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
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
Neutrino Supernova Mechanism
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
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−.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
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 ′.
James Lattimer Gravitational Collapse Supernovae
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
+ · · ·
#.
James Lattimer Gravitational Collapse Supernovae
Analytic vs. Numerical Solution
James Lattimer Gravitational Collapse Supernovae
Analytic vs. Numerical Solution
James Lattimer Gravitational Collapse Supernovae
Proto-Neutron Stars
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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
).
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
Model Simulations
MB/M MB/M
Pons et al.
James Lattimer Gravitational Collapse Supernovae
Model Simulations
James Lattimer Gravitational Collapse Supernovae
Model Signal
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse Supernovae
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.
James Lattimer Gravitational Collapse 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.
James Lattimer Gravitational Collapse Supernovae
Direct Urca Threshold
Klahn et al., Phys. Rev. C74 (2006) 035802
James Lattimer Gravitational Collapse Supernovae
Neutron Star Cooling
Page, Steiner, Prakash & Lattimer (2004)
Cas A
J1119-6127
James Lattimer Gravitational Collapse Supernovae
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 .
James Lattimer Gravitational Collapse Supernovae
Page, Lattimer, Prakash & Steiner (2009)
Minim
alC
ooling
Paradigm
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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) =
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae
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
James Lattimer Gravitational Collapse Supernovae