.
∼
which the variations are larger would require a more de- Instead of fully modeling the thermal evolution of the
tailedcalculation.Wesimulatetwodifferentoutburstrecur- NS core, we presume that it is isothermal, and is char-
rence timescales (1 yr and 30 yr) and three time-averaged acterised by a single temperature Tcore which is equal to
accretionrates(10−10,10−11,and10−12 M⊙/yr).Asweare the temperature of the crust at the interface with the core,
primarily interested in illustrating the response of the NS T(rcore).The core temperatureobeys
(cid:13)c 0000RAS,MNRAS000,000–000
4 Greg Ushomirsky and Robert E. Rutledge
CcoredT(drtcore) =4πrc2oreK ∂∂Tr(cid:12)rcore −Lν, (2) lvoawrisabuislittyoosfuqruveieysctehnet tehnetrirmearlaenmgeissoiofnp.oTsshibisilaitpiepsrofaocrhtahle-
(cid:12) †
where Ccore is the total heat c(cid:12)apacity, and Lν is the total crustal conductivity.
neutrino emissivity of the core (see below). This approxi- Atthetop ofthecrust (ρ≈108 gcm−3) weutilize the
mate treatment is justified because of the core’s large ther- results of Potekhin et al. (1997) for the relation between
‡
mal conductivity and heat content. In effect, the tempera- the core temperature to the surface temperature. They
ture of the core changes by a negligible amount compared extended the original calculations of Gudmunsson et al.
tothetemperaturevariationsinthecrust,andthecoreacts (1983) by considering the opacities of accreted envelopes,
as an energy sink at the bottom edge of the crust. With rather than iron envelopes. Because of the smaller opac-
this approximation, we do not need to integrate the heat ity of a light-element envelope, a ∼50% smaller core tem-
equationinthecore,andinstead useEq.(2)asaboundary perature is necessary to carry a given flux (see also Bland-
condition for Eq. (1) at thebottom edge of the crust. ford et al. 1983). The thermal time at the top of the crust,
To survey the parameter space, we use two different wheretheouterboundarycondtionisapplied,is<1dayin
∼
prescriptions for core cooling. “Standard” cooling models our models, so our simulations cannot follow variability on
use modified Urca (Friman & Maxwell 1979; Yakovlev & shorter timescales.
Levenfish1995)ande-eneutrinobremsstrahlung(Kaminker
& Haensel 1999), while “rapid” cooling models assume the
presenceofapioncondensate(Shapiro&Teukolsky1983)⋆. 3.2 Time-Averaged Quiescent Luminsity
Nucleonsuperfluiditysignificantlyreducesneutrinoemissiv-
Before describing the results of our time evolution simula-
ity in the core (see Yakovlev et al. 1999 for an in-depth re-
tions, we discuss the thermal luminosity of neutron stars
view). In our calculation we use superfluid parameters col- assuming steady accretion at the rate hM˙i corresponding
latedbyBrown(2000).Maxwell(1979) arguedthatattem-
to the time average over the outburst recurrence interval
peratures well below the superfluid transition temperature
τrec. This provides a reasonable estimate of the average lu-
T , this suppression is exponential. However, detailed cal-
c minosity level of neutron star transients. The variations of
culations, summarized by Yakovlev et al. (1999) show that
the luminosity discussed in § 3.3 are excursions about this
this suppression is weaker by a power-law factor in T/T ,
c averagelevel.Asoutlinedin§3.1, wesurveytheparameter
leading to lower core temperatures than one would expect
space by considering two different strengths of core neu-
based on the simple exponential reduction in emissivity.
trino emission (standard and enhanced), and two different
Coreheatcapacityisthesumofcontributionsfromelec-
assumptions regarding theconductivity of thecrust.
trons, neutrons, and protons. However, superfluidity alters
TheaveragecoretemperatureandluminosityoftheNS
the heat capacity of the nucleons (see Yakovlev et al. 1999
are set by the balance between the heat input due to non-
for an in-depth review). At T ≪ Tc, the reduction of the equilibriumreactionsinthecrust(attherateQnuchM˙i/mb)
heatcapacityissimilartothesuppressionofneutrinoemis-
and heat loss due to neutrino emission from the core (L )
sivity.Maxwell (1979) gives a fittingformula for 1S0 super- andphotonluminosityfromthesurface(L ).Atthelowcoνre
γ
flcouried),(awphpilliecaLbelveentofisnheu&trYonaksoivnleNvS(1cr9u9s4t)agnivdepfirtottionngsfoinrmthue- temperaturescharacteristicofNStransients(Tcore ∼<108 K)
modified Urca neutrino emission (standard cooling case) is
lae for the 3P2 superfluid as well (applicable to neutrons substantially suppressed by nucleon superfluidity and can-
in the core). At temperatures of interest (∼ 107 −108 K)
notcompetewithphotonlossesfromthesurfaceofthestar,
electrons dominate theheat capacity.
L ≪ L . Therefore, most of the heat deposited by the
ν γ
Throughout the crust, heat is transported by degener-
nonequilibrium reactions is radiated from the surface, i.e.,
ate,relativisticelectrons,andthethermalconductivityK is
thequiescent luminosity is just (BBR98)
verysensitivetothepurityofthecrust.Ifthecrustisapure
crystal,electron-phononscattering(Baiko&Yakovlev1995) L ≈ QnuchM˙i (3)
sets the conductivity. However, crusts of accreting neutron q mb
stars are unlikely to be pure crystals, since they are com- = 8.7×1033 hM˙i Qnuc/mb erg s−1.
posed of (likely quite impure) products of nuclear burning (cid:18)10−10 M⊙ yr−1(cid:19)(cid:18)1.45 MeV(cid:19)
in the upper atmosphere. In fact, calculations of Schatz
et al. (1999) suggest that the typical values of the im- This estimate is independent of the conductivity of the
purity parameter Qimp = Yimp(Zimp −hZi)2, where Zimp crust. The core temperature, however, depends on the
and Yimp are the charge and the fraction of the impurities, crustal conductivity. If the crustal conductivity is large
could be comparable to the average charge hZi2. Follow- (i.e., set by electron-phonon scattering), the crust is nearly
ing Brown (2000), we set the absolute lower bound on the
conductivity by using electron-ion scattering (Yakovlev &
† Thepresenceofimpuritiesdoesnotdramaticallyaffecttheheat
Urpin 1980), which has the same form as electron-impurity
capacity of the ioniclattice. Whilethe phonon spectrum willbe
scattering with Qimp =Z2. The much smaller conductivity affected by the presence of impurities, the heat capacity should
in the case of electron-impurity scattering with large Qimp stillapproachtheDebyelimit.
leadstomuchlongercrustalthermaltime,andmuchsmaller ‡ Potekhinetal.(1997)expressthesurfacetemperatureinterms
of the temperature in the isothermal region of the crust, which
they assume to be at ρ > 1010 g cm−3. However, since the au-
⋆ Instead,onecouldassumethattheNSmassishighenoughto thors assume that the crust has high thermal conductivity, the
allowdirectUrcaneutrinoemission(see,e.g.,Colpietal.(2001)), temperature difference between 1010 g cm−3 and 108 g cm−3 is
astheemissivitiesaresimilar. negligibleintheirmodels.
(cid:13)c 0000RAS,MNRAS000,000–000
Time-Variable Emission from Transiently Accreting Neutron Stars 5
Figure1. Timeevolutionofthetemperatureofthecrustforamodelwithstandardcoolingandlowthermalconductivitycrustfollowing
a1-day outburst and hM˙i=10−10M⊙ yr−1,with arecurrence timeof 30 years. The x axis is the density, covering the entire crustof
theneutronstar;y axisisthetime,andz axisisthetemperature.
isothermal, and the core temperature, Tcore ≈ 1.2 × trino emission makes Lγ comparable to Lν, a good fit
108 K (hM˙i/10−10M⊙ yr−1)0.41, can be obtained by sim- to the results of our calculations for the particular case
ply using the relation of Potekhin et al. (1997). If thecrust of enhanced neutrino emission from a pion condensate is
has low thermal conductivity, then a substantial tempera- Tcore ≈2.7×107 K (hM˙i/10−10M⊙ yr−1)0.09, regardless of
ture gradient is needed to carry the heat from the inner theconductivityofthecrust.Thethermalluminosity,how-
crust to the surface, and the core temperature is Tcore ≈ ever,dependsontherelation between thecoretemperature
2.2×108 K (hM˙i/10−10M⊙ yr−1)0.45 (obtained by fitting and that at the top of the crust. When the conductivity is
our detailed calculations). determined by electron-phonon scattering (i.e., K is large),
On the other hand, if rapid cooling processes are al- the crust is nearly isothermal, and, using the relation of
lowed in the core, then, despite the suppression of neu- Potekhin et al. (1997), we find
tbryinnooenm-eiqssuiiolnibrbiyumsupreearcfltuioidnisty,ismroasdtiaotfedtheawhaeyatbdyepnoesuitterid- L ≈2.2×1032 hM˙i 0.22 Qnuc/mb 0.22 erg s−1,(4)
nos, L ≫ L . The core temperature is then set by bal- q (cid:18)10−10 M⊙ yr−1(cid:19) (cid:18)1.45 MeV(cid:19)
ν γ
ancing the energy input from the crustal reactions with which agrees well with our detailed calculations. On the
neutrino-luminosity. Except at very low accretion rates (∼< other hand, if the crustal conductivity is low, the relation
5×10−12M⊙ yr−1), where superfluid suppression of neu- between the core and the surface temperature is nontrivial,
(cid:13)c 0000RAS,MNRAS000,000–000
6 Greg Ushomirsky and Robert E. Rutledge
since an appreciabletemperature gradient between neutron thestar.Moreover,whilethetemperaturepeaksinthecrust
drip and the core is needed to carry the flux from the non- due to individual reactions are well-separated at the early
equilibrium reactions into the core. In this case, the crust times,theybecomeblendedtogetherbythetimetheyreach
is warmer than in the electron-phonon scattering case, and thesurface.
thethermal luminosity is Figure 1 clearly suggests that, in order to produce ap-
preciable variability in the thermal emission from the sur-
L ≈7×1032 hM˙i 0.8 Qnuc/mb 0.8 erg s−1(5) face,theenergydepositionbythenon-equilibriumreactions
q (cid:18)10−10 M⊙ yr−1(cid:19) (cid:18)1.45 MeV(cid:19) should be sufficient to heat the crust locally by δT ∼ T.
The typical outburst fluences indicate that only ∆M <
(whichweobtainedbyfittingtheresultsofourdetailedcal- ∼
1023 grams of material is accreted duringan outburst,and
culations).Comparingequation(3)with(4)and(5),wesee
hence < 1041 erg of energy is deposited in the crust. This
that when enhanced neutrino cooling processes are allowed ∼
amount is rather small, which allows us to place some in-
in the NS core, only less than 10% of the non-equilibrium
teresting constraints on the heat capacity of NS crust. The
energyreleaseisradiatedfromthesurface,whiletheremain-
thermal time at the depths where this energy is deposited
ing>90%oftheheatisemittedinneutrinos.Regardlessof
is at least comparable to and usually much greater than
the crustal conductivity, the core temperature in the rapid
a typicaloutburst duration. Thus,during theoutburst, the
cooling case is set entirely by the core neutrino emission.
heatisnotconductedaway,butprimarilyheatsthecrustlo-
However,dependingontheconductivityofthecrust,atem-
cally.Neglectingheatdiffusion,thenuclearenergydeposited
perature gradient between the core and the surface may or
in the region near neutron drip will heat this region by an
may not be present, resulting in different surface tempera-
amount
tures and photon luminosities (cf. Eqs. (4) and (5)). Thus,
neutron stars with rapid cooling processes active in the core −1 −1
C p
appear dimmer than those with just standard cooling, and, δT ∼104 K d (6)
(cid:18)kB/baryon(cid:19) (cid:18)1030 erg cm−3(cid:19)
whentheir cores arevery colddueto rapidneutrino cooling,
NSs with high conductivity crusts will appear dimmer than × Qnuc ∆M
those with lower conductivity crusts. (cid:16)1 MeV(cid:17)(cid:18)1023 g(cid:19)
whereC istheheatcapactiyinunitsofk (Boltzmanncon-
B
stant)perbaryonandp isthepressureatwhichtheenergy
3.3 Time-Variable Emission d
is deposited. As outlined in § 3.1, the typical crustal tem-
Aswenowdescribe, theapproximatesteady-stateluminos- peratureis∼107−108 K.Therefore,iftheheatcapacityof
ityisonlyapartofthestory.Dependingonthemicrophysics the neutron star crust is ∼ k per baryon, the crust relax-
B
of thecrust and core, thetime-dependentthermalluminos- ation luminosity is only a small perturbation on theoverall
ityoftheneutronstarmayeitheralwaysbeveryclosetothe cooling of thecore. Inorder toproduceobservablevariabil-
steady-state estimate (δL/L < 1%), or vary wildly around ity of quiescent thermal emission, the NS crust must have
it (δL/L>1). C ≪k perbaryon.
∼ B
InFigure1wedisplaythetimeevolutionofthetemper- Figure2showscontributionstotheheatcapacityinthe
atureintheNScrustforamodelwithlowthermalconduc- crust, for a representative case of hM˙i = 10−11M⊙ yr−1.
tivity, standard neutrino emission, hM˙i = 10−10M⊙ yr−1, In the outer crust, the heat capacity of the ionic latice,
andτrec=30yrs.Othermodelsaresimilarqualitatively,but, C ∼< 10−2kB per baryon (dotted line, Shapiro & Teukol-
of course, differ in the magnitude of variability. The slices sky1983;vanRiper1991;Chong&Cheng1994),dominates.
of the figure in the x−z plane represent the temperature Stronglydegenerate,relativisticelectrons(dashedline)con-
in the crust as a function of density, and the y axis is the tributeanegligibleamounttotheheatcapacityeverywhere
time (in days) since the beginning of a 1-day outburst. At exceptinthedeepinnercrust(ρ>2×1013 gcm3).Atden-
∼
the early times, the temperature rises locally at the densi- sities above neutron drip, free neutrons, if not superfluid,
tieswhere theenergy is deposited.It reachesthemaximum would have appreciable heat capacity, approaching k per
B
after 1 day (i.e., at the end of the outburst). In the en- baryon(dot-dashedline). Thus,iftheneutronsintheinner
suing cooling period, the heights of the temperature peaks crust are not superfluid,their heat capacity is so large that
decrease, and their widths increase due to heat diffusion. theenergyreleaseof<1041 ergcharacteristicoftheheating
∼
After ∼ 6 days the “heat wave” due to the very first reac- duringanoutburstisnotsufficienttoheatthecrustsubstan-
tion, at ρ= 1.5×109 g cm−3 reaches the top of the crust. tially.However,itiscommonlybelievedthattheneutronsin
This temperature increase at the top of the crust is then theinnercrustforma1S0superfluid,andtheirheatcapacity
directlytranslatedintotheincreaseinthethermalluminos- isgreatlyreduced(Maxwell1979;Yakovlevetal.1999).Su-
ity of the NS (Potekhin et al. 1997). At much later time perfluiditycompletelysuppressesfreeneutronheatcapacity
(∼ 1000 days after the outburst) the heat wave due to the atdensities>8×1011 gcm3 (solidlineinFigure2).Inthis
reactions intheinnercrust,at ρ>1012 gcm−3 reachesthe case,theionsdominatetheheatcapacityintheinnercrust,
∼
surface. Even though the energy release in these reactions and C <10−2k per baryon. Therefore, observing the evo-
∼ B
is much larger, the change in the outer boundary tempera- lutionofquiescentluminosityprovidesatestofsuperfluidity
ture and, hence, the variation of the thermal luminosity is ofneutrons:iflate-timeevolutionisobserved(δT/T >10−3),
muchsmaller,becausethetemperaturepeakspreadsoutto thentheneutronsmustbesuperfluid.Whilethismaynotbe
a width corresponding to roughly the thermal time at the a controversial conclusion, it is worth noting that the only
depositiondepthasittravelstowardsthesurfaceofthestar, existing observational evidence for superfluidity of crustal
and because some of the heat is conducted into the core of neutron gas is the interpretation of pulsar glitches as be-
(cid:13)c 0000RAS,MNRAS000,000–000
Time-Variable Emission from Transiently Accreting Neutron Stars 7
100
104
103
n) 10−1
o
y
r 102
a
))
b ss
/ yy
kB dada 101
(
y 10−2 ( (
t ee
ci mm 100
a ii
p t t
a alal
c mm

t
a rr
He 10−3 hehe 102
TT
101
10−4
109 1010 1011 1012 1013 1014
Density (g cm−3)
100
10 9 10 10 10 11 10 12 10 13 10 14
Figure 2. Heat capacity of neutron star matter as function of
depth, forhM˙i=10−11 M⊙ yr−1,inunitsofkB perbaryon(in Density (g cm−3)
these units,theheat capacity ofidealmonoatomic gas would be
C = 3/2A). Dashed line: degenerate relativistic electrons, dot-
Figure3. Thermaldiffusiontimetothesurface(solidlines)and
ted line: ionic lattice contribution; solid line: degenerate, non-
tothecore(dottedlines)asfunctionofdensityfor(readinglines
relativistic,superfluidneutrons;dash-dottedline:degeneratenon-
relativistic normal neutrons. Note that the heat capacity of de- toptobottom) hM˙i=10−10,10−11,and10−12 M⊙ yr−1.The
toppanel corresponds to modelswithstandardcorecooling and
generate neutrons firstrises,and then drops dramaticallyatthe
onsetofsuperfluidity,atρ≈7×1011gcm−3.Theheat-capacities low crustal thermal conductivity, while the bottom panel is for
athM˙i=10−10 and10−12 M⊙ yr−1 arequalitativelysimilar. modelswithstandardcorecoolingandhighcrustalthermalcon-
ductivity. Thefilledbars inthe top panel indicate schematically
the regions of nuclear energy deposition. The height of the bars
is proportional to the deposited energy, with the highest bar at
ing initiated in the crust. Variability of quiescent thermal ρ≈1.5×1012 g cm−3 corresponding to 0.47 MeV per accreted
emission from NS transients, if disentangled from accretion baryondeposited.
in quiescence or other sources of variability outlined in § 1,
would be a completely separate direct observational confir-
§
mation of thisview. ity. Black verticalbarsin thetoppanelshowthelocations
The spreading and blending of the temperature peaks of the energy deposition due to non-equilibrium reactions,
due to different reactions, clearly evident in Figure 1, as with the height of the bars proportional to the energy de-
well as therelative magnitudeof heat diffusion towards the posited. Clearly, the energy depositions in the inner crust
surface and the core can be easily understood in terms of happenat verysimilar depths,suchthatthedifferencesbe-
the thermal time at the place where the heat is deposited. tween the thermal times of the reactions are much smaller
Tomakethediscussion moreprecise, wedefinethethermal than the thermal times themselves. By the time the heat
diffusion timescale (Henyey& L’Ecuyer 1969; BBR98) from these reactions diffuses to the surface, the differences
between them are blended. It is also clear from this figure
r2 ρc 1/2 2 thatifthethermaldiffusiontimetothecoreissmallerthan
τth =(cid:20)Zr1 (cid:16) Kv(cid:17) dr(cid:21) . (7) that to the surface, then most of the heat will flow to the
core, rather than to thesurface.
Roughly speaking, τth is the time it takes a heat impulse In Figures 4 and 5 we survey the dependence of the
to diffuse from r1 to r2. In Figure 3, we plot the thermal transientemissionfollowinganaccretionoutburstonvarious
diffusion timescales as a function of density in the crust to model parameters. Figure 4 shows results for a crust with
the NS surface (solid line) and the core (dotted lines). The
toppanelisforthemodelwithelectron-impurityscattering
conductivityinthelimitofveryimpurecrust,whilethebot- § Thethermaltimesareshorterfortheenhancedcoolingmodels
tompanelisforthemodelwithelectron-phononconductiv- (notshownhere)owingtothelowercrustaltemperatures.
(cid:13)c 0000RAS,MNRAS000,000–000
8 Greg Ushomirsky and Robert E. Rutledge
lowthermalconductivity(electron-impurityscatteringwith cence from systems with longer recurrence timescales (>10
∼
Qimp = Z2), while Figure 5 shows results for high thermal yr).
conductivity(electron-phononscattering).Thesefiguresare
plottedwithluminositynormalizedbythethermalluminos-
ityjustpriortotheoutburst,whichisthelowestluminosity
4 COMPARISON WITH OBSERVATIONS
of the outburst cycle. This tendsto exaggerate visually the
range of luminosities. It also does not display the absolute Is the phenomenon we describe here observed in transient
differences in the luminosities due to different hM˙i, which NSs in quiescence? If so, both the time-averaged luminos-
are discussed in § 3.2. However, it does make clear the rel- ity level, and the variability around it need to be self-
evant factors of variability which can be produced over a consistently accounted for. This constraint provides an ob-
recurrencecycleduetothismechanism,andthatisourgoal servationaldiscriminant forthevariouscorecooling scenar-
here. ios in NS transients.
Asisclearfrom thefigures,thevariability isthesmall- The observed magnitude of variability (factors of ∼
est (< 1%) for models with standard cooling and short (1 3− 5) cannot be achieved in models with standard neu-
∼
year, in our case) outburst recurrence times. The variabil- trino cooling, which vary by less than 50% in all cases we
ity is largest for models with rapid core cooling and long investigate here. On the other hand, rapid neutrinocooling
(30 year) outburst recurrence time. Under the same condi- modelsarecapableofproducingintensityvariabilitybyfac-
tions, the variability of the models with high crustal ther- tors of up to 20 on timescales of months-years; this may be
malconductivity(Figure5)isalwayslargerthanthatofthe invokedfortheobservedvariabilityof4U2129+37(afactor
lowconductivitymodels(Figure4).Allofthesefeaturesare of 3.4 overmonths),and thelonger time-scale variability of
straightforward to understand in light of the above discus- Cen X−4(factor of 2-5over6years, an additionalfactor of
sion. Since the energy deposited into the crust is not large 3 over 10 years). The factor of 2-3 variability in Aql X−1
(< 1041 erg), the colder the crust, the larger the transient over months-years can be similarly explained, again requir-
∼
thermal response. Models with rapid core cooling and high ing rapid cooling to be operative in its core. However, to
thermal conductivity have systematically lower crust tem- explain the factor of ∼3 variation over a period of 4-8 days
peratures (see § 3.2), and hence have larger amplitude of in the quiescent X-ray flux of Cen X−4 observed by Cam-
variability. pana et al. (1997), one must invoke an unobserved, short
(< 1 day) outburst, preceeding those observations by only
Asdiscussed above,thefirst humpin thelight curveis ∼
∼ few days, which seems unlikely, although it cannot be
due to reactions in the outer crust, while the second hump
excluded dueto thelack of X-ray coverage.
is due to those in the inner crust. This is a generic feature
However, as argued by BBR98, the inferred quiescent
of all models except for the low-K, short-recurrence ones
luminosity of Aql X−1 can be naturally understood if the
(left column of Figure 4). As is evident from the top panel
neutrino losses from its core are negligible, i.e. if only
of Figure 3, thethermal diffusion time from theinnercrust
standard cooling processes operate (cf. our Eq. [3]). Time-
tothesurfaceis>103 days,i.e., longerthantherecurrence
∼ averaged L of Aql X−1 is inconsistent with rapid cooling,
q
time.Thus,thesecondpeakinthelightcurvedoesnotmake
regardless of the assumed crustal conductivity (cf. Eq. [4]
it to the NS surface before thenext outburst, and only one
and [5]). This is also the case for 4U 1608-52. For these
peakisobservedinthequiescencelightcurve.Thismismatch
sources, variability in quiescence must be due to processes
betweenthethermaldiffusiontimescaleandtheoutburstre-
otherthanthermalrelaxation ofthecrust(outlinedin§1).
currencetimescaleexplainswhyonlyonelight-curvepeakis
OftheobservedrecurrentNStransients,onlyCenX−4
observedintheτrec=1yr,lowthermalconductivitysimula- has time-averaged L much lower than given by Eq. (3).
q
tions (Figures 4a & b) while two peaks are observed in all
BBR98 conjectured that this discrepancy is due to the fact
other simulations. For long τrec sources, the time between that only Qnuc ∼ 0.1 MeV, rather than ∼ 1 MeV is de-
the outburst and the second peak in the lightcurve is an ¶
posited in its crust, while Colpi et al. (2001) argued that
excellent observational indicator of the thermal time in the
innercrust. Qnuc ∼ 1 MeV, but rapid neutrino cooling is operating in
CenX−4.Clearly,analysisofvariability ofCenX−4inqui-
As we see in Figure 3, the thermal time of the inner escence(orothertransientswithlongrecurrencetimes)pro-
crust in low-K models is ≈103−104 days, while it is only videsadiscriminantbetweenthevariouscorecoolingmech-
∼ 102 days for the high-K models. As a result, for low-K anisms. If enhanced neutrino cooling is responsible for the
models with τrec=1 yr, the crust does not completely cool relatevelylowLq ofCenX−4,thenitsquiescentluminosity
down,andthereforeremainsataroughlyconstanttempera- must vary in a well-defined way (see panels d of Figures 4
ture.Ontheotherhand,forτrec=30yr,thecrustdoeshave and 5). In the contrary case of standard cooling with low
enoughtimetocooldown,andhencethevariabilityismuch Qnuc, the baseline thermal luminosity of Cen X−4 should
greater(tensofpercenttofactors offew).Inaddition,it is beconstanttobetterthan50%,andanyvariabilitymustbe
clearfromEq.(6)thatitis∆M =hM˙iτrec,ratherthanhM˙i accountedforbytheprocessesoutsidetheNScrust.Finally,
or τrec individually that is important for theoverall magni- itisdifficulttoreliablypredicttheexpectedquiescentlumi-
tudeof variability.Fora givenhM˙i,models with larger τrec nosity of 4U 2129+47, due to its unusual accretion history
exhibit larger variability, simply because the amount of en-
ergydepositedintothecrustislarger.Thisexplainswhythe
τrec=1 yrlight curves are (relatively) less variable than the ¶ The low quiescent luminosity could also potentially be ex-
τrec=30 yr light curves. In general, one therefore expects a plained by an overestimation of the time-average accretion rate
greatermagnitudeof(relative)intensityvariabilityinquies- ofCenX−4,duetothesmallnumberofoutbursts observed.
(cid:13)c 0000RAS,MNRAS000,000–000
Time-Variable Emission from Transiently Accreting Neutron Stars 9
1.40
a) normal cooling c) normal cooling,
t =1 yr t =30 yrs
1.010 rec rec
1.30
1.008
1.006 1.20
1.004
1.10
1.002
1.000 1.00
100 101 102 100 101 102 103 104
alalalal
nitinitinitiniti
LLLLiiii
//// 5
LLLL
b) enhanced cooling d) enhanced cooling
t =1 yr t =30 yrs
rec rec
1.040
4
1.030
3
1.020
2
1.010
1.000 1
100 101 102 100 101 102 103 104
DDDDaaaayyyyssss SSSSiiiinnnncccceeee OOOOuuuuttttbbbbuuuurrrrsssstttt
Figure4. ThermalemissionluminosityfromtheNSphotosphere,duetocrustalheatingandNScoretemperature(neglectingaccretion
luminosity) as a function of time since a one-day-long “outburst” – for crustal conductivity due to electron-impurity scattering. The
luminosity is normalized by its value justprior to the outburst (i.e., by the lowest value over the recurrence time). Panels a and c are
for models with modified Urca (“standard”) cooling, while panels b and d are for models with accelerated cooling (in this case, pion
condensate). Theassumedoutburstrecurrencetimeisindicatedineachpanel.Thetimeaverageaccretionratesusedare:hM˙i=10−10
(solidline),10−11 (dotted line),10−12 (brokenline)M⊙ yr−1.
(Pietsch et al. 1986), and so we draw no additional conclu- andthepersistentemissionfromSAXJ1808.4−3658 (Stella
sions for this source. et al. 2000; Dotani et al. 2000), can potentially be used for
measuring the late-time emission predicted here. However,
If transiently accreting pulsars in quiescence fully ex-
theoretical modeling of the quiescent emission in this case
clude accretion onto the NS surface due to the propeller
is complicated by the need to understand the effects of the
effect, then observations of the thermal pulse during quies-
magneticfield,transverseheatflowinthecrust,andwhether
cence will be due only to crustal and core emission. More-
or not the crust is replaced byaccretion in these systems.
over, optical observations can indicate when the accretion
diskisabsentinthesesources(Negueruelaetal.2000), and
observationswhichfollowthepulsedintensityprofileofsuch
anobject overseveralyearsfollowing anoutburstcan read-
5 CONCLUSIONS
ily constrain the NS atmospheric luminosity. The quiescent
emission of transient pulsars, such as the pulsed emission Wehaveinvestigatedthetime-variableluminosityofatran-
recently detected from A0535+35 (Negueruela et al. 2000) sientlyaccretingNS,followinga“delta-function”-likeaccre-
(cid:13)c 0000RAS,MNRAS000,000–000
10 Greg Ushomirsky and Robert E. Rutledge
1.50
1.014 a) normal cooling c) normal cooling
t =1 yr t =30 yrs
rec rec
1.012 1.40
1.010
1.30
1.008
1.006 1.20
1.004
1.10
1.002
1.000 1.00
10−1 100 101 102 10−1 100 101 102 103 104
alalalal
nitinitinitiniti
LLLLiiii
//// 1.50 20
LLLL
b) enhanced cooling d) enhanced cooling
t =1 yr t =30 yrs
rec rec
1.40
15
1.30
10
1.20
1.10 5
1.00
10−1 100 101 102 10−1 100 101 102 103 104
DDDDaaaayyyyssss SSSSiiiinnnncccceeee OOOOuuuuttttbbbbuuuurrrrsssstttt
Figure 5. SameasFigure4,except theassumedcrustalconductivity isduetoelectron-phonon scattering.
tionoutburst(onedayinduration),forthreedifferentvalues conductivityandhM˙i.Forrapidcorecooling,themagnitude
ofhM˙i,forNScoreswithbothstandardandrapidcorecool- oftheluminosityswings variesbetween∼fewpercent,toas
ing (specifically, for ν emissivity from modified Urca and much as a factor of 20, and is largest for long τrec and high
from pion condensate), and for high crustal thermal con- crustal thermal conductivity.
ductivity dominated by electron-phonon scattering (for the Itispossibletoobserveeitheronepeakinthelightcurve
case of a pure crystal crust) or for low conductivity due (due to the blended emission from the several reactions in
to electron-impurity scattering with large impurity fraction theoutercrust) ortwo peaks (thesecond is dueto blended
(forthemorelikelycaseofveryimpurecrustexpectedinan emission from reactions in the inner crust). Observing the
accreting NS). second peak requires a recurrence time long enough to per-
Themagnitudeandtime-dependenceoflate-timeemis- mit the inner-crust emission to reach the surface. For low
sion due to heat deposited by non-equilibrium reactions in crustal thermal conductivity, this thermal time is ∼ 10 yr,
the crust depends intimately on the microphysics active in sothesecondpeakisonlyobservedintransientswithlonger
thecrustandthecore,thetime-averageaccretionrate(and recurrence times. In the electron-phonon scattering domi-
thusthetime-averagequiescentluminosityitself)and–most nated crust, with thermal time of < 1 yr, the second peak
∼
importantly – on the recurrence timescale of the transient may be observed in short τrec sources. In general, only the
system.Inthecaseofstandardcorecooling, themagnitude energydepositedatdepthswherethethermaltimeissmaller
of the luminosity swings (peak-to-peak) is ∼<1% for τrec=1 than the outburst recurrence interval will produce observ-
yr,andupto15-40%forτrec=30yr,independentofthecrust able variability in the quiescent thermal emission.
(cid:13)c 0000RAS,MNRAS000,000–000