Astronomy meets QCD

Astronomy meets QCD

Sergei Popov

(SAI MSU)

Phase diagram

Astrophysical point of viewAstrophysical appearence of NSsis mainly determined by:

• Spin• Magnetic field• Temperature• Velocity• Environment The first four are related to the NS structure!

See a recent review on astrophysical contraints on the EoS in 0808.1279.

Structure and layers

Plus an envelope and an atmosphere...

Neutron starsRadius: 10 kmMass: 1-2 solarDensity: about the nuclearStrong magnetic fields

NS interiors: resume

(Weber et al. ArXiv: 0705.2708)

R=2GM/c2

P=ρR~3GM/c2

ω=ωKR∞=R(1-2GM/Rc2)-1/2 Lattimer & Prakash (2004)

Quark stars

Quark stars can be purely quark (with and without a tiny crust) or hybrid

Proposed already in 60-s (Ivanenko and Kurdgelaidze 1965). Then the idea was developed in 70-s.

Popular since the paper by Witten (1984) Renessance in recent 10 years or so.

See a recent astrophysical review in 0809.4228.

How to recognize a quark star?

Mass-radius relation. Smaller for smaller masses.

Cooling. Probably can cool faster due to additional channels.

Surface properties. If “bare” strange star – then there is no Eddington limit.

ikikik Tc

GRgR

4

8

2

1

)( )4(

1 1

(3)

4 )2(

21

41 1 (1)

1

22

2

1

22

3

22

PP

c

P

dr

dP

cdr

d

rdr

dm

rc

Gm

mc

Pr

c

P

r

mG

dr

dP

{ Tolman (1939)Oppenheimer-Volkoff (1939)

TOV equation

EoS

EoS

(Weber et al. ArXiv: 0705.2708 )

Effective chiral model ofHanauske et al. (2000)

Relativistic mean-field modelTM1 of Sugahara & Toki (1971)

Particle fractions

Au-Au collisions

Experimental results and comparison

(Danielewicz et al. nucl-th/0208016)

1 Mev/fm3 = 1.6 1032 Pa

Astrophysical measurement Mass

Radius

Red shift (M/R)

Temperature

Moment of inertia

Gravitational vs. Baryonic mass

Extreme rotation

In binaries, especially in binaries with PSRs.In future – lensing.

In isolated cooling NS, in bursters in binaries,in binaries with QPO

Via spectral line observations

In isolated cooling NSs and in sometransient binaries (deep crustal heating)

In PSRs (in future)

In double NS binaries, if good contraintson the progenitor are available

Millisecond pulsars (both single and binary)

NS Masses

Stellar masses are directly measured only in binary systems

Accurate NS mass determination for PSRs in relativistic systems by measuring PK corrections

White

dw

arfs

Neu

tro

n s

tars

Brown dwarfs,Giant planets

Maximum-massneutronstar

Minimum-massneutron star

Maximum-masswhite dwarf

km 250~

1.0~ Sun

R

MM

km 129~

)5.25.1(~ Sun

R

MM

c

Neutron stars and white dwarfs

Minimal massIn reality, minimal mass is determined by properties of protoNSs.Being hot, lepton rich they have much higher limit: about 0.7 solar mass.

Stellar evolution does not produce NSs with barion mass less thanabout 1.4 solar mass (may be 1.3 for so-called electron-capture SN).

Fragmentation of a core due to rapid rotation potentially can lead to smallermasses, but not as small as the limit for cold NSs

Page & Reddy (2006)

BHs ?

However, now somecorrections are necessary

Compact objects and progenitors.Solar metallicity.

(Woosley et al. 2002)

There can be a range of progenitormasses in which NSs are formed,however, for smaller and larger progenitors masses BHs appear.

Mass spectrum of compact objects

(Timmes et al. 1996, astro-ph/9510136)

Results of calculations(depend on the assumed modelof explosion)

A NS from a massive progenitor

(astro-ph/0611589)

Anomalous X-ray pulsar in the associationWesterlund1 most probably has a very massive progenitor, >40 MO.

So, the situation with massive progenitorsis not that clear.

NS+NS binaries

Pulsar Pulsar mass Companion mass

B1913+16 1.44 1.39B2127+11C 1.35 1.36B1534+12 1.33 1.35J0737-3039 1.34 1.25J1756-2251 1.40 1.18J1518+4904 <1.17 >1.55J1906+0746 1.25 1.35

(PSR+companion)/2

J1811-1736 1.30J1829+2456 1.25

Secondary companion in double NS binaries can give a good estimateof the initial mass (at least, in this evolutionary channel).

0808.2292

GC

Non-recycled

In NS-NS systems we can neglect all tidal effects etc.

Also there arecandidates, for examplePSR J1753-2240 arXiv:0811.2027

PSR J1518+4904

[Janssen et al. arXiv: 0808.2292]

Surprising results !!!Surprising results !!!

Mass of the recycled pulsar is <1.17 solar massesMass of its component is >1.55 solar masses

Central values are even more shocking:

0.72+0.51-0.58 and 2.00+0.58

-0.51

V~25 km/s, e~0.25The second SN was e--capture?

NS+WD binariesSome examplesSome examples

1. PSR J0437-4715. WD companion [0801.2589, 0808.1594 ]. The closest millisecond PSR. MNS=1.76+/-0.2 solar.Hopefully, this value will not be reconsidered.

2. The case of PSR J0751+1807. Initially, it was announced that it has a mass ~2.1 solar [astro-ph/0508050]. However, then in 2007 at a conference the authors announced that the

resultwas incorrect. Actually, the initial value was 2.1+/-0.2 (1 sigma error).New result: 1.24 +/- 0.14 solar[Nice et al. 2008, Proc. of the conf. “40 Years of pulsars”]

3. PSR B1516+02B in a globular cluster. M~2 solar (M>1.72 (95%)). A very light companion. Eccentric orbit. [Freire et al. arXiv: 0712.3826] Joint usage of data on several pulsars can give stronger constraints on thelower limit for NS masses.

It is expected that most massive NSs get their additional “kilos” due to

accretion from WD companions [astro-ph/0412327 ].

Pulsar masses

[Nice et al. 2008]

With WD companions With NS companions

PSR 0751+1807

(Trumper)

Massive NS: 2.1+/-0.3 solar masses – Now shown to be wrong (!) [see Nice et al. 2008]

Binary pulsars

Relativistic corrections and measurable parameters

For details seeTaylor, Weisberg 1989ApJ 345, 434

Mass measurementsPSR 1913+16

(Taylor)

Double pulsar J0737-3039

(Lyne et al. astro-ph/0401086)

Masses for PSR J0737-3039

(Kramer et al. astro-ph/0609417)

The most precise values.

Neutron stars in X-ray binariesStudy of close binary systems gives an opportunity to obtain mass estimate forprogenitors of NSs (see for example, Ergma, van den Heuvel 1998 A&A 331, L29).For example, an interesting estimate was obtained for GX 301-2. The progenitor mass is >50 solar masses.On the other hand, for several other systems with both NSs and BHsprogenitor masses a smaller: from 20 up to 50. Finally, for the BH binary LMC X-3 the progenitor mass is estimated as >60 solar.So, the situation is tricky.Most probably, in some range of masses, at least in binary systems, stars canproduce both types of compact objects: NSs and BHs.

Mass determination in binaries:mass function

mx, mv - masses of a compact object and of a normal star (in solar units), Kv – observed semi-amplitude of line of sight velocity of the normal star (in km/s),P – orbital period (in days), e – orbital eccentricity, i – orbital inclination (the angle between the prbital plane and line of sight).

One can see that the mass function is the lower limit for the mass of a compact star.

The mass of a compact object can be calculated as:

So, to derive the mass it is necessary to know (besides the line of sight velocity)independently two more parameters: mass ration q=mx/mv, and orbital inclination i.

Recent mass estimates

ArXiv: 0707.2802

Massive NSs

We know several candidates to NS with high masses (M>1.8 Msun) in X-ray binaries:

Vela X-1, M=1.88±0.13 or 2.27±0.17 Msun (Quaintrell et al., 2003)

4U 1700-37, M=2.4±0.3 Msun (Clark et al., 2002)

2S 0921-630/V395 Car, M=2.0-4.3 Msun [1] (Shahbaz et al., 2004)

We will discuss formation of very massive NS due to accretion processes in binary systems.

What is «Very Massive NS» ?1.8 Msun < Very Massive NS < 3.5 Msun

• 1.8Msun: (or ~2Msun) Upper limit of Fe-core/young NS according to modeling of Supernova explosions (Woosley et al. 2002).

• ~3.5Msun: Upper limit of rapidly rotating NS with Skyrme EOS (Ouyed 2004).

Evolution

For our calculations we use the “Scenario Machine’’ code developed at the SAI. Description of most of parameters of the code can be found in (Lipunov,Postnov,Prokhorov 1996)

Results

1 000 000 binaries was calculated in every Population Synthesis set

104 very massive NS in the Galaxy (formation rate ~6.7 10-7 1/yr) in the model with kick

[6 104 stars and the corresponding formation rate ~4 10-6 1/yr for the zero kick].

State of NS with kick

zero kick

Ejector 32% 39%

Propeller+Georotator 2% 8%

Accretor 66% 53%astro-ph/0412327

Results II

Mass distribution of very massive NS

Dashed line: Zero natal kick of NS ( just for illustration).Solid line: Bimodal kick similar to (Arzoumanian et al. 2002).

Luminosity distribution of accreting very massive

NS

Outside of the star

RrRR

MMMM

g

b

/1/

2.0~ sun

r

r

drdrr

rdtc

r

rds

c

GMr

r

r

rc

GM

MrmRr

gr

gg

gg

1

1 1

2 ,1

21e

(1) и (3) изconst )( При

222

1

222

222

Bounding energy

Apparent radius

redshift

Baryonic vs. Gravitational massPodsiadlowski et al. [astro-ph/0506566] proposed that in the double pulsarJ0737-3039 it is possible to have a very good estimate of the initial massof one of the NSs (Pulsar B). The idea was that for e--capture SN baryonic massof a newborn NS can be well fixed.

[Bisnovatyi-Kogan et al. In press]

However, in reality it is necessaryto know the baryonic mass with1% precision. Which is not easy.

Equator and radiusds2=c2dt2e2Φ-e2λdr2-r2[dθ2+sin2θdφ2]

In flat space Φ(r) and λ(r) are equal to zero.

• t=const, r= const, θ=π/2, 0<Φ<2π l=2πr

• t=const, θ=const, φ=const, 0<r<r0 dl=eλdr l=∫eλdr≠r0

0

r0

Gravitational redshift

)( e

0

e ,e

r

dt

dNr

dt

dN

d

dNdtd

r

r

rcGm

2

2

21

1e

It is useful to use m(r) – gravitational mass inside r –instead of λ(r)

Frequency emitted at r

Frequency detected byan observer at infinity

This function determinesgravitational redshift

NS Radii

A NS with homogeneous surface temperature and local blackbody emission

42 4 TRL

42

2 /

4TDR

D

LF

From X-rayspectroscopy

From dispersion measure

Limits from RX J1856

(Trumper)

Radius determination in bursters

See, for example,Joss, Rappaport 1984,Haberl, Titarchuk 1995

Explosion with a ~ Eddington liminosity.

Modeling of the burst spectrumand its evolution.

Burst oscillations

[Bhattacharyya et al. astro-ph/0402534]

Fitting light curves of X-ray bursts.Rc2/GM > 4.2 for the neutron star in XTE J1814-338

Fe K lines from accretion discs

[Cackett et al. arXiv: 0708.3615]

Measurements of the inner disc radius provide upper limits on the NS radius.

Ser X-1 <15.9+/-14U 1820-30 <13.8+2.9-1.4GX 349+2 <16.5+/-0.8(all estimates for 1.4 solar mass NS)

Suzaku observations

Mass-radius diagram and constraints

(astro-ph/0608345, 0608360)

Unfortunately, there are nogood data on independentmeasurements of massesand radii of NSs.

Still, it is possible to putimportant constraints.Most of recent observationsfavour stiff EoS.

Rotation!

Most rapidly rotating PSR716-Hz eclipsing binary radio pulsar in the globular cluster Terzan 5

(Jason W.T. Hessels et al. astro-ph/0601337)

Previous record (642-Hz pulsar B1937+21)survived for more than 20 years.

Interesting calculationsfor rotating NS have beenperformed recently by Krastev et al.arXiv: 0709.3621

Rotation starts to be important from periods ~3 msec.

QPO and rapid rotationXTE J1739-285 1122 HzP. Kaaret et al.astro-ph/0611716

(Miller astro-ph/0312449)

1330 Hz – one of thehighest QPO frequency

The line corresponds tothe interpretation, thatthe frequency is thatof the last stable orbit,6GM/c2

Rotation and composition

(Weber et al. arXiv: 0705.2708)

Computed for a particular model:density dependent relativistic Brueckner-Hartree-Fock (DD-RBHF)

(equatorial) (polar)

Rotation and composition

(Weber et al. arXiv: 0705.2708) 1.4 solar mass NS (when non-rotating)

hyperonhyperon quark-hybridquark-hybrid quark-hybridquark-hybrid(quarks in CFL)(quarks in CFL)

Mass-radius relation Main features

• Max. mass• Diff. branches (quark and normal)• Stiff and soft EoS• Small differences for realistic parameters• Softening of an EoS with growing mass

Rotation is neglected here. Obviously, rotation results in:• larger max. mass• larger equatorial radius

Spin-down can result in phase transition.Haensel, Zdunikastro-ph/0610549

Gravitational redshift• Gravitational redshift may provide M/R in

NSs by detecting a known spectral line, E∞ = E(1-2GM/Rc2)1/2

• Fe and O lines in EXO 0748-676, M/R ~ 0.22 (Cottam et al 2002)

Combination of different methods

(Ozel astro-ph/0605106)

EXO 0748-676

Limits on the EoS from EXO 0748-676

(Ozel astro-ph/0605106)

Stiff EoS are better.Many EoS for strangematter are rejected.But no all! (see discussionin Nature).

X- hydrogene fractionin the accreted material

However, redshift measurements forEXO 0748-676are strongly critisized now.

Limits on the moment of inertia

Spin-orbital interaction

PSR J0737-3039(see Lattimer, Schutzastro-ph/0411470)

The band refers to ahypothetical 10% error.This limit, hopefully,can be reached in several years of observ.

NS Cooling

NSs are born very hot, T > 1010 K At early stages neutrino cooling dominates The core is isothermal

LLdt

dTC

dt

dEV

th

1)( , 4 2/142 TTTRL ss

Neutrino luminosity

Photon luminosity

Cooling depends on:

(see Yakovlev & Pethick 2004)

1. Rate of neutrino emission from NS interiors2. Heat capacity of internal parts of a star3. Superfluidity4. Thermal conductivity in the outer layers5. Possible heating

Depend on the EoSand composition

Neutrinocooling stage

Photoncooling stage

Main neutrino processes

(Yakovlev & Pethick astro-ph/0402143)

Questions

• Additional heating

• Heating in magnetars (and probably in the M7)

• For example, PSR J0108-1431 [0805.2586] Old (166 Myr) close by (130 pc) PSR with possibly thermal (90 000K) emission)

Cooling curvesCooling curves for hadron (ie. normal) stars.

There are many-many models.

(Kaminker et al. 2001)

Data on cooling

Page, Geppertastro-ph/0508056

Two tests

Age – Temperature

&

Log N – Log S

Standard test: temperature vs. age

Kaminker et al. (2001)

Log N – Log S

Log of flux (or number counts)

Lo

g o

f th

e n

um

ber

of

sou

rces

bri

gh

ter

than

th

e g

iven

flu

x

-3/2 sphere: number ~ r3

flux ~ r-2

-1 disc: number ~ r2

flux ~ r-2

calculations

Log N – Log S as an additional test Standard test: Age – Temperature

Sensitive to ages <105 yearsUncertain age and temperatureNon-uniform sample

Log N – Log SSensitive to ages >105 years (when applied to close-by NSs)Definite N (number) and S (flux)Uniform sample

Two test are perfect together!!!

astro-ph/0411618

List of models (Blaschke et al. 2004)

Model I. Yes C A Model II. No D B Model III. Yes C B Model IV. No C B Model V. Yes D B Model VI. No E B Model VII. Yes C B’ Model VIII.Yes C B’’ Model IX. No C A

Blaschke et al. used 16 sets of cooling curves.

They were different in three main respects:

1. Absence or presence of pion condensate

2. Different gaps for superfluid protons and neutrons

3. Different Ts-Tin

Pions Crust Gaps

Model I Pions. Gaps from Takatsuka &

Tamagaki (2004) Ts-Tin from Blaschke, Grigorian,

Voskresenky (2004)

Can reproduce observed Log N – Log S

Model II

No Pions Gaps from Yakovlev et al.

(2004), 3P2 neutron gap suppressed by 0.1

Ts-Tin from Tsuruta (1979)

Cannot reproduce observed Log N – Log S

Model III

Pions Gaps from Yakovlev et al.

(2004), 3P2 neutron gap suppressed by 0.1

Ts-Tin from Blaschke, Grigorian, Voskresenky (2004)

Cannot reproduce observed Log N – Log S

Model IV

No Pions Gaps from Yakovlev et al.

(2004), 3P2 neutron gap suppressed by 0.1

Ts-Tin from Blaschke, Grigorian, Voskresenky (2004)

Cannot reproduce observed Log N – Log S

Model V

Pions Gaps from Yakovlev et al.

(2004), 3P2 neutron gap suppressed by 0.1

Ts-Tin from Tsuruta (1979)

Cannot reproduce observed Log N – Log S

Model VI

No Pions Gaps from Yakovlev et al.

(2004), 3P2 neutron gap suppressed by 0.1

Ts-Tin from Yakovlev et al. (2004)

Cannot reproduce observed Log N – Log S

Model VII

Pions Gaps from Yakovlev et al. (2004),

3P2 neutron gap suppressed by 0.1.

1P0 proton gap suppressed by 0.5

Ts-Tin from Blaschke, Grigorian, Voskresenky (2004)

Cannot reproduce observed Log N – Log S

Model VIII

Pions Gaps from Yakovlev et al.

(2004), 3P2 neutron gap suppressed by 0.1. 1P0 proton gap suppressed by 0.2 and 1P0 neutron gap suppressed by 0.5.

Ts-Tin from Blaschke, Grigorian, Voskresenky (2004)

Can reproduce observed Log N – Log S

Model IX

No Pions Gaps from Takatsuka

& Tamagaki (2004) Ts-Tin from Blaschke,

Grigorian, Voskresenky (2004)

Can reproduce observed Log N – Log S

HOORAY!!!!

Log N – Log S can select models!!!!!Only three (or even one!) passed the second test!

…….still………… is it possible just to update the temperature-age test???

May be Log N – Log S is not necessary?Let’s try!!!!

Different sensitivities of two tests Effects of the

crust (envelope) Fitting the crust it

is possible to fulfill the T-t test …

…but not the second test: Log N – Log S !!!

(H. Grigorian astro-ph/0507052)

Sensitivity of Log N – Log S

Log N – Log S is very sensitive to gaps Log N – Log S is not sensitive to the crust if it is applied to

relatively old objects (>104-5 yrs) Log N – Log S is not very sensitive to presence or absence of

pions

We conclude that the two test complement each other

Model I (YCA) Model II (NDB) Model III (YCB) Model IV (NCB) Model V (YDB) Model VI (NEB)Model VII(YCB’) Model VIII (YCB’’) Model IX (NCA)

Mass constraint• Mass spectrum has to be taken into account when discussing data on cooling• Rare masses should not be used to explain the cooling data• Most of data points on T-t plot should be explained by masses <1.4 Msun

In particular:• Vela and Geminga should not be very massive

Subm. to Phys. Rev .Cnucl-th/0512098(published as a JINR [Dubna] preprint)

Hybrid starsWe use models of HySsintroduced by Grigorian et al. (2005)Phys. Rev. C 71, 045801astro-ph/0511619

2SC phase

μc = 330 MeV

Mass spectrum

Model I

Model II

Model III

Model IV

Resume for HySs

One model among four was able to pass all tests.

Cooling of X-ray transients“Many neutron stars in close X-ray binaries are transient accretors (transients);They exhibit X-ray bursts separated by long periods (months or evenyears) of quiescence. It is believed that the quiescence corresponds to a lowlevel,or even halted, accretion onto the neutron star. During high-state accretionepisodes, the heat is deposited by nonequilibrium processes in the deep layers(1012 -1013 g cm-3) of the crust. This deep crustal heating can maintain thetemperature of the neutron star interior at a sufficiently high level to explain apersistent thermal X-ray radiation in quiescence (Brown et al., 1998).”

(quotation from the book by Haensel, Potekhin, Yakovlev)

Cooling in soft X-ray transients

[Wijnands et al. 2004]

MXB 1659-29~2.5 years outburst

~1 month

~ 1 year

~1.5 year

Aql X-1 transient

A NS with a K star.The NS is the hottestamong SXTs.

Deep crustal heating and cooling

ν

γ

γ

γγ

γ

Accretion leads to deep crustal heating due to non-equilibrium nuclear reactions.After accretion is off:• heat is transported inside and emitted by neutrinos• heat is slowly transported out and emitted by photons

See, for example, Haensel, Zdunik arxiv:0708.3996

New calculations appeared very recently 0811.1791 Gupta et al.

Time scale of cooling(to reach thermal equilibrium of the crust and the core) is ~1-100 years.

To reach the state “before”takes ~103-104 yrs

ρ~1012-1013 g/cm3

Pycnonuclear reactionsLet us give an example from Haensel, Zdunik (1990)

We start with 56FeDensity starts to increase

56Fe→56Cr56Fe + e- → 56Mn + νe

56Mn + e- → 56Cr + νe

At 56Ar: neutron drip56Ar + e- → 56Cl + νe

56Cl → 55Cl +n55Cl + e- → 55S + νe

55S → 54S +n54S → 52S +2n

Then from 52S we have a chain:52S → 46Si + 6n - 2e- + 2νe

As Z becomes smallerthe Coulomb barrier decreases.Separation betweennuclei decreases, vibrations grow.40Mg → 34Ne + 6n -2e- + 2νe

At Z=10 (Ne) pycnonuclear reactions start.

34Ne + 34Ne → 68Ca36Ne + 36Ne → 72Ca

Then a heavy nuclei can react again:72Ca → 66Ar + 6n - 2e- + 2νe

48Mg + 48Mg → 96Cr96Cr → 88Ti + 8n - 2e- + 2νe

A simple model

[Colpi et al. 2001]

trec – time interval between outburststout – duration of an outburstLq – quiescent luminosityLout – luminosity during an outburst

Dashed lines corresponds to the casewhen all energy is emitted from a surface by photons.

Deep crustal heating~1.9 Mev per accreted nucleonCrust is not in thermal equilibrium with the core.After accretion is off the crust cools down andfinally reach equilibrium with the core.

[Shternin et al. 2007]

KS 1731-260

Testing models with SXT

[from a presentation by Haensel, figures by Yakovlev and Levenfish]

SXTs can be very important in confronting theoretical cooling models with data.

Theory vs. Observations: SXT and isolated cooling NSs

[Yakovlev et al. astro-ph/0501653]

Conclusions Mass

Radius

Red shift (M/R)

Temperature

Moment of inertia

Gravitational vs. Baryonic mass

Extreme rotation

In binaries, especially in binaries with PSRs.In future – lensing.

In isolated cooling NS, in bursters in binaries,in binaries with QPO

Via spectral line observations

In isolated cooling NSs and in sometransient binaries (deep crustal heating)

In PSRs (in future)

In double NS binaries, if good contraintson the progenitor are available

Millisecond pulsars (both single and binary)