1.1 The Chandrasekhar Limit

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A Star is a self-gravitating ball of hydrogen atoms supported by thermal

pressure P _ nkT where n is the number density of atoms. In equilibrium,

E = Egrav + Ekin (1.1)

is a minimum. For a star of mass M and radius R

Egrav _ 􀀀

GM2

R

(1.2)

Ekin _ nR3 hEi (1.3)

where hEi is average kinetic energy of atoms. Eventually, fusion at the

core must stop, after which the star cools and contracts. Consider the

possible _nal state of a star at T = 0. The pressure P does not go to zero

as T ! 0 because of degeneracy pressure. Since me _ mp the electrons

become degenerate _rst, at a number density of one electron in a cube of

side _ Compton wavelength.

n􀀀1=3

e _

~

hpei

; hpi = average electron momentum (1.4)

Can electron degeneracy pressure support a star from collapse

at T = 0?

Assume that electrons are non-relativistic. Then

hEi _ hpei2

me

: (1.5)

6

So, since n = ne,

Ekin _

~2R2r2=3

e

me

: (1.6)

Since me _ mp, M _ neR3me, so ne _

M

mpR3 and

Ekin _

~2

me

_

M

mp

_5=3

| {z }

constant for

_xed M

1

R2 : (1.7)

Thus

E _ 􀀀

_

R 􀀀

_

R2 ; _; _independent of R: (1.8)

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E

R

Rmin _

~2M􀀀1=3

Gmem5=3

p

Rmin

The collapse of the star is therefore prevented. It becomes a White Dwarf

or a cold, dead star supported by electron degeneracy pressure.

At equilibrium

ne _

M

mpR3

min

_

meG

~2

􀀀

Mm2p

_2=3

_3

: (1.9)

But the validity of non-relativistic approximation requires that hpei _ mec,

i.e.

hpei

me

=

~n1=3

e

me _ c (1.10)

or ne _

_mec

~

_2

: (1.11)

7

For a White Dwarf this implies

meG

~2

􀀀

Mm2p

_2=3

_

mec

~

(1.12)

or M _

1

m2p

_

~c

G

_3=2

: (1.13)

For su_ciently large M the electrons would have to be relativistic, in

which case we must use

hEi = hpei c = ~cn1=3

e (1.14)

) Ekin _ neR3 hEi _ ~cR3n4=3

e (1.15)

_ ~cR3

_

M

mpR3

_4=3

_ ~c

_

M

mp

_4=3 1

R

(1.16)

So now,

E _ 􀀀

_

R

+

 

R

: (1.17)

Equilibrium is possible only for

 = _ ) M _

1

m2p

_

~c

G

_3=2

: (1.18)

For smaller M, R must increase until electrons become non-relativistic,

in which case the star is supported by electron degeneracy pressure, as we

just saw. For larger M, R must continue to decrease, so electron degeneracy

pressure cannot support the star. There is therefore a critical mass MC

MC _

1

m2p

_

~c

G

_3=2

) RC _

1

memp

_

~3

Gc

_1=2

(1.19)

above which a star cannot end as a White Dwarf. This is the Chandrasekhar

limit. Detailed calculation gives MC ' 1:4M_.

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