Measuring the Masses and Radii of Neutron Stars Through X-Ray Spectral Observations Tolga Güver

Measuring the Masses and Radii of Neutron Stars through X-ray Spectral Observations Tolga Güver

Feryal Özel, Dimitrios Psaltis, Slavko Bogdanov, Craig Heinke, Sebastien Guillot, Gordon Baym, Herman Marshall, Matteo Guainazzi, Maria Diaz-Trigo, Antonio Cabrera-Lavers, Pat Slane Neutron Stars

• Strongest gravitational ﬁeld among all objects in the Universe that still have surface, • Extreme Densities, • Strongest magnetic ﬁelds.

Observations of neutron stars offer unique insight to the behaviour of matter and radiation in these extraordinary conditions. 1992ApJ...398

Equation of State to Mass and Radius 1992ApJ...398 1992ApJ...398

Lindblom 1992, ApJ, 398, 569 Rep. Prog. Phys. 76 (2013) 016901 Equation of State to Mass andF Ozel¨Radius 1992). In principle, the P –ρ relation can be obtained from astrophysical measurements of neutron star masses and radii by inverting this mapping. In practice, however, this requires a measurement of radii for neutron stars that span the entire range of masses between, e.g. 0.2–2 M . Neutron stars with masses much smaller than the Chandrasekhar⊙ mass of the progenitor cores cannot be formed astrophysically (see the discussion in Ozel¨ et al 2012b), severely limiting the applicability of this direct inversion. Even though the full functional form of the P –ρ relation cannot be mapped out from astrophysical observations, it has been shown that, for most model equations of state, neutron star masses and radii allow us to infer the pressure of ultradense matter at a few appropriately chosen densities above ρs. In particular, the radii at 1.4 M lead primarily to ⊙ the determination of the pressure at 2ρs (Lattimer and Prakash 2001), the slope of the mass–radius relation is most strongly affected by the pressure at 4ρs, and the maximum mass of neutron stars is dictated by the pressure at 8ρs (Read et al 2009a, 2009b, Ozel¨ and Psaltis 2009). Therefore,∼ measuring the masses and radii of even a small number of neutron stars can provide significant input to the microphysics calculations Özel, 2013 ¨ Figure 10. Mass–radius relations for a selection of neutron star (Ozel et al 2010a, Steiner et al 2010). equations of state. Each color-shaded region corresponds to a Figure 10 shows the mass–radius relations for a number different calculation and represents a range of model parameters of equations of state representing the different approaches investigated in the corresponding study. APR is the nucleonic discussed above. For each equation of state, the shaded equations of state of Akmal et al (1998) with the expansion in terms of 2- and 3-body interactions. MS is a field-theoretical calculation region represents the range of uncertainty in the mass–radius with meson exchange interactions (Muller¨ and Serot 1996). GS relations that are obtained for different input parameters within represents field-theoretical calculations that incorporate a those calculations. The curves labeled APR correspond to a condensate of kaons (Glendenning and Schaffner-Bielich 1999). nucleonic equations of state with the expansion in terms of ABPR is a hybrid model based on the APR equation of state but 2- and 3-body interactions and are characterized by radii that incorporates a transition to quark matter at densities larger than 2–3ρs (Alford et al 2005). BBB represents a are nearly independent of the stellar mass (Akmal et al 1998). Brueckner–Hartree–Fock∼ model. The equations of state that include The MS region is an example of a field-theoretical calculation strange quark matter are shown as the shaded region labeled SQM with meson exchange interactions; as in the case of APR (Prakash et al 1995). relations, the radii are very weakly dependent on mass (Muller¨ & Serot 1996). BBB is a representative Brueckner–Hartree– each observable, and those in the theoretical models all Fock model based on similar potentials as those incorporated in play a role in determining the overall accuracy of the radii the APR equation of state and predict comparable dependence determinations. Below, I discuss the neutron star radius and of mass on radius. The equations of state by Glendenning compactness determinations by different techniques in various and Schaffner-Bielich (1999), which are the field-theoretical groups of sources. The first technique utilizes spectral data calculations that incorporate a condensate of kaons, possess an and includes radii measurements in accreting neutron stars inflection point at a characteristic density in the P –ρ relation. during quiescence and during thermonuclear bursts. The This manifests itself as a characteristic kink in the mass–radius second technique is based on modeling pulse profiles obtained relations of GS as shown in figure 10 and reduces the predicted from timing data and leads primarily to constraints on the maximum mass neutron stars can support. Moreover, for the neutron star compactness (mass-to-radius ratio) in accreting mass range of astrophysical interest, the radii become smaller millisecond pulsars, millisecond radio pulsars, and stars that with increasing stellar mass. Hybrid neutron stars with quark show flux oscillations during thermonuclear bursts. matter cores are represented by the region labeled ABPR, which is based on the APR equation of state but incorporates 4.2. Spectral measurements a transition to quark matter at densities larger than 2–3ρs (Alford et al 2005). Finally, the equations of state that∼ include Numerous observations of accreting neutron stars during strange quark matter are shown as the shaded region SQM, quiescence led, so far, to constraining determinations of characterized by a positive slope in the mass–radius relation apparent radii in a handful of sources. In particular, sources in (Prakash et al 1995). globular clusters, to which distances can be measured through Measurements of neutron star radii and compactness independent means, have primarily been selected for these have been achieved through observations and modeling studies. In addition, sources that show modest or negligible of several different classes of sources. The number of non-thermal components in their quiescent spectra and exhibit independent observables, the uncertainties associated with little variability between different quiescent episodes serve as

21 Spectroscopic Measurements

Time Resolved X-ray Spectroscopic Measurements of Thermonuclear X-ray Bursts from Low Mass X-ray Binaries

Cooling Neutron Stars in Quiescent Low Mass X-ray Binaries Time Resolved X-ray Spectroscopy of X-ray Bursts

2 Galloway et al.

and pressures, helium burning is also extremely unsta- ble, and a rapid and intense helium burst follows. At the highest accretion rates (0.1 M˙ ), stable helium burning caused by thermonuclear Edd becomes viable on the surface of the neutron star, which burning of the accreted depletes the primary fuel reserves and causes bursts to material. occur less frequently, or not at all. Hanawa & Fujimoto (1982) pointed out that the luminosities of neutron star LMXBs that have been observed to burst imply accretion rates from 0.01 0.1 M˙ Edd, so that an individual source that varies∼ in− luminosity should exhibit changes in its bursting behavior between Case 2 and Case 1. This has proved difficult to test. The accretion rate varies on time scales of weeks to months Fig. 1.— Example lightcurves of bursts observedGalloway by RXTE et al. .The2008 top panel shows a long burst from GS 1826−24 on 1998 June 8 (e.g. Muno et al. 2002c), much longer than the average 04:11:45 UT. The lower left panel shows a burst observed from observation. The majority of sources either exhibit few 4U 1728−34 on 1999 June 30 19:50:14 UT, while the burst at bursts, or several bursts are seen in a single luminos- lower right was observed from 4U 2129+12 in the globular cluster ity state (Lewin et al. 1993). Only a few sources have M15 on 2000 September 22 13:47:41 UT. The persistent (pre-burst) level has been subtracted (dotted line). Note the diversity of burst been observed to burst over a range of luminosities, and profiles, which arises in part from variations in the fuel composi- these have produced some perplexing results. As the ac- tion; bursts with a slow rise and decay are characteristic of mixed cretion rate increases, the column of material above the H/He fuel, while bursts with much faster rises likely burn primar- burning layer builds more quickly, and thus the time re- ily He. Both bursts in the lower panels exhibited photospheric radius-expansion. quired to reach the critical temperature for ignition is expected to decrease. This is best seen in the burst rate increase observed in GS 1826 24 as the persistent flux increased by a factor of 2− (Galloway et al. 2004b). ≈ et al. 1981; Ayasli & Joss 1982; Fushiki & Lamb 1987; On the other hand, increases in persistent flux over a Fujimoto et al. 1987; Bildsten 1998; Cumming & Bild- wider range often result in a decrease in burst rate, as in 4U 1820 30 (Clark et al. 1977). This contrary result sten 2000; Narayan & Heyl 2003; Woosley et al. 2004). − The frequency, strength, and time scales of thermonu- may be attributed to steady helium burning at the high- clear bursts depend on the composition of the burning est accretion rates, which reduces the amount of fuel for material, as well as the metallicity (here referring to the X-ray bursts. This causes bursts to occur less often, as CNO mass fraction, Z ) of the matter accreted onto also seen in EXO 0748 676 (Gottwald et al. 1986) and CNO 4U 1705 44 (Langmeier− et al. 1987), or to cease alto- the neutron star, the amount of hydrogen burned be- − tween bursts, and the amount of fuel left-over from the gether, as observed in GX 3+1 (Makishima et al. 1983). previous burst. Variations from source to source are also However, no correlation was found between persistent expected because of differences in the core temperatures flux and burst recurrence times in Ser X-1 (Sztajno et al. of the neutron stars and the average accretion rate onto 1983), 4U 1735 44 (Lewin et al. 1980; van Paradijs et al. 1988b), and 4U− 1636 536 (Lewin et al. 1987). This sug- the surface (Ayasli & Joss 1982; Fushiki & Lamb 1987; − Narayan & Heyl 2003). Theoretical ignition models such gests either that an additional mechanism contributes to as that described by Fujimoto et al. (1981) predict how the frequency of bursts, or that the persistent flux is not burst properties in an individual system change as the a good measure of the accretion rate in these sources. accretion rate onto the neutron star varies. They iden- The change in the composition of the burning layer ˙ tify three principal regimes of bursting, depending on as M increases also affects the properties of the bursts. the accretion rate (M˙ ) usually expressed as a fraction Helium burning occurs via the triple-α process, which is ˙ −8 −1 moderated by the strong nuclear force and proceeds very of the Eddington rate MEdd ( 1.3 10 M⊙ yr or quickly at the temperatures and densities of the burning 8.8 104 g cm−2 s−1 locally, averaged≡ × over the surface of × ˙ layer. Hydrogen burning proceeds more slowly, because a 10 km NS). At the lowest accretion rates (< 0.01MEdd, it is limited by a β-decay that is moderated by the weak referred to as Case 3 by Fujimoto et al. 1981), the tem- force. Therefore, faster, more intense bursts character- perature in the burning layer is too low for stable hy- istic of a helium-rich burning layer should occur at rela- drogen burning; the hydrogen ignites unstably, in turn tively low accretion rates (Case 2), while hydrogen-rich triggering helium burning, which produces a type I X- bursts with slower rise and decay times should occur at ray burst in a hydrogen-rich environment. At higher higher rates (Case 1). Surprisingly, most sources behave ˙ accretion rates (0.01

Draft version September 11, 2015 A Preprint typeset using LTEXstyleemulateapjv.03/07/07

DATA SELECTION CRITERIA FOR SPECTROSCOPIC MEASUREMENTS OF NEUTRON STAR RADII WITH X-RAY BURSTS Feryal Ozel¨ 1,DimitriosPsaltis1,TolgaGuver¨ 2 Draft version September 11, 2015

ABSTRACT Data selection and the determination of systematic uncertainties in the spectroscopic measurements of neutron star radii from thermonuclear X-ray bursts have beenthesubjectofnumerousrecent studies. In one approach, the uncertainties and outliers were determined by a data-driven Bayesian mixture model, whereas in a second approach, data selection was performed by requiring that the observations follow theoretical expectations. We show here that, due to inherent limitations in the data, the theoretically expected trends are not discernible in the majority of X-ray bursts even if they are present. Therefore, the proposed theoretical selection criteria are not practical with the current data for distinguishing clean data sets from outliers. Furthermore, when the data limitations are not taken into account, the theoretically motivated approach selects a small subset of bursts with properties that are in fact inconsistent with the underlying assumptions of the method. We conclude that the data-driven selection methods do not suﬀer from the limitations of this theoretically motivated one. Subject headings: dense matter — equation of state — stars:neutron — X-rays:stars — X-rays:bursts — X-rays:binaries

did not behave like the majority. (arbitrary) Luminosity spheres, the deviationA = 2 of4 the1 − spectrum2 , from a blackbody (2) 2R D f ! Rc " In the absence of errors in the determinationIn an alternate of the approach, observable Poutanen quantities, et the al. last (2014; two see is typically" quantifiedc in terms of the color correction where G isR the gravitational constant, c is the speed of light, k is the opacity to electron equations can be solved for the massalso and Kajava radius of et the al. neutron 2014)star. developed However, a because theoretically of the mo- "factor es tivated method to select bursts. They used the burst-scattering, and fc is the color correction factor. particular dependences of FTD and A on the neutron star mass and radius (see also Fig. 1 fc ≡ Tc/Teff , (1) in Ozel¨ 2006), the loci of mass-radius points that correspond to each observable intersect, in Electronic address: E-mail: [email protected] 2 In the absencewhere ofT errorsis the in color the determination temperature of obtained the observab fromle quantities, the spec- the last two 1 Eddington Flux = Touchdownc Flux general, at two distinct positions. Moreover,Department the diverse of Astronomy, nature of University uncertainties of Arizona, associated 933equations N. cantral be solved fits and for theT massis the and e radiusffective of the temperature neutron star. of However, the atmo- because of the Cherry Ave., Tucson, AZ 85721, USA Temperature (keV) eff to each of the observables requires a2 formal assessment of the propagation of errors, whichparticular dependencessphere. The of FTD colorand correctionA on the neutron factor star depends mass and on radius the eff (seeec- also Fig. 1 Istanbul University, Science Faculty, Department of Astronomy¨ we present here. and Space Sciences, Beyazit, 34119, Istanbul, Turkey in Ozel 2006),tive the temperature loci of mass-radius of the points atmosphere, that correspond its composition, to each observable and intersect, in general, at two distinct positions. Moreover, the diverse nature of uncertainties associated We assign a probability distribution function to each of the observable quantities and Özel et al. 2016 to each of the observables requires a formal assessment of the propagation of errors, which denote them by P (D)dD, P (F )dF ,andP (A)dA. Because the various measurements TD TD we present here. that lead to the determination of the three observables are independent of each other, the total probability density is simply given by the product We assign a probability distribution function to each of the observable quantities and denote them by P (D)dD, P (FTD)dFTD,andP (A)dA. Because the various measurements

P (D, FTD,A)dDdFTDdA = that lead to the determination of the three observables are independent of each other, the total probability density is simply given by the product P (D)P (FTD)P (A)dDdFTDdA . (3)

P (D, FTD,A)dDdFTDdA = Our goal is to convert this probability density into one over the neutron-star mass, M, P (D)P (FTD)P (A)dDdFTDdA . (3) and radius, R. We will achieve this by making a change of variables from the pair (FTD,A) to (M,R) and then by marginalizing over distance. Formally, this implies that Our goal is to convert this probability density into one over the neutron-star mass, M, 1 and radius, R. We will achieve this by making a change of variables from the pair (FTD,A) P (D, M, R)dDdMdR = P (D)P [F (M,R,D)] 2 TD to (M,R) and then by marginalizing over distance. Formally, this implies that FTD,A 1 P [A(M,R,D)]J dDdMdR , (4) P (D, M, R)dDdMdR = P (D)P [F (M,R,D)] ! M,R " 2 TD F ,A P [A(M,R,D)]J TD dDdMdR , (4) ! M,R " Effects of Rotation and Temperature Correction to Eddington Limit 3

Özel et al. 2016 Fig. 1.— Contours of constant apparent angular size (blue) and touchdown ﬂux (green) for a M =1.4 M⊙ and R =10kmneutronstar spinning at 600 Hz and a spectral temperature during the touchdown moment of Eddington limited bursts calculated using equation (8) for this mass and radius. These curves include the corrections totheapparentareaduetoneutronstarspinandthetemperature correction to the Eddington limit, respectively. The red curves are the corresponding contours when these corrections are not takenintoaccountand would have led to no solutions for the neutron star mass and radius.

In Figure 1, we show in blue the contour of constant apparent angular size on the mass-radius plane for a M =1.4 M⊙ and R = 10 km neutron star spinning at 600 Hz. To highlight the effect of the rotational corrections, we also plot in red the corresponding contour obtained under the Schwarzschild approximation used in the previous studies. As discussed in Bauböck et al. (2015), the rotational effects lead to larger angular sizes for the same neutron star mass and radius. 2.2. Temperature Corrections to the Eddington Limit The atmospheres of neutron stars during thermonuclear bursts are dominated by electron scattering. During strong bursts, the radiation forces lift the photosphere above the neutron star surface and allow for a measurement of the Eddington critical luminosity. When this luminosity is measured at the touchdown point, i.e., when the photosphere has returned to the neutron star surface, it is related to the neutron star mass and radius via GMC 2GM 1/2 F = 1 , (3) td k D2 − Rc2 es ! " where k 0.2(1 + X)cm2 g−1 (4) es ≡ is the electron scattering opacity and X is the hydrogen mass fraction of the atmosphere. Because of the energy dependence in the Klein-Nishina cross sectionandthefactthatthephotonsexchangeenergy with the electrons at each scattering, the Eddington flux depends on the temperature of the atmosphere. Paczynski (1983) derived an approximation for this temperature correction, which was further refined by Suleimanov et al. (2012) by taking into account the angular dependence of the scattering processes. With the approximate formula given in the latter study, the Eddington flux becomes − GMC 2GM 1/2 kT ag 2GM ag/2 F = 1 1+ c 1 , (5) td k D2 − Rc2 38.8keV − Rc2 es ! " # ! " ! " $ where g a =1.01 + 0.067 eff (6) g 1014 cm s−2 and % & − GM 2GM 1/2 g = 1 . (7) eff R2 − Rc2 ! " In equation (5), the correction to the Eddington flux depends on the color temperature when the atmosphere reaches that limit, and this color temperature, in turn, depends on the mass and radius of the neutron star and the composition of the atmosphere via − g c 1/4 GMc 1/4 2GM 1/8 T = f T = f eff = f 1 , (8) Edd,c c Edd,eff c σ k c σ k R2 − Rc2 ! B es " ! B es " ! " Draft version September 11, 2015 A Preprint typeset using LTEXstyleemulateapjv.03/07/07

DATA SELECTION CRITERIA FOR SPECTROSCOPIC MEASUREMENTS OF NEUTRON STAR RADII WITH X-RAY BURSTS Feryal Ozel¨ 1,DimitriosPsaltis1,TolgaGuver¨ 2 Draft version September 11, 2015

ABSTRACT Data selection and the determination of systematic uncertainties in the spectroscopic measurements of neutron star radii from thermonuclear X-ray bursts have beenthesubjectofnumerousrecent studies. In one approach, the uncertainties and outliers were determined by a data-driven Bayesian mixture model, whereas in a second approach, data selection was performed by requiring that the observations follow theoretical expectations. We show here that, due to inherent limitations in the data, the theoretically expected trends are not discernible in the majority of X-ray bursts even if they are present. Therefore, the proposed theoretical selection criteria are not practical with the current data for distinguishing clean data sets from outliers. Furthermore, when the data limitations are not taken into account, the theoretically motivated approach selects a small subset of bursts with properties that are in fact inconsistent with the underlying assumptions of the method. We conclude 7 that the data-driven selection methods do not suﬀer from the limitations of this theoretically motivated one. Subject headings: dense matter — equation of state — stars:neutron — X-rays:stars — X-rays:bursts — X-rays:binaries

1. INTRODUCTION ing neutron star atmosphere models of Suleimanov et al. Thermonuclear flashes on neutron stars have been (2012) to calculate the evolution of the blackbody nor- used in the past decade to perform spectroscopic malization and chose only those bursts that followed measurements of neutron star radii and masses (e.g., their predicted trends. When applied to the source ¨ ¨ 4U 1608−52, they found that this approach selects a very Majczyna et al. 2005; Ozel 2006; Ozel et al. 2009, small fraction of bursts as–5– acceptable ones and leads to 2010; Poutanen et al. 2014). The rich burst dataset substantially larger inferred neutron star radii. (Galloway et al. 2008) from the Rossi X-ray Timing Ex- In this paper, we show that the bursts selected by plorer (RXTE) has not only allowed the measurement 3. Determinationthe data-driven approach of the Neutron of Güver Star et al. Mass (2012a,b) and Radius are of the macroscopic properties of half a dozen of neutron not in conflict with the theoretical expectations of the stars but also initiated detailed studies of systematic un- Thermonuclear X-ray BurstsIn anas approachtools Suleimanovfor similar mass-radius to etOzel¨ al. (2012) (2006), measurements atmosphere we use the models. spectroscopic The theo- measurements of the certainties in these measurements. Even though the ma- retical models predict that the blackbody normalization touchdown flux FTD and the ratio A during the cooling tails of the bursts, together with the jority of sources showed bursting behavior that is highly evolves rapidly in a narrow range of fluxes close to the reproducible (Güver et al. 2012a,b), a subsetmeasurement shows burst of theEddington distance limit.D to the However, source in in order most to bursts, determine the instru- the neutron star mass properties and evolution that are more complex.M and radius R.ment The observed limitations spectroscopic of RXTE do quantities not allow depend resolving on the this stellar parameters A number of studies explored different approaches –5– according to the relationsfast predicted evolution. When this limitation is not to identifying outliers in the burst samples of differ- taken into account in burst selection,1 as/2 was the case ent sources that contaminate the statistical inferences GMc 2GM 3.in Determination PoutanenF etTD al.= of (2014), the Neutron this1 − procedure Star Mass biases and the Radius se- (1) k D2 ! Rc2 " in the measurements. In Güver et al. (2012a,b), we used lection toward a peculiares set of bursts. We further show a data-driven approach that employs a Gaussian mix- and In an approachthat similar this subset to Ozel¨ of bursts (2006), are we not use photospheric the spectroscopic radius measurements ex- of the ture Bayesian inference to reject outliers. In particular, 2 −1 arXiv:1509.02924v1 [astro-ph.HE] 9 Sep 2015 pansion bursts, whichR is a requirement2GM in the theoretical touchdown flux FTD and the ratio A during the cooling tails of the bursts, together with the we looked at the cooling tails of X-ray bursts in the flux- A = 2 4 1 − 2 , (2) motivated selectionD criteriafc ! of PoutanenRc " et al. (2014) and temperature diagram, which would have yieldedmeasurement identical of theKajava distance et al.D (2014).to the Becausesource in of order both to its determine practical the lim- neutron star mass tracks among bursts of the same source inwhere theM and absenceG radiusis theR gravitational.itations The observed and constant, its spectroscopic biasedc is results, the speed quantities we argue of light, depend thatkes thisis on the the data opacity stellar to parameters electron of any astrophysical complexities. The degree of scatter scattering,according to and thefc relationsselectionis the color method correction does factor. not lead to reliable radius mea- we observed in the cooling tracks allowed us to measure surements. 1/2 the level of systematic uncertainty in the measurements,In the absence of errors in the determinationGMc of2GM the observable quantities, the last two FTD = 2 1 − 2 (1) e.g., due to obscuration, reflection off the accretionequations disk, can be solved2. APPLYING for the mass THE and THEORETICALkes radiusD ! of theRc SELECTION neutron" star. CRITERIA However, because of the or uneven burning on the stellar surface, as well as to TO RXTE BURSTS particularand dependences of FTD and A on the neutron star− mass and radius (see also Fig. 1 remove a small percentage of cooling tracks that clearly In theoretical modelsR2 of bursting2GM neutron1 star atmo- did not behave like the majority. in Ozel¨ 2006), the loci of mass-radiusA = points that1 − correspond, to each observable intersect,(2) in spheres, the deviationD2f of4 ! the spectrumRc2 " from a blackbody In an alternate approach, Poutanen et al.general, (2014; at see two distinctis typically positions. quantified Moreover,c in terms the diverse of the n colorature correction of uncertainties associated also Kajava et al. 2014) developed a theoreticallytowhere eachG of mo-is the the observables gravitationalfactor requires constant, a formalc is the assessment speed of of light, the propagationkes is the opacity of errors, to electron which tivated method to select bursts. They used the burst- wescattering, present here. and fc is the color correctionfc factor.≡ Tc/Teff , (1) Electronic address: E-mail: [email protected] where T is the color temperature obtained from the spec- 1 WeIn the assign absence a probability of errorsc distribution in the determination function to of each the observab of the observablele quantities, quantities the last and two Department of Astronomy, University of Arizona, 933 N. tral fits and T is the effective temperature of the atmo- Cherry Ave., Tucson, AZ 85721, USA eff 2 denoteequations them can− by beP solved(D)dD for, P the(F mass)dF and,and radiusP ( ofA) thedA. neutron Becausestar. the However, various measurements because of the IstanbulFig. University, 4.— Same Science as in Faculty, Figure Department 3 but for SAX of Astro J1748.9nomy 2021.sphere. The colorTD correctionTD factorGüver depends& Özel 2013 on the effec- and Space Sciences, Beyazit, 34119, Istanbul, Turkeythatparticular lead to dependences the determinationtive temperature of FTD ofand the ofA the threeon atmosphere, the observables neutron its star are composition, mass independent and radius and of each (see other, also Fig. the 1 totalin Ozel¨ probability 2006),3.1.2. the density lociSAX of is mass-radius simplyJ1748.9 given−2021 points by the that product correspond to each observable intersect, in general, at two distinct positions. Moreover, the diverse nature of uncertainties associated The transient neutron star X-ray binary SAX J1748.92021 is located in the globular cluster NGC 6440, which is a P (D, F ,A)dDdF dA = massive and old cluster in the Galacticto each bulge. of the Two observables optical andrequires oneTD a near formal-IR assessment studiesTD give of th consistente propagation and of well-constrained errors, which we present here. +1.5 P (D)P (FTD)P (A)dDdFTDdA . (3) distances to NGC 6440: Kuulkers et al. (2003) reported 8.4−1.3 kpc, Harris et al. (2010) found 8.5 kpc, while Valenti et al. (2007) found 8.2 0.6 kpc using near-IRWe assign data. a probability In this last distribution study, the function distance to each uncerta of theintyobservable is improved quantities and and takes ± into account the systematic errorsdenote introducedOur them goal isby toP the convert(D) methoddD, thisP (F probabilityTD of) cdFomparingTD,and densityP the(A) intodA properties. one Because over the of the NGC neutron-star various 6440, measurements including mass, M, its metallicity and age, to the referenceandthat cluster. radius, lead toR We the. We adopt determination will achieve here the this of lat the byter makingthree distance observables a change and of are its variables uncertainty. independent from the of eachpair other, (FTD,A the) SAX J1748.9 2021 has a spin frequencytototal (M,R probability) of and 420 then Hz, density by detected marginalizing is simply during given over intermittent by distance. the product Formally,pulsations this im observedplies that in the persistent emission (Altamirano− et al. 2008). The same study also found a binary orbital period of 8.7 hr. Because there is no specific information about the evolutionary state of the donor, we take a1 flat prior in the hydrogen mass fraction P (D,P M,(D, R F)dDdMdRTD,A)dDdF= TDPdA(D=)P [F (M,R,D)] between 0 and 0.7. 2 TD P (D)P (FTD)P (A)dDdFTDdA . (3) The top left panel of Figure 4 shows the evolution of the flux and temFTDperature,A during the cooling tails of four P [A(M,R,D)]J dDdMdR , (4) bursts observed from SAX J1748.9 2021, while the top right panel shows! M,R the 68%" and 95% confidence contours in the measurement of the blackbody normalization− Our goal is to vs. convert temperature this probabilityduring density the touchdown into one over phasesthe neutron-star in its two mass, Eddington-M, limited bursts. and radius, R. We will achieve this by making a change of variables from the pair (FTD,A) The lower panels of Figure 4 showto the (M,R 68%) and confidence then by marginalizing contour in over mass distance.and radius Formally, as well this as im theplies posterior that likelihood marginalized over mass using the Bayesian framework and priors discussed above. 1 P (D, M, R)dDdMdR = P (D)P [F (M,R,D)] 3.1.3. EXO 1745−248 2 TD F ,A EXO 1745 248 is located in Terzan 5, one of the most metal-richP [A(M,R,D globular)]J TD clusterdDdMdRs in the , Galaxy. The distance(4) to Terzan 5 was− obtained using HST/NICMOS data (Ortolani et al. 2007).! TheM,R sources" of uncertainty in the distance measurement were discussed in detail in Ozel¨ et al. (2009). We adopt here the same flat likelihood over distance centered at 6.3 kpc with a width of 0.63 kpc. No burst oscillations or persistent pulsations have ever been observed from EXO 1745 248. As before, we adopt a flat prior over its spin frequency between 250 and 650 Hz when calculating the spin corrections− to the apparent angular Possible Sources of Uncertainties

Is it really statistically appropriate to ﬁt the X-ray spectra during burst decay with Planckian functions ? (Güver et al. 2012a)

Do we observe the whole neutron-star surface during the decay of an X- ray burst? (Güver et al. 2012a)

How are the maximum ﬂuxes of X-ray bursts related to the Eddington Limit ? (Güver et al. 2012b)

Are the ﬂux measurements absolutely correct (Güver et al. 2016) HowThe Astrophysical good Journal,747:76(16pp),2012March1 are the blackbodyGuver,¨ Psaltis, ﬁts & Ozel¨ ?

1000 1000 KS 1731-260 4U 1728-34

# of bursts : 24 # of bursts : 90

# of spectra : 1312 # of spectra : 2532 100 100

10 10 Number of Spectra Number of Spectra

1 1

0 1 2 3 4 0 1 2 3 4 X2/dof X2/dof

1000 4U 1636-536

# of bursts : 162

# of spectra : 4579 100

10 Number of Spectra

0 1 2 3 4 X2/dof Figure 1. Histograms show the distributions of X2/dof values obtained from fitting 1309, 2519, and 4596 X-ray burst spectra observed from the sources KS 1731 260, The systematic4U 1728 34, and 4U uncertainty 1636 536, respectively. The solidrequired lines show the expected toχ 2render/dof distributions the for the number observed of degrees of freedom spectra used during the fits. −as The a vertical dashed− lines correspond− to the highest values of X2/dof that we considered as statistically acceptable for each source. The vast majority of spectra are well blackbodydescribed is by blackbody~3-5% functions. (A color version of this figure is available in the online journal.) A similar amount (~3-5%) of X-ray spectra are just not consistent with a blackbody function. 1984; Bildsten 1995;Spitkovskyetal.2002), and the excitation star is !0.2keV,evenfortheburststhatshowthelargestburst of non-radial modes on the stellar surface (Heyl 2004;Piro& oscillation amplitudes. All of the above strongly suggestGüver that et al. 2012a Bildsten 2005;Narayan&Cooper2007)areallexpectedtolead the expected flux anisotropy during the cooling tail of an X- to some variations in the effective temperature of emission at ray burst is !5%–10% and, therefore, the expected systematic different latitudes and longitudes on the stellar surface. uncertainties in the inferred apparent stellar radius will be half The characteristics of burst oscillations observed during of that value, since the latter scales as the square root of the flux. the cooling tails of X-ray bursts, however, imply that any Afinalinherentsystematicuncertaintyinthespectroscopic variations in the surface temperatures of neutron stars can measurement of the apparent surface area of a neutron star arises only be marginal. Indeed, any component of the variation in from the dependence of the color correction factor on the ef- the surface temperature that is not symmetric with respect fective temperature of the atmosphere. In Figure 2,weshow to the rotation axis leads to oscillations of the observed flux the predicted evolution of the color correction factor as a func- at the spin frequency of the neutron star. Such oscillations have tion of the color temperature of the spectrum as measured by been observed in the tails of bursts from many sources (Galloway an observer at infinity, for calculations by different groups for et al. 2008a). The rms amplitudes of burst oscillations in the tails different neutron-star surface gravities and atmospheric compo- of bursts can be as large as 15%, although the typical amplitude sitions (Madej et al. 2004;Majczynaetal.2005;Suleimanov is 5% (Muno et al. 2002). The stringent upper limits on the et al. 2011). To make the models comparable to the observed observed≃ amplitudes at the harmonics of the spin frequencies evolution of the blackbody normalizations presented in the next can be accounted for only if the temperature anisotropies are section, we plot the color correction factor against the most dominated by the m 1modeinwhichexactlyhalfthe directly observed quantity, i.e., the color temperature at infin- neutron star is hotter than= the other half (Muno et al. 2002). ity. When the color temperature at infinity is less than 2.5 keV, Moreover, the rather weak dependence of the rms amplitudes on purely helium or low-metallicity models show 0%–8% evolution photon energy (Muno et al. 2003)requiresthatanytemperature of the color correction factor per keV of color temperature at in- variation between the hotter and cooler regions of the neutron finity, while above 2.5 keV,they show an evolution of 12%–20%

5 The Astrophysical Journal,747:76(16pp),2012March1 Guver,¨ Psaltis, & Ozel¨

KS 1731-260 30 5 20

10 Number of spectra )

-2 0

80 cm -1 60

40 erg s -8 20 Number of spectra

Flux (10 1 25 20

Number of spectra 5 How reproducible are the0 cooling tails ? 0 100 200 300 2 2 3 3 2 1 2 1 Normalization (Rkm / D10kpc) Color Temperature (keV)

10 4U 1728-34 10

5 Number of spectra )

-2 0

40 cm -1 30

20 erg s -8 10 Number of spectra

50 Flux (10 40

1 30

Number of spectra 10

0 0 100 200 300 2 2 3 23 1 2 1 Normalization (Rkm / D10kpc) Color Temperature (keV) 10 25 20 Güver et al. 2012a 4U 1636-536 15

5 Number of spectra 5 )

-2 0

80 cm -1 60

40 erg s -8 20 Number of spectra

Flux (10 50

1 40

Number of spectra 10 0 0 100 200 300 400 500 2 2 3 32 1 2 1 Normalization (Rkm / D10kpc) Color Temperature (keV) Figure 3. Left: the flux–temperature diagram for all the spectra in the cooling tails of bursts from (top) KS 1731 260, (Middle) 4U 1728 34, and (bottom) 4U 1636 536 that have statistically acceptable values of χ 2/dof. The diagonal lines correspond to the best-fit blackbody− normalization and its uncertainty,− as reported in the rightmost− column of Table 3. Right: the distribution of measured normalization values of the blackbody spectra in three of the flux intervals we chose. The normalization values for the vast majority of spectra fall within a narrowly peaked distribution, with only a number of outliers toward lower (for 4U 1728 34) or higher values (for 4U 1636 536) of the normalization. This justifies the assumption that the entire neutron-star surface is emitting during the cooling tail of− a burst with marginal temperature variations− in latitude or longitude. (A color version of this figure is available in the online journal.)

7 The Astrophysical Journal,747:76(16pp),2012March1How reproducible are the cooling tails ? Guver,¨ Psaltis, & Ozel¨

200 300 300

KS 1731-260 250 4U 1728-34 250 4U 1636-536 ) ) ) 150 2 10kpc 2 10kpc 10kpc 2 200 200 / D / D / D km 2 2 km 2 km

100 150 150

100 100 50 Normalization (R Normalization (R Normalization (R 50 50

0 0 0 0.5 1.0 2.0 5.0 1 2 5 10 0.5 1.0 2.0 5.0 -8 -1 -2 -8 -1 -2 -8 -1 -2 Flux (10 erg s cm ) Flux (10 erg s cm ) Güver et al. 2012a Flux (10 erg s cm ) Figure 5. Dependence of the parameters of the intrinsic distribution of blackbody normalizations on X-ray flux during the cooling tails of thermonuclear X-ray bursts in KS 1731 260, 4U 1728 34, and 4U 1636 536, when the outliers have been removed. Each dot represents the most likely centroid of the intrinsic distribution, while each error− bar represents− its most likely− width, as calculated using the procedure outlined in Section 4.2 and depicted in Figure 4.Ineachpanel,thesolidand dashed horizontal lines show the best-fit normalization and its systematic uncertainty inferred using the flux bins that do not correspond to near-Eddington fluxes and are denoted by filled circles on the error bars. (A color version of this figure is available in the online journal.)

Table 3 Blackbody Normalizationsa

8 1 2 b Source Flux Interval (10− erg s− cm− )Average 0.5–1 1–2 2–3 3–4 4–5 5–6 6–7 7–8 8–9 9–10 4U 1636 536 112.6 124.6 126.9 142.7 144.2 145.7 145.7 149.5 ...... 130.7 − 18.7 16.1 12.4 22.4 18.4 15.1 14.1 4.6 20.9 ± ± ± ± ± ± ± ± ± 4U 1702 429 151.0 167.6 180.4 184.2 185.7 171.4 151.0 119.3 98.2 ... 176.6 − 13.9 10.1 9.9 4.1 3.9 8.1 8.1 17.2 12.4 11.6 ± ± ± ± ± ± ± ± ± ± 4U 1705 44 80.2 86.9 86.9 82.4 ...... 83.9 − 9.9 7.1 10.9 7.4 9.1 ± ± ± ± ± 4U 1724 307 98.7 108.3 120.4 127.4 125.4 102.8 ...... 113.8 − 7.1 4.6 12.6 3.7 19.4 15.4 15.4 ± ± ± ± ± ± ± 4U 1728 34 114.1 126.1 137.4 145.0 135.2 137.4 133.7 126.1 117.8 93.7 134.4 − 15.6 13.4 10.9 8.9 18.2 18.4 16.4 13.1 15.4 13.1 14.9 ± ± ± ± ± ± ± ± ± ± ± KS 1731 260 72.6 89.9 93.0 95.2 95.2 ...... 96.0 − 6.1 5.4 4.1 9.6 1.9 7.9 ± ± ± ± ± ± 4U 1735 44 71.6 71.6c 72.6c ...... 72.1c − +2.0 +2.4 +1.3 4.6 1.5 1.9 1.0 ± − − − 4U 1746 37d 12.9 16.3 17.3 15.6 15.2 15.2 19.9 ...... 15.7 − 1.9 0.5 0.4 2.4 2.4 1.6 6.4 2.4 ± ± ± ± ± ± ± ± SAX J1748.9 2021 92.7 87.7 91.7 87.7 ...... 89.7 − 11.9 6.6 7.6 15.4 9.6 ± ± ± ± ± SAX J1750.8 2900 126.9 110.8 106.8 97.2 86.2 ...... 93.2c − 40.5 19.9 10.1 8.1 8.6 9.4 ± ± ± ± ± ± Notes. a The parameters of the intrinsic distribution of blackbody normalizations in different flux intervals for all the sources we considered in this manuscript. All normalizations are in units of (km/10 kpc)2. b Taking into account all flux intervals for which the color temperature of the spectrum is !2.5keV;seethetextfordetails. c The range of normalizations in this flux interval is consistent with no systematic uncertainties; the quoted uncertainties are purely statistical. d For 4U 1746 37, all flux intervals are in units of 10 9 erg s 1 cm 2. − − − − we obtained from the data at different flux intervals as a systematic uncertainties in that measurement, which account for systematic uncertainty in the measurement. In order to achieve the potential decline of the normalization with decreasing flux. this, we applied the Bayesian Gaussian mixture algorithm to In Figure 5,weidentifythefluxintervalsweusedforeach the combined data set for each source for all flux intervals source with a filled circle in the middle of each error bar. that correspond to a color temperature !2.5keV.Thisway, We also depict with a solid line the most probable value of we computed the most probable value for the blackbody the normalization in this wide flux range and with dashed normalization for each source in a wide flux range, as well as the lines the range of systematic uncertainties. Our results are

10 e 5 ent ective ﬀ itions of the this paper, which

–5– erences in the flux erent defin ff ntrates, which leads ff is too small to assess S We employ a Bayesian 3. Determination of the Neutron Star Mass and Radius (ii) In an approach similar to Ozel¨ (2006), we use the spectroscopic measurements of the

Evolutiontouchdown During ﬂux FTD and thethe ratio ACoolingduring the cooling tails Tails of the bursts, together with the 4. et G¨uver al. (2010) discussed The Astrophysical Journal,747:76(16pp),2012March1 Guver,¨ Psaltis, & Ozel¨

measurement of the distance D to the source in order to determine the neutron star mass We add an uncertainty of 1% in the d.o.f. in the range 1-8 in the spectral touchdown ﬂuxes in the most proliﬁc

M and radius R. The observed spectroscopic quantities depend on the stellar parameters / 2 1000 300 b). The spectra from these two bursts (iv) 6 kpc. Harris et al. (1996; 2010 revision) χ

according to the relations . measuring neutron star apparent angular umbers of the bursts used in this study in 0 ay et al. (2008a). 4U 1702-429 GMc 2GM 1/2 tamination of the surface emission, either

4U 1702-429 4U 1702-429 ± 1% on the systematic di # of bursts : 6 F250TD = 1 − (1) rocedures used and described in et Güver al. asurements and increases the uncertainties in ) 2 2 4 brightest sources. . t include the 1% flux calibration uncertainties, hat have been brought up to the photosphere, rst observed from this source. As discussed in kesD ! Rc " /d.o.f. We also make use of the touchdown flux rce distances and atmospheric compositions for ∼ bursts. 2 # of spectra : 285 ) rtainty in this quantity following the analysis of ect the measured burst fluxes. -2 ling tails of the two normal bursts observed from

5 2 10kpc χ t al. 2012b). In the remainder of this paper, we surements of the angular sizes and the touchdown

and een used to measure neutron star radii using their al. 2003) and in the near-IR bands (Valenti et al. ﬀ e above range (see equation 10). e Suleimanov et al. (2011) study are signiﬁcantly

100 −1 ted studies on the entire RXTE burst dataset and 200 2 rticular deﬁnition of the color correction factor and 30 or esulting in R 2GM -star atmospheres by Suleimanov et al. (2012) for e / D cm ing the neutron star radius. -1 ble 1).

A = 1 − , (2) − 2 km

2 4 2 he ﬂux calibration uncertainties. We summarize all the D fc ! Rc "

where G is the gravitational constant,150 c is the speed of light, kes is the opacity to electron erg s

-8 2 scattering, and fc is the color correction factor. 10 4U 1820 In the absence of errors in100 the determination of the observable quantities, the last two Number of Spectra 6. The open and ﬁlled circles correspond to two di . about 0.7 of the critical Eddington ﬂux. Flux (10

equations can be solved for the mass and radius of the neutron star. However, because of the 3.1.1. 05 (see Figure 2). Note that the color correction factors do evolv 1 14 . Thermonuclear Bursters 2). The two horizontal lines show the range of values we use in Normalization (R 0 particular dependences of FTD and50 A on the neutron star mass and radius (see also− Fig. 1 3 ± . in Ozel¨ 2006), the loci of mass-radius points that correspond to each observable intersect, in 307, for which Suleimanov et al. (2011) reported a radius measurem 1 3.1. 4 − . 4 kpc and the second gives 8

general, at two distinct positions.0 Moreover, the diverse nature of uncertainties associated . =14 0 , which, in principle, can a

0 1 2 3 4 3.0 2.5 to2.0 each of1.5 the observables1.0 requires a formal1 assessment2 of the propagation5 of errors,ﬀ which ± 2 -8 -1 -2 e g

X /dof Color Temperature (keV) Flux (10 erg s cm ) 6

we present here. .

We assign a probability distribution function to each of the observable quantities and Chandra 1000 5 200 denote them by P (D)dD, P (FTD)dFTD,andP (A)dA. Because the various measurements 4U 1705-44 Güver+ 2012, Özel+ 2015 we apply the appropriate deadtime correction to the observed cou that lead4Utheoretical 1705-44 to the determinationdata from Suleimanov+ of the 2012 three observables4U are 1705-44 independent of each other, the # of bursts : 44 ) (i) OBSERVATIONS AND RADIUS MEASUREMENTS OF INDIVIDUAL SOURCE total probability density is simply given by the product When the number of Eddington-limited bursts of an individual source # of spectra : 694 ) 150 -2 2 10kpc ective gravitational acceleration or on the pa

100 3. ﬀ (iii) / D cm P (D, FTD,A)dDdFTDdA = -1 2 km 2 P (D)P (FTD)P (A)dDdFTDdA . (3) 100 erg s 30 is an ultracompact binary in the metal-rich globular cluster NGC 662 Color correction factors from the models of He-rich neutron -8 − Our goal is to convert this probability density into one over the neutron-star mass, M, 207 to determine the apparent angular size (see et G¨uver al. 2012

10 10% systematic uncertainties in the apparent angular sizes and the − and radius, R. We will achieve this by making a change of variables from the pair (FTD,A) ∼

Number of Spectra 1

Flux (10 to (M,R) and then by marginalizing50 over distance. Formally, this implies that erent from blackbodies and from model atmosphere spectra, r Normalization (R 4U 1820 We reanalyze the data on these six sources uniformly, following the p Since the earliest measurements, et Güver al. (2012a, b) conduc There are five sources for which thermonuclearWe burst also data include have b in this analysis 4U 1724 Fig. 2.— 1 ff apparent angular sizes and themeasurements touchdown in fluxes to Table account 1individual for and t sources discuss below. thewhich Note additional we that details add the of inTable uncertainties the quadrature A2 in sou when of Table the inferring 1 Appendix, the do radii. no following the We numbering also system list used the in ID Gallow n to a small increase in the angular sizes and touchdown fluxes for the calibration between RXTE and (2012 a,b). Specifically, Gaussian-mixture model to quantifyfluxes. the intrinsic This scatter isthe in typically the inferred larger mea radii. than the formal uncertainties in the me measured from these bursts et (Güver al. 2012a) whenfound determin bursters. In addition, et Güver al. (2015) placed an upper limit of the scatter in etGüver al. the (2012a) touchdown on flux, the we sample take of an sources with 11% limited systematic number unce of two distance measurements performed2007). in The first the gives optical a (Kuulkers distance et of 7 show the expected thermal shape and result in acceptable values f fits (see alsoby in’t the Zand accretion &which flow makes or Weinberg the by 2010). results4U atomic unreliable. 1724 lines This Instead, from indicates we the significant make ashes use con of of the the burst coo t significantly at verysizes high and during very theconsider low cooling a flux tails flat levels, of prior which distribution thermonuclear we of bursts exclude the (see when color e Güver correction factor in th apparent angular sizes, touchdown fluxes, and distancesbased (see on Ta the etGüver spectral al. evolution (2012b), duringdi the the spectra cooling from tail of that one long long burst bu used in th color-correction factor exploredaccurately by reproduces Suleimanov et the al. model (201 resultsvery weakly for on fluxes the lessare than e approximately constant in the range 1 P (D, M, R)dDdMdR = P (D)P [F (M,R,D)] gravitational accelerations in the range log 1 2 TD 0 F ,A P [A(M,R,D)]J TD dDdMdR , (4) 0 1 2 3 4 3.0 2.5 2.0 1.5 1.0 0.5 ! 1.0M,R " 2.0 X2/dof Color Temperature (keV) Flux (10-8 erg s-1 cm-2)

1000 300 4U 1724-307 4U 1724-307 4U 1724-307 # of bursts : 2 5 250 )

# of spectra : 58 ) -2 100 2 10kpc 200 / D / cm -1 km 2

2 150 erg s -8 10 100 Number of Spectra

Flux (10 1 Normalization (R Normalization 50 1 0 0 1 2 3 4 3.0 2.5 2.0 1.5 1.0 0.5 1.0 2.0 5.0 X2/dof Color Temperature (keV) Flux (10-8 erg s-1 cm-2) Figure 6. Left: the distribution of the X2/dof values obtained from fitting the spectra during the tails of thermonuclear X-ray bursts observed from the sources 4U 1702 429, 4U 1705 44, and 4U 1724 307 together with the theoretically expected distribution; the vertical dashed line marks the maximum value of X2/dof beyond which− we consider− the blackbody model− to be inconsistent with the data. Middle: the flux–temperature diagrams of the cooling tails of bursts from the same sources; the solid and dashed lines correspond to the most probable values of the blackbody normalizations throughout the bursts and their systematicuncertainties. Right: the dependence of the parameters of the intrinsic blackbody normalization on X-ray flux; the solid and dashed lines correspond to the most probable values of the normalizations and their systematic uncertainties for the flux bins that are marked by a filled circle. Error bars without filled circles appear at near-Eddington fluxes where the color correction factor increases, causing the apparent decline in the normalization. (A color version of this figure is available in the online journal.) summarized in Table 3. The blackbody normalizations for KS sources from Table 1.Thelasttwosources,4U0513 40 and Aql 1731 260, 4U 1728 34, and 4U 1636 536 are 96.0 7.9 X-1, show large variations in the cooling tails of individual− bursts (km/−10 kpc)2,134.4− 14.9 (km/10 kpc)−2,and130.7 ±20.9 and are discussed in detail in Appendixes C and D,respectively. (km/10 kpc)2,respectively,withtheuncertaintiesdominated± ± Moreover, in Appendixes A and B we also discuss a number of entirely by systematics. bursts from 4U 1702 429 and a long burst from 4U 1724 307, which we did not include− in the analysis. − 5. THE APPARENT RADII OF X-RAY BURSTERS As in the case of the three sources discussed in the previous Figures 6–8 and Tables 2 and 3 show the results obtained section, the vast majority of the spectra observed during the after applying the procedure outlined in Section 4 to seven more cooling tails of X-ray bursts are very well described by black-

11 Uncertainties in the apparent radii of neutron stars

SAX J1750.8-2900

SAX J1748.9-2021

4U 1746-37

4U 1735-44

4U 1731-260

4U 1728-34

4U 1724-307

4U 1705-44

4U 1702-429

4U 1636-536

-20 -15 -10 -5 0 5 10 Systematic Uncertainty in Apparent Radius (%) Güver et al. 2012a Photospheric Radius Expansion Events

Güver et al. 2012b Fig. 6.— The evolution of the spectral parameters of two bursts from 4U1608−52. The left panel shows one selected by Poutanen et al. (2014) as a PRE burst, whereas the right panel shows a bona fide PRE burst according to Güver et al. (2012a). The blackbody normalization in the first case never exceeds the asymptotic value at the cooling tail and its touchdown flux is substantially below the touchdown flux of the second burst. This strongly argues against the classification of the bursts selected by Poutanen et al. (2014) as PRE bursts. (2014) that required selected bursts to follow trends ex- by Poutanen et al. (2014) for 4U 1608−52 using the the- pected from the bursting neutron star atmosphere mod- oretically motivated criteria without taking into account els of Suleimanov et al. (2012). The two approaches the data limitations discussed above. We showed that, led to non-overlapping data selections. Furthermore, contrary to the implicit assumption in the method, the Poutanen et al. (2014) and Kajava et al. (2014) have bursts selected are not PRE bursts. Indeed, at no point used the theoretically motivated model to argue that the during any of these bursts does the radius of the pho- data-driven selection is not reliable because it is in con- tosphere exceed the asymptotic radius measured during flict with the theoretical expectations. the cooling tails of the same bursts. Furthermore, the in- We showed that the theoretically expected trends be- ferred touchdown fluxes of these bursts is 30% below the tween different spectroscopic quantities, such as the touchdown fluxes of the securely identified PRE bursts blackbody normalization, the color temperature, and the that reach the Eddington limit. blackbody flux, get substantially smeared in the data be- The quantitative analyses presented in this paper leads cause of three effects. First, the inherent limitations of us to the conclusion that this theoretically motivated the RXTE data do not allow resolving the rapid spec- selection procedure is neither practical, given the limi- tral evolution that occurs at the end of the photospheric tations of the data, nor unbiased, given that it selects radius expansion episodes. Second, even a mild scat- bursts that are inconsistent with its own assumptions. ter in the emitting area, e.g., due to uneven burning, However, in the future, with the use of an X-ray detec- an evolving photosphere, partial obscuration of the sur- tor with a larger collecting area and significantly fewer face, and/or reflection off of the accretion disk, lead to a limitations when observing sources with high count rates, much larger scatter in the measured spectroscopic quan- this could lead to useful radius measurements. tities that mask the theoretically expected trends. Fi- nally, counting statistics lead to correlated measurement uncertainties between the spectroscopic quantities, fur- We thank all the participants of the “The Neutron Star ther smearing any trends. We quantitatively showed that Radius” conference in Montreal for helpful discussions these effects are at a level to prevent the detection of and Gordon Baym, Sebastien Guillot, and Craig Heinke theoretical trends in the majority of the burst data and for comments on the manuscript. FO¨ acknowledges sup- preclude using these criteria to argue against the data- port from NSF grant AST 1108753. TG acknowledges driven selection methods. support from Istanbul University Project numbers 49429 Finally, we studied the sub-sample of bursts selected and 48285.

REFERENCES Galloway, D. K., Muno, M. P., Hartman, J. M., Psaltis, D., & Kajava, J. J. E., Nättilä, J., Latvala, O.-M., et al. 2014, MNRAS, Chakrabarty, D. 2008, ApJS, 179, 360 445, 4218 Güver, T., Ozel,¨ F., & Psaltis, D. 2012a, ApJ, 747, 77 Majczyna, A., & Madej, J. 2005, Acta Astronomica, 55, 349 Güver, T., Psaltis, D., & Ozel,¨ F. 2012b, ApJ, 747, 76 Majczyna, A., Madej, J., Joss, P. C., & Ró˙zańska, A. 2005, A&A, 430, 643 Touchdown Moment

It is impossible to resolve the rapid change of the color correction factor around touchdown Özel et al. 2015 The Astrophysical Journal,747:77(12pp),2012March1 Guver,¨ Ozel,¨ & Psaltis

250 1.2 4U 1636-536 4U 1636-536 ) 8.19•10 7.00•10 ) 1.0 -2 6.40•10 -8 -8 2 10kpc 200

-8 cm /D 5.21•10 1.41•10 4.62•10 -1 2 km 5.81•10 0.8

-8 -7

-8 1.29•10 -8 10% -7

150 1.18•10 erg s 1.06•10 0.6 -8

-7 -7

9.38•10 8.19•10 -8 0.4 Touchdown Flux Measurements100 -8 5% 7.00•10 6.40•10

-8 Width (10 5.81•10 -8 Normalization (R -8 3.43•10 2.83•10 0.2 2.24•10 -8 4.62•10 5.21•10 -8 4.02•10 -8 -8 -8 50 -8 2.2 2.4 2.6 2.8 3.0 6.0 6.5 7.0 7.5 8.0 -8 -1 -2 kTBB (keV) Touchdown Flux (10 erg s cm ) The Astrophysical Journal,747:77(12pp),2012March1 Guver,¨ Ozel,¨ & Psaltis

250 1.2 180 1.2 4U 1636-536 4U 1636-536 4U 1728-34 4U 1728-34 ) )

8.19•10 ) 7.00•10 ) 160 1.0

1.0 -2 -2 1.04•10 6.40•10 -8 8.83•10 2 -8 10kpc 2 10kpc 200 -7 -8 8.05•10 -8 1.82•10 cm /D cm /D 140 5.21•10 1.41•10 -1 4.62•10 -1 2 km -8 -7 0.8 2 km 5.81•10 0.8 7.27•10 10% -7 -8 1.67•10 -8 -8 1.29•10 -8

12010% -7 -7 erg s 150 1.18•10 erg s 1.51•10 0.6 1.06•10 0.6 -8 -8 1.35•10 -7 -7 -7 -7 9.38•10 100 1.20•10 8.19•10 -8 5% 0.4 -7 0.4 -8 5% 1.04•10 100 80 -7 7.00•10 6.40•10 Width (10 Width

-8 Width (10 5.81•10 -8 4.92•10 Normalization (R Normalization (R 7.27•10 8.83•10 -8 4.14•10 6.48•10 3.43•10 0.2 3.35•10 -8 8.05•10 0.2 5.70•10 -8 -8 2.83•10 -8 2.24•10 -8 4.62•10 5.21•10 -8 -8 -8 4.02•10 -8 -8 -8 -8 -8 60 50 -8 2.2 2.4 2.6 2.8 3.0 6.0 6.52.6 2.87.0 3.07.5 3.2 8.03.4 7.0 7.5 8.0 8.5 9.0 9.5 10.0 -8 -1 -2 -8 -1 -2 kTBB (keV) Touchdown Flux (10kTBB erg (keV) s cm ) Touchdown Flux (10 erg s cm ) Figure 6. Left panels: 68% and 95% confidence contours of the blackbody normalization and temperature obtained from fitting the X-ray spectra at the touchdown moments of each PRE burst observed from 4U 1636 536 and 4U 1728 34. The dotted red lines show contours of constant bolometric flux. Right panels: 68% and 180 1.2 − − 4U 1728-34 95% confidence contour4U of the1728-34 parameter of an assumed underlying Gaussian distribution of touchdown fluxes. The width of the underlying distribution reflects the systematic uncertainty in the measurements. The dashed red lines show the width when the systematic uncertainty is 5% and 10% of the mean touchdown flux. ) 160 ) 1.0 Güver et al. 2012b

(A color-2 version of this ﬁgure is available in the online journal.) 1.04•10 8.83•10 2 10kpc

-7 8.05•10 -8 1.82•10 cm /D 140 -1

2 km -8 -7 0.8 7.27•10 10% 1.67•10 For each burst, the bolometric flux at touchdown is obtained variations in the Compton upscattering in the converging inflow -8 120 -7 from the combination of the blackbody temperature and normal- prior to touchdown are some of the physical mechanisms that 1.51•10 erg s 0.6 1.35•10 ization.-8 Figure 6 shows the 68% and 95% confidence contours can contribute to the intrinsic spread. -7 100 -7 of the blackbody normalization and temperature inferred from For the high temperatures observed during the touchdown 1.20•10 fitting the X-ray5% spectra obtained during the touchdown moment phases of the bursts, most of the burst spectrum falls within the -7 0.4 1.04•10 80 -7 for 4U 1728 34 and 4U 1636 536. We also plot in these fig- RXTE energy range, resulting in bolometric corrections that are

Width (10 Width − − 4.92•10 ures contours of constant bolometric flux, shown as dotted (red) at most 7% (Galloway et al. 2008a). Therefore, any uncertainties Normalization (R 7.27•10 8.83•10 4.14•10 6.48•10 3.35•10 -8 8.05•10 0.2 5.70•10 -8 -8 -8 -8 -8 60 -8 -8 lines. Even though the uncertainties in the normalization and in the bolometric correction can only introduce minimal spread temperature are correlated, the bolometric flux in each burst is to the width of the observed touchdown fluxes. Uncertainties in 2.6 2.8 3.0 3.2 3.4 well constrained.7.0 Furthermore,7.5 8.0 as8.5 Figure9.06 shows,9.5 the individual10.0 the determination of the touchdown moment are also expected -8 -1 -2 kTBB (keV) confidence contoursTouchdown from each Flux burst (10 appear erg tos be cm in very) good to be of the same magnitude since the fluxes in the nearby Figure 6. Left panels: 68% and 95% confidence contours of the blackbody normalizationstatistical agreement and temperature with obtained each other from fitting for both the X-ray sources. spectra at the touchdowntime bins differ typically by less than 10% (see, e.g., Figures 2 moments of each PRE burst observed from 4U 1636 536 and 4U 1728 34. TheThe dotted distribution red lines show of contours inferred of bolometric constant bolometric fluxes flux. at Righttouchdown panels: 68% andand 4). Our 10% limit on the pre-burst persistent flux bounds the 95% confidence contour of the parameter of an assumed− underlying Gaussian− is distribution expected of to touchdown have a fluxes. finite The width width both of the because underlying of distribution measure- reflects theuncertainties introduced by our subtraction of the background. systematic uncertainty in the measurements. The dashed red lines show the widthment when uncertainties the systematic and uncertainty because is of5% the and possible 10% of the variations mean touchdown in the flux. WecanalsoestimatetheexpectedvariationsduetotheCompton (A color version of this figure is available in the online journal.) physical conditions that determine the emerging flux during a upscattering in the converging flow: this effect scales as v/c and PRE burst. The measurement uncertainties include formal un- can, therefore, introduce an uncertainty at most of the order certainties from counting statistics, the uncertainties in the bolo- of 10% (van Paradijs & Stollman 1984). On the other hand, For each burst, the bolometric flux at touchdown is obtainedmetric correction,variations in the the subtraction Compton upscattering of the background in the converging emission, inflowvariations in the isotropy or the composition of the bursts can, from the combination of the blackbody temperature and normal-and theprior determination to touchdown of the are touchdown some of the moment. physical Anisotropies mechanisms thatin principle, generate larger spread in the touchdown fluxes. ization. Figure 6 shows the 68% and 95% confidence contoursin the bursts,can contribute variations to the in the intrinsic composition spread. and the reflection Our goal is to quantify the widths of the underlying dis- of the blackbody normalization and temperature inferred fromoff the accretionFor the flow high (e.g., temperatures Galloway observed et al. 2004 during, 2006 the), touchdown and tributions of touchdown fluxes, which we will call systematic fitting the X-ray spectra obtained during the touchdown moment phases of the bursts, most of the burst spectrum falls within the for 4U 1728 34 and 4U 1636 536. We also plot in these fig- RXTE energy range, resulting in bolometric corrections that are − − ures contours of constant bolometric flux, shown as dotted (red) at most 7% (Galloway et al. 2008a). Therefore, any uncertainties6 lines. Even though the uncertainties in the normalization and in the bolometric correction can only introduce minimal spread temperature are correlated, the bolometric flux in each burst is to the width of the observed touchdown fluxes. Uncertainties in well constrained. Furthermore, as Figure 6 shows, the individual the determination of the touchdown moment are also expected confidence contours from each burst appear to be in very good to be of the same magnitude since the fluxes in the nearby statistical agreement with each other for both sources. time bins differ typically by less than 10% (see, e.g., Figures 2 The distribution of inferred bolometric fluxes at touchdown and 4). Our 10% limit on the pre-burst persistent flux bounds the is expected to have a finite width both because of measure- uncertainties introduced by our subtraction of the background. ment uncertainties and because of the possible variations in the WecanalsoestimatetheexpectedvariationsduetotheCompton physical conditions that determine the emerging flux during a upscattering in the converging flow: this effect scales as v/c and PRE burst. The measurement uncertainties include formal un- can, therefore, introduce an uncertainty at most of the order certainties from counting statistics, the uncertainties in the bolo- of 10% (van Paradijs & Stollman 1984). On the other hand, metric correction, the subtraction of the background emission, variations in the isotropy or the composition of the bursts can, and the determination of the touchdown moment. Anisotropies in principle, generate larger spread in the touchdown fluxes. in the bursts, variations in the composition and the reflection Our goal is to quantify the widths of the underlying dis- off the accretion flow (e.g., Galloway et al. 2004, 2006), and tributions of touchdown fluxes, which we will call systematic

6 Uncertainties in the touchdown ﬂux measurements

AQL X-1

SAX J1748.9-2021

SAX J1750.8-2900

4U 1735-44

4U 1731-260

4U 1728-34

4U 1724-307

4U 1636-536

4U 0513-401

-40 -20 0 20 40 Systematic Uncertainty in Apparent Radius (%) Güver et al. 2012b Recent measurements

Özel et al. 2016 Systematics of Cooling Neutron Star Mass-Radius Measurements

X7 and X5 in 47 Tuc (Heinke et al. 2003, 2006), U24 in NGC 6397 (Guillot et al. 2011; Heinke et al. 2014), source 26 in M28 (Becker et al. 2003; Servillat et al. 2012), NGC 2808 (Webb & Barret 2007; Servillat et al. 2008), M13 (Gendre et al. 2003a; Webb & Barret 2007; Catuneanu et al. 2013), ω Centauri (Rutledge et al. 2002; Gendre et al. 2003b; Heinke et al. 2014), and M30 (Lugger et al. 2007; Guillot & Rutledge 2014).

Possible Sources of Systematic Uncertainties

Atmospheric Composition

Non-thermal Component

Interstellar Extinction 14

Fig. 10.— The 68% conﬁdence contours in mass and radius for the quiescent neutron star in ! Cen, inferred by Heinke et al. (2014; H14) and by Guillot & Rutledge (2014; G14) using di↵Currenterent assumptions regarding Mass-Radius the interstellar extinction (wabs: Measurements Morrison & McCammon 1983; tbabs: Wilms et al. 2000), the presence of a power-law spectral component, and for di↵erent distances to the globular cluster (4.8 kpc vs. 5.3 kpc) .

Fig. 11.— The combined constraints at the 68% conﬁdence level over the neutron star mass and radius obtained from (Left) all neutronÖzel et al. 2016 stars with thermonuclear bursts (Right) all neutron stars in low-mass X-ray binaries during quiescence.

Heinke et al. (2014) also explored the e↵ect of assuming di↵erent spectral indices in modeling the power-law component. Even though the low counts preclude an accurate measurement of this parameter, the specific value has a small e↵ect on the radius measurement, which can be folded in as a sytematic uncertainty. Finally, because of the low temperature of the surface emission from qLMXBs, the spectral modeling is a↵ected significantly by the assumed model of the interstellar medium to account for the low-energy extinction. Heinke et al. (2014) explored di↵erent models for the interstellar extinction in their analysis of the qLMXBs in ! Cen and NGC 6397 and found statistically consistent results, with small di↵erences in the central values but larger di↵erences in the uncertainties. In the left panel of Figure 10, we show the e↵ect of di↵erent assumptions on the power-law index, the distance, and the interstellar extinction model on the inference of the mass and radius of the neutron star in !Cen. In particular, one of the larger e↵ects arises from the use of the two common interstellar extinction models they consider (the earlier Morrison & McCammon 1983 model with solar abundances, referred to as wabs in the spectral fitting software XSPEC, and the more recent Wilms et al. 2000 model, with ISM abundances from the same paper, referred to as tbabs with wilms in XSPEC). The wabs model (employed by Guillot et al. 2013) leads to somewhat larger radii for the same distance. In the present study, we repeat the analysis of Guillot et al. (2013) individually for all the sources in M13, M28, NGC 6304, NGC 6397, M30, and !Cen. (Note that for the last two sources, the observations were reported in Guillot & Rutledge 2014). In all of the spectral fits, we allow for a power-law component with a fixed photon index =1but The Astrophysical Journal, 820:28 (25pp), 2016 March 20 Özel et al.

Table 2 Properties of Quiescent LMXBs

a b c d Source NH kTeff P.L. Norm. Distance Radius (10cm22- 2)(eV)(10----7112 keV s cm )(kpc)(km) +0.04 +27 +3.6 M13 0.02-0.02p 81-12 4.2-3.1p 7.1±0.4 (1) 10.9±2.3 +0.03 +35 +4.9 M28 0.30-0.03 128-13 8.3-4.7p 5.5±0.3 (2) 8.5±1.3 +0.03 +30 +5.4 M30 0.02-0.02p 96-13 9.3-5.3p 9.0±0.5 (3), (4) 11.6±2.1 +0.04 +24 +1.3 ωCen 0.15-0.04 80-10 0.8-0.7p 4.59±0.08 (5), (6) 9.4±1.8 +0.15 +33 +2.7 NGC6304 0.49-0.13 100-17 2.4-1.9p 6.22±0.26 (7) 10.7±3.1 +0.02 +17 +1.8 NGC6397 0.14-0.02 66-7 3.3-1.8 2.51±0.07 (8) 9.2±1.8

Notes. a NGC6397 was ﬁtted with a Helium atmosphere model (nsx in XSPEC). b p indicates that the posterior distribution did not converge to zero probability within the hard limit of the model. c References: (1Comparison)Harris (1996, 2010 revision), (2)Servillat et al.with(2012), (3)Carretta Quiescent et al. (2000), (4)Lugger et al. (2007 ), (LMXBs5)Watkins et al. (2013), (6)see also the discussion in Heinke et al. (2014), (7)Guillot et al. (2013) and references therein, (8)Heinke et al. (2014). d The radius and its 68% uncertainty obtained by marginalizing the mass–radius likelihood of each source over the observed mass distribution, as in Figure 12.

Figure 11. The combined constraints at the 68% conﬁdence level over the neutron star mass and radius obtained from (left) all neutron stars with thermonuclear bursts (right) all neutron stars in low-mass X-ray binaries during quiescence. et al. 2012b). Speciﬁcally, we write N PR( ∣ data)(= CÖzelò P etip RMal. , 2016∣ data)() P MdM () 11 i=1 where C is an appropriate normalization constant, PRMi ( ,∣ data) is the two-dimensional posterior likelihood over mass and radius for each of the N sources (as given, e.g., in Equation (9) for the bursters), and Pp(M) is the Gaussian likelihood with a mean of 1.46M and a dispersion of 0.21M for the mass distribution inferred by Özel et al. (2012b) for the descendants of these systems. The left panel of Figure 12 shows the individual terms of the product in the equation above; i.e., the posterior likelihoods over radius for each of the 12 sources. They are all well approximated by Gaussian distributions that peak between 9 and 12 km and typical uncertainties of ∼2 km. The right panel of Figure 12 shows the posterior likelihood over the single radius in this mono- parametric equation of state, which is peaked at a radius of 10.3 km with an uncertainty of 0.5 km. As expected, given that all radii are statistically consistent with each other, combining the data of the 12 sources led to a reduction in the uncertainty by a factor of 12 3.5. The result is a level of uncertainty that is comparable to what is required to severely constrain the neutron star equation of state, as we will show in detail in the next section.

5. THE NEUTRON STAR EQUATION OF STATE FROM RADII AND LOW-ENERGY EXPERIMENTS We now make use of the one-to-one mapping between the neutron star mass–radius relation and the pressure-density relation of cold dense matter to put direct constraints on the neutron-star equation of state. In this procedure, we take the most general approach

15 Mass-Radius Constraints for X7 and X5 in 47 Tuc 9

TABLE 2 Summary of mass-radius measurements for X7 and X5 Mass-Radius Constraints for X7 and X5 in 47 Tuc 7

a b b 2 NH Teff MRχ /d.o.f. 20 −2 6 Model (10 cm )(10K) (M⊙)(km) X5 nsatmos +0.21 +1.27 +1.3 <4.9 2.74−1.48 0.5−0 10.5−10.5 95.1/88 +0.30 +1.7 <5.2 2.61−1.23 (1.4) 9.7−2.0 95.2/89 +1.56 +0.62 <5.0 1.40−0.15 0.84−0.35 (12) 96.6/89 X7 nsatmos +1.50 +0.28 +1.8 <2.2 1.39−0.19 1.46−1.46 10.8−10.8 88.2/70 +1.32 +0.8 <2.3 1.39−0.09 (1.4) 11.0−0.7 88.2/71 +0.06 +0.42 <2.3 1.28−0.08 1.09−1.09 (12) 88.7/71 nsx +1.07 +1.89 +2.3 (He) <3.3 1.06−0.06 0.50−0.50 14.8−0.6 82.9/70 +0.03 +1.7 <3.3 1.18−0.07 (1.4) 14.5−0.9 83.0/71 +0.94 +0.31 <3.3 1.20−0.19 2.01−0.16 (12) 83.2/71 a 20 −2 For NH,thelowerboundinthefitswasfixedto1.3 × 10 cm , the value for 47 Tuc. All fits reached this lower limit so only the 90% confidence upper bound is quoted. b The values quoted are redshift-corrected, i.e., as measuredattheneu- Fig. 6.— The total Chandra ACIS-S subarray spectrum of X5 X7tron star surface.- X5 in 47 Tuc nsatmos c All quoted uncertainties correspondfitted with to 90% an confidenceabsorbed H level atmosphere.Valuesin model ( )convolved parentheses were held fixed duringwith the a pile-up fit. model (top) and the best-fit residuals (bottom).

Fig. 8.— The total Chandra ACIS-S subarray spectrum of X7 ﬁtted with an absorbed atmosphere model convolved with a pile-up model. The bottom panel shows the best ﬁt residuals expressedin terms of σ.

3.7. Photon Pile-up As noted previously, Chandra ACIS data of even mod- erately bright sources is susceptible to severe event pile- up owing to the combination of slow readout and high count rate. While we mitigated most of this negative effect in the new subarray observations by using a faster detector readout mode, it still affects the data at a low level. The spectroscopic mass-radius measurement technique for qLMXBs is highly sensitive to the shape of the thermal spectrum. As a consequence, even a small artif- ical distortion in the spectral shape can bias the M − R measurement. For pile-up specifically, a portion of the photons piled at lower energies are either rejected entirely (if their event grades are consistent with those of Fig. 9.— The mass-radius+0.8 constraints obtained for X7 from theFig.Chandra 7.— TheACIS-S mass-radius+0.9 subarray constraints data assuming obtained the fornsatmos X5 byHatmosphere fitting cosmic rays) or recorded as a singe photon displaced to (left) and nsx11.1He atmosphere-0.7 (right) km models. @ 68%1.4 and M 95%⊙ confidencethe Chandra contour9.6ACIS-Ssareshownobtainedfromtheposteriorlikelihoodover 1/8 subarray-1.1 km data @ with 1.4 a piled-up M and⊙ ab- Mhigher energies. As reported in §2, this occurs for ∼1% and R (see text). A 3% systematic uncertainty in the calibrationsorbed and a hydrogen model for atmosphere 1% pile-up modelare included (nsatmos in the). The analyses. 68% and See 95% Table 2 for the best fit parameters. confidence levels are shown, obtained from the posterior likelihood of the photons in the X7 and X5 spectra, which we would over M and R (see text). The blue and red lines correspond, re- naively expect to be negligible. spectively, to the fits with and withoutBogdanov a 3% systematic et al. uncer 2016tainty To assess the impact of this seemingly small effect 2013). The low temperature of SAX J1808.4−3658 sug-instrumentaling processes calibration based uncertainties. on other The qLMXBs, shaded gray while area in a the large ra- gests it has a higher mass than that of other quies-upper leftdius marks (>14 the km)region would excluded conflict by causality both constrain with thets. See measure-on our results, we repeated our spectroscopic fits with cent NSs, with the latter presumably near the typi-Table 2ments for best of fit the parameters. radius of X5 presented here, and with mea-and without a pile-up model component (i.e., pileup in cal mass of 1.4 M⊙ and current evidence pointing to surements of other NSs (Guillot et al. 2011; Heinke et al.XSPEC). Figure 3 illustrates the results for X7, showing the 68% confidence contours in the M − R plane with SAX J1808.4−3658 having MNS ! 1.6M⊙ (at ∼2σ) 2014; Ozel¨ et al. 2015), including the radius derived for (Wang et al. 2013). In comparison, the relatively high and without pile-up. Despite the small degree of pile- ACIS detectorsthe NS in is NGC limited 6397 by for a combination either H or ofHe the atmospheres. un- temperature of X7 suggests it has a comparable, if not up, the impact on the results is substantial. In addition certaintiesBased in on the this quantum line of reasoning, efficiency nearwe argue the thatread a out, hydrogen lower, mass, i.e., around 1.4M⊙ or lighter. This is con- to enlarging the confidence intervals, correcting for pile- the quantumcomposition efficiency for the non-uniformity atmosphere of across X7 is the more detec- plausible. trary to our He atmosphere fits which yield M "1.7 M⊙ up displaces them towards somewhat larger M and R. (see the right panel of Figure 9) and would result intor a resulting from charge transfer inefficiencies, and the This arises because photon pile-up artificially hardens depth of the5. contaminant on the ACIS filter (important X7 temperature that is at least as low as that of SAX IMPLICATIONS FOR THE NEUTRON STAR the intrinsic source spectrum, which produces a higher ∼ J1808.4−3658, contrary to observations. primarily below 2keV).EQUATION OF STATE best-fit temperature and hence a smaller inferred stellar Following Guillot et al. (2013), we adopt a 3% system- We find that a He atmosphere model would require ei- Most existing mass-radius measurements of neutronradius. The enlargement of the confidence contours arises atic error to account for the instrument response uncer- ther a quite large radius or a high mass. A high mass star have uncertainties that are too large to offer usefulprincipally from the statistical uncertainty in the pile-up tainties. We note that an in-depth evaluation of Chan- would be at odds with our current understanding of cool- constraints when considered individually. Nevertheless,parameter α, which gives the probability of rejection of dra calibration uncertainties in the context of qLMXB piled events. Some of the displacement in the contours NS M − R measurements based on the prescriptions may also be due to adding another parameter to the fit by Drake et al. (2006), Lee et al. (2011), and Xu et al. when applying pile-up correction. In light of these find- (2014) will be presented in a subsequent publication. ings, to ensure robust constraints on the NS M and R, Radius16 measurements assuming a mass distribution

Fig. 12.— (Left) The posteriorÖzel likelihood et al. 2016 over the radius obtained by marginalizing the two dimensional likelihoods over the neutron star mass, with a prior equal to the observationally inferred distribution of recycled pulsar masses, for all twelve sources in our sample. The peak probabilities are highly clustered in the 9-12 km range. (Right) The combined posterior likelihood assuming that all sources in our sample have the same radius and masses drawn from the observationally inferred distribution of recycled pulsar masses. We use this inferrence only as an illusration of the fact that using radius measurements for twelve sources leads to a highly accurate constraint on the neutron-star equation of state.

In Ozel¨ et al. (2010), we used the framework devised in Ozel¨ & Psaltis (2009) to convert the mass-radius measurements of three sources to posterior likelihoods over the pressures at these three ﬁducial densities. In order to incorporate the mass-radius measurements of the twelve sources presented in Section 3, we will follow here the Bayesian approach outlined below. (See Steiner et al. 2010 for a similar Bayesian inference approach.) To calculate the posterior likelihood over the pressures P1(⇢1), P2(⇢2), and P3(⇢3)usingthelikelihoodsPi(M,R) for twelve sources, we write P (P ,P ,P data) = CP(data P ,P ,P )P (P 1)P (P )P (P ), (12) 1 2 3 | | 1 2 3 p p 2 p 3 where Pp(P 1), Pp(P2), and Pp(P3) are the priors over the three pressures and

N P (data P ,P ,P )= P (M ,R P ,P ,P ) (13) | 1 2 3 i i i | 1 2 3 i=1 Y To obtain (Mi,Ri) from the pressures P1,P2,P3, we also need to specify, and marginalize over, the central density of the star ⇢c,i.e., 1 P (M ,R P ,P ,P )=C P (M ,R P ,P ,P , ⇢ )P (⇢ )d⇢ . (14) i i i | 1 2 3 1 i i i | 1 2 3 c p c c Z0 Because there is a one-to-one correspondence between the central density ⇢c and mass, we can write the integral over the mass instead as

Mmax P (M ,R P ,P ,P )=C P (M,R(M) P ,P ,P )P (M)dM, (15) i i i | 1 2 3 2 i | 1 2 3 p ZMmin where we take Mmin to be 0.1M and Mmax to be the maximum mass for the equation of state specified by that P1,P2,P3 triplet. Here, Pp(M) is the prior likelihood over the mass of each neutron star, which we take to be constant. We use a variety of physical and observational constraints to define the priors on P1,P2, and P3. (i) We require that the equation of state be microscopically stable, i.e., P3 P2 P1, and that P1 be greater than or 14 3 equal to the pressure of matter at ⇢0 = 10 g cm that is specified by the SLy equation of state (see Ozel¨ & Psaltis 2009). (ii) We impose the physically plausible condition of causality that @P c2 = c2 (16) s @✏ 

when evaluated at all three ﬁducial densities; here, cs is the sound speed and ✏ is the energy density. (iii) We require that the maximum stable mass for each equation of state corresponding to a P1,P2,P3 triplet exceeds Resulting Observational EoS Mass-Radius Constraints for X7 and X5 in 47 Tuc 11 14 measurements lead to a neutron star of 9.9 - 11.2 km @ M = 1.5 M⊙

Bogdanov et al. 2016 Fig. 10.— The mass-radius relation (solid blue curve) corresponding to the most likely triplet of pressures that agrees with the current neutron star data. These include the X5 and X7 radius measurements shown in this work, as well the neutron star radii measurements for the twelve neutron stars included in Ozel¨ et al. (2015), the low energy nucleon-nucleon scattering data, and the requirement that the EoS allow for a M>1.97M⊙ neutron star. The ranges of mass-radius relations corresponding to the regions of the (P1,P2,P3)parameter space in which the likelihood is within e−1/2 and e−1 of its highest value are shown in dark and light blue bands, respectively. The results for both ﬂat priors in P1,P2, and P3 (left panel) and for ﬂat priors the logarithms of these pressures (right panel) are shown.

existing M −R measurements from other qLMXB as well ments. This work was funded in part by NASA Chandra as bursting neutron stars, we obtain increasingly more grants GO4-15029A and GO4-15029B awarded through robust constraints on the neutron star equation of state. Columbia University and the University of Arizona and Specifically, we find that the preferred equation of state issued by the Chandra X-ray Observatory Center, which that is empirically derived from the measurements of all is operated by the Smithsonian Astrophysical Observa- fourteen sources predicts radii between 9.9 and 11.2 km tory for and on behalf of NASA under contract NAS8- around M =1.5M⊙ (corresponding to the range where 03060. COH acknowledges support from an NSERC Dis- the likelihood falls to e−1 of its maximum value). This covery Grant and a Humboldt Fellowship, and the hos- also implies a relatively low pressure around twice nu- pitality of the Max Planck Institute for Radio Astron- clear saturation density, which most directly affects neu- omy, Bonn, Germany. TG was supported by Scientific tron star radii. We find that such an equation of state Research Project Coordination Unit of Istanbul Univer- can easily produce ∼ 2M⊙ neutron stars that is observed sity, Project numbers 49429 and 57321. This research through radio pulsar timing. This preferred equation of has made extensive use of the NASA Astrophysics Data state is softer than some purely nucleonic equations of System (ADS), the arXiv, and software provided by the state that are tuned to fit experiments at low densities, Chandra X-ray Center (CXC) in the application package such as AP4, and may point to new degrees of freedom CIAO. appearing around ∼ 2 ρsat in neutron-rich matter. Facilities: CXO (ACIS).

We thank W.C.G. Ho for numerous helpful com-

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We have a better understanding on the systematic effects in the measurements of masses and radii of neutron stars but our measurements still rely on several assumptions. Independent measurements are necessary to conﬁrm these results.

NICER mission will allow us to both continue these measurements but also allow for measurement using pulse proﬁles, providing an independent estimate.

Better estimates on the distances from GAIA

More and deeper observations with Chandra are needed to obtain high signal to noise ratio data of LMXBs in quiescence.

Further observations especially in the 0.5 - 25 keV range should be performed and an archive like the RXTE/PCA has should be established.