Stellar model 2:
PROBLEM SET
Linear density model
r
PROBLEM 1. Show that (0) = c and (R) = 0 when (r) = c 1 –
R
is this reasonable?
r
R
PROBLEM 2.
Show that dm = 4c 1 –
PROBLEM 3.
Show that
PROBLEM 4.
Show that the total mass is M = m(R) =
Why
r2 dr.
r
3
3
4
m(r) =  4c r2 – r dr = 4c r – r .
3
R
4R
0
PROBLEM 5. Show that c = 4<>.
constant density model?
PROBLEM 6.
.
1
3
cR3.
How does this compare to c in the
Show that the following integral
R
r3
r4
P(R) - P(r) = -G4c
–
c 1 – 1 dr2
3
4R
R r
r
can be written as
R
r2
1
r
0 - P(r)= -G4c2
–
1 –
dr
3
4R
R
r
PROBLEM 7.
Show that the above integral can be written
P(r) = G4c2
PROBLEM 8.
R2
R3
R3 .
R4 .
–
–
+
6
9R
12R
16R2
r2
r3
r3 .
r4 .
+
6
9R
12R
16R2
.
Show that the above integral cleans up to
P(r) = G4c2
5R2
r2
r3
r3 .
r4 .
+
+
.
144
6
9R
12R
16R2
5
PROBLEM 9. Show that Pc = 36 Gc2R2. Hint: P(0)  Pc. Why would a star
having a linear density be more likely to sustain fusion reactions than
a star having a constant density?
PROBLEM 10. Write Pc in terms of M and R.
Hint:
Substitute out c.
PROBLEM 11. Find Pc in the sun, assuming a linear density model. Which
model do you think fits the sun best - constant density, or linear
density? Why do you think so?