Solutions To Stellar Structure Problems

 a) Derive a third equation for stellar structure where dL/dr is the subject. (Hint: Make use of the energy generation form for dL/dM).

b) From one or more equations of stellar structure, obtain an estimate for the Sun’s central temperature and pressure. (Take the solar radius R = 7 x 10⁸ m and the solar mass M = 2 x 10³⁰ kg, and the density r = 1400 kgm^-3. )

Solutions:

a) We use: dL/dM = e  and dM/dr = 4p r² r

Then by the chain law for derivatives:

(dL/dM) (dM/dr) = dL/dr = e  (4p r² r)

 

b) We approximate: dP/dr = - G M(r) r dr/ r²

To: P/R = G M r / r²

P = (6.7 x 10^-11 Nm²kg^-2)( 2 x 10³⁰ kg)( 1400 kgm^-3)/ R

Where R = 7 x 10 ⁸ m

Then: P = 2.6 x 10 ¹⁴ Pa

For the estimate of the central temperature we use:

T =  m P / r R

Where we know m = 0.57 for a fully ionized H-plasma.

Then: T =

 (0.57)( 2.6 x 10 ¹⁴ Pa)/(1400 kgm^-3)( (8.3 x 10³ JK^-1)

 And then:

T = 1.2 x 10 ⁷ K