Gravitational Collapse Supernovae · Gravitational Collapse Supernovae James Lattimer Department of...

Gravitational Collapse Supernovae

James Lattimer

Department of Physics & Astronomy449 ESS Bldg.

Stony Brook University

April 13, 2017

Nuclear Astrophysics [email protected]

James Lattimer Gravitational Collapse Supernovae

Neutrino Opacities

In the Fe cores of massive progenitor stars prior to gravitational collapse, theelectron fraction is Ye = Z/A ' 0.42. During collapse, Ye decreases further,and ultimately will reach values near Ye ' 0.04 in the final neutron star.However, the short mean free path of neutrinos in dense matter delays thisreduction.

The weak interaction cross section is

σo = 4π“mec

~

”4„

GF

mec2

«2

= 1.76 · 10−44 cm2.

The four major neutrino-matter interactions are

1. Neutral current free nucleon scattering (ν + nZ−→ ν + n, ν + p

Z−→ ν + p)

σn =σo

4

„Eν

mec2

«2

= 1.7 · 10−44 E 2ν

MeV2 cm2 ND

σn =π2σo

64(1+2g 2

A)

„kBT

mec2

«2EνpF c

mnc2

εF= 2.1·10−45 T 2Eν

MeV3

ns

ncm2 ED


2. Neutral current heavy nucleus scattering (ν + (Z ,A)Z−→ ν + (Z ,A)),

σA =σo

16

„Eν

mec2

«2 hA + Z(4 sin2 θW − 2)

i2

' 4.2× 10−45N2 E 2ν

MeV2 cm2

3. Charged current nucleon absorption (νe + nW−→ p + e−, νe + p

W−→ n + e+)

σa =σo

2(1 + 3g 2

a )Ye

„Eν

mec2

«2

= 1.9 · 10−43YeEnu

2

MeV2 cm2 ND

σa =3π2σo

128(1 + 3g 2

A)

„kBT

mcc2

«2mnc

2

εF

„Ye

1− Ye

«1/3

= 1.43 · 10−42

„Ye

1− Ye

«1/3T 2

MeV2

“ns

n

”2/3

cm2 ED

4. Charged and neutral current electron scattering (ν + e−W ,Z−−−→ ν + e−)

σe = 0.1σo

„Eν

Mec2

«2Eνµe' 2.0 · 10−47

„ns

nYe

«1/3E 3ν

MeV3 cm2

The largest source of opacity during collapse is from coherent scattering.Neutrino mean free path λν =< σn >−1 (ρ12 = nmn/(1012 g cm−3)) is

λν '60

ρ12

“6Xn + 5Xp + A(1− xN )2XA

”−1„

10 MeV

Eν

«2

km.


Neutrino Trapping

Effectively, λν equals the core’s size when

λν '„

3M

4πρ

«1/3

= 87.3

„M

1.4 M

«1/3

ρ−1/312 km

which occurs when ρ ' 3 · 1010 g cm−3.

But trapping only occurs later, when the neutrino diffusion timescale is smallerthan the collapse timescale.

Diffusion in spherical symmetry: The flux is driven by a density gradient.

Fν = −cλν3

∂nν∂r

The diffusion equation is:

∂nν∂t

= − 1

r 2

∂r 2Fν∂r

=cλν3r 2

∂

∂r

»r 2 ∂nν∂r

–.

For simplicity, assume λν is a constant, and seek a separable solution of thetype nν = noψ(r)φ(t). We find

1

φ

dφ

dt=

cλν3r 2ψ

d

dr

»r 2 dψ

dr

–= −α

where α is a constant.James Lattimer Gravitational Collapse Supernovae

Solving each differential equation, we find

φ = φoe−αt , ψ =

sinβr

βr, β =

r3α

cλν.

Note that the radial equation is the same as the Lane-Emden equation forpolytropic index n = 1. If one takes the radius of the neutrinosphere to beR ' π/β, since ψ = 0 there, the diffusion timescale is

τd =1

α=

3R2

π2cλν' 0.013ρ12

„M

1.4 M

Yν0.06

«2/3

s.

The collapse timescale can be estimated from self-similar collapse models,

τc = f

„3

8πGρ

«1/2

' 0.0077ρ−1/212 s

where f =√

153 for γ = 4/3 and f =√

33 for γ = 1.30, which we used in theabove. Equating these timescales gives the trapping density

ρ12,trap ' 0.71

„M

1.4 M

Yν0.06

«−4/9


Entropy During the Collapse

Following trapping, we will find the collapse proceeds approximatelyadiabatically, since significant heat is unable to be lost on collapse timescales.

The entropy prior to collapse has contributions from nuclei, free neutrons andprotons, and electrons. Assume T ' 0.7 MeV, ρ ' 109 g cm−3 and Ye ' 0.42.

Nuclear entropy originates from translation and excited states. Per nucleus:

SH,trans

kB=

5

2+ ln

"„56mbkBT

2π~2

«3/21

nH

#' 17,

SH,ex

kB= 56

π2

2

T

EF' 4.8

where nH is the number density of nuclei (assumed to be iron) and EF ' 35MeV is the Fermi energy of nuclear matter.

The electron entropy, per electron, is Se = π2kBT/µe ' 1.1kB .

The dilute vapor of nucleons (mostly neutrons) has an entropy per nucleon

Sn,p

kB=

5

2+ ln

"„mbkBT

2π~2

«3/22

nn,p

#' (12.9, 36)

where nn is the neutron density. The total entropy per baryon is thus

s = XHSH,trans + SH,ex

56+ SeYe + SnXn + SpXp ' 0.92kB .

For comparison, the solar center has s ' 16.5kB .


Thermodynamics During Collapse

The first law of thermodynamics can be written

Q = kBTs +X

i

µi Yi =< Eν,esc >“Ye + Yν

”,

where i = (H, n, p, e). The heat change Q = dQ/dt is due to escapingneutrinos. In nuclear statistical equilibriumX

i

µi Yi = Ye(µe − µ) + Yνµν ,

sokBTs = −Ye (µe − µ− µν)−

“Ye + Yν

”(µν− < Eν,esc >) .

There is entropy generation from being out of beta equilibrium, and fromneutrino energy losses. Early on, neutrinos freely escape, and µν = 0. Whenneutrinos are trapped, < Eν,esc >= µν because only neutrinos at the top of theFermi sea will escape. Thus

kBTs = −Ye (µe − µ− < Eν,esc >) , or kBTs = −Ye (µe − µ− µν)

depending on trapping. Initially, entropy changes, but once trapping ensues and

beta equilibrium follows, the entropy is frozen.


Ye Changes During Collapse

Changes in composition (Ye) are due to electron captures on heavy nuclei andfree protons.

Electron captures on heavy nuclei are very sensitive to the energy available forelectron capture, ∆ = µe − µ, perhaps as the 3rd or 4th power. As the electronfraction decreases, the heavy nuclei become more neutron rich and moremassive. But shell closures will effectively halt electron captures on heavynuclei some time after the collapse begins.

Following this, electron captures are dominated by those on free protons

Ye =3

5

„µe

mec2

«2

nYeXpσoc ' 488ρ12YeXpµ2e s−1

The free proton abundance is Xp ∝ eµp/kB T .

In the liquid drop model, the dominant nucleus has A = (as + Ss I2)/(2acx

2);

µp = −B +2

3

as

A1/3+

acxA2/3

3(6− x) + (1−2x)

»Sv (2x − 3) +

2Ss

3A1/3(4x − 5)

–.

Thus ∂µp/∂x ' 90 MeV showing that µp is very sensitive to Ye ' x . As Ye

decreases, µp falls, which decreases both Xp and Ye . Therefore, electron

captures are highly self-regulating. Ye falls slowly with increasing density.


Electron captures on nuclei produce neutrinos with average energy εν ' 3∆/5,where ∆ = µe − µ− 3MeV is the maximum energy available for captures. (3MeV represents a typical excitation energy in the daughter nucleus.)

Before trapping, when neutrinos escape freely, this leads to an entropy increasekBTs ' −Ye [(2/5)(µe − µ) + 1.8MeV] ' −4YeMeV.

Electron capture on free protons produces neutrinos with a larger εν ' 5µe/6.Before neutrino trapping, this leads to an entropy losskBTs ' −Ye(µe/6− µ) ' 9Ye MeV.

After the density exceeds 1012 g cm−3, neutrinos become trapped. Inversecapture reactions build so that Yν → −Ye , and the net lepton numberYL = Ye + Yν becomes frozen. Although Ye decreases further with increasingdensity, Yν rises to compensate.

After trapping, neutrinos become degenerate with µν → µe − µ as betaequilibrium finally becomes established, and we have Ts ' 0.

There is only a small window early in the collapse where the entropy canchange, and changes due to the two modes of electron capture largely cancel.Effectively, the entropy per baryon remains near unity throughout collapse.

Calculations show that at the end of collapse the trapped lepton fraction

YL ' 0.38, with Ye ' 0.32, whereas initially Ye ' 0.42.


Core Bounce and Shock Formation

The inner core collapses homologously and subsonically, and when the centraldensity reaches nuclear densities, the resulting bounce affects the entirehomologous core. The velocity gradient at the core’s edge steepens into ashock, which moves outwards from the edge of the core, whose interior remainsunshocked, at about r = 30 km. The shock effectively damps inner coreoscillations, which forms a proto-neutron star.

However, the shock has to do work, not only reversing the motion of matter itencounters, but also dissociating nuclei at a cost of nearly 8 MeV per baryon.As a result, the shock at least temporarily stalls when it reaches a distance ofr = 100 to 200 km.

The energy of the shock originates from binding energy released by theformation of the unshocked core. In general, one expects this binding energywill scale with the dimensions of the inner core as GM2

ic/Ric ∝ M5/3ic . For

polytropes, masses scale as K 3/2. For matter near ns , whose pressure isdominated by leptons, we expect K ∝ Y

4/3L . Thus, the binding energy should

scale as Y10/3

L .


Core Binding Energy

Below ns , the effective polytropic exponent is about 4/3, and if the inner corehad a sub-nuclear central density, we would expect its total energy to beapproximately zero. However, the central density exceeds ns and the effectivepolytropic exponent at those densities is much larger due to nuclear repulsion.Indeed, self-similar results for γ <∼ 4/3 indicate the core mass will exceed theeffective Chandrasekhar mass by about 10%.

The mass in the inner core residing at densities greater than ns leads tobinding, as can be seen by considering nested polytropes. Consider theequation of state satisfying

p = Kρ1+1/n; ε = np ρ < ρt

p = Kρ1/n−1/n11 ρ1+1/n1 ; ε = n1p + (n − n1)pt ρ > ρt

For this equation of state, one can manipulate the identities dΩ = VdP anddU = εdV , where E = U + Ω, to find

E =n − 3

5− n

GM2

R+

»n1 − 3

5− n1− n − 3

5− n

–GM2

t

Rt+3Pt

»Mt

ρt− Vt

– »n − 1

5− n− n1 − 1

5− n1

–.

The mass, radius and volume interior to the point where ρ = ρt are Mt ,Rt and

Vt , respectively. In the idealized case where n = 3 and n1 = 0, one sees that

E = −(3/5)GM2t /Rt ; the high-density core provides binding.


Prompt Success of the Shock

Although one can’t immediately estimate Mt or Rt , it is possible to note thatin hydrostatic equilibrium, the energy change from adding the mass dM isdE = −GMdM/R, so one expects the binding energy to be

BE = GMCh(M −MCh)/R ' 0.1GM2Ch/RCh ' (5− 10) · 1051 erg s−1.

As we established earlier, BE ∝ Y10/3

L,fin , with YL,fin the final lepton fraction.

To lowest order, a successful shock has to be able to dissociate the remainderof the iron core that accretes through the shock onto the core. The rapidsteepening of the density gradient beyondthe edge of the iron core will result in asuccessful shock, if only the shock is notcarried inward by the infalling matterbefore it can propagate that far.

Dissociation of iron nuclei comes ata cost of about 9 MeV/nucleon or1.8 · 1052 erg/M. The amount ofmass to be dissociated is the massof the initial iron core minus themass of the final homogous core, i.e.Mdis ∝ Y 2

e,init − Y 2L,fin. We have seen

that Ye,init ' 0.42 and YL,fin ' 0.38.


Long Term Shock Success

It’s long been realized that neutrinos are important in the supernovamechanism. The unshocked core, whose entropy is about s ∼ 1kB per baryon,has T ' 20 MeV. Within, neutrinos are trapped, but beyond about Rν ∼ 30km, neutrinos escape with a roughly thermal distribution with a temperatureabout Tν ∼ 4− 5 MeV. The shock has stalled at Rs ∼ 100− 200 km.

The dominant means of exchanging energy between neutrinos and matter isthrough charged current nucleon absorption, for which the cross section,assuming non-degeneracy, is

σa =σo

2(1 + g 2

A)Ye

„Eν

mec2

«2

.

With Ye = 1/2, this results in an opacity

κa =σa

mn= κo

E 2ν

MeV2 = 6.7 · 10−20 E 2ν

MeV2 cm g−1.

Neutrino energy deposition can’t cause an explosion, since the Eddington limit

Lν,Edd =4πcGM

κa= 1.0 · 1057 MeV2

E 2ν

erg s−1 ∼ 1054 erg s−1,

assuming < Eν >∼ 30 MeV, is about 100 times larger than the expected Lν .


Neutrino Supernova Mechanism


Neutrino Heating and Cooling

The neutrinos emitted from the newly-formed neutron star will be thermal withtemperature Tν ; the average mean square energy is

< E 2ν >=

R∞0

fνE3νE

2νdEνR∞

0fνE 3

νdEν=

F5(0)

F3(0)T 2ν =

310π2

147T 2ν = 20.8T 2

ν

where F ’s are Fermi integrals. νe ’s and νe ’s are emitted roughly equally.

The dominant coupling is through nucleon absorptions of, and thermalradiation through, both νe ’s and νe ’s. The net heating rate per gram is

q =7ac

16κo

F5(0)

F3(0)T 6ν

"f

4

“ rνr

”2

−„

T

Tν

«6#.

We assumed µe/T ∼ 0. The 7/16 isfrom Fermi statistics; a = π2/[15(~c)3].The heating (cooling) rate isproportional to T 6

ν (T 6): T 4 from thethermal distribution and T 2 from theenergy dependence of the cross section.f is a geometrical factor which is 4 inthe opaque limit (r = rν) and 1 in thefree-streaming limit (r →∞).

rν

rs


Neutrino Re-Heating

Where T is small, there is net heating. The maximum temperature is reachedwhen heating and cooling balance, and then, using r7 = r/107 cm,

T = Tν

√f rν2r

!1/3

' 0.5Tν

r1/37

.

Ordinarily, one expects that as the protoneutron star loses neutrinos, that theaverage neutrino energy and Tν should decrease. However, because the loss ofneutrinos leads to a loss of electrons as well, because of beta equilibrium, this isaccompanied by a loss of pressure, leading to compressional heating. Therefore,while leptons are being lost near the neutrinosphere, Tν actually rises, which ishelpful.

There is also an overall decrease in density at a given radius, as the infallingmatter thins with time, so the optical depth decreases, revealing highertemperature regions nearer the core, which also leads to an increasing Tν withtime.

Early after shock formation, perhaps 20 ms after bounce, the matter accreted

through the shock has a density ρ ' 1010 g cm−3 and is electron-rich. The

electron capture timescale is short, however, τcap ∼ 5(2ρ10Ye)−5/3 ms. There is

thus net electron and pressure loss, so the mantle (above the core, behind the

shock) sinks and accretes onto the core.James Lattimer Gravitational Collapse Supernovae

Conditions for Shock Success

Self-similar arguments show that the pre- and post-shock densities vary as

ρpre ∝ ρpost ∝ r−3/2t1−3γ/2 ∝ r−3/2t−1

with γ ∼ 4/3. As time goes on, τcap decreases. τcap ' t when ρ10 ' 0.1,halting electron captures.

Rarefaction of matter also leads it to become radiation-dominated, whichoccurs when, using T = Tmax ,

ρ = ρrd = 4 · 109

„T

2.5 MeV

«3

g cm−3 ' 4 · 109

„Tν

5 MeV

«31

r7g cm−3.

As long as matter is matter pressure-dominated, the specific internal energy iskept approximately constant at a given radius, since T ' Tmax . Thegravitational specific energy

−Eg '=GM

r' 2 · 1019 M

1.5M

1

r7erg g−1,

is independent of ρ. However, the specific internal energy ofradiation-dominated matter Erd ∝ T 4/ρ increases with time.

A critical density is reached when Erd = |Eg |. Using T = Tmax , this is

ρcrit = 7.4 · 108

„T

5 MeV

«4

r−1/37 g cm−3.


Time Is On the Supernova’s Side

In a similar way, one can show that the radiation pressure will exceed the rampressure, the pressure of matter ahead of the shock, ρprev

2pre , at the same

density.

Once matter is radiation-dominated, it no longer needs to be pushed to escape:it’s total energy is neutral.

A crucial assumption in the above is that T ' Tmax . This assumptiondepends on the heating timescale, Eg/q.

τH 'Eg

q' 1054 erg s−1

Lν

Mr7

1.5M

„5 MeV

Tν

«2

ms ' 10− 100 ms.

The neutrino luminosity is Lν = (7π/16)acr 2νT

4ν .

Thus, while heating is not instantaneous, it is faster than the timescales forincreases in Tν , decreases in pre- and post-shock densities, electron captureturn-off, ram pressure decrease, specific energy increases, etc.

Everything depends on the neutrino flux and temperature.


Competition Between Accretion and Heating

The bounce shock stalls within about 20 ms of its creation at a distance ofrs ∼ 100− 200 km. Whether it is pushed back onto the core or successfullypropagates outwards depends on the rate of mass accretion (−M) and neutrinoheating of the matter behind the shock. Matter in front of the shock has toolow density to absorb much energy from neutrinos.

The dynamical timescale is τd = rs/vs where vs is the velocity of matter behindthe shock. If τd is smaller than for significant changes in M, a quasisteady-state is achieved. Once again we examine the Eulerian equations ofhydrodynamics:

ρdv

dt= ρv =

∂v

∂t+ ρv

∂v

∂r= −Gmρ

r 2− ∂P

∂r,

−M =∂m

∂t+ 4πr 2ρv ,

Ts +X

i

µi Yi = T

»∂s

∂t+ v

∂s

∂r

–+X

i

µi

»∂Yi

∂t+ v

∂Yi

∂r

–= q

=∂ε

∂t+ P

∂(1/ρ)

∂t+ v

»∂ε

∂r+ P

∂(1/ρ)

∂r

–= q,

where i = n, p, e. Ye = Yp = −Yn. We assume M is a constant.


For definiteness, we approximate the dilution factor f in q as

D =f

4

r 2ν

r 2=

1

2

»1 +

r 2ν

r 2

–"1−

r1− r 2

ν

r 2

#.

The rate of change of the particle fractions follow closely from the form of q:

q =7ac

16κo

F5(0)

F3(0)T 6ν

»D − T 6

T 6ν

„F5(ηe)

F5(0)Ye +

F5(−ηe)

F5(0)(1− Ye)

«–,

v∂Ye

∂r=

7ac

16κo

F4(0)

F2(0)T 5ν

»D(1− 2Ye)− T 5

T 5ν

„F4(ηe)

F4(0)Ye −

F4(−ηe)

F4(0)(1− Ye)

«–.

We assumed that µνe = −µνe = 0, so

Lνe = Lνe =7ac

16πr 2νT

4ν ,

and ηe = µe/T . We also assumed the absence of heavy nuclei in thepost-shock region. The terms proportional to D account for νe and νe captureson nucleons, and the others account for the inverse reactions of electron andpositron captures, respectively, on nondegenerate free nucleons.

We neglected energy transfers from ν-lepton and ν-nucleon scattering, which is

an order 10% effect.


Steady-State Solution

Assuming steady-state, ∂/∂t = 0 and ∂/∂r = d/dr , and a constant massm(r) = M, we have

1

v

dv

dr= −1

ρ

dρ

dr− 2

r, v

dv

dr= −GM

r 2− 1

ρ

dP

dr, ρ

dε

dr=

P

ρ

dρ

dr+ q.

ε and P are functions of ρ,Ye and T . We can eliminate dv/dr .

dρ

dr(c2

s − v 2) =2ρ

r

„v 2 − GM

r

«− A

CV

q

v− B

dYe

dr,

CVdT

dr(c2

s − v 2) =2ρC

r

„v 2 − GM

r

«+“c2

T − v 2” q

v−

−dYe

dr

"“c2

T − v 2”„ ∂ε

∂Ye

«ρ,T

+ C

„∂P

∂Ye

«ρ,T

#,

We used

A =

„∂P

∂T

«ρ,Ye

, B =

„∂P

∂Ye

«ρ,T

− A

CV

„∂ε

∂Ye

«ρ,T

, C =P

ρ2−„∂ε

∂ρ

«T ,Ye

,

CV =

„∂ε

∂T

«ρ,Ye

, c2T =

„∂P

∂ρ

«T ,Ye

, c2s =

„∂P

∂ρ

«s,Ye

= c2T + AC .


Steady-State

As in the self-similar collapse flow, there is a critical, or sonic, point where|v | = cs .

The inner boundary condition is set at the neutrinosphere, r = rν , T = Tν ,L = Lν = πr 2

ν(7ac/16)T 4ν .

The outer boundary condition is set at the shock front, r = rs . Subscript s (f )refers to behind (in front of) the shock. The Rankine-Hugoniot shock jumpconditions, using pf << ps , are

ρsv2s + Ps = ρf v

2f ,

v 2s

2+ εs +

Ps

ρs=

v 2f

2, vf = −

r2GM

rs.

The last follows from self-similar results.

We take M and M to be constant, so that v = −M/(4πr 2ρ) everywhere. Thus

ρf =M

4πr 2s

rrs

2GM, ρsvs = ρf vf = − M

4πr 2s.

The distance between rs and rν is established by the conditionZ rs

rν

κνeρdr = 2/3,

where κνe = κo [F4(0)/F2(0)](Tν/MeV)2 is the opacity of νe + n→ p + e−.


Steady-State Isothermal Flows

Some insight is gained by considering an isothermal steady-state flow. Thesteady-state equations

vdv

dr= −GM

r 2− 1

ρ

dP

dr,

1

v

dv

dr= −1

ρ

dρ

dr− 2

r

can be made dimensionless by using x = rc2T/(2GM) and the Mach number

M = v/cT , where M and c2T = (∂P/∂ρ)T are assumed constant:„M− 1

M

«dMdx

=2

x− 1

2x2.

One begins the integration at xν and Mν .

The Bondi solution goes through the sonic point Msp = −1 when xsp = 0.25and ρ = ρsp = −Mc3

T/(πG 2M2). At other points, the Bondi flow satisfies

M2Bondi − lnM2

Bondi = x−1 − 3 + 4 ln 4x , ρ = ρsp(−16MBondix2)−1.

The Bondi solution separates flows that are always subsonic from those thatbecome supersonic.

Shocks are possible when the Rankine-Hugoniot conditions are satisfied:

ρ−M− = ρ+M+, ρ−(M−)2 + ρ− = ρ+(M+)2 + ρ+

where +(−) refer to up-(down-)stream of the shock. This gives M+M− = 1.James Lattimer Gravitational Collapse Supernovae

Dotted lines show downstream conditions at the shock where the upstream isBondi flow (blue) or free-fall (red/green). Dashed lines show the Bondi flow.

Solid lines showdownstream flows forfixed ρν , rν , M andM as cT is increasedfrom left to right.

Where solid linescross dotted lines,shocks are possible inthe flow. ImpliescT ≤ ccrit

T , xcrit ≤ 14

for shocks to exist.

When upstream isfree-fall,M+ = −x−1/2 andthe solution of theRankine-Hugoniotrelations gives (reddots)

M = −1M s−1, M = 1.4M, rν = 30 km, ρν = 3 · 1010 g cm−3

M− =Mff =

√x−1 − 4− x−1/2

2. In this case, xcrit

ff =3

16and Mcrit

ff = −3−1/2.


The parameters ρν , rν , M and M determine ccritT .

M(xν)√

xν =M

4πρνr 2ν

rrν

2GM, ccrit

T =

r2GM

rνxν .

The region with shocks corresponds to conditions resulting in stalled supernovashocks.

However,changingconditions canlead to theexpulsion of theshock and asuccessfulexplosion.

In this case,increasing ccrit

T ,i.e., increasingT , or loweringM can increasethe chances of asuccessfulsupernovashock.

M = 1.4M, rν = 30 km, ρν = 3 · 1010 g cm−3


The Full Solution: Shock Radius

In a full steady-statesolution, theneutrino luminosityand mass accretionrate determine thelocation of theshock, in concertwith theRankine-Hugoniotshock jumpconditions.

For Lν below themaximum, there aretwo solutions for theshock radius.

Only the lower oneis relevant in thesupernova context.

M = M s−1

M = 1.4M, rν = 50 km


The Full Solution: Critical Luminosity

Just as observed in the isothermal steady-state case, there is a maximumneutrino luminosity for a given mass accretion rate that will support a shock.

The criticalluminositydepends onassumptionsfor rν ,Mand theoptical depthbetween rνand rs .


An Analytic Solution

Studies show that a number of simplifying approximations can be made.

I The EOS satisfies h = 4P/ρ with no dependence on Ye , so dYe/dr = 0.

I The density varies as r−3: ρ = ρν(rν/r)3.

I The neutrino heating rate ' aLν/r2, i.e., f = 1 and a = 155πκoT

2ν/294.

I The neutrino cooling rate is ' br−4. A parcel falling to rν , if no heating ispresent, will radiate the gravitational potential energy

ηGM

rν=

Z ∞rν

4πr 2ρ

M

b

r 4dr =

πρνb

Mrν=⇒ b = η

GMM

πρν

where η ∼ 0.4 is the cooling efficiency.

I The net heating rate is then q = aLν/r2 − b/r 4.

I Neglect the terms vdv/dr in the Euler equations and those with v 2s in the

Rankine-Hugoniot conditions at rs , so the outer boundary conditionsbecome h(rs ) = v 2

f /2 = GM/rs and P(rs ) = Mvf /(4πr 2s ).

I Ignore the subsequent Rankine-Hugoniot inconsistency ρsvs 6= P(rs )/vf .

The Euler equations become, using dh = dp/ρ or h = (ε+ p)/ρ,

P(rν)− P(rs ) = −Z rν

rs

GMρ

r ′2dr ′,

h(rν)− h(rs ) =GM

rν− GM

rs+

Z rν

rs

4πr ′2ρ

Mqdr ′.


Analytic Solution Details

Performing the integrals and applying outer boundary conditions yield

A

„rs

rν

«4

+ B

„rs

rν

«2

+ C = 0,

A =2πrνρν

M

„aLν −

b

2r 2ν

«, B =

Mvf

πρνr 2ν

− 2πρνrν

MaLν , C =

πρνb

rνM− GM

rν.

Ignoring the weak rs dependence in B, solving the quadratic shows there aretwo critical points. „

rs

rν

«2

=−B ±

√B2 − 4AC

2A.

The first is when A = 0 or Lν = b/(2ar 2ν), where the+ solution is finite and the

− solution is infinite. This gives the minimum Lν that allows two values of rs .

The second point is when B2 = 4AC , where the two solutions merge. Thisgives the maximum Lν that gives steady-state solutions. One finds

Lcritν =

GM|M|πρνar 2

ν

241 + η − Mvf

2πρνrνGM−

s(1 + η)

„1− Mvf

πρνrνGM

«35 .Lcritν ≈

ηGM|M|2πρνr 2

νa

"1 +

η

4

„1 +

Mvf

πρνrνGMη

«2

+ · · ·

#.


Analytic vs. Numerical Solution


Proto-Neutron Stars


Proto-Neutron Star Evolution

Consider the Newtonian case. The electron neutrino number flux Fν andν energy flux Lν change the lepton number YL and entropy s:

ndYL

dt= n

dYe

dt+

d(Yν − Yν)

dt=

1

r2

∂

∂rr2Fν

nTds

dt= − 1

4πr2

∂(Lν + Lν)

∂r− n

∑n,p,e,ν

µidYi

dt.

In the diffusion approximation, fluxes are driven by density gradients:

Fν = −∫ ∞

0

c

3

(λN

∂nν(E )

∂r− λN

∂nν(E )

∂r

)dE ,

Lν = −∫ ∞

0

4πr26∑i

cλiE

3

∂εi (E )

∂rdE .

λN and λiE ’s are mean free paths for number and energy transport,

respectively. nν(E ) and εi (E ) are the νe number and energy densities ofspecies i = e, µ, τ and their antiparticles at neutrino energy E .There are two main sources of opacity:

1. ν-nucleon absorption, affects only νe , νe .λν ' λν ' λ0(E0/E )2;λ0 ≈ 5 cm; E0 ≈ 1 MeV

2. ν − e scattering, affects all flavors. λiE ' λi

1(E0/E )2;λ1 ≈ 4 km


Proto-Neutron Star Evolution

A newly formed neutron star has an unshocked interior (s ∼ 1kB ) with asignificant trapped lepton fraction YL ∼ 0.38. The densities exceed the nuclearsaturation density ns ' 0.16 fm−3 and the central temperature is kBTc ∼ 20MeV. Weak interaction timescales are very short, so beta equilibrium is anexcellent assumption.

Recall that beta equilibrium is just minimization of the total energy withrespect to the charge density, i.e., ∂(Eb + EL)/∂x = −µn + µp + µe − µν = 0.The baryon enegy, expanded around n = nb and x = Yp = Ye = 1− Yn = 1/2:

Eb ∼ −B +Ks

18

„n

ns− 1

«2

+ S(n)(1− 2x)2 + · · ·

∂Eb/∂x = −µn + µp = −4S(n)(1− 2x).

The leptons are relativistic and degenerate, with Ye + Yν = YL, so

∂EL/∂x = µe − µν = ~c(3π2n)1/3hY 1/3

e − (2Yν)1/3i.

(3π2n)1/3hx1/3 − (2(YL − x))1/3

i= 4S(n)(1− 2x).

At the end of deleptonization, YL = Ye ∼ 0.04 and Yν ∼ 0, so

dYL/dYν ∼ 0.34/.06 = 5.8.


Proto-Neutron Stars – Analytic Analysis

Neutrino fluid

nν(E ) =E 2

2π3(~c)3fν(E ), fν(E ) =

[1 + e(E−µν)/T

]−1

n(Yν − Yν) =

∫(nν + nν)dE =

µ3ν + π2T 2µν6π2(~c)3

, εν(E ) = E (nν + nν)

Diffusion approximation

Fν = −c

3

∫ ∞0

[λN

∂nν(E )

∂r− λN

∂nν(E )

∂r

]dE

'− cλ0E20 T

6π2(~c)3

∂T [F0(µν/T )− F0(−µν/T )]

∂r= − cλ0E

20

6π2(~c)3

∂µν∂r

,

Lν = −4πr2

∫ ∞0

6∑i

cλiE

3

∂εi (E )

∂rdE

' − r2c

3π(~c)3

3∑i

λi1E

i20

∂T 2[F1(µν/T ) + F1(−µν/T )]

∂r

= − r2c

3π(~c)3

3∑i

λi1E

i20

∂

∂r

(µ2ν +

π2T 2

3

).


The Deleptonization of a Proto-Neutron Star

Energy transport dominated by degenerate electron neutrinos propagatingthrough degenerate matter.Number transport equation

∂nYL

∂t=∂YL

∂Yν

1

6π2(~c)3

∂µ3ν

∂t=

1

r2

∂

∂rr2Fν

= − cλ0E20

6π2(~c)3

1

r2

∂

∂r

(r2 ∂µν

∂r

)∂YL

∂Yν' 5, µν = µν,0ψ(x)φ(t), x = x1r/R

τDdφ2

dt= − 1

x2ψ3

∂

∂x

[x2 ∂ψ

∂x

]= −1

Spatial solution is Lane-Emden function of index 3, ψ3.

φ =

√1− t

τD, ψ3(x1) = 0 =⇒ x1 = 6.897

τD =6

cλ0

∂YL

∂Yν

(µν,0E0

)2(R

x1

)2

≈ 5− 10 s


Heating During Deleptonization

Beta equilibrium:∑j

µjdYj = (−µn + µp + µe − µν)dYe + µνdYL = µνdYL

Energy transport equation:

nT∂s

∂t= − 1

4πr2

∂Lν∂r− n

∑j

µj∂Yj

∂t' −nµν

∂YL

∂t

The diffusion term is small because neutrinos ultimately escape withenergy Eν << µν .

Entropy is dominated by baryons, with s ' aT where a ' 0.06 MeV−1.

s

a

ds

dt' −µν

∂YL

∂t' −3

∂YL

∂YνYν,0µν,0

(µνµν,0

)3∂µν/µν,0

∂t

s2f − s2

i '3a

2

(∂YL

∂Yν

)µν,0Yν,0

si ∼ 1, sf ∼ 2.5, Tf ' 50 MeV


Core Cooling

Following deleptonization, µν << T

Lν =− 4πr2 c

6π2(~c)3

∂

∂r

∫ ∞0

6∑i

λiE E i2

0 EfνdE

=− 4πr2 cλ1E20

π2(~c)3

∂

∂r

π2T 2

3.

In the energy transport equation we can still assume baryon degeneracy,but can now neglect composition changes:

nT∂s

∂t=− 1

4πr2

∂Lν∂r− n

∑j

µjdYj

dt' − 1

4πr2

∂Lν∂r

naT∂T

∂t=

cλ1E20

3(~c)3

1

r2

∂

∂rr2 ∂T 2

∂r.

Assume T = T0φ(t)ψ(x), with T0 ∼ 50 MeV.


Core Cooling

τc

φ

dφ

dt= −1 =

1

ψ2x2

∂

∂xx2 ∂ψ

2

∂x.

ψ2 is Lane-Emden function of index 1: ψ2 = ψ1 = sin x/x

ψ1(x1) = 0 =⇒ x1 = π,

(x∂ψ1

∂x

)x1

= −1.

φ = exp

(t0 − t

τc

), τc =

3(~c)3an

cλ1E 20

(R

x1

)2

≈ 10− 30 s

Emergent Luminosity:

Lν(R, t) = −4πRcλ1E

20 T 2

0

3(~c)3

(x∂ψ2

∂x

)x1

φ2(t) =cF3(0)

2(~c)3R2Tν,eff (t)4

= L0φ(t)2 ≈ 1.2 · 1051φ(t)2 erg s−1

Tν,eff (t) =

(2L0(~c)3

cF3(0)R2

)1/4√φ(t) ' 4

√φ(t) MeV, < Eν >∼ 3Tν .


Model Simulations

MB/M MB/M

Pons et al.


Model Simulations


Model Signal


Conclusions From Proto-Neutron Star Simulations

I The initial neutrino luminosity exceeds 1053 erg s−1.

I There are two main stages of proto-neutron star evolution: coredeleptonization and core cooling.

I The duration of core deleptonization is of order 10 seconds due totrapping.

I The average emitted neutrino energy is of order 10–15 MeV, with anapparent neutrino temperature of about 4 MeV.

I The central temperature initially rises mostly due to neutrinodiffusion, and less so from adiabatic compression.

I The average emitted neutrino energy initially increases, due to risingcore temperatures and increasing transparency.

I During core cooling, the neutrino luminosity is of order 1051 erg s−1.

I Core transparency occurs after about 50 seconds, followed by a steepdrop in neutrino luminosity. Even a supernova in our Galaxy will bedifficult to detect in neutrinos after this time.


Neutrinos from SN 1987AThe neutrino observation from SN 1987A was a milestone event not dissimilarto the observations of gravitational radiation.

# t (s) Ee (MeV) θ Eνe (MeV)Kamiokande II

1 0.0 20.0± 2.9 18± 18 21.3± 2.92 0.107 13.5± 3.2 15± 27 14.8± 3.23 0.303 7.5± 2.0 108± 32 8.9± 2.054 0.324 9.2± 2.7 70± 30 10.6± 2.755 0.507 12.8± 2.9 135± 23 14.4± 3.056 1.541 35.4± 8.0 32± 16 36.9± 8.17 1.728 21.0± 4.2 30± 18 22.4± 4.258 1.915 19.8± 3.2 38± 22 21.2± 3.259 9.219 8.6± 2.7 122± 30 10.0± 2.810 10.433 13.0± 2.6 49± 26 14.4± 2.6511 12.439 8.9± 1.9 91± 39 10.3± 1.95

IMB1 0.0 38± 9.5 74± 15 40.5± 10.12 0.42 37± 9.3 52± 15 38.9± 9.63 0.65 40± 10 56± 15 42.1± 10.44 1.15 35± 8.8 63± 15 37.0± 9.25 1.57 29± 7.3 40± 15 30.5± 7.46 2.69 37± 9.3 52± 15 38.9± 9.67 5.01 20± 5 39± 15 21.4± 5.18 5.59 24± 6 102± 15 26.1± 6.4

Eνe =Ee + ∆m + (∆2

m −m2e )/2mp

1− (Ee − pe cos(θ))/mp

∆m = (mn −mp)c2

pe and Ee are e+ momentum and energy.


Neutrino Energies and Opacities

The energy distribution of emitted neutrinos is assumed to be ∝ E 2ν f (Eν/T )

where f (Eν/T ) is the Fermi function with zero chemical potential. T is theeffective temperature of the source neutrinos.

The average neutrino energy at the source is then

< Eν >s =

R∞0

E 3ν f (Eν/T )dEνR∞

0E 2ν f (Eν/T )dEν

=F3(0)

F2(0)T ' 3.15T

where Fi is the standard Fermi integral.

The neutrino absorption opacity is that due to νe + p → n + e+, which is

κ(Eν) = κoEepe

where κo = 9 · 10−44 cm2 per proton.

The main source of protons are the two hydrogen protons in an H2O detector;

the 8 protons in O are too deeply bound to interact effectively with the

supernova anti-neutrinos.


Detector Characteristics

The detectors had fiducial masses M, low-energy thresholds H, efficienciesW (Eν), and observational errors ∆(Eν):

Detector M (kt) H W (Eν) ∆(Eν)

Kamioka 2.14 7 1− 4.9e−Eν/3.6 0.20 + 0.19EνIMB 5.0 20 1− 3e−Eν/16 −0.38 + 0.26Eν

Generally, W (H) ∼ 0.5.

Define the detector response function R(Eν , x) to be the differential conditionalprobability for measuring an observed neutrino energy Eν , given a true neutrinoenergy x . The distribution of observed energies is then

N(Eν) ∝Z ∞

H

x2f (x/T )κ(x)W (x)R(Eν , x)dx .

Since the combination of opacity, efficiency and incoming spectrum is a steep

function of the energy, neglect of the detectgor response introduces a

systematic bias. The sense of the bias is such that the true energy of a

neutrino is closer to the median energy of the distribution if response is

ignored. It is hopeless to determine the true energy of a single event, but a

collection of events can be corrected for the bias.James Lattimer Gravitational Collapse Supernovae

Errors

The errors, according to experimentalists, are Gaussian, so

R(Eν , x) =1√

2π∆(Eν)exp

»− (Eν − x)2

2∆(Eν)2

–.

A standard approximation is to treat R to be a function of x rather than Eν ,and so to use the integrated response functionsZ ∞

H

R(Eν , x)dEν ≡ A(x) = P

„x − H

∆

«,

Z ∞H

EνR(Eν , x)dEν ≡ xB(x) = xP

„x − H

∆

«+ ∆2R(x ,H).

Here P(z) = [1 + erf(z/√

2)]/2 is the cumulative normal distribution function.In the limit of negligible errors, A and B are step functions: zero for x < H,unity for x ≥ H.

Two popular methods of analyzing the detected neutrinos are the simpler

moment method and the more realistic maximum likelihood method.


Moment Method

The average detected energy is

< Eν >d =

R∞H

x3f (x/T )κ(x)W (x)B(x)dxR∞H

x2f (x/T )κ(x)W (x)A(x)dx= T

G5(T )

G4(T ).

We introduced Gi as a modified truncated Fermi integral, whose indices aredetermined by the fact that κ ∝ x2.

The rate of energy absorption by the detector is proportional to G5, while thedetection rate of neutrions is proportional to G4.

In the event that H = 0, W = 1, and errors are negligible, Gi (T ) = Fi (0) and

F5(0)/F4(0) ' 5. The above equation is solved iteratively for T .


Maximum Likleihood Method

Here, one minimizes

Λ = −2n∑

i=1

ln N(Eν,i )

where n is the number of events, with respect to the fitted parameters, inthis case T .

The normalization constant is set so that the total count is independentof fitted parameters: ∫ ∞

h

N(Eν)dE (ν) = n.

Only shape parameters of the spectrum, such as T , can be determined,but not the overall flux.

For simultaneous fits to both detectors, the sum of the two integrals ofthe two detectors is set to the total count. The fit is therefore sensitiveto the relative counts in the two detectors but not to the total number ofneutrinos observed.


Neutrino Luminosity and Count Rate

Once the temperature is determined, the anti-neutrino luminosity is found:

Lνe (t) = 4πR2νe

(t)cF3(0)

8π2(~c)3T (t)4,

where Lνe ,Rνe (the radius of the νe neutrinosphere) and T are functions oftime.The count rate in a detector is

dn

dt=

„Rν(t)

D

«2

Np

Z ∞H

x2f (x/T )W (x)κ(x)dx =κoNpLνe (t)

4πD2

G4(t)T (t)

F3(0)

=M„

50 kpc

D

«2Lνe (t)

7.5× 1052 erg s−1

G4(t)T (t)

F3(0)s−1.

D is the distance to the detector and Np = (2/3) · 1032M is the number ofavailable protons (i.e., 2/18) in an H2O detector of mass M, in kilotons.

There is not enough data to determine anything but the average temperature.If T = constant, the total emitted νe energy is

Eνe =

ZLνe dt = 7.5× 1052

„D

50 kpc

«2n

MT

F3(0)

G4ergs.


More Complex Neutrino Spectra

Characterizing the source spectra with an effective chemical potential has littleinfluence.

The integrals Fi and Gi peak near the energy x = iT , so e(Eν−µν )/T >> 1.Thus a factor eµν/T becomes common to all these integrals, and so cancels inmost of the preceding expressions.

However, if one were to try to derive a value for Rνe from the time integratedLνe using

Lνe (t) = 4πR2νe

(t)cF3(µν/T )

8π2(~c)3T (t)4,

the unbalanced F3 integral would depend sensitively on µν/T . Compared to

the case of µν = 0, the derived radius becomes smaller by a factor of 0.63

(0.27) if µν/T is 1 (3). Therefore, no sensible estimate of Rνe is possible for

the case of SN 1987A.


Results

Method of Analysis TKamioka TIMB Eνe ,Kamioka Eνe ,IMB

with detector response

Moment 2.8± 0.5 4.0± 0.7 6.9+3.5−3.8 5.7+10.0

−2.8

Relative counts 5.0+1.25−1.0 2.6+1.9

−1.6

Maximum likelihood 2.8± 0.4 4.2± 1.0 6.3+4.0−3.1 4.5+12.0

−3.4

Maximum likelihood (combined) 3.7± 0.4 4.8+2.1−1.8

without detector response

Moment 2.8± 0.6 4.6± 0.8 6.1+5.0−3.0 2.9+2.6

−1.6

Relative counts 4.8+1.2−1.0 2.6+2.2

−1.7

Maximum likelihood 2.8± 0.4 4.7± 0.8 5.9+3.4−2.8 2.7+2.6

−1.0

Maximum likelihood (combined) 3.8± 0.4 4.3+2.0−1.8

Temperatures are in MeV and νe energies are in 1052 erg.

For comparison, in the case of negligible errors and unity efficiency,T '< Eν >d /5. For Kamioka and IMB, this would imply T = 3.3 and 6.7MeV, respectively.

From Eνe ' 19.1 · 1051n/(MT ) ergs for D = 50 kpc, one obtains Eνe 30.3 and4.6 · 1051 ergs, respectively.

Since A(x) > B(x), including detector response decreases T and increases Eνe .

Detector response is more important for the IMB detector, reducing the

disparity between the results for the two detectors.James Lattimer Gravitational Collapse Supernovae

Other Results

I The total binding energy of the newly-formed neutron star can beestimated from 6Eνe if it is assumed each of the 6 neutrino flavors carryaway an equal energy. This amounts to (1.8− 4.1) · 1053 erg, whichtranslates to an estimated protoneutron star mass of (1.1− 1.65)M.

I It has been suggested that the bunched structure of Kamiokande neutrinoarrival times indicate a late secondary collapse possibly connected to theformation of a Bose condensate or a mixed phase of deconfined quarksand hadrons. However, if it is assumed that there is a Poisson distributionof counts about a mean number, a gap of about 7 seconds with at least 3neutrinos detected following the gap occurs 5% of the time. The gap istherefore not statistically significant. Also, the IMB neutrinos don’t showa similar structure.

I If neutrinos have mass, their arrival will be delayed inversely with energy:

∆t = 2.6

„D

50 kpc

«“mν

eV

”2„

MeV

Eν

«2

s.

There is no mass for which all 11 Kamiokande events can be emitted overa duration < 12 s. If mνe > 20 eV, the duration is > 20 s. Analysis showsthat mνe < 30 eV is all that can be definitively established, butaccelerator experiments already have shown that mνe < 10 eV.


I A finite neutrino magnetic moment µ would provide an additional path forν−emission by allowing right-handed ν’s (νR ) and left-handed ν’s (νL).These are unable to interact with matter and hence freely stream out ofthe protoneutron star. If µ, and the production rate of flipped-helicity ν’s

rflip ' 2.4× 105

„1010µ

µB

«2

Yenc

4nos−1,

are large enough, this path dominates cooling, and the emission durationwould no longer match observations. µB is the Bohr magneton.

µ < 3 · 10−11µB implies a mean free path c/rflip < Rν , not allowed. Forr−1

flip < 1s, µ < 3 · 10−13µB , 100 times better than other experiments give.

I Could helicity flips efficiently transport νs to the mantle, powering asupernova explosion? This requires

RµBdr ≥ 1 and B > Bc , where

Bc 'GF ne

µ√

2= 3.3× 106ρ

µB

1012µG.

The electron density is ne and the weak interaction constant is GF . If

B(r) = BNS 1012/r 2 G, ρ(r) = 1032/r 3 g cm−3,

BNS < 1018 G by the Virial Theorem. B > Bc only for radii greater than

RB =3.3× 1026

BNS

µB

1012µcm.

which is much greater than the stalled shock radius. So helicity flips canhave no effect on supernovae.


The Urca Processes

Gamow & Schonberg proposed thedirect Urca process dominates neutronstar cooling. Nucleons at the top ofthe Fermi sea β−decay at finite T .n→ p + e− + νe ,p → n + e+ + νe

Energy conservation guaranteed byβ−equilibriumµn − µp = µe

Momentum conservation requires|kFn| ≤ |kFp|+ |kFe |.Charge neutrality requires kFp = kFe ,therefore |kFp| ≥ 2|kFn|.Degeneracy implies ni ∝ k3

Fi , thusx ≥ xDU = 1/9 is required.

With muons (n > 2ns)xDU = [1 + (1 + 2−1/3)3]−1 ' 0.148

If x < xDU , bystander nucleonsneeded: modified Urca process.(n, p) + n→ (n, p) + p + e− + νe ,(n, p) + p → (n, p) + n + e+ + νe

Neutrino emissivities:εMU ' (T/µn)2

εDU ∼ 10−6εDU .

β−equilibrium composition:xβ ' (3π2n)−1 (4Esym/~c)3

' 0.04 (n/ns)0.5−2.


Direct Urca Threshold

Klahn et al., Phys. Rev. C74 (2006) 035802


Neutron Star Cooling

Page, Steiner, Prakash & Lattimer (2004)

Cas A

J1119-6127


Minimal Cooling Paradigm

I Minimal Cooling Paradigm: Neutron star cooling including effects ofsuperfluidity, such as Cooper-Pair breaking and formation, but no “rapid”neutrino cooling processes such as direct Urca involving nucleons orexotica.

I If some observations are inconsistent with the MCP, then according toSherlock Holmes, rapid cooling must occur for these exceptions.

I All sources are consistent with the MCP only IF

I tight conditions are placed on the magnitude and density dependenceof the neutron 3P2 gap, AND

I some neutron stars have heavy Z envelopes and others have light Zenvelopes, AND

I ALL core-collapse supernova remnants with no observable thermalemission contain black holes.

I Highly suggestive that rapid cooling occurs in some neutron stars (ofhigher masses?)

I A possible constraint on Esym(n) or ncentral .


Page, Lattimer, Prakash & Steiner (2009)

Minim

alC

ooling

Paradigm


Superfluidity in Neutron Star MatterNuclei show the existence ofsuperfluid-like gaps in the excitationspectra of even-even nuclei. Theseare the energies of Cooper pairs thatmust break to form the excitation.

Only certain spin-angular momentumcombinations exist for Cooper pairs.

Scattering phase shifts show theexistence of attractive interactions inthe 1S0 and 3P2 channels.

Mass number A150 170 190 210 230 250

0.1

0.5

1.0

E (MeV)

odd−even/even−odd nuclei

even−even nuclei

G A

P

Phase shift (in degrees)

o

20o

10o

0o

−10o

−20o

−30o

E (MeV)labN−N

E (MeV)F

(10 g cm )ρ 14 −3

100 200 300 400 500 600

25 50

1 2

75

4

100

6 8

125

10 12

150

30

10

P30

G1 4

D1 2

P31

S01

P32

S

L = 0 L > 0

Spin−singlet pairs

S = 1

Spin−triplet pairs

L = 0 L > 0

S = 0


Superfluid Properties in Neutron Stars

No particles in asuperfluid Fermi liquidhave energies betweenεF −∆ and εF + ∆due to formation ofCooper pairs.

In BCS theory, thezero-temperatue gapis proportional to thecritical temperature∆ ' 1.75kBTc .

Theoreticalpredictions of Tc (n)vary widely.

1.4 2.01.8 1.0 1.4 2.01.8

SF

3bf

HGRR

NijI

CDB

AV182bf

AO T

Cru

st

Core NijII

a

b

c

a2

Cru

st

Core

1.0

k

ε

Normal Fermi Liquid Superfluid Fermions

ε

Fk

ε

2∆

ε(k)

kF

(k)

ε

k

F εF


Superfluid Effects on Neutron Star CoolingThere are three major effects:

I Suppression of thespecific heat CV forT << Tc .

I Suppression of modifiedUrca neutrino emissivityfor T << Tc .

I Enhancement of neutrinoemission due toCooper-pair formationand breaking for T <∼ Tc .

32

S10

P32

S10P3

2

Co

ntr

ol

fun

ctio

n R

νT/Tc T/Tc

1.0

0.5

0.0

1

2

3

p n

2.43

2.19

0.2 0.4 0.6 0.8 1.00

0.2 0.4 0.6 0.8 1.0

Specific Heat

Co

ntr

ol

fun

ctio

n R

Neutrino Emission

C

S10 P

PB

F

10 P3

2

ν

_

n

pairnn Cooper

n

ν

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

c

1.0

T/T

Contr

ol

funct

ion R

S

10 3 1 0Model gap: c b a

9maxT (10 K) =cn 1 .6 0.2.49cnmaxT (10 K) =


Transitory Rapid Cooling

MU emissivity: εMU ∝ T 8

PBF emissivity (f ∼ 10):εPBF ∝ F (T ) T 7 ∝ T 8 ' f εMU

Specific heat: CV ∝ T

Neutrino dominated cooling:dEth/dt = CV dT/dt = −Lν

=⇒ T ∝ (t/τ)−1/6

τPBF = τMU/f

(d ln T/d ln t)transitory

' (1− 10)(d ln T/d ln t)MU

' (1− 25)(d ln T/d ln t)MU (p SC)

Very sensitive to n 1S0 criticaltemperature (TC ) and existenceof proton superconductivity

TC

core

tem

per

atu

re

surf

ace

tem

per

atu

re

No p superconductivity With p superconductivity

Page et al. 2009


Cas A

Remnant of Type IIb(gravitational collapse,no H envelope) SN in1680 (Flamsteed).

3.4 kpc distance

3.1 pc diameter

Strongest radio sourceoutside solar system,discovered in 1947.

X-ray source detected(Aerobee flight, 1965)

X-ray point sourcedetected(Chandra, 1999)

1 of 2 known CO-richSNR (massiveprogenitor and neutron star?) Spitzer, Hubble, Chandra


Cas A Superfluidity

X-ray spectrumindicates thin Catmosphere,Te ∼ 1.7× 108 K(Ho & Heinke 2009)

10 years of X-raydata show coolingat the rated ln Te

d ln t = −1.23± 0.14

(Heinke & Ho 2010)

Modified Urca:(d ln Te

d ln t

)MU ' −0.08

We infer thatTC ' 5± 1× 108 KTC ∝ (tC L/CV )−1/6

Page et al. 2010


Gravitational Collapse Supernovae · Gravitational Collapse Supernovae James Lattimer Department of...

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