Ways to constrain neutron star equation of state models using relativistic disc lines
aa r X i v : . [ a s t r o - ph . H E ] S e p Mon. Not. R. Astron. Soc. , 1– ?? (2009) Printed 8 November 2018 (MN L A TEX style file v2.2)
Ways to constrain neutron star equation of state modelsusing relativistic disc lines
Sudip Bhattacharyya ⋆ Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai 400005, India
ABSTRACT
Relativistic spectral lines from the accretion disc of a neutron star low-mass X-raybinary can be modelled to infer the disc inner edge radius. A small value of thisradius tentatively implies that the disc terminates either at the neutron star hardsurface, or at the innermost stable circular orbit (ISCO). Therefore an inferred discinner edge radius either provides the stellar radius, or can directly constrain stellarequation of state (EoS) models using the theoretically computed ISCO radius forthe spacetime of a rapidly spinning neutron star. However, this procedure requiresnumerical computation of stellar and ISCO radii for various EoS models and neutronstar configurations using an appropriate rapidly spinning stellar spacetime. We havefully general relativistically calculated about 16000 stable neutron star structures toexplore and establish the above mentioned procedure, and to show that the Kerrspacetime is inadequate for this purpose. Our work systematically studies the methodsto constrain EoS models using relativistic disc lines, and will motivate future X-rayastronomy instruments.
Key words: accretion, accretion discs — equation of state — methods: numerical— relativity — stars: neutron — X-rays: binaries
The core density of a neutron star is typically 5 − ∼ K) can-not be probed by heavy-nuclei collision experiments or withobservations of the early universe (Lattimer and Prakash(2007) and references therein). Plausibly the only way toprobe this degenerate matter is to constrain the theoreti-cally proposed equation of state (EoS) models of neutronstar cores (Shapiro and Teukolsky 1983). For a given EoSmodel and an assumed central density, the stable struc-ture of a non-spinning neutron star can be computed bysolving the Tolman-Oppenheimer-Volkoff (TOV) equation(Shapiro and Teukolsky 1983). These stable stars trace asingle curve in the mass ( M ) − equatorial-radius ( R ) space.Therefore, measurement of mass and radius of the same non-spinning neutron star would constrain the EoS models. If theneutron star spins, then the EoS models can be constrainedby the reliable measurements of three independent parame-ters, such as mass, radius and spin-frequency ( ν spin ) of the same neutron star (e.g., Fig. 1 of Bhattacharyya (2010)).This is extremely difficult because of a number of unknownsystematics. Low-mass X-ray binaries (LMXBs) can be par- ⋆ E-mail: [email protected] ticularly promising systems for such measurements, becauseseveral complementary methods are available for them. Arecent review by Bhattacharyya (2010) describes how thesimultaneous application of these methods can reduce thesystematic uncertainties.One such method involves broad relativistic spectrallines from the inner portion of the accretion disc. Thestrongest among these fluorescent spectral emission linesis the one for the n = 2 → n = 1 transition of theiron atom (or ion), and is observed from many accretingsupermassive and stellar-mass black hole systems (seeReynolds and Nowak (2003); Miller (2007); and referencestherein). Recently, Bhattacharyya and Strohmayer (2007)has, for the first time, established that the broad ironlines from neutron star LMXBs also originate from theinner accretion discs. This discovery was soon confirmed byCackett et al. (2008) using data from a different satellite.After these initial reports, the inner disc origin of broadiron line has been confirmed for several other neutronstar LMXBs (e.g., Pandel et al. (2008); D’A´ı et al.(2009); Cackett et al. (2009); Papitto et al. (2009);Reis et al. (2009); di Salvo et al. (2009); Iaria et al.(2009); Cackett et al. (2010)).The broad iron line is affected by the strong gravityand spin of the compact star (neutron star or black hole).Before discussing how this relativistic line can be used to c (cid:13) Bhattacharyya S. constrain the neutron star parameters, let us see how itinfers the black hole spin. The line shape, especially theextent of the red wing, carries the signature of the discinner edge radius in the unit of stellar mass ( r in c /GM ).This is because the red wing is primarily affected by lon-gitudinal Doppler effect due to the oribital motion of thedisc matter, which broadens the line, and gravitational red-shift, which shifts the line towards lower energies. Since,for a black hole the disc can extend up to the innermoststable circular orbit (ISCO), the black hole spin parame-ter j = Jc/GM ( J : total angular momentum; M : mass),which determines the ISCO location, can be estimated byfitting the line shape with an appropriate relativistic modelfor Kerr spacetime (Laor 1991; Beckwith and Done 2004;Dovˇciak et al. 2004; Miller 2007).A neutron star system is usually more complex thana black hole syetem because of the following reasons. (1)While a spinning black hole is defined by only two parame-ters (mass and spin), and the spacetime around it is the Kerrspacetime having analytical expressions, the structures andspacetimes of rapidly spinning neutron stars, usually har-boured by LMXBs, may have to be numerically calculatedfrom at least two parameters apart from the EoS model(Cook et al. 1994). (2) Unlike a black hole, a neutron starhas a hard surface which can have observable effects. There-fore, for a neutron star, the disc may be terminated eitherby ISCO or by the stellar surface (Thampan et al. 1999).An example of the competition between these two effectshas been shown in the Fig. 1 of Miller et al. (1998) andFig. 1 of Bhattacharyya et al. (2000), which demonstratethat usually circumferential radius r in of disc inner edgefirst decreases and then increases with the increase of stellarspin for a given EoS model and mass. This complicates themeasurement of Jc/GM and other stellar parameters us-ing the inferred circumferential r in c /GM , if it is not knownwhether the disc terminates at ISCO or at the stellar surface.Therefore, theoretical computations of r in as a function ofstellar parameters for various EoS models are essential. Suchcomputations will also be useful to determine if a measured r in c /GM directly gives the neutron star radius-to-mass ra-tio Rc /GM . Note that the computations of r in and stellarparameters will involve the numerical calculations of rapidlyspinning neutron star structures, and hence Kerr spacetimecannot be used. Moreover, in order to constrain the EoSmodels, directly measurable neutron star parameters (e.g., Rc /GM , M , ν spin ; see Bhattacharyya (2010)) should beexpressed as functions of r in c /GM . Since Jc/GM cannotusually be measured directly, a Jc/GM vs. r in c /GM plotmay not be very useful to constrain the EoS models. In thispaper, such plots (Figs 1, 2 and 3) have been shown forcomparisons with Kerr spacetime and to gain insight ( § r in c /GM relations canbe utilized only if r in c /GM < ∼
6, because a much largervalue of r in c /GM might imply the truncation of the discby other effects (see § r in c /GM <
6, because such values will confirm the effect ofneutron star spin on a corotating disc. Therefore, a crucialquestion is whether observations show that r in c /GM < ∼ r in c /GM values fall intoa small range of 6 −
15 for most of the neutron star LMXBswith established relativistic disc lines. Moreover, although these authors could not measure a value of r in c /GM lessthan 6 (as they used non-spinning, i.e., Schwarzschild space-time), many of their fitted r in c /GM values across thesources pegged at the lower limit 6. This happenned for bothphenomenological and reflection models, and even when theCompton broadening was taken into account (Cackett et al.2010). Such pegged best-fit values strongly suggest that r in c /GM < § § §
4, we describe our method, give the resultsand provide a discussion respectively. Note that, since theaccretion disc is believed to be thin, we have not considered c (cid:13) , 1– ?? onstraining neutron star EoS models nonequatorial orbits in our calculations. We have also notconsidered counterrotating orbits, because that would im-ply r ISCO c /GM >
6, which could be easily confused withtruncations caused by other effects (see § In this section, we briefly describe the procedure to computerapidly spinning neutron star structures, and their equilib-rium sequences. The spacetime around such a star can be de-scribed by the following metric (using c = G = 1; Bardeen(1970); Cook et al. (1994)):d s = − e γ + ρ d t + e α (d r + r d θ ) + e γ − ρ r sin θ (d φ − ω d t ) , (1)where the metric potentials γ , ρ , α , and the angular speed( ω ) of the stellar fluid relative to the local inertial frameare all functions of r and θ . For a given EoS model,and assumed values of stellar central density and polar-radius to equatorial-radius ratio, Einstein’s field equationscan be solved to find out r and θ dependence of γ , ρ , α and ω , as well as to obtain the stable stellar structure(Cook et al. 1994; Datta et al. 1998; Bhattacharyya et al.2000, 2001a,b,c; Bhattacharyya 2002). This equilibrium so-lution can then be used to calculate bulk structure param-eters (e.g., M , R , J ) of the spinning neutron star. Notethat, henceforth, R (= r e e ( γ e − ρ e ) / ) will denote equato-rial circumferential radius of the neutron star (equation B6of Cook et al. (1994)). The equations of motion of a testparticle in the spacetime around such a star are given inThampan and Datta (1998). For example, the radial equa-tion of motion is ˙r ≡ e α + γ + ρ (d r/ d τ ) =˜E − ˜V , where, d τ is the proper time, ˜E is the specific energy and a constantof motion, and ˜V is the effective potential. ˜V is given by˜V = e γ + ρ [1 + l /r e γ − ρ ] + 2 ω ˜E l − ω l , where l is the specificangular momentum, a constant of motion. We determine theradius of ISCO using the condition ˜V ,rr = 0, where a commafollowed by one r represents a first-order partial derivativewith respect to r and so on (Thampan and Datta 1998).Thus we can compute both circumferential radius of ISCO(denoted by r ISCO ) and R , and set the theoretical value of r in to the larger of them (see § ν spin sequences of equilibrium structures keeping ν spin = constant, and changing other parameters. Since ν spin is not an input for computing structures, a number of iter-ations are usually needed to compute the structure for adesired ν spin value. The other sequences (e.g., M sequence, Rc /GM sequence; figures of §
3) have been calculated byinterpolations of ν spin sequences.We have used four representative EoS models of widelyvarying stiffness properties. This ensures sufficient gener-ality of our results. We briefly describe these models be-low. Model A (Sahu et al. 1993): This very stiff EoS modelwith maximum non-spinning mass M max ≈ . M ⊙ is a fieldtheoretical EoS for neutron-rich matter in beta equilibriumbased on the chiral sigma model. Model B (Akmal et al.1998): This stiff EoS model with M max ≈ . M ⊙ is the Ar-gonne v model of two-nucleon interaction, with the three-nucleon interaction (Urbana IX [UIX] model) and the ef-fect of relativistic boost corrections. Model C (Baldo et al.1997): This intermediate EoS model with M max ≈ . M ⊙ is a microscopic EoS for asymmetric nuclear matter, derived from the Brueckner-Bethe-Goldstone manybody theory withexplicit three-body terms. Model D (Pandharipande 1971):This very soft EoS model with M max ≈ . M ⊙ assumes anadmixture of hyperons with the hyperonic potentials simi-lar to the nucleon-nucleon potentials, but altered suitablyto represent the different isospin states. We have computed ν spin sequences ( §
2) for 15 ν spin val-ues in the range of 0 −
750 Hz for each EoS model.About 16000 neutron star structures have been calcu-lated to establish our results, and we give example fig-ures in this section using a fraction of our computed num-bers. The code to compute these structures and r ISCO val-ues is well tested (Datta et al. 1998; Thampan and Datta1998; Bhattacharyya et al. 2000, 2001a,b,c; Bhattacharyya2002). So far Kerr spacetime has been used to model the ironlines from spinning neutron star systems. Therefore, first weexamine how our
Jc/GM vs. r in c /GM plot deviates fromthe corresponding Kerr curve, in order to find out if Kerrcalculations for iron lines can give acceptable constraintson neutron star parameters. Fig. 1 shows when the equa-torial radius of the neutron star is smaller than the ISCOradius, the deviation is relatively small, but is not negligi-ble, depending on the values of ν spin and M and the chosenEoS model. But when the neutron star equatorial radius islarger than the ISCO radius (see, for example, Miller et al.(1998)), the deviation is large because stellar equatorial ra-dius increases with the increase of ν spin , and such a situ-ation does not occur for black holes (Kerr spacetime). Forexample, for the EoS model A, the stellar equatorial ra-dius is greater than the ISCO radius for all ν spin values for M = 1 . M ⊙ , and hence the deviation is always very large.However, for M = 2 . M ⊙ , the stellar equatorial radius is lessthan the ISCO radius for smaller values of ν spin (Fig. 2). Inthis case, the curves for both EoS models A and B are closerto the Kerr curve compared to these curves for M = 1 . M ⊙ .However, for M = 2 . M ⊙ , stable neutron star structures donot exist for EoS models C and D. Therefore, from Fig. 1 andFig. 2 we find that the deviation is more for (1) higher ν spin ,(2) lower M , and (3) stiffer EoS models. The first two ef-fects can also be seen in Fig. 1 of Miller et al. (1998). Theseeffects are expected for r in c /GM = Rc /GM , because allthese three points result in the increase of R . Let us now tryto understand these points for r in c /GM = r ISCO c /GM .The first point is understandable, because for ν spin = 0the spacetime outside even a neutron star is Schwarzschild,which is the non-spinning special case of Kerr. The secondpoint could be understood from the fact that for lower M the neutron star is usually less compact (that is the hardsurface is farther from the centre), causing its spacetime todeviate more from that of a black hole. The third point maybe explained from the lesser stellar compactness for a stifferEoS model for given M and ν spin . However, we find that if M/M max is kept fixed instead of M , the differences amongthe Jc/GM vs. r in c /GM curves for various EoS modelsare small (Fig. 3). This indicates that M max may be a suit-able parameter to characterize an EoS model.After finding that the Kerr spacetime is usually notgood enough to model r in c /GM , and since this spacetime c (cid:13) , 1– ?? Bhattacharyya S. cannot be used for neutron star parameter calculation, wehave explored ways to constrain stellar EoS models using ap-propriate general relativistic computations for neutron stars(see § r in c /GM and a few stellar parameters, which can be measured fromindependent methods. These parameters are ν spin , Rc /GM and M . Whenever ν spin can be measured (using regular pul-sations or burst oscillations; Bhattacharyya (2010)), it ismeasured very accurately. Therefore, in the Figs. 4, 5, 6and 7, we have fixed ν spin . Rc /GM and M , on the otherhand, may be constrained in a range (using thermonuclearX-ray bursts, binary orbital motions, etc.; Bhattacharyya(2010)), and hence we have used them as dependent vari-ables in these figures. Figs. 4 and 5 show Rc /GM vs. r in c /GM plots for two values of ν spin and four EoS mod-els. For a given ν spin value, and if the stellar equatorial ra-dius is less than the ISCO radius, each EoS model tracesa distinct curve. These curves, which are more separatedfrom each other for higher ν spin , can be used to constrainEoS models from the r in c /GM value inferred from the ironline fitting, and the Rc /GM value measured independently(see Bhattacharyya (2010)). These figures show that evenif Rc /GM is not well constrained, a suitable upper limitof r in c /GM can reject softer EoS models. If the stellarequatorial radius is larger than the ISCO radius, an obliquestraight line is found for all EoS models, and an inferred r in c /GM value directly gives the Rc /GM value. Figs. 4and 5 clearly show the value of r in c /GM (for a given ν spin ),above which r in c /GM can be used to directly infer the EoS-model-independent Rc /GM value. Figs. 6 and 7 show evenif, instead of Rc /GM , M is known from independent mea-surement (Bhattacharyya 2010), r in c /GM inferred fromiron line can be used to constrain the EoS models. Even foran unknown M , a suitable upper limit of r in c /GM can beused to reject softer EoS models. Fig. 8 explores how fora known M value, r in c /GM inferred from iron line andindependently constrained Rc /GM can be used to con-strain EoS models. Similar result is shown in Fig. 9, where Rc /GM is known and M is reasonably constrained. How-ever, for these two procedures (Figs. 8 and 9), measurementsof Rc /GM and M appear to be more important than aninferred r in c /GM . In this paper, we explore ways to constrain neutron star EoSmodels by comparing the inferred values of r in c /GM withthe theoretical values. A r in c /GM value may be inferredby fitting the relativistic disc lines with appropriate mod-els (Bhattacharyya 2010). We have shown that r in c /GM calculated from Kerr spacetime is inadequate to distinguishbetween EoS models even when r in c /GM = r ISCO c /GM ,and can largely differ from the correct value for rapidly spin-ning neutron star spacetime when r in c /GM = Rc /GM .We have numerically computed a huge number of r in c /GM values, assuming that the disc terminates either at ISCO orat the stellar hard surface. This should be at least approxi-mately true for r in c /GM < ∼
6, although systematics may beintroduced due to the effects of magnetic field and radiativepressure. Such systematics would imply that any inferred r in c /GM value is basically an upper limit of r ISCO c /GM Figure 1.
Neutron star angular momentum parameter(
Jc/GM ) vs. disc inner edge radius to stellar mass ratio( r in c /GM ). The solid curve is for Kerr spacetime ( § §
2) for M = 1 . M ⊙ , and ν spin ranging from 0 Hz to 750 Hz.The negative slope of the curves implies that the disc terminatesat ISCO, while the positive slope implies that it terminates atthe stellar surface. This figure shows that the realistic r in c /GM values for neutron stars can significantly deviate from the Kerrvalues. Figure 2.
Neutron star angular momentum parameter(
Jc/GM ) vs. disc inner edge radius to stellar mass ratio( r in c /GM ). Similar to Fig. 1, but for M = 2 . M ⊙ . Note thatstable neutron star structures do not exist for this high mass forthe softer equation of state models C and D ( § or Rc /GM , which can still be used to reject softer EoSmodels (Figs. 4, 5, 6 and 7). These systematics can be re-duced by detailed modelling of these effects, as well as byindependent observations. We have studied the relations be-tween r in c /GM and several directly measurable neutronstar parameters (Bhattacharyya 2010), in order to estab-lish new ways to constrain EoS models. We have found thatthis iron line method could be effective to constrain EoSmodels, if the neutron star spin frequency is independentlymeasured (Bhattacharyya 2010). This work is timely andimportant, as it provides motivation for future X-ray mis-sions, and because of the rapid progress in the disc line fieldvia observations with XMM-Newton , Suzaku and
Chandra . c (cid:13) , 1– ?? onstraining neutron star EoS models Figure 3.
Neutron star angular momentum parameter(
Jc/GM ) vs. disc inner edge radius to stellar mass ratio( r in c /GM ). Similar to Fig. 1, but for M = 0 . × M max . Here M max is the maximum mass for a non-spinning stable neutronstar for a given EoS. This figure shows that a fixed M/M max gives similar curves, and hence M max may be a suitable parame-ter to characterize an EoS model. Figure 4.
Neutron star equatorial-radius-to-mass ratio( Rc /GM ) vs. disc inner edge radius to stellar mass ratio( r in c /GM ) for various EoS models ( §
2) for ν spin = 200 Hz.Note that the oblique straight line portions of the curves are for r in c /GM = Rc /GM . This figure shows how a r in c /GM valueinferred from iron line can be used to constrain the EoS modelsfor a known ν spin value ( § ACKNOWLEDGMENTS
We thank A. Thampan for the rapidly spinning neutron starstructure computation code, and A. Gopakumar and B. Iyerfor discussion. This work was supported in part by US NSFgrant AST 0708424.
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