Best Answer:
If your density profile is linear, you've only run through one iteration of the equations of stellar structure, and your density function has nowhere near converged yet. Run through a few more times, and it'll get better.

But...

dM/dr = 4π r² ρ(r)

ρ(r) = ρ(c) (1−r/R)

M(r) = ∫dM = 4π ∫ r² ρ(r) dr

M(r) = 4π ρ(c) ∫ r² (1−r/R) dr

M(r) = 4π ρ(c) ∫ (r²−r³/R) dr

M(r) = 4π ρ(c) { r³/3 − r⁴/(4R) }

M(r) = π ρ(c) { (4/3) r³ − r⁴/R }

dP/dr = −G M(r) ρ(r) / r²

P(c) = ∫dP = −G ∫ [ M(r) ρ(r) / r² ] dr

ρ(r) = ρ(c) (1−r/R)

M(r) = π ρ(c) [ (4/3) r³ − r⁴/R ]

P(c) = −G ∫ { π ρ(c) [ (4/3) r³ − r⁴/R ] ρ(c) (1−r/R) / r² } dr

P(c) = −Gπ ρ²(c) ∫ { [ (4/3) r − r²/R ] (1−r/R) } dr

You can see from this step that you had one factor of r too many in your integrand.

P(c) = −Gπ ρ²(c) ∫ { (4/3) r − (7/3) r²/R + r³/R² } dr

P(c) = −Gπ ρ²(c) { (2/3) R² − (7/9) r³/R + (1/4) r⁴/R² } | 0,R

P(c) = Gπ ρ²(c) { (2/3) R² − (7/9) R² + (1/4) R² }

P(c) = Gπ ρ²(c) { (24/36) R² − (28/36) R² + (9/36) R² }

P(c) = (5/36) Gπ ρ²(c) R²

M(r) = π ρ(c) { (4/3) r³ − r⁴/R }

M = (π/3) ρ(c) R³

ρ(c) = 3M/(πR³)

P(c) = (5/36) Gπ [3M/(πR³)]² R²

P(c) = (5/36) Gπ R² 9M²/(π²R⁶)

P(c) = (5/4) GM²/(πR⁴)

I can understand the temptation to rest content with a linear density profile. But you won't get a really good model unless you iterate through the equations of stellar structure until the density profile's graph stops changing appreciably. Of course, then the density profile equation is somewhat more complicated, and the task of calculating the core temperature, density and pressure is a little harder than it was before. But that's what numerical methods are for, yes?

Source(s):

Anonymous
· 9 years ago

Asker's rating