Structures of White Dwarfs

And Neutron Stars

INPE Advanced Course on Compact Objects

Course III--Lecture 2

Structure of White Dwarfs and the Chandrasekhar Limit:

Electron degeneracy pressure:

Remember that pressure of a degenerate gas is given by

!

Pe =1

3" 2h3

v(p)p3dp

0

pF

#

!

dPe

dr= "#(r)

GM(r)

r2

We solve this equation along with the continuity and force equations:

Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic case (v~c) to give

!

Pe

=(3" 2

)2 / 3

5

h2

me

ne

5 / 3, v << c

!

Pe

=(3" 2

)1/ 3

4hcn

e

4 / 3, v ~ c

To obtain

!

R"#c

1$n

2n

!

M "#c

3$n

2n

Structure of White Dwarfs and the Chandrasekhar Limit:

We can plug in some numbers for low-density white dwarfs

!

" =5

3, n =

3

2

and the constants to obtain

!

R =1.122 "104#c

106gcm

$3

%

& '

(

) *

$1/ 6

km

!

M = 0.4964"c

106gcm

#3

$

% &

'

( )

1/ 2

M

!

R = 3.347 "104#c

106gcm

$3

%

& '

(

) *

$1/ 6

km

!

M =1.457M

!

" =4

3, n = 3

In the relativistic limit

Remember there was no dependence on me, ρc, or R in the extreme relativistic limit.

Chandrasekhar limit

Chandrasekhar Limit for White Dwarfs: A Quick Treatment

The existence of a maximum mass for degenerate stars is very fundamental. Let’s understand it in two ways:

I.

!

M ~ "c

3#n

2n

So as matter under extreme density gets more and more relativistic, mass can no longer increase by increasing the central density but asymptotes to a constant.

II. Another way to look at it is the Fermi energy at the quantum limit where the volume per fermion is 1/n = R3/N (Pauli principle), momentum per fermion is

!

hn1/ 3

so that

!

EF~

hcN1/ 3

R

while the gravitational energy per baryon is

!

EG~ "

GMmB

R

Setting EF + EG = 0 gives

!

Nmax

" 2 #1057

Mmax

" Nmaxm

B"1.5M

Important Note on the Chandrasekhar Limit:

White dwarfs and neutron stars have maximum masses for different reasons!!

1. MCh for degenerate neutron gas is ~0.7 M !

2. Neutron stars have a maximum mass because of general relativity (as we will see)

3. White dwarfs do not reach the Chandrasekhar mass (the absolute maximum) becauseinverse β-decay kicks in at lower densities.

4. Neutron stars can exceed “their Chandrasekhar limit” because there are othersources of pressure (not just pressure of degenerate neutrons)

White Dwarf Cooling

Interiors of white dwarfs are roughly isothermal because of high thermal conductivity of degenerate matter.

No heat generation ==> outer layers are in radiative equilibrium, photons carrying the thermal flux

There is also local thermodynamic equilibrium (electrons and photons are thermalized)

Finally, hydrostatic equilibrium holds for the star

Solve photon diffusion equation (along with hydrostatic equilibrium + EOS)

!

L = "4# r2c

3$%

d

dr(aT

4)

where opacity κ is provided mainly by free-free and bound-free transitions.

For values appropriate for a white dwarf, we find

!

L " 2 #106ergs$1M

M

%

& '

(

) * T*

7 / 2

!

T*"10

6#10

7K

How long does it take the White Dwarf to Cool?

!

U =3

2kT

*

M

Amu

!

L =dU

dt

Combining this with our expression for L and solving for the cooling time gives

!

" #L

M

$

% &

'

( )

*5 / 7

Or about ~109 yrs for typical white dwarf luminosities.

Two (most important) effects that we neglected:

1. When T falls below the melting temperature Tm, the liquid crystallizes and releases q ~ kTm per ion.

2. Crystallization also changes the heat capacity, adding additional 1/2 kT per mode from the lattice potential energy.

The overall effect is to increase the thermal lifetime of the white dwarf.

Observations of White Dwarf Cooling

• Very detailed studies of white dwarfs in globular clusters are carried out

• Detailed cooling models are applied to, e.g., HST data

• One such study of NGC 6397 (Hansen et al. 2007) finds a cluster age of Tc=11.47 ± 0.47 Gyrs.

magnitude

N

A typicalluminosity

function

Observations of White Dwarf Cooling

• Sloan Digital Sky Survey discovers “ultracool” WDs

• At some arbitrarily low T, we start calling them “black dwarfs”

• Spectral fits (and in some cases binary companions) allow us to determine WD masses as well

from Kepler et al. 07

Neutron Stars

Density Regimes in Neutron Stars

1. Atmosphere (ρ ≈ 104 g /cm3):

2. 104 ≤ ρ ≤ 107 g /cm3 :

3. 107 ≤ ρ ≤ 1011 g /cm3 :

5. ρ ≈ 5x1012 g /cm3:

Matter in gaseous form, filamentary if B ≥ 1010 G)

Matter as in white dwarfs. A lattice of nuclei embedded in a degenerate relativistic electron gas.

Inverse β-decay transforms protons into n in nuclei. As nuclei get n-rich, the most stable configuration is no longer A=56 but shifts to higher values.

4. 1011 ≤ ρ ≤ 5x1012 g /cm3 :

Nuclei become so heavy (A~122) and so neutron-rich (n/Z=83/39) that they “drip” neutrons, forming a free neutron gas.

Mixture of degenerate n gas, ultrarelativstic electrons and heavy nuclei. Pn ~ Pe at this density.

6. ρ ≥ 5x1012 g /cm3:

Nuclei disappear, p, e, and n exist in β-equilibrium.

These density regimes are found in the “crust” of the neutron star, which is ~few hundred km thick and makes up a few percent of the star’s mass.

7. 1013 ≤ ρ ≤ 5x1015 g /cm3 :

Free neutrons dominate.

8. ρ ≈ 1015 g /cm3:

???

Neutron Star Structure and Equation of State

Structure of a (non-rotating) star in Newtonian gravity:

!

dM(r)

dr= 4" r2#(r)

!

dP(r)

dr= "

GM(r)

r2

#(r)

!

M(r) = 4" r2#(r)dr0

r

$(enclosed mass)

Need a third equation relating P(r) and ρ(r ) (called the equation of state --EOS)

!

P = P(")

Solve for the three unknowns M, P, ρ

Equations in General Relativity:

!

dM(r)

dr= 4" r2#(r)

!

dP(r)

dr= "

G M(r) + 4# r3P(r)[ ]

r21"2GM(r)

rc2

$

% & '

( )

*(r) +P

c2

+

, -

.

/ 0

} Oppenheimer-Volkoff Equations

Two important differences between Newtonian and GR equations:

1. Because of the term [1-2GM(r)/c2] in the denominator, any part of the star with r < 2GM/c2 will collapse into a black hole

2. Gravity ≠mass density Gravity = mass density + pressure (because pressure always involves some form of energy)

Unlike Newtonian gravity, you cannot increase pressure indefinitely to support an arbitrarily large mass

Neutron stars have a maximum allowed mass

Equation of State of Neutron Star Matter

We saw for degenerate, ideal, cold Fermi gas:

P ~ { ρ5/3 (non-relativistic neutrons)

ρ4/3 (relativistic neutrons)

Solving Oppenheimer-Volkoff equations with this EOS, we get:

R~M-1/3 As M increases, R decreases

--- Maximum Neutron Star mass obtained in this way is 0.7 M

--- There are lots of reasons why NS matter is non-ideal

(so that pressure is not provided only by degenerate neutrons)

(there would be no neutron stars in nature)

Some additional effects we need to take into account :(some of them reduce pressure and thus soften the equation of state, others increase pressure and harden the equation of state)

I. β-stability

Neutron matter is formed by inverse β-decay

p + e → n + νe

And is also unstable to β-decay

escape

n → p + e + νe _

escape

In every neutron star, β-equilibrium implies the presence of ~10% fraction of protons,and therefore electrons to ensure charge neutrality.

The presence of protons softens the EOS and reduces the maximum mass

II. The Strong Force

The force between neutrons and protons (as well as within themselves) has a strong repulsive core

II. The Strong Force

At very high densities, this interaction provides an additional source of pressure. The shape of The potential when many particles are present is very difficult to calculate from first principles,and two approaches have been followed:

a) The potential energy for the interaction between 2-, 3-, 4-, .. particles is parametrized and and the parameter values are obtained by fitting nucleon-nucleon scattering data.

b) A mean-field Lagrangian is written for the interaction between many nucleons and its parameters are obtained empirically from comparison to the binding energies of normal nucleons.

III. Isospin Symmetry

The Pauli exclusion principle makes it energetically favorable for a system of nucleons to have approximately equal number of protons and neutrons. In neutron stars, there is a significant difference between the neutron and proton fraction and this costs energy. This interaction energy is usually added to the theory using empirical formulae that reproduce the (A,Z) relation of stable nuclei.

IV. Presence of Bosons, Hyperons, Condensates

As we saw, neutrons can decay via the β-decay

n → p + e + νe _

yielding a relation between the chemical potentials of n, p, and e:

!

µn "µp = µe

And they can also decay through a different channel

n → p + π _

when the Fermi energy of neutrons exceeds the pion rest mass

!

EF ,n

" m#c2"140MeV

The presence of pions changes the thermodynamic properties of the neutron star interior significantly.

WHY?

Because pions are bosons and thus follow Bose-Einstein statistics ==> can condense to the ground state. This releases some of the pressure that would result from adding additional baryons and softens the equation of state. The overall effect of a condensate is to produce a “kink” in the M-R relation:

It is very difficult for π— to be present in the centers of neutron stars. How about other particles?

Nucleon reactions of the form

!

N + N" N + # +$

n + n" n + %+

+$&

are possible and lead to the creation of other particles with different decay properties.

For example, for the K mesons,

!

K0 " 2#

$% " µ% + & µ " e% + &

e+ & µ + & µ

$+ + µ% " 2# + & µ

which means that K0 and K+ will spontaneously decay, but

!

µ"# = µ

e#

so K- can be present.

Hyperons in neutron-star matter

Hyperons and the masses of neutron stars

V. Quark Matter or Strange Matter

Exceeding a certain density, matter may preferentially be in the form of free (unconfined) quarks. In addition, because the strange quark mass is close to u and d quarks, the “soup” may contain u, d, and s.

Quark/hybrid stars: typically refer to a NS whose cores contain a mixed phase of confined and deconfined matter. These stars are bound by gravity. Strange stars: refer to stars that have only unconfined matter, in the form of u, d, and s quarks. These stars are not bound by gravity but are rather one giant nucleus.

Mass-Radius Relation for Neutron Stars

•We will discuss how accurate M-R measurements are needed to determine the correct EOS. However, even the detection of a massive (~2M) neutron star alone can rule out the possibility of boson condensates, the presence of hyperons, etc, all of which have softer EOS and lower maximum masses.

Stars withcondensates

Strange Stars

Normal Neutron Stars

Effects of Stellar Rotation on Neutron Star Structure

Spin frequency (in kHz)

Usin

g Co

ok e

t al.

1994

Effects of Magnetic Field on Neutron Star Structure

Magnetic fields start affecting NS equation of state and structure when B ≥ 1017 G. by contributing to the pressure. For most neutron stars, the effect is negligible.