Neutron Stars and Dense Matter - Arizona State University

Neutron Stars and Dense Matter

Edward Brown Michigan State University

Joint Institute for Nuclear Astrophysics

artwork courtesy T. Piro

© 1938 Nature Publishing Group




How are they formed?

Massive Star | Final Days

Essential Cosmic Perspective, Bennett et al.

Stellar Death

Pressure in core becomes too high, electrons capture onto protons:

This forces core to contract, making pressure even higher…

Collapse stops when nuclear density reached

F+ Q → O+ ν̄F

1987a | neutrinos detected

Hirata et al. ’88Hirata et al. ’88

1987a | neutrinos detected

Hirata et al. ’88Hirata et al. ’88

neutron star | basics

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ideal gas | Mmax = 0.75 Msun

FEBRUARY 15, 1939 PH YS ICAL REVIEW VOLUM E 55

On Massive Neutron Cores

J. R, OPPENHEIMER AND G. M. VOLKOFFDepartment of Physics, University of California, Berkeley, California

(Received January 3, 1939)

It has been suggested that, when the pressure within stellar matter becomes high enough,a new phase consisting of neutrons will be formed. In this paper we study the gravitationalequilibrium of masses of neutrons, using the equation of state for a cold Fermi gas, and generalrelativity. For masses under —,Q only one equilibrium solution exists, which is approximatelydescribed by the nonrelativistic Fermi equation of state and Newtonian gravitational theory.For masses —,'Q &m&-,'Q two solutions exist, one stable and quasi-Newtonian, one morecondensed, and unstable. For masses greater than 4 Q there are no static equilibrium solutions.These results are qualitatively confirmed by comparison with suitably chosen special casesof the analytic solutions recently discovered by Tolman. A discussion of the probable eEectof deviations from the Fermi equation of state suggests that actual stellar matter after theexhaustion of thermonuclear sources of energy will, if massive enough, contract indefinitely,although more and more slowly, never reaching true equilibrium.

I. INTRoDUcTIQN~OR the application of the methods commonly

used in attacking the problem of stellarstructure' the distribution of energy sources andtheir dependence on the physical conditionswithin the star must be known. Since at the timeof Eddington's original studies not much wasknown about the physical processes responsiblefor the generation of energy within a star,various mathematically convenient assumptionswere made in regard to the energy sources, andthese led to different star models (e.g. theEddington model, the point source model, etc.).It was found that with a given equation of statefor the stellar material many important propertiesof the solutions (such as the mass-luminositylaw) were quite insensitive to the choice ofassumptions about the distribution of energysources, but were common to a wide range ofmodels.In 1932 Landau' proposed that instead of

making arbitrary assumptions about energysources chosen merely for mathematical con-venience, one should attack the problem by firstinvestigating the physical nature of the equi-librium of a given mass of material in which noenergy is generated, and from which there is noradiation, presumably in the hope that such an~A. Eddington, The Internal Constitution of the Stars

(Cambridge University Press, 1926); B. Stromgren,Ergebn. Exakt. Naturwiss. 10, 465 (1937);Short summaryin G. Gamow, Phys. Rev. 53, 595 (1938).' L. Landau, Physik. Zeits. Sowjetunion 1, 285 (1932).

3

investigation would afford some insight into themore general situation where the generation ofenergy is taken into account. Although such amodel gives a good description of a white dwarfstar in which most of the material is supposed tobe in a degenerate state with a zero point energyhigh compared to thermal energies of even 10'degrees, and such that the pressure is determinedessentially by the density only and not by thetemperature, still it would fail completely todescribe a normal main sequence star, in whichon the basis of the Eddington model the stellarmaterial is nondegenerate, and the existence ofenergy sources and of the consequent temperatureand pressure gradients plays an important part indetermining the equilibrium conditions. Thestability of a model in which the energy sourceshave to be taken into account is known to dependalso on the temperature sensitivity of the energysources and on the presence or absence of atime-lag in their response to temperature changes.However, if the view which seems plausible atpresent is adopted that the principal sources ofstellar energy, at least in main sequence stars, arethermonuclear reactions, then the limiting caseconsidered by Landau again becomes of interestin the discussion of what will eventually happento a normal main sequence star after all theelements available for thermonuclear reactionsare used up. Landau showed that for a modelconsisting of a cold degenerate Fermi gas thereexist no stable equilibrium configurations for

74

ideal gas | Mmax = 0.75 Msun










3


74










3


74

with nuclear eq. of state | Mmax > 2 Msun

What can neutron stars teach us about dense

matter?

Interest to Nuclear Physics

NSAC report ‘07

(SOM)]. Different theoretical formulationsconcerning the energy density would lead todifferent pressures (that is, to different EOSsfor nuclear matter) in the equilibrium limit, inthese simulations, and in the actual collisions.

At an elapsed time of 3 ! 10"23 s in thereaction, the central density (in Fig. 1b#) ex-ceeds 3 $0. The corresponding back panel,labeled (b), indicates a central pressure great-er than 90 MeV/fm3 (1 MeV/fm3 % 1.6 !1032 Pa; that is, 1.6 ! 1027 atmospheres).These densities and pressures are achieved byinertial confinement; the incoming matterfrom both projectile and target is mixed andcompressed in the high-density region wherethe two nuclei overlap. Participant nucleonsfrom the projectile and target, which followsmall impact parameter trajectories (at x,y &0), contribute to this mixture by smashinginto the compressed region, compressing itfurther. The calculated transverse pressure inthe central region reaches ' 80% of its equi-librium value after ' 4 ! 10"23 s (Fig. 1c#)and is equilibrated for the later times in Fig.1. Equilibrium is lost at even later times, butonly after the flow dynamics are essentiallycomplete.

Spectator nucleons, which are those thatavoid the central region by following largeimpact parameter trajectories (with large !x!( 6 fm), initially block the escape of com-pressed matter along trajectories in the reac-tion plane and force the matter to flow out ofthe compressed region in directions perpen-dicular to the reaction plane (Fig. 1, b to d).Later, after these spectator nucleons pass,nucleons from the compressed central regionpreferentially escape along in-plane trajecto-ries parallel to the reaction plane that are nolonger blocked. This enhancement of in-plane emission is beginning to occur to alimited extent in Fig. 1e at this incident en-ergy of 2 GeV per nucleon. This later in-plane emission becomes the dominant direc-tion at higher incident energies of 5 GeV pernucleon, where the passage time is consider-ably less. Thus, emission first develops out ofplane (along the y axis in Fig. 1) and thenspreads into all directions in the x-y plane.

The achievement of high densities andpressures, coupled with their impact on themotions of ejected particles, provide the sen-sitivity of collision measurements to theEOS. The directions in which matter expandsand flows away from the compressed regiondepend primarily on the time scale for theblockage of emission in the reaction plane bythe spectator matter and the time scale for theexpansion of the compressed matter near x &y & z & 0. The blockage time scale can beapproximated by 2R/()cmvcm), where R/)cm isthe Lorentz contracted nuclear radius, andvcm and )cm are the incident nucleon velocityand the Lorentz factor, respectively, in thecenter-of-mass reference frame. The block-

age time scale therefore decreases monoton-ically with the incident velocity. The expan-sion time scale can be approximated by R/cs

where cs % c*+P/+e is the sound velocity inthe compressed matter and c is the velocity oflight. The expansion time scale therefore de-pends (via cs) on the energy density e and onthe nuclear mean field potential U accordingto Eqs. 2 and 3 and the associated discussion.This provides sensitivity to the density de-pendence of the mean field potential, which isimportant because uncertainties in the densitydependence of the mean field make a domi-nant contribution to the uncertainty in theEOS. More repulsive mean fields lead tohigher pressures and to a more rapid expan-sion when the spectator matter is still present.This causes preferential emission perpendic-ular to the reaction plane where particles canescape unimpeded. Less repulsive meanfields lead to slower expansion and preferen-tial emission in the reaction plane after thespectators have passed.Analyses of EOS-dependent observ-

ables. The comparison of in-plane to out-of-plane emission rates provides an EOS-depen-dent experimental observable commonlyreferred to as elliptic flow. The sidewaysdeflection of spectator nucleons within thereaction plane, due to the pressure of thecompressed region, provides another observ-able. This sideways deflection or transverseflow of the spectator fragments occurs pri-marily while the spectator fragments are ad-jacent to the compressed region, as shown inFig. 1b’ to 1d’. The velocity arrows in Fig.1d’ and 1e’ suggest that the changes in thenucleon momenta that result from a sideways

deflection are not large. However, thesechanges can be extracted precisely from theanalysis of emitted particles (31). In general,larger deflections are expected for more re-pulsive mean fields, which generate largerpressures; and conversely, smaller deflec-tions are expected for less repulsive ones.

In terms of the coordinate system in Fig.1, matter to the right (positive x) of thecompressed zone, originating primarily fromthe projectile, is deflected along the positive xdirection; and the matter to the left, from thetarget, is deflected to the negative x direction.Experimentally, one distinguishes spectatormatter from the projectile and the target bymeasuring its rapidity y, a quantity that in thenonrelativistic limit reduces to the velocitycomponent vz along the beam axis (35). Forincreasing values of the rapidity, the meanvalue of the x component of the transversemomentum increases monotonically (12, 14–16, 31). Denoting this mean transverse mo-mentum as , px( and corresponding trans-verse momentum per nucleon in the detectedparticle as , px/A( , we find that larger valuesfor the pressure in the compressed zone, dueto more repulsive EOSs, lead to larger valuesfor the directed transverse flow F defined(12) by

F !d-px/A.

d/ y/ycm0"

y/ycm ! 1

(4)

where ycm is the rapidity of particles at rest inthe center of mass and A is the number ofnucleons in the detected particle. (F can beviewed qualitatively as the tangent of themean angle of deflection in the reactionplane. Larger values for F correspond to larg-

Fig. 1. Overview ofthe dynamics for aAu 1 Au collision.Time increases fromleft to right, the cen-ter of mass is at r % 0,and the orientation ofthe axes is the samethroughout the figure.The trajectories ofprojectile and targetnuclei are displacedrelative to a “head-on” collision by an im-pact parameter of b %6 fm (6 ! 10"13 cm).The three-dimensionalsurfaces (middle pan-el) correspond to con-tours of a constantdensity $ ' 0.1 $0. The magenta arrows indicate the initial velocities of the projectile and target(left panel) and the velocities of projectile and target remnants following trajectories that avoid thecollision (other panels). The bottom panels show contours of constant density in the reaction plane(the x-z plane). The outer edge corresponds to a density of 0.1 $0, and the color changes indicatesteps in density of 0.5 $0. The back panels show contours of constant transverse pressure in the x-yplane. The outer edge indicates the edge of the matter distribution, where the pressure isessentially zero, and the color changes indicate steps in pressure of 15 MeV/fm3 (1 MeV/fm3 %1.6 ! 1032 Pa; that is, ' 1.6 ! 1027 atmospheres). The black arrows in both the bottom and theback panels indicate the average velocities of nucleons at selected points in the x-z plane and x-yplanes, respectively.

R E S E A R C H A R T I C L E S

www.sciencemag.org SCIENCE VOL 298 22 NOVEMBER 2002 1593

equation of state from heavy nucleus collisions Danielewicz et al. (2002) Science

equation of state from heavy nucleus collisions Danielewicz et al. (2002) Science

er deflections.) The open and solid points inFig. 2 show measured values for the directedtransverse flow in collisions of 197Au projec-tile and target nuclei at incident kinetic ener-gies Ebeam/A, ranging from about 0.15 to 10GeV per nucleon (29.6 to 1970 GeV totalbeam kinetic energies) and at impact param-eters of b ! 5 to 7 fm (5 " 10#13 to 7 "10#13 cm) (13–16 ). The scale at the top ofthis figure provides theoretical estimates forthe maximum densities achieved at selectedincident energies. The maximum density in-creases with incident energy; the flow dataare most strongly influenced by pressurescorresponding to densities that are somewhatless than these maximum values.

The data in Fig. 2 display a broad maxi-mum centered at an incident energy of about2 GeV per nucleon. The short dashed curvelabeled “cascade” shows results for the trans-verse flow predicted by Eq. 1, in which themean field is neglected. The disagreement ofthis curve with the data shows that a repulsivemean field at high density is needed to repro-duce these experimental results. The othercurves correspond to predictions using Eq. 1and mean field potentials of the form

U ! $a% " b%&)/[1'(0.4%/%0)&–1] ' (Up

(5)

Here, the constants a, b, and & are chosen toreproduce the binding energy and the satura-tion density of normal nuclear matter whileproviding different dependencies on densityat much higher density values, and (Up de-scribes the momentum dependence of themean field potential (28 , 33, 34 ) (see SOMtext). These curves are labeled by the curva-

ture K § 9 dp/d%)s/% of each EOS about thesaturation density %0. Calculations with largervalues of K, for the mean fields above, gen-erate larger transverse flows, because thosemean fields generate higher pressures at highdensity. The precise values for the pressure athigh density depend on the exact form chosenfor U. To illustrate the dependence of pres-sure on K for these EOSs, we show thepressure for zero temperature symmetricmatter predicted by the EOSs with K ! 210and 300 MeV in Fig. 3. The EOS with K !300 MeV generates about 60% more pres-sure than the one with K ! 210 MeV atdensities of 2 to 5 %0 (Fig. 3).

Complementary information can be ob-tained from the elliptic flow or azimuthalanisotropy (in-plane versus out-of-planeemission) for protons (24 , 25 , 36 ). This isquantified by measuring the average value* cos2+, , where + is the azimuthal angle ofthe proton momentum relative to the x axisdefined in Fig. 1. (Here, tan+ ! py/px , wherepx and py are the in-plane and out-of-planecomponents of the momentum perpendicularto the beam.) Experimental determinations of* cos2+, include particles that, in the cen-ter-of-mass frame, have small values for therapidity y and move mainly in directionsperpendicular to the beam axis. Negative val-ues for * cos2+, indicate that more protonsare emitted out of plane (+ - 90°or + -270°) than in plane (+ - 0°or + - 180°), andpositive values for * cos2+, indicate thereverse situation.

Experimental values for * cos2+, for in-cident kinetic energies Ebeam/A ranging from0.4 to 10 GeV per nucleon (78.8 to 1970 GeVtotal beam kinetic energies) and impact pa-rameters of b! 5 to 7 fm (5 x 10#13 to 7 "10#13 cm) (17 –19 ) are shown in Fig. 4. Neg-ative values for * cos2+, , reflecting a pref-erential out-of-plane emission, are observedat energies below 4 GeV/A, indicating thatthe compressed region expands while the

spectator matter is present and blocks thein-plane emission. Positive values for* cos2+, , reflecting a preferential in-planeemission, are observed at higher incident en-ergies, indicating that the expansion occursafter the spectator matter has passed the com-pressed zone. The curves in Fig. 4 indicatepredictions for several different EOSs. Cal-culations without a mean field, labeled “cas-cade,” provide the most positive values for* cos2+, . More repulsive, higher-pressureEOSs with larger values of K provide morenegative values for * cos2+, at incident en-ergies below 5 GeV per nucleon, reflecting afaster expansion and more blocking by thespectator matter while it is present.

Transverse and elliptic flows are also in-fluenced by the momentum dependencies(Up of the nuclear mean fields and the scat-tering by the residual interaction within thecollision term I indicated in Eq. 1. Experi-mental observables such as the values for* cos2+, measured for peripheral collisions,where matter is compressed only weakly andis far from equilibrated (28 ), now providesignificant constraints on the momentum de-pendence of the mean fields (21, 28 ). This isdiscussed further in the SOM (see SOM text).The available data (30 ) constrain the mean-field momentum dependence up to a densityof about 2 %0. For the calculated resultsshown in Figs. 2 to 4, we use the momentumdependence characterized by an effectivemass m* ! 0.7 mN, where mN is the freenucleon mass, and we extrapolate this depen-dence to still higher densities. We also makedensity-dependent in-medium modificationsto the free nucleon cross-sections followingDanielewicz (28 , 32) and constrain these

Fig. 2. Transverse flow results. The solid andopen points show experimental values for thetransverse flow as a function of the incidentenergy per nucleon. The labels “Plastic Ball,”“EOS,” “E877,” and “E895” denote data takenfrom Gustafsson et al. (13 ), Partlan et al. (14 ),Barrette et al. (15 ), and Liu et al. (16 ), respec-tively. The various lines are the transport the-ory predictions for the transverse flow dis-cussed in the text. %max is the typical maximumdensity achieved in simulations at the respec-tive energy.

Fig. 3. Zero-temperature EOS for symmetricnuclear matter. The shaded region correspondsto the region of pressures consistent with theexperimental flow data. The various curves andlines show predictions for different symmetricmatter EOSs discussed in the text.

Fig. 4. Elliptical flow results. The solid and openpoints show experimental values for the ellip-tical flow as a function of the incident energyper nucleon. The labels “Plastic Ball,” “EOS,”“E895,” and “E877” denote the data of Gutbrodet al. (17 ), Pinkenburg et al. (18 ), Pinkenburg etal. (18 ), and Braun-Munzinger and Stachel (19 ),respectively. The various lines are the transporttheory predictions for the elliptical flow dis-cussed in the text.

R E S E A R C H A R T I C L E S

22 NOVEMBER 2002 VOL 298 SCIENCE www.sciencemag.org1594

EOS | mass-radius

radius (km)

mas

s(M�)

7 8 9 10 11 12 13 14 15

0.0

0.5

1.0

1.5

2.0

2.5

APR98

MDI x = 0

EOS | mass-radius

radius (km)

mas

s(M�)

7 8 9 10 11 12 13 14 15

0.0

0.5

1.0

1.5

2.0

2.5

APR98

MDI x = 0

GW170817 | the multi-messenger era

www.ligo.caltech.edu

http://www.ligo.caltech.edu

www.ligo.caltech.edu

http://www.ligo.caltech.edu

How do we observe dense matter in the wild?

Detection

Detection

Pulsar masses Demorest et al. 2010

Figure 3: Neutron star (NS) mass-radius diagram. The plot shows non-rotating mass versus physical radius for several typical NS equations of state(EOS)[25]. The horizontal bands show the observational constraint from ourJ1614−2230 mass measurement of 1.97±0.04 M⊙, similar measurements fortwo other millsecond pulsars[3, 26], and the range of observed masses fordouble NS binaries[2]. Any EOS line that does not intersect the J1614−2230band is ruled out by this measurement. In particular, most EOS curves in-volving exotic matter, such as kaon condensates or hyperons, tend to predictmaximum NS masses well below 2.0 M⊙, and are therefore ruled out.

10

no radius information

detection | X-rays

accreting neutron stars

artwork courtesy T. Piro

a flare detected

Belian et al. 1969

27

28

29

nuclear-powered variability

H, He (hours–days)


Galloway et al. 2008

MINBAR catalog: A sample of 4192 X-ray bursts from 48 sources





deep crustal heating (102–103 yrs)

Crust reactions can be studied at the Facility for Rare Isotope Beams at Michigan State University

Be (4)B (5)

C (6)N (7)

O (8)F (9)

Ne (10)Na (11)Mg (12)

Al (13)Si (14)

P (15)S (16)

Cl (17)Ar (18)

K (19)Ca (20)

Sc (21)Ti (22)

V (23)Cr (24)

Mn (25)Fe (26)

Co (27)Ni (28)

Cu (29)Zn (30)

Ga (31)Ge (32)

As (33)Se (34)

Br (35)Kr (36)

Rb (37)Sr (38)

Y (39)Zr (40)

Nb (41)Mo (42)Tc (43)

Ru (44)Rh (45)

Pd (46)Ag (47)

Cd (48)

6 8 10 12 14

16 18

20

22

24 26

28

30 32 34 36

3840

42 4446

4850

52

54

5658

6062 64 66

68 70

border of experimental m

asses

FRIB reach for neutron drip

FRIB reach for masses

JINA/JINA-CEE experiments across the crust nuclear landscape

search for gs-gs pairs

charge exchange for transition strength

proposed NSCL 61V β– decay study

fusion reactions towards n-rich nuclei ANL fusion measurements

mass measurements completed NSCL TOF

figure courtesy M. Caplan

transient | observations

Wijnands et al; Cackett et al; Degenaar et al; Fridriksson et al; Díaz Trigo et al; Homan et al.

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10 Homan et al.

TABLE 7FITS TO COOLING CURVES WITH AN EXPONENTIAL DECAY TO A CONSTANTa

Source τ (days) A (eV) B (eV) Data ReferencesMAXI J0556–332 161±5 151±2 184.5±1.5 this work (model I)

197±10 137±2 174±2 this work (model II)IGR J17480–2446 157±62 21.6±4 84.3±1.4 Degenaar et al. (2013)EXO 0748–676 172±52 18±3 114.4±1.2 Degenaar et al. (2014)XTE J1701–462 230±46 35.8±1.4 121.9±1.5 Fridriksson et al. (2011)KS 1731–260 418±70 39.8±2.3 67.7±1.3 Cackett et al. (2010a)MXB 1659–29 465±25 73±2 54±2 Cackett et al. (2008)

a kT∞eff(t) = A×e−t/τ +B, where t is time since the end of the outburst in days.

below our estimated distance range (e.g., 20 kpc) we findtemperatures (134–218 eV for model I and 131–195 eV formodel II) that are substantially higher than those observed inXTE J1701–462 during its first∼500 days (125–163 eV). Theshort cooling timescale observed inMAXI J0556–332 impliesa high thermal conductivity of the crust, similar to the othercooling neutron stars that have been studied.Given the similarities between the outbursts of MAXI

J0556–332 and XTE J1701–462, it is interesting to comparethese two systems in more detail, as it may help us under-stand what causes the neutron-star crust in MAXI J0556–332to be so hot. MAXI J0556–332 was in outburst for ∼480days with a time-averaged luminosity of ∼1.7×1038(d45)2erg s−1, while XTE J1701–462 was in outburst for∼585 dayswith a time-averaged luminosity ∼2.0×1038(d8.8)2 erg s−1(Fridriksson et al. 2010). The total radiated energies of theMAXI J0556–332 and XTE J1701–462 outbursts are there-fore 7.1×1045(d45)2 erg and 1.0×1046(d8.8)2 erg, respec-tively. Despite the fact that the radiated energies and time-averaged luminosities of the two systems are comparable, theinitial luminosity of the thermal component (which reflectsthe temperature at shallow depths in the crust at the end of theoutburst) is an order of magnitude higher in MAXI J0556–332 than in XTE J1701–462. This suggests the presence ofadditional shallow heat sources in the crust of MAXI J0556–332 and/or that the shallow heat sources in MAXI J0556–332were more efficient per accreted nucleon.The high observed temperatures are difficult to explain with

current crustal heating models. Bringing the initial tempera-tures down to those seen in XTE J1701–462 requires a dis-tance of ∼10–15 kpc (depending on the assumed model).Such distances are problematic for several reasons. First itimplies that Z source behavior in MAXI J0556–332 is ob-served at much lower luminosities (by factors of 9 or more)than in other Z sources. Second, fits to the quiescent spec-tra with such a small distance are of poor quality. Finally, asmaller distance does not solve the fact that crustal heating ap-pears to have been much more efficient per accreted nucleonthan in other sources. A reduction in distance by a factor of 3results in a reduction in luminosity and presumably then, byextension, the total mass accreted onto the neutron star andtotal heat injected into the crust by a factor of 9. Given thatwe inferred ∼30% less mass accreted onto the neutron star inMAXI J0556–332 during its outburst than in XTE J1701–462for our preferred distance of ∼45 kpc, this would mean ∼13times less mass accreted onto MAXI J0556–332 than XTEJ1701–462 yet similar initial temperatures.The nsamodel that we used to fit the thermal emission from

the neutron star in MAXI J0556–332 did not allow us to ex-plore values of the neutron-star parameters other than Mns =1.4M⊙ and Rns = 10 km, as these parameters are advised toremain fixed at those values (Zavlin et al. 1996). While otherneutron-star atmosphere models allow for changes inMns andRns, none of the available models are able to handle the hightemperatures observed during the first ∼200 days of quies-cence. It is, of course, possible that the properties of theneutron star in MAXI J0556–332 are significantly differentfrom those in the other cooling neutron-star transients thathave been studied. Lower temperatures would be measuredif one assumed a lower Mns and/or a larger Rns. To estimatethe effects of changes in neutron-star parameters we used thensatmosmodel to fit the spectrum of observation 11, initiallyassuming Mns = 1.4M⊙ and Rns =10 km. While keeping thedistance from this fit fixed, and changing Mns to 1.2M⊙ andRns to 13 km (values that are still reasonable), the measuredtemperature was reduced by only ∼10%. Such changes arenot large enough to reconcile the temperatures measured inMAXI J0556–332 with those of the other sources.An alternative explanation for the high inferred tempera-

tures could be that part of the quiescent thermal emission iscaused by low-level accretion. Indications for low-level ac-

FIG. 5.— Evolution of the effective temperature of the quiescent neutronstar in MAXI J0556–332, based on fits with model II (purple stars). Temper-ature data for five other sources are shown as well. The solid lines representthe best fits to the data with an exponential decay to a constant. See Table 7for fit parameters and data references.

Plot from Homan et al. ‘14

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code to generate following plots is posted at https://github.com/nworbde/dStar

crust cooling | low conductivity

Cools too slow

Make the crust more pure (improves heat conduction)

The crust cools in a few years

Make the crust more pure (improves heat conduction)

The crust cools in a few years

But the neutron star is too cold

just after accretion ends

Add a heat source

crust cooling | implications

●

●

●

●

●● ●

t (d)

kT1 e↵

(eV)

1 10 102 103

60

80

100

120

140

160

𝝉 ~ 102 d: low specific heat (n pairing); high thermal conductivity (pure lattice)

- ≈ (�.�.F7V−�)× E./EU

Shternin et al. ’07; Brown & Cumming ’09; Page & Reddy ’13; Turlione ’13; Deibel et al. ’17; Merritt et al. ’16; Waterhouse et al. ’16; Parikh et al. ‘17

5DPSF = ��.,

MXB1659-29 (Brown & Cumming ’09)

findings | questionsNeutron stars have crusts

These crusts cool quickly

Why is the crust so pure?

Have we found pasta?

What is the source of the shallow heat?

What can we infer about the neutron star core?

The Superburst Mystery KS 1731–260 (Kuulkers 2002)



findings | questionsNeutron stars have crusts

These crusts cool quickly

Why is the crust so pure?

Have we found pasta?

What is the source of the shallow heat?

What can we infer about the neutron star core?

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Neutron Stars and Dense Matter - Arizona State University