1-1 Expressions and Formulas

Stellar model 1: Constant density model REFERENCE: Bowers and Deeming

The simplest form a star of mass M and radius R could take is that of a constant density star in hydrostatic equilibrium. To analyze such a model we need the mass-continuity equation, and the equation of hydrostatic equilibrium, given here, along with a reasonable boundary condition: dm [Eq. 1] dr = (r)4r2, dP Gm ( r )  ( r) [Eq. 2] dr = - r 2 , [Eq. 3] P(R) = 0. From Eq. 1 we have [Eq. 4] dm = 4r2dr, where we have removed the explicit r dependence from the density function. This model assumes that the density is constant. If the mass of the star is M then 3M  [Eq. 5] = 4  R 3. Integrating Eq. 4 from 0 to an arbitrary value of r we obtain 3 r 4  r 3 Mr 3 [Eq. 6] m(r) =  4r2dr = 3 = R . 0 Eq. 5 was substituted at the end. Eq. 6 tells us how much of the star's total mass is enclosed in a sphere of radius r  R. Recall that this was just the mass that caused a gravitational attraction toward the center of the star, from Newton's shell theorem.

Now we can solve Eq. 2, the equation of hydrostatic equilibrium. We will integrate from an arbitrary r to the surface, so we can apply our boundary condition Eq. 3. For now we'll leave the density as : R G (  4  r 3 )  dr 2 P(R) - P(r) = - 3 r r

R G 4   2rdr 0 - P(r)= - 3 r   2 2 2 2 G ( R - r ) [Eq. 7] P(r) = 3 .

Eq. 7 is written in terms of 2, but we can write it in terms of other powers of : GM  ( R 2 - r 2) [Eq. 8] P(r) = 2 R 3 , or 3 GM 2 ( R 2 - r 2) [Eq. 9] P(r) = 8  R 6 .

At the center of the star the pressure P(0)  Pc will be the highest, and Eq. 8 and Eq. 9 can be written GM  [Eq. 10] Pc = 2 R ,

3 GM 2 [Eq. 11] 8R4 Stellar model 1: Constant density model REFERENCE: Bowers and Deeming

Pc = .

We will use Eq. 5, Eq. 10 and Eq. 11 for our most basic estimates concerning dwarfs and neutron stars. Of course, we must not forget that this model assumes a constant density--which is not the usual state of a star. In fact, for the purposes of our simplified analysis, we can write Eq. 10 as Pc = GM/R, dispensing with the factor of 3/2. More in keeping with the form of Eq. 2, the equation of hydrostatic equilibrium for this stellar model will be written

P GM  R = R 2, where 3M  = 4  R 3 .