Embed Size (px)
Citation preview
Beyond the Chandrasekhar limit:Structure and formation of compact stars
Dipankar BhattacharyaIUCAA, 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
log ρ
log
P
P ∝ n pF vpF ∝ n1/3
At low pF, v ∝ pF
v → c as n increases
Degeneracypressure
Electrons in stars:n ∝ ρ even when relativisticsince mass is contributedby protons and neutrons
log n
log ρ
log
P
Gravity: GM2/3ρ4/3
Pc ! GM2/3!4/3⇒dPdr=
GM(r)!(r)r2
log ρ
log
P
Gravity: GM2/3ρ4/3
⇒ Pc ! GM2/3!4/3dPdr=
GM(r)!(r)r2
log ρ
log
P
Gravity: GM2/3ρ4/3
⇒ Pc ! GM2/3!4/3dPdr=
GM(r)!(r)r2
log ρ
log
P
Gravity: GM2/3ρ4/3
⇒ Pc ! GM2/3!4/3dPdr=
GM(r)!(r)r2
log ρ
log
P
Gravity: GM2/3ρ4/3
Mlim
⇒ Pc ! GM2/3!4/3dPdr=
GM(r)!(r)r2
log ρ
log
P
Gravity: GM2/3ρ4/3
⇒ Pc ! GM2/3!4/3dPdr=
GM(r)!(r)r2
Mlim = 5.7 μe-2 M☉
1.4M☉ for μe = 2
Beyond the limit: collapse to morecompact configuration:e.g. Neutron Staror Black Hole
(Chandrasekhar 1931, 1935)
The upper mass limit of Neutron stars
Neutron Degeneracy:- Made mostly of neutrons, replace μe by 1- Mass and pressure from the same species, P ∝ ρc2 in relativity
- GR important; TOV eqn
dPdr=
G(M + 4!r3P/c2)(" + P/c2)r2(1 ! 2GM/rc2)
⇒ 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 starssuffers from serious uncertainties
Situation may improve only with - improvements in QCD theory - high-energy accelerator experiments - constraints from astronomical obs:
- state of matter at very high density is essentially unknown
M, R, Ω, oscillations
1977ApJS...33..415A
log R (km)
M/M
☉
Arnett & Bowers 1977
2
3
4
5
Kalogera & Baym ‘96ρf (g/cc)
Mm
ax (
M☉
)
cs = c at ρ > ρf
Model-independent upper mass limit
Rhoades & Ruffini ‘74
Glendenning 1997
Strange Quark MatterAt high density, when EFu,d > ms c2 , some non-strange quarks may become strange, reducing energy.
The resulting quark matter may have energy/baryon < 930 MeV, making this themost 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
NS
SS
1.6 ms
0.5 ms
0 5 10 15 200
0.5
1
1.5
2M
/M☉
R (km) Glendenning 1997
Strange Stars
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5
Mass (
Sola
r units)
Central Density (1015
g/cm3)
APR
PC
815
800
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
Pola
r m
ag fie
ld (
10
15 G
)
Mass (Solar units)
815813
Quark core
Nuclear
Bhattacharya & Soni 2007
Hybrid Stars
8 10 12 14R (km)
0_
0.5_
1_
1.5_
2_
2.5_
M(Mo.)
APR + Phenomenological QM EoS
APR only
c = 0.3 !
c=2-6n
0
c = 0 !
c=2n
0
c = 0!
c=3n
0
Alford et al 2004
Hybrid Stars
Zhang et al 2010
List of 61 NS mass estimates
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 NSdecided by fallback
Zhang, Woosley & Hagar ‘07
1.2 1.4 1.6 1.8 2Remnant Mass (M☉)
Frac
tion
of re
mna
nts
Which stars make BH? & long GRBs?
depends on NS mass limit
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 NSdecided by fallback
NS mass may grow by accretion
Hulse-Taylor binary: PSR B1913+16MPSR = 1.441 M☉Mcomp = 1.387 M☉
Double Pulsar: PSR J0737-3039MA = 1.337 M☉MB = 1.250 M☉
89.1
89.12
89.14
89.16
89.18
89.2
89.22
89.24
0.48 0.49 0.5 0.51 0.52
Inclin
atio
n A
ng
le (
de
g)
Companion Mass (solar)
1.8 1.85 1.9 1.95 2 2.05 2.1 2.15
Pro
ba
bili
ty D
en
sity
Pulsar Mass (solar)1.8 1.9 2.0 2.2
Pulsar Mass Probability Distribution
Mass (M☉)
Prob
abilit
y
PSR J 1614 -2230Demorest et al 2010
1.97±
0.0
4 M☉
Demorest et al 2010
NS
SS
1.6 ms
0.5 ms
0 5 10 15 200
0.5
1
1.5
2M
/M☉
R (km) Glendenning 1997
Rotation
1994ApJ...424..823C
Cook, Shapiro & Teukolsky 1994
1994ApJ...424..823C
Cook, Shapiro & Teukolsky 1994
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5 3 3.5
M/M☉
Ω /1
03 r
ad s
-1
Mass shed
Rad
ial in
stab
ility
Cook, Shapiro & Teukolsky 1994
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5 3 3.5
M/M☉
Ω /1
03 r
ad s
-1
EOS - 1
EOS - 2
Cook, Shapiro & Teukolsky 1994
NS in LMXBs
Radio ms PSRs
Ω (rad/s)
Spin distribution of millisecond neutron stars
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5 3 3.5
M/M☉
Ω /1
03 r
ad s
-1
EOS - 1
EOS - 2
Cook, Shapiro & Teukolsky 1994
-3 -2 -1 0 1
log P (s)
8
9
10
11
12
13
14
log B
(G
)
Spin-up line
Hubble line
Dea
th line
Graveyard
-3 -2 -1 0 1
log P (s)
8
9
10
11
12
13
14
log B
(G
)
Spin-up line
Hubble line
Dea
th line
Graveyard
Recycled Pulsa
rs
Millisecond pulsars are spun up by accretion
Cen X-3
Finger et al 1998
X-ray Flux
Spin
-up
rate
EXO 2030+375
Wilson et al 2002
Spin-up limited by magnetic field: Pmin ∝ B6/7
-3 -2 -1 0 1
log P (s)
8
9
10
11
12
13
14
log B
(G
)
Spin-up line
Hubble line
Dea
th line
Graveyard
Recycled Pulsa
rs
Millisecond pulsars are spun up by accretion
Spin-up limited by magnetic field: Pmin ∝ B6/7
At low B, spin-up may be halted by gravitational waves
Required mass accretion ΔM ≳ 0.1 M☉ (P/2ms)-4/3 Max. accretion rate ~ 10-8 M☉/yr
Gravitational wave instabilityChandrasekhar - Friedman - Schutz
(1970) (1978)
http://www.sissa.it/RelAstro/cfs.html
! " # $ % & '( '' ') '*
(+(
(+)
(+!
(+#
(+%
'+(
!"#$"%&'%"()"*+,
-*./012$2/,&$232/
!,!
-./ 01
r-mode instability
Andersson 1998
No. 1, 1999 r-MODE INSTABILITY AND ACCRETING COMPACT STARS 309
the viscosity damping time). We thus !nd that the modegrows if the period is shorter than
PcB 2.8
A R10 km
MB1@24A T
107 KB1@3
ms (6)
for a normal Ñuid star, and
PcB 2.3
A R10 km
B3@2A T107 K
B1@3ms (7)
when we use the viscosity due to electron-electron scat-tering in a superÑuid. Interestingly, these critical periods arenot strongly dependent on the mass of the star. Further-more, the uncertainties in equations (1), (2), and (5) havelittle e†ect on the critical period. For example, the uncertainfactors of 2 in and individually lead to an uncertaintytgw tsvof 12% in When combined, the uncertainties suggestP
c.
that we may be (over)estimating the critical period at the25% level. Considering uncertainties associated with thevarious realistic equations of state for supranuclear matterand the many approximations on which our present under-standing of the r-mode instability is based, we feel that it isacceptable to work at this level of accuracy.
3. IMPLICATIONS FOR MSPs
We will now discuss the possibility that the r-mode insta-bility may be relevant for the period evolution of the fastestobserved pulsars. All observed MSPs have periods largerthan the 1.56 ms of PSR 1937]21, and it is relevant to askwhether there is a mechanism that prevents a neutron starfrom being spun up farther (e.g., to the Kepler limit) byaccretion. Speci!cally, we are interested in the possibilitythat the r-mode instability plays such a role. Before pro-ceeding with our discussion, we recall that Andersson et al.(1999) have already pointed out that the instability hasimplications for the formation of MSPs (albeit in an indirectway). Speci!cally, the strength of the r-mode instabilityseems to rule out the scenario in which MSPs (withP \ 5È10 ms) are formed as an immediate result ofaccretion-induced collapse of white dwarfs. Continued acc-retion would be needed to reach the shortest observedperiods. 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 expectthe instability to be relevant also for older (and in conse-quence much colder) neutron stars. Even though the criticalperiod is much shorter for a cold star, our estimates (eqs.[6] and [7]) are still above the Kepler period (B0.8 ms forour canonical star), which suggests that the instability couldbe relevant. As an attempt to answer the question, we willconfront our rough approximations with observed data forMSPs and the neutron stars in LMXBs.
3.1. T he MSPsIn this section we discuss the r-mode instability in the
context of the recycled MSPs. These stars are no longeraccreting, and supposing that they have been cooling forsome time, they should not be a†ected by the instability atpresent. Our main question is whether the observed data isin conÑict with a picture in which the r-mode instabilityhalted accretion-driven spin-up at some point in the past.
Our estimates show that the rotation will be limited bythe Kepler frequency (using ms for a canonicalPK B 0.8star) if the interior of the star is colder than T B 2 ] 105 K.
Also, it is straightforward to show that in order to ““ ruleout ÏÏ the instability (to lead to a critical period equal to theKepler period at, e.g., temperature 4 ] 108 K), the dissi-pation coefficient of the shear viscosity (or any other dissi-pation mechanism) must be almost 6 orders of magnitudestronger than equation (4).
Our inferred critical periods (eqs. [6] and [7], for acanonical neutron star) are illustrated and compared withobserved periods and upper limits on the surface tem-peratures (from ROSAT observations ; see data given byReisenegger 1997) for the fastest MSPs in Figure 1. In the!gure we also indicate the associated upper limits on thecore temperatures as estimated using equation (8) of Gud-mundsson, Pethick, & Epstein (1982).
The illustrated r-mode instability estimates would be inconÑict with the MSP observations if the interior tem-perature of a certain star were such that it was placed con-siderably below the critical period for the relevanttemperature. Basically, an accreting star whose spin islimited by the r-mode instability would not be able to spinup far beyond the critical period, since the instability wouldradiate away any excess accreted angular momentum. Asthe accretion phase ends, the star will both cool down andspin down (the timescales for these two processes, photoncooling and magnetic dipole braking, are such that an MSPwould evolve almost horizontally toward the left in Fig. 1).
Given the uncertainties in the available data, we do notthink the possibility that the r-mode instability may haveplayed a role in the period evolution of the fastest MSPscan be ruled out. First of all, it must be remembered that theROSAT data only provide upper limits on the surface tem-perature, and the true temperature may well be consider-ably lower than this. If the true core temperatures of thefastest spinning pulsars were roughly 1 order of magnitude
FIG. 1.ÈInferred critical period for the r-mode instability at tem-peratures relevant to older neutron stars (solid lines). The upper line is for anormal Ñuid star, while the lower one is for a superÑuid star (only takingelectron-electron scattering into account ; see text for discussion). The datais for a neutron star with M \ 1.4 and R \ 10 km. The Kepler limit,M
_which corresponds to P B 0.8 ms for our canonical star, is shown as ahorizontal dashed line. We compare our theoretical result with (1) theobserved periods and temperatures of the most rapidly spinning MSPs (seeReisenegger 1997 for the data) : the surface temperatures are indicated assolid vertical lines : the dashed continuation of each line indicates theestimated core temperature ; (2) the observed/inferred periods and tem-peratures for accreting neutron stars in LMXBs ; and (3) the recently dis-covered 2.49 ms X-ray pulsar SAX J1808.4-3658.
Andersson et al 1999
Constraining R
- X-ray burst continuum spectra & luminosities- Quasi-periodic oscillations in X-ray intensity- Relativistic iron lines in X-ray spectra
Galloway et al 2008
X-ray Bursts
Galloway et al 2008
Ozel et al 2010
4U 1608-24EXO 1745-2484U 1820-30
Strohmayer 1996
Time (sec)
Burst Oscillation
Kaaret et al 2007
Burst OscillationXTE J 1739-285
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5 3 3.5
M/M☉
Ω /1
03 r
ad s
-1
EOS - 1
EOS - 2
Cook, Shapiro & Teukolsky 1994
11 Apr
18 Apr (x 0.01)
25 Apr (x 0.001)
27 Apr (x 0.0001)
Pulse frequency
Wijnands & van der Klis 1998
SAX J 1808.4 -3658
LMXB power spectra
van der Klis 2008
LMXB power spectra
!"#$%&'()*+,-.
!"#$%&'&(%)$*+
,/$("$0*"0*.('$01*2"*1'45"(67*89
, :&;('$0*.(4'*<4'4=&(&'.7*>-?
, :&;('$0*.(4'*.<"0@
van der Klis et al 1995
LMXB power spectra
Sco X-1
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)
Cackett et al 2008
Relativistic Iron Lines
Suzakuspectra
1991ApJ...376...90L
Laor 1991
Cackett et al 2008
Combined constraints
Seismology2004 hyperflare of SGR 1806-20
Superposed on the 7.5-s rotation, high freq QPOs seen
Stohmayer & Watts 2006
0 100 200 300Time (s)
100
1000
10000
1e+05C
ounts
/s
1840 Hz720, 976, 2384 Hz
625 Hz150 Hz
92 Hz
18, 26 Hz
29 Hz
Stohmayer & Watts 2006
Seismology2004 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 starsmay 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
Chandrasekhar, S. (1932)Zeitschrift für Astrophysik, Vol. 5, p.321-326
Today, we are beginning to be able to address this question using astronomical observations themselves
Thank you
