Beyond the Chandrasekhar Limit: Structure and Formation of Compact Stars

Beyond the Chandrasekhar limit: Structure and formation of compact stars

Dipankar Bhattacharya IUCAA, Pune Plan of the talk:

A. Physics of mass limits - White Dwarfs - Neutron Stars B. Observational constraints on NS equation of state End states of stellar evolution: - no energy generation - source of pressure support other than thermal

White Dwarfs: pressure source: electron degeneracy Degeneracy ∝ pressure At low pF, v pF → ∝ v c as n increases P n pF v ∝ 1/3 pF n log P Electrons in stars: ∝ n ρ even when relativistic since mass is contributed by protons and neutrons

loglog ρn dP GM(r)ρ(r) = ⇒ P GM2/3ρ4/3 dr r2 c ∼ log P

Gravity: GM2/3ρ4/3

log ρ dP GM(r)ρ(r) = ⇒ P GM2/3ρ4/3 dr r2 c ∼ log P

Gravity: GM2/3ρ4/3

log ρ dP GM(r)ρ(r) = ⇒ P GM2/3ρ4/3 dr r2 c ∼ log P

Gravity: GM2/3ρ4/3

log ρ dP GM(r)ρ(r) = ⇒ P GM2/3ρ4/3 dr r2 c ∼ log P

Gravity: GM2/3ρ4/3

log ρ dP GM(r)ρ(r) = ⇒ P GM2/3ρ4/3 dr r2 c ∼

Mlim log P

Gravity: GM2/3ρ4/3

log ρ dP GM(r)ρ(r) = ⇒ P GM2/3ρ4/3 dr r2 c ∼ log P

Gravity: GM2/3ρ4/3

log ρ μ -2 Mlim = 5.7 e M☉ μ 1.4M☉ for e = 2 (Chandrasekhar 1931, 1935)

Beyond the limit: collapse to more compact conﬁguration: e.g. Neutron Star or Black Hole The upper mass limit of Neutron stars

Neutron Degeneracy: μ - Made mostly of neutrons, replace e by 1 - Mass and pressure from the same species, P ∝ ρc2 in relativity dP G(M + 4πr3P/c2)(ρ + P/c2) - GR important; = dr r2(1 2GM/rc2) TOV eqn − ⇒ Mlim = 0.69 M☉ (Oppenheimer & Volkoff 1939)

In reality, strong interaction between nucleons determine the equation of state, and hence Mlim Unlike white dwarfs, the equation of state of neutron stars suffers from serious uncertainties - state of matter at very high density is essentially unknown

Situation may improve only with - improvements in QCD theory - high-energy accelerator experiments - constraints from astronomical obs: M, R, Ω, oscillations

1977ApJS...33..415A

M/M☉ log R (km) Arnett & Bower s 1977 5 Model-independent upper mass limit

4 ) ☉

(M ρ ρ cs = c at > f

max 3 M

ρ Rhoades & Rufﬁni ‘74 f (g/cc) Kalogera & Baym ‘96 Glendenning 1997 Strange Quark Matter 2 At high density, when EFu,d > ms c , some non-strange quarks may become strange, reducing energy. The resulting quark matter may have energy/baryon < 930 MeV, making this the most stable phase of matter (Bodmer 1971, Witten 1984)

- Any matter coming into contact with SQM should get converted to quark matter

- At high density strange stars may form; more compact than neutron stars Strange Stars 2 1.6 ms

0.5 ms 1.5 ☉ 1 SS M/M NS

0.5

0 0 5 10 15 20 R (km) Glendenning 1997 Hybrid Stars 2.5 3.5

Nuclear 3 2 APR G) 15 QuarkPC core 2.5 1.5 2 815 1 1.5 800 1 Mass (Solar units) 0.5 813 815 Polar mag field (10 0.5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Central Density (1015 g/cm3) Mass (Solar units)

Bhattacharya & Soni 2007 Hybrid Stars APR + Phenomenological QM EoS 2.5_ APR only

2_ c = 0.3 !c=2-6n0 1.5_ M (Mo.) c = 0 1_ ! =3n c = 0 c 0 ! =2n 0.5_ c 0

0_ 8 10 12 14 R (km)

Alford et al 2004 List of 61 NS mass estimates

Zhang et al 2010 Mgrav < Mbary

1.4 M☉ core → 1.25 M☉ NS

If pre-collapse core is n-enriched, MCh is reduced, giving MNS < 1.2 M☉

Final mass of newly born NS decided by fallback Which stars make BH? & long GRBs?

depends on NS mass limit emnants action of r r F Zhang, Woosley & Hagar ‘07

1.2 1.4 1.6 1.8 2 Remnant Mass (M☉) Mgrav < Mbary

1.4 M☉ core → 1.25 M☉ NS

If pre-collapse core is n-enriched, MCh is reduced, giving MNS < 1.2 M☉

Final mass of newly born NS decided by fallback

NS mass may grow by accretion Hulse-Taylor binary: PSR B1913+16

MPSR = 1.441 M☉

Mcomp = 1.387 M☉

Double Pulsar: PSR J0737-3039

MA = 1.337 M☉

MB = 1.250 M☉ 89.24 Pulsar Mass Probability Distribution PSR J 1614 -2230 89.22

☉ Demorest et al 2010

89.2

89.18 1.97±0.04 M

89.16 obability r Probability Density Inclination Angle (deg) 89.14 P

89.12

89.1 Mass (M☉) 0.48 0.49 0.5 0.51 0.52 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 Companion Mass (solar) 1.8 1.9 Pulsar Mass2.0 (solar) 2.2 Demorest et al 2010 Rotation 2 1.6 ms

0.5 ms 1.5 ☉ 1 SS M/M NS

0.5

0 0 5 10 15 20 R (km) Glendenning 1997 1994ApJ...424..823C Cook, Shapir o & T euk olsky 1994 1994ApJ...424..823C Cook, Shapir o & T euk olsky 1994 12

-1 8 ad s

r 6 3 Mass shed

4 Ω /10

2 Radial instability

0 0 0.5 1 1.5 2 2.5 3 3.5 M/M☉ Cook, Shapiro & Teukolsky 1994 12

10 EOS - 1

-1 8 EOS - 2 ad s

r 6 3

4 Ω /10

0 0 0.5 1 1.5 2 2.5 3 3.5 M/M☉ Cook, Shapiro & Teukolsky 1994 Spin distribution of millisecond neutron stars

NS in LMXBs

Radio ms PSRs

Ω (rad/s) 12

10 EOS - 1

-1 8 EOS - 2 ad s

r 6 3

4 Ω /10

0 0 0.5 1 1.5 2 2.5 3 3.5 M/M☉ Cook, Shapiro & Teukolsky 1994 14

11 log B (G)

10 Spin-up line

Graveyard

9 Hubble line

Death line 8

-3 -2 -1 0 1 log P (s) 14Millisecond pulsars are spun up by accretion

11 log B (G)

10 Spin-up line rs Graveyard

9 Hubble line Recycled Pulsa

Death line 8

-3 -2 -1 0 1 log P (s)

Cen X-3

Finger et al 1998 EXO 2030+375 ate Spin-up r

X-ray Flux

Wilson et al 2002 ∝ 6/7 Spin-up limited by magnetic ﬁeld: Pmin B 14Millisecond pulsars are spun up by accretion

11 log B (G)

10 Spin-up line rs Graveyard

9 Hubble line Recycled Pulsa

Death line 8

-3 -2 -1 0 1 log P (s) ∝ 6/7 Spin-up limited by magnetic ﬁeld: Pmin B Required mass accretion ΔM ≳ 0.1 M☉ (P/2ms)-4/3 Max. accretion rate ~ 10-8 M☉/yr

At low B, spin-up may be halted by gravitational waves Gravitational wave instability Chandrasekhar - Friedman - Schutz (1970) (1978)

http://www.sissa.it/RelAstro/cfs.html r-mode instability .HSOHUIUHTXHQF\ ,QVWDELOLW\OLPLW

! !

ORJ 7

Andersson 1998 No. 1, 1999 r-MODE INSTABILITY AND ACCRETING COMPACT STARS 309 the viscosity damping time). We thus Ðnd that the mode Also, it is straightforward to show that in order to ““ rule grows if the period is shorter than out ÏÏ the instability (to lead to a critical period equal to the Kepler period at, e.g., temperature 4 ] 108 K), the dissi- R 39@24 1.4 M 1@24 T 1@3 P B 2.8 _ ms (6) pation coefficient of the shear viscosity (or any other dissi- c A10 kmB A M B A107 KB pation mechanism) must be almost 6 orders of magnitude stronger than equation (4). for a normal Ñuid star, and Our inferred critical periods (eqs. [6] and [7], for a R 3@2 T 1@3 canonical neutron star) are illustrated and compared with P B 2.3 ms (7) observed periods and upper limits on the surface tem- c 10 km 107 K A B A B peratures (from ROSAT observations; see data given by when we use the viscosity due to electron-electron scat- Reisenegger 1997) for the fastest MSPs in Figure 1. In the tering in a superÑuid. Interestingly, these critical periods are Ðgure we also indicate the associated upper limits on the not strongly dependent on the mass of the star. Further- core temperatures as estimated using equation (8) of Gud- more, the uncertainties in equations (1), (2), and (5) have mundsson, Pethick, & Epstein (1982). little e†ect on the critical period. For example, the uncertain The illustrated r-mode instability estimates would be in factors of 2 in t and t individually lead to an uncertainty conÑict with the MSP observations if the interior tem- gw sv of 12% in Pc . When combined, the uncertainties suggest perature of a certain star were such that it was placed con- that we may be (over)estimating the critical period at the siderably below the critical period for the relevant 25% level. Considering uncertainties associated with the temperature. Basically, an accreting star whose spin is various realistic equations of state for supranuclear matter limited by the r-mode instability would not be able to spin and the many approximations on which our present under- up far beyond the critical period, since the instability would standing of the r-mode instability is based, we feel that it is radiate away any excess accreted angular momentum. As acceptable to work at this level of accuracy. the accretion phase ends, the star will both cool down and spin down (the timescales for these two processes, photon 3. IMPLICATIONS FOR MSPs cooling and magnetic dipole braking, are such that an MSP We will now discuss the possibility that the r-mode insta- would evolve almost horizontally toward the left in Fig. 1). bility may be relevant for the period evolution of the fastest Given the uncertainties in the available data, we do not observed pulsars. All observed MSPs have periods larger think the possibility that the r-mode instability may have than the 1.56 ms of PSR 1937]21, and it is relevant to ask played a role in the period evolution of the fastest MSPs whether there is a mechanism that prevents a neutron star can be ruled out. First of all, it must be remembered that the from being spun up farther (e.g., to the Kepler limit) by ROSAT data only provide upper limits on the surface tem- accretion. SpeciÐcally, we are interested in the possibility perature, and the true temperature may well be consider- that the r-mode instability plays such a role. Before pro- ably lower than this. If the true core temperatures of the ceeding with our discussion, we recall that Andersson et al. fastest spinning pulsars were roughly 1 order of magnitude (1999) have already pointed out that the instability has implications for the formation of MSPs (albeit in an indirect way). SpeciÐcally, the strength of the r-mode instability seems to rule out the scenario in which MSPs (with P \ 5È10 ms) are formed as an immediate result of accretion-induced collapse of white dwarfs. Continued accretion would be needed to reach the shortest observed periods. In other words, all MSPs with periods shorter than (say) 5È10 ms should be recycled. Our main question here is whether it is realistic to expect the instability to be relevant also for older (and in conse- quence much colder) neutron stars. Even though the critical period is much shorter for a cold star, our estimates (eqs. [6] and [7]) are still above the Kepler period (B0.8 ms for our canonical star), which suggests that the instability could be relevant. As an attempt to answer the question, we will confront our rough approximations with observed data for MSPs and the neutron stars in LMXBs. 3.1. T he MSPs FIG. 1.ÈInferred critical period for the r-mode instabilityAnderssonat tem- et al 1999 In this section we discuss the r-mode instability in the peratures relevant to older neutron stars (solid lines). The upper line is for a context of the recycled MSPs. These stars are no longer normal Ñuid star, while the lower one is for a superÑuid star (only taking accreting, and supposing that they have been cooling for electron-electron scattering into account; see text for discussion). The data is for a neutron star with M \ 1.4 M_ and R \ 10 km. The Kepler limit, some time, they should not be a†ected by the instability at which corresponds to P B 0.8 ms for our canonical star, is shown as a present. Our main question is whether the observed data is horizontal dashed line. We compare our theoretical result with (1) the in conÑict with a picture in which the r-mode instability observed periods and temperatures of the most rapidly spinning MSPs (see halted accretion-driven spin-up at some point in the past. Reisenegger 1997 for the data): the surface temperatures are indicated as solid vertical lines: the dashed continuation of each line indicates the Our estimates show that the rotation will be limited by estimated core temperature; (2) the observed/inferred periods and tem- the Kepler frequency (using P B 0.8 ms for a canonical K peratures for accreting neutron stars in LMXBs; and (3) the recently dis- star) if the interior of the star is colder than T B 2 ] 105 K. covered 2.49 ms X-ray pulsar SAX J1808.4-3658. Constraining R

- X-ray burst continuum spectra & luminosities - Quasi-periodic oscillations in X-ray intensity - Relativistic iron lines in X-ray spectra X-ray Bursts

Galloway et al 2008 Galloway et al 2008 4U 1608-24 EXO 1745-248 4U 1820-30

Ozel et al 2010 Burst Oscillation

Time (sec)

Strohmayer 1996 Burst Oscillation XTE J 1739-285

Kaaret et al 2007 12

10 EOS - 1

-1 8 EOS - 2 ad s

r 6 3

4 Ω /10

0 0 0.5 1 1.5 2 2.5 3 3.5 M/M☉ Cook, Shapiro & Teukolsky 1994 LMXB power spectra 11 Apr SAX J 1808.4 -3658

18 Apr (x 0.01) Pulse frequency

25 Apr (x 0.001)

27 Apr (x 0.0001)

Wijnands & van der Klis 1998 LMXB power spectra

van der Klis 2008 !"#$%&'()*+,-. LMXB power spectra

Sco X-1

!"#$%&'&(%)$*+ , /$("$0*"0*.('$01*2"*1'45"(67*89 , :&;('$0*.(4'*<4'4=&(&'.7*>-? , :&;('$0*.(4'*.<"0@

van der Klis et al 1995 Upper KHz frequency

= Keplerian freq at inner edge of accretion disk

Leads to constraints on the radius

Max. known upper QPO freq: 1330 Hz (4U0614+091) Relativistic Iron Lines

Suzaku spectra

Cackett et al 2008 1991ApJ...376...90L Laor 1991 Combined constraints

Cackett et al 2008 Seismology 2004 hyperﬂare of SGR 1806-20

Superposed on the 7.5-s rotation, high freq QPOs seen

Stohmayer & Watts 2006 1e+05 720, 976, 2384 Hz 1840 Hz

625 Hz 150 Hz 92 Hz 29 Hz 10000 18, 26 Hz Counts/s

1000

100 0 100 200 300 Time (s)

Stohmayer & Watts 2006 Seismology 2004 hyperﬂare of SGR 1806-20

Superposed on the 7.5-s rotation, high freq QPOs seen Interpreted as pure crustal modes, constrain crust thickness to 10-13% of stellar radius (Strohmayer & Watts 2006)

In future, global oscillation modes of neutron stars may be detectable by gravity wave observatories (Andersson et al 2010) The future: - enlarge the sample of binary MSPs: e.g. radio searches in Fermi error boxes, other sensitive all-sky pulsar surveys - better study of x-ray burst spectra - high resolution LMXB timing - iron-line studies Suzaku, Astrosat, IXO, LOFT.... - atomic lines from NS surface? - oscillations: magnetar QPO, Grav. Wave 1932ZA...... 5..321C question using astr T oday, wear e beginningtobeableaddr onomical obser Zeitschrif Chandr asekhar, S.(1932) t fürAstr v ations themselv ophysik, V ess this ol. 5,p.321-326 es Thank you