130_notes.dvi

To understand the collapse of stars, we must compare this to thepressure of gravity. We compute
this approximately, ignoring general relativity and, more significantly, the variation of gravitational
pressure with radius.

E = −

∫R

0

GMinside 4 πr^2 ρ

r

dr

= −

∫R

0

G(^43 πr^3 ρ)4πr^2 ρ

r

dr

= −

(4π)^2

15

Gρ^2 R^5 =−

3 GM^2

5 R

The mass of the star is dominated by nucleons.

M=NMN

Putting this into our energy formula, we get.

E=−

3

5

G(NMN)^2

(

4 π

3

)^13

V−

(^13)
We can now compute the pressure.
Pg=−

∂E

∂V

=−

1

5

G(NMN)^2

(

4 π

3

) (^13)
V−
(^43)
The pressures must balance. For awhite dwarf, the pressure from electrons is.
Pe=
π^3 ̄h^2
15 me

(

3 Ne

π

) (^53)
V−
(^53)
We can solve for the radius.

R=

(

3

4 π

) (^23)
π^3 ̄h^2
3 GmeMN^2

(

3

π

) (^53)
N
(^53)
e
N^2
There are about two nucleons per electron
N≈ 2 Ne
so the radius becomes.
R=

(

81 π^2

512

)^13

̄h^2

GmeMN^2

N−

(^13)
The radius decreases as we add mass. For one solar mass,N= 10^57 , we get a radius of 7200 km,
the size of the earth. The Fermi energy is about 0.2 MeV.
Awhite dwarfis the remnant of a normal star. It has used up its nuclear fuel, fusing light elements
into heavier ones, until most of what is left is Fe^56 which is the most tightly bound nucleus. Now
the star begins to cool and to shrink. It is stopped by the pressure of electrons. Since thepressure