Astro 160: The Physics of Stars

where c n is again a dimensionless function that you should write down. Also show that the centralpressure of a polytrope can be written asP c = d n GM 2/3 ρ 4/3cwhere d n depends on a n and c n . The values of a n ,d n , and d n can be determined by numerically solving theLane-Emden equation. The most useful cases for our purposes are γ = 4/3 (n = 3) and γ = 5/3 (n = 3/2)polytropes. For n = 1.5, a n = 5.99 and c n = 0.77 while for n = 3, a n = 54.183 and c n = 11.05. We will usethese quite a bit during this course. Note how, as mentioned in class, the results for the central pressure anddensity of polytropes above are very similar to what you would get from an order of magnitude estimate,except that for polytropes we get an exact correct numerical factor given by a n , c n and d n .We know thatand alsoa =given these relationships we can now find[K = P c ρ −γca 2 4πGn+1 = Kρ1/n−1 cbut since we know that γ = 1/n+1 we findand finally we findP c ρ −γc ρ1/n−1 c(n+1)Kρc1/n−1 ] 1/24πG= P c ρ −γc ρc1/n−1= P c ρc−(1+1/n) ρc1/n−1 = P c ρ −2cusing this results we can now deriveP c = a2 4πGρ 2 cn+1which becomesP c = a2 4πGρ 2 cn+1P c = a2 4πGa 2 nn+1looking at these two expressions we can see that= GM2R 4 c n( ) 3M 24πR 3 = GM2R 4 c nc n = 9a2 a 2 n4πR 2 1n+1now looking atand from (b) we findP c = d n GM 2/3 ρ 4/3cP c = a2 4πGρc 2/3 ρc4/3 ( )= a2 4πG an 3M 2/3n+1 n+1 4πR 3 ρc4/316