The Cooling ofNeutron Stars

Dany PageInstituto de Astronomía, UNAM, Mexico

KIAS - APCTP International Symposium in Astro-Hadron Physic Seoul, Korea, 10 - 14 November 2003

Neutrino Emission ScenariosPrologue ...

The previously denominated “Standard Cooling Model”

Nucleon pairing introducesanother neutrino process due to the

FORMATION and BREAKING of COOPER PAIRSFlowers, Ruderman & Sutherland, Ap. J. 205 (1976), 541

Voskresenskii & Senatorov, Zh. Eksp. Teor. Fiz. 90 (1986), 1505 [JETP 63 (1986), 885]

Voskresenskii & Senatorov, Yad. Fiz. 45 (1987), 657 [Sov. J. Nucl. Phys. 45 (1987), 411]

Minimal Coolingof

Neutron Stars

Dany PageInstituto de Astronomía, UNAM

Ongoing collaboration with:

•J.H. Lattimer (SUNY Stony Brook)

•M. Prakash (SUNY Stony Brook)

•A. Steiner (UM, Mineapolis)

Revised version of the “Standard Model”

PART I

Motivation:Many new observations of cooling neutron stars

with CHANDRA and XMM-NEWTON.

Some have low estimates of Te

Do we have any strong evidence for the presence of some “exotic” component in the

core of some of these neutron stars ?

ATMOSPHERE: a few cm thick.Determines the spectrum: distribution of

observable flux as a function of photon energy Measurement of “surface” temperature

ENVELOPE: a few tens of meter thick.Blanket which controls the outgoing heat flux

Luminosity

CRUST: only important for the early cooling, little effect later on.

OUTER CORE: n, p, e, essential for neutrino emission, and

thermal energy content

INNER CORE: mystery. Assumed not to exist for now.

The Supranuclear

Equation of State (EOS)

for the

Minimal Model

•APR: Akmal & Pandharipande, Phys. Rev. C56 (1997), 2261

Akmal, Pandharipande & Ravenhall, Phys. Rev. C58 (1998), 1804 [AV18 potential + UIX 3body interaction + vb boost]

•WFF3: Wiringa, Fiks & Fabrocini, Phys. Rev. C38 (1988), 1010 [UV14 potential + TNI 3body interaction]

•BPAL21 & BPAL31: Bombaci, Prakash, Ainsworth & Lattimer, Phys. Rep. 280, 1 (1997) [Parametric EOS which reproduces saturation properties, with S ~ n1/2]

Selection criteria for the supranuclear EOS: • The only present baryons are neutrons and protons. (No meson condensate, no hyperons, no quark matter, no ...)• The proton fraction is sufficiently low that DURCA is not allowed.

Point 2 eliminates most Effective Field Theoretical (EFT) models andrelativistic Dirac-Brückner-Hartree-Fock (DBHF) models

PRESSURE vs. DENSITY

0 1 2 3 4 5 6

nB/n0n0 = saturation density

Neutron Star MASS vs. RADIUS

At 1.4 Mo :

R ~ 11 – 12 km

At MMax:

R ~ 9.5 – 10.5 km

NUCLEON EFFECTIVE MASS

Conclusions:

Within the Minimal Model the EOS is pretty well defined.

• 1.4 Mo neutron stars have radii ~ 11 - 12 km• MMax neutron stars have radii ~ 9.5 – 10.5 km

The Envelope: (outer boundary condition)

•Sensitivity Strip

•Magnetic field

•Chemical composition

Temperature profile in the envelope:the “sensivity strip”

Gudmundsson, Pethick & Epstein, Ap. J. 259 (1982), L19 and Ap. J. 272 (1983) 286

2/1 s

8b

6s

TT

and K 10 TK when 10T

b∝

≈≈

“Te – Tb relationship” for dipolar and dipolar+quadrupolar fields

Page & Sarmiento, 1996

Menv = 0

Light elements in the envelope

Chabrier, Potekhin & Yakovlev, ApJ 477 (1997), L99

Menv = 10-17 Mo

Light elements in the envelope

Chabrier, Potekhin & Yakovlev, ApJ 477 (1997), L99

Menv = 10-15 Mo

Light elements in the envelope

Chabrier, Potekhin & Yakovlev, ApJ 477 (1997), L99

Menv = 10-13 Mo

Light elements in the envelope

Chabrier, Potekhin & Yakovlev, ApJ 477 (1997), L99

Menv = 10-11 Mo

Light elements in the envelope

Chabrier, Potekhin & Yakovlev, ApJ 477 (1997), L99

Menv = 10-9 Mo

Light elements in the envelope

Chabrier, Potekhin & Yakovlev, ApJ 477 (1997), L99

Menv = 10-7 Mo

Light elements in the envelope

Chabrier, Potekhin & Yakovlev, ApJ 477 (1997), L99

€

dEth

dt = CV

dTdt

= −Lν − Lγ

€

dTdt

= − qν 0

cV 0

×T 7 ⇒ t − t0 = A 1T 6 − 1

T06

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ⇒ T ∝ t−1/6

€

CV = 43 πR3 cV0 ×T

Lν = 43 πR3 qν 0 ×T 8

Lγ = 4πR2 Te4 ∝T 2+α [α <<1]

Neutrino Cooling era: L >> L

Photon Cooling era: L << L

€

dTdt

∝ −T 1+α ⇒ t − t0 = B 1T α − 1

T0α

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ⇒ T ∝ t−1/α

Basic Cooling: neutrino vs photon cooling eras

)(TTT ee =

Effect of envelope chemical compositions

Light elements envelope

Iron-like envelope

€

Lγ = 4πR2 Te4 ∝T 2+α

Neutron and Proton

Pairing

Predictions for the NEUTRON 1S0 gap

WAP: Wambach, Ainsworth & Pines, Nulc. Phys. A555 (1993), 128

CCDK: Chen, Clark, Davé & Khodel, Nucl. Phys. A555 (1993), 59

SCLBL: Schulze, Cugnon, Lejeune, Baldo & Lombardo, Phys. Lett. B375 (1996), 1

SFB: Schwenk, Friman & Brown, Nucl. Phys. A717 (2003), 191

Crust-core transition

Important feature:Medium polarization effects reduce Tc by a factor three

Predictions for the PROTON 1S0 gap

T: Takatsuka, Prog. Thero. Phys. 50 (1970), 905

CCY: Chao, Clark & Yang, Nucl. Phys. A179 (1972), 320

AO: Amundsen & Osgaard, Nucl. Phys. A437 (1985), 487

BCLL: Baldo, Cugnon, Lejeune & Lombardo, Nucl. Phys. A536 (1992), 349

CCDK: Chen, Clark, Davé & Khodel, Nucl. Phys. A555 (1993), 59

EEHO: Elgaroy, Engvik, Horth-Jensen & Osnes, Nucl. Phys. A604 (1996), 466

Important features:

All vanish at pF >1.3 fm-1

and most at pF > 1 fm-1

Expected maximum Tc ~ 1 - 2 x 109 K

Medium polarization effects seem to reduce Tc by a factor three

Predictions for the NEUTRON 3P2 gap

0: Hoffberg, Glassgold, Richardson & Ruderman, Phys. Rev. Lett. 24 (1970), 775

1: Amundsen & Osgaard, Nucl. Phys. A442 (1985), 4163

2: Takatsuka, Prog. Theor. Phys. 48 (1972), 1517

a, b, c:Baldo, Elgaroy, Engvik, Horth-Jensen & Schulze,

Phys. Rev. C58 (1998), 1921

Important feature:

WE DO NOT REALLY KNOW WHAT IT IS

Medium polarization effects were expected to increase the 3P2 gap while they probably strongly suppress it.

Specific Heat

and its

Suppression by Pairing

Distribution of Cv in the core among constituents

At T=109 K

€

CV = N(0) π 2

3 kB2T N (0) = m* pF

π 2h3

€

CVpaired =

CVnormal × M (T /Tc )

≈ CVnormal × e−Δ(T ) /kT

Pairing and neutrino emission:

•Supression

•Cooper pair formation and destruction

Suppression of MURCA et al. by pairingnormalpaired )phase;/( qTcTSq =

Neutrino emission through the formation and breaking of Cooper pairs

Flowers, Ruderman & Sutherland, Ap. J. 205 (1976), 541

Voskresenskii & Senatorov, Zh. Eksp. Teor. Fiz. 90 (1986), 1505 [JETP 63 (1986), 885]

Voskresenskii & Senatorov, Yad. Fiz. 45 (1987), 657 [Sov. J. Nucl. Phys. 45 (1987), 411]

89

21MUrca

79

22Coop

T 10

T )phase;/( 10

≈

≈

q

TcTFq

Cooper pair neutrino luminosities for p 1S0 and n 3P2 gaps (APR 1.4 Mo)

Cooper Pair Neutrino Luminosities vs MURCA and Photons in complete realistic evolutionary calculations (APR 1.4 Mo)

Neutron 3P2 gap “a” Neutron 3P2 gap “b” Neutron 3P2 gap “c”

Proton 1S0 gap from Amundsen & Ostgaard

Variations on a theme:

•Varying the star´s mass

•Varying the EOS

•Cranking up the MURCA rate

Varying the star´s mass

EOS: APR

Varying the EOS

Cranking up the MURCA rate

(à la Friman & Maxwell)

Putting things together:

Minimal Model(and all its uncertainties)

vs.

DATA(and all their uncertainties)

Everything together:

All possible neutron and proton gapsLight element envelopes

Heavy element envelopes

Heavy element envelopes

All possible neutron and proton gaps

Predictions for the NEUTRON 3P2 gap

Heavy elements envelopesNeutron 3P2 gap = 0 All possible n & p 1S0 gaps

Heavy elements envelopesNeutron 3P2 gap = "a" (Tc ~109 K) All possible n & p 1S0 gaps

Heavy elements envelopesNeutron 3P2 gap = "b" (Tc ~3x109 K) All possible n & p 1S0 gaps

Heavy elements envelopesNeutron 3P2 gap = "c" (Tc ~1010 K) All possible n & p 1S0 gaps

Heavy elements envelopes All possible n & p gaps

Light element envelopes

All possible neutron and proton gaps

Light elements envelopesNeutron 3P2 gap = 0 All possible n & p 1S0 gaps

Light elements envelopesNeutron 3P2 gap = "a" (Tc ~109 K) All possible n & p 1S0 gaps

Light elements envelopesNeutron 3P2 gap = "b" (Tc ~3x109 K) All possible n & p 1S0 gaps

Light elements envelopesNeutron 3P2 gap = "c" (Tc ~1010 K) All possible n & p 1S0 gaps

Light elements envelopesAll possible n & p gaps

Light element envelopes

Iron envelopes

Summary: Temperature vs Time

Summary: Luminosity vs Time

CONCLUSIONSabout the

THEORY • EOS quite well determined

• The mass of the star has little impact

• The dominant neutrino emission process is from the formation and breaking of Cooper pairs from the neutron 3P2 gap (unless this gap is very small)

• Possibility of the presence of light elements in the envelope allows to accomodate a range of Te at a given age

CONCLUSIONSabout

COMPARISON with DATA • Neutron 3P2 pairing with Tc ~ 109 K and various envelope composition may be marginally acceptable.

CONCLUSIONSabout

COMPARISON with DATA • Neutron 3P2 pairing with Tc > 3x109 K and various envelope composition seems to be marginally inacceptable.

CONCLUSIONSabout

COMPARISON with DATA • Neutron 3P2 pairing with Tc ~ 0 is inacceptable and would requiere a more elaborate model

but a vanishing neutron 3P2 gap is a

serious problem

Fast Coolingof

Neutron Stars

PART II

ATMOSPHERE: a few cm thick.Determines the spectrum: distribution of

observable flux as a function of photon energy Measurement of “surface” temperature

ENVELOPE: a few tens of meter thick.Blanket which controls the outgoing heat flux

Luminosity

CRUST: only important for the early cooling, little effect later on.

OUTER CORE: n, p, e, essential for neutrino emission, and

thermal energy content

INNER CORE: mystery. ==> Strong neutrino emission

Neutrino Emission Scenarios

Fast Cooling with Direct Urca Process

“The Cooling of Neutron Stars by the Direct Urca Process”, Page & Applegate, ApJ 394, L17 (1992)

Critical mass for Durca:

1.35 Mo

Notice: the 1.4 Mo star has a "Durca pit" of 0.04 Mo !

<- Arbitrary, we DO NOT KNOW what it really is

Fast Cooling with Direct Urca Process

“The Cooling of Neutron Stars by the direct Urca Process”, Page & Applegate, ApJ 394, L17 (1992)

With pairing (e.g., n 3P2) the cooling can be temporarily stopped at practically any temperature, depending on the value of Tc in the "Durca pit"

Fast Cooling with a Kaon Condensate

“Strangeness Condensation, Nucleon Superfluidity, and Cooling of Neutron Stars”, Page & Baron, ApJ 354 L17 (1990)

Fast Neutrino Emission Scenarios

Q =

T erg s-1 cm-3

[K- condensate]

T erg s-1 cm-3

- condensate

T erg s-1 cm-3

[Direct URCA]

From: D. Page, “Thermal Evolution of Isolated Neutron Stars”,in The Many Faces of Neutron Stars [NATO ASI, Lipari, 1996]

“Prospects of Detecting Baryon and Quark Superfluidity from Cooling Neutron Stars”, Page, Prakash, Lattimer & Steiner, PRL 85, 2048 (2000)

A "Maximal Model"Direct Urcas with Nucleons, Hyperons and Quarks

€

H (t) = J44 ×1040 t + τ 0

100 yrs ⎛ ⎝ ⎜

⎞ ⎠ ⎟−3/2

erg s-1

J44= differential angular momentum in the frictionally coupled inner crust neutron superfluid, in units of 1044 g cm2 rad s-1

Fast Cooling with a Kaon Condensate with frictional heating and light element envelopes

“Fast Cooling of Neutron Stars: Superfluidity versus Heating and Accreted Envelope”, Page, ApJ 479, L43 (1997)