1) Begin by finding the mean molecular weight of the material given that:
X = 0.9 and we use:
m = 4/ (3 + 5X) = 4/ (3 + 5(0.9)) = 0.533
The gas pressure, P = (r/ mH) kT = r RT/ m
= (20.2 kg m-3) (8.3 x 103 JK-1)(3.14 x 106 K)/ 0.533
P = 9.88 x 10 11 Nm-2
The radiation pressure is: P R = aT4/ 3
Where a = 7.55 x 10 -16 Jm-3K-4
So that P R = 1/3 (7.55 x 10 -16 Jm-3K-4 )( 3.14 x 106 K)4
And: P R = 2.44 x 10 10 Nm-2
The total pressure is:
PT = r RT/ m + aT4/ 3 =
9.88 x 10 11 Nm-2 + 2.44 x 10 10 Nm-2 » 10 12 Nm-2
Thus, the overall contribution of radiation pressure is only on the order of 0.02 or about 2% of the total pressure.
The thermal energy per kg is: U = P/ (g - 1) r
Or: U = (10 12 Nm-2 )(5/3 – 1) (20.2 kg m-3)
U = 7.4 x 10 10 J kg-1
Then by the chain rule for derivatives:
(dL/dM) (dM/dr) =
dL/dr = e (4p r2 r)
dL/dr = e (4p r2 r)
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
(Note: The actual central pressure is about an order of magnitude larger.)
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)
T = 1.2 x 10 7 K
(Compare to 15 million degrees K, using a more detailed approach)
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