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PHY688, White Dwarfs

James Lattimer

Department of Physics & Astronomy449 ESS Bldg.

Stony Brook University

February 7, 2017

Nuclear Astrophysics [email protected]

James Lattimer PHY688, White Dwarfs

White Dwarfs

I 1783: First white dwarf discovered by Hershel – 40 Eri B.I 1844: Bessel used astrometry to infer unseen companions to Sirius

and Procyon.I 1862: Clark discovered Sirius B.I 1910: Boss determined the mass of Sirius B to be about 1 M�.I 1910: Russell, Pickering and Fleming find A spectra for 40 Eri BI 1915: Adams determined an A spectral type for Siriuis B.I 1916: Opik estimated its density as 25,000 g/cm3, ”impossible”.I 1917: van Maanan’s star, an isolated white dwarf, discovered.I 1922: Luyten first used term white dwarf, popularized by Eddington.I 1924: Eddington predicted gravitaional redshifting from Sirius BI 1925: Confirmation of gravitational redshift from Sirius B by Adams.I 1926: R.H. Fowler resoved compactness paradox with quantum

mechanics.I 1929: The concept of maximum mass first published by AndersonI 1931: the Chandrasekhar mass determined.I By 1939, 18 white dwarfs known. By 1950, more than 100; by 1999,

2000; and now, about 10,000 due to the Sloan Digital Sky Survey.


The Maximum Mass

Gravitational energy of a uniform sphere: Eg ∼ −GM2/RKinetic energy of electrons: Ek = NpcPauli exclusion principle:

∆p∆x ∼ ~; p ∼ ~n1/3 ∼ ~(

N

V

)1/3

∼ ~N1/3

R

Therefore

Ek =N4/3~c

R=

(M

mb

)4/3 ~c

R

Ek + Eg minimized for finite value of R only when Ek = |Eg |, or when

M ∼ m−2b

(~c

G

)3/2

∼ 1.5 M�

Radius estimate Ef ∼ mc2 or p = mc :

R ∼(

~3

Gc

)1/21

mbm= 10, 000 km (m = me) or 3 km (m = mb)


Varieties of White Dwarfs

I DA: Hydrogen surfaceI DB: Helium surfaceI DO: Ionized helium surfaceI PG 1159: C/O surface ~~~~~~~~~~~~~


Mass Distribution of White Dwarfs

Kepler et al. (2007)

r DAr DB


Correction of Mass Distributions

Kepler et al. (2016)


Where Do White Dwarfs Come From?


White Dwarf Cooling

I White dwarf interior transports energy primarily by conduction.I Degenerate electrons have long mean free paths, so conductivity is

very large.I A good approximation is that the interior is isothermal.I Near surface, opacity increases and isothermality breaks down,

surface is diffusive:dT

dr= − 3

4ac

κρ

T 3

L

4πr2.

I Hydrostatic equilibrium at surface:

dP

dr= −Gm(r)ρ

r2,

dP

dT=

4ac

3

4πGM

κoL

T 6/5

ρ.

I We assumed Kramer’s opacity,κ = κoρT

−7/2 and m(r) ' M.I Also, at surface: P = (Nok/µ)ρT :

PdP =4ac

3

4πGM

κoL

kNo

µT 15/2dT .


White Dwarf Cooling

I Integrating from P = 0 to P, and T = 0 to T :

ρ =

√4

17

4ac

3

4πGM

κoL

µ

kNoT 13/4.

I There will be a transition from this at a depth where the interiorbecomes degenerate. Equating degenerate and non-degeneratepressure expressions we find the transition to be at the density

ρ∗ ' 2.4× 10−8Y−1e T

3/2∗ g cm−3.

I The temperature T∗ is the temperature of the isothermal interior.

I Solving for L

L = 5.7× 105 µY 2e

Z (1 + X )

M

M�T

7/2∗ ≡ CMT

7/2∗ .

I This is similar to the blackbody law L = 4πR2σT 4eff , suggesting

Teff ∝ T7/8∗ .


White Dwarf Cooling

I Luminosity is the change in internal energy with time, L = −dU/dt.I The internal energy is dominated by ions, since the electrons are

degenerate:

U =3

2

Nok

AMT∗.

I Integrating:

t = to +3

5

kNo

AC

(T−5/2∗ − T

−5/2∗o

).

I When T∗o >> T∗, the cooling time is

τ =3

5

Nok

AT−5/7∗ =

3

5

NokT∗A

M

L.

L

L�' 8.4× 10−4 M

M�

(t

109 yr

)−7/5

.

I If the production rate of white dwarfs is uniform, the space densityof white dwarfs per unit interval of log L is

φ(L) ∝(

d log L

dt

)−1

∝ t, log φ = − 5

12log L + constant.


Early Neutrino Cooling

The early cooling is dominated by neutrino emission through the plasmonprocess

γ∗ → νe + νe

which operates when the white dwarf is not completely degenerate.

Photons acquire a finite effective mass in a plasma where they moveslower than c . The resulting plasmons (γ∗) are unstable. The plasmonmass is ~ωP , where

ω2P =

4α

π

c3

~2

∫ ∞0

p2

ε

(1− v2

3

)[fF (µ,T ) + fF (−µ,T )] .

In the case when the gas is non-relativistic and partially degenerate,

ωP =

√4πe2ne

me.

Typically, the emissivity (erg cm−3 s−1) is

Q ∝(

~ωP

kBT

)15/2

e−~ωP/(kB T ).


Cooling Comparison

with ν

without ν

Hansen (2016)

→

cool

ing

beg

ins

→ cool

ing


Additional Physics

I Crystallization

Γ =Z 2e2

akT' 180− 240

T∗106 K

' 2.5

(Z

6

)5/3(ρ

106 gcm−3

)1/3(180

Γ

), L ' 10−4 L�

This releases additional internal energy and slows cooling.

WDdiamond


Additional Physics

I Convection (Schwarzschild criterion)

1

κ

(T

P

)4(∂ ln T

∂ ln P

)ad

<3

64πG

L

M

Convection zone isa small fraction oftotal mass, so L/Mis a constantdetermined by theinterior.

Surface convectionzone appears aboutwhen crystallizationoccurs.

L/M = 10−3, 10−4, 10−5


White Dwarf Evolution

End of neutrino cooling

Crystallization

ConvectionIsochron, Gyr

Hook due to H2 collision-induced absorption

Richer et al. (2007)


Models and Comparison to Observations

Neutrino cooling

Crystallization

Crystallization

Mestel

Mestel


Comparison

Ubaldi (2014)


Self-Gravitationally Lensed Eclipsing WD Binary

Kruse & Agol (2014)

KOI-3278


Comparison

Kruse & Agol (2014)

Lensing alone

Lensing +

Eclipse

Eclipse alone

KOI-3278


Eclipsing Doppler-Beamed System

Bloemen et al. (2010)

KPD 1946+4340

Fλ = F0λ (1− Bvr/c)

B = 5 + d ln Fλ/d lnλ


Neutronization

At high density, both neutronization and pyconuclear reactions occur.Neutronization is due to the increase in the electron Fermi energy withdensity.

The energy of nuclei increases with asymmetry. Compared to the ironpeak, the energy per baryon of asymmetric nuclei varies as

∆E ∼ 20

(N − Z

A

)2

MeV = 20(1− 2Yp)2 MeV.

The electron energy per baryon is

Ee =3Yp

4~c(3π2ρN0Yp)1/3.

Thus, to keep the total energy minimized with respect to the protonfraction means

∂(∆E + Ee)

∂Yp= −80(1− 2Yp) + ~c(3π2ρN0Yp)1/3 = 0

or

Yp '1

2− ~c

160

(3π2

2ρN0

)1/3

' 0.475

when ρ = 109 g cm−3. James Lattimer PHY688, White Dwarfs

Gravitational Collapse

At 109 g cm−3, the Fermi energy of an electron is 3.695 MeV, the threshold forthe inverse beta-decay 56Fe+e− →56Mn+νe .

The Mn nucleus immediately electron captures: 56Mn+e− → 56Cr+νe , andthe Cr nucleus is stable until 1010 g cm−3.

Lighter nuclei have other thresholds: 4He is at 20.6 MeV, 12C is at 13.4 MeVand 16O is at 10.4 MeV, and 20Ne is at 7.0 MeV.

The net effect is to soften the EOS: since MCh = 5.808Y 2p , it decreases.

If neutronization occurs a massive white dwarf gravitationally collapses. InNewtonian gravity, the maximum mass is reached for infinite central density.

In general relativity, the maximum white dwarf mass decreases and the centraldensity at maximum, ρc,max , is finite. For He and C white dwarfs,Mmax = 1.397M�, and neutronization does not occur in these cases becausethe neutronization thresholds are above ρc,max = 2.36× 1010 g cm−3.

However, for O and Ne white dwarfs, the neutronization threshold densities are,

respectively, 1.90× 1010 g cm−3 and 6.21× 109 g cm−3, leading to maximum

masses of 1.396M� and 1.389M�.


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