Question

A star of mass, M, and radius, R, has a density profile that increases linearly with radius in the star (r) via the expressions: rho(r)=rho(1-r²/R²).

Here rhoc = central density
A. Find the mass stratification, i.e., m(r)
B. Find the Pressure stratification, i.e., P(r)
C. Derive the relation between total mass of the star M and R
D. What is the average density of the star in units of the core density

Answer 1

A. To find the mass stratification, use the equation m(r) = (4/3)πρ(1-r²/R²)R³. B. To find the pressure stratification, use the equation P(r) = P(1-r²/R²). C. Derive the relation between total mass of the star, M, and R by integrating the mass stratification equation over the entire star. D. The average density of the star in units of the core density can be found by dividing the total mass by the volume of the star.

Step-by-step explanation:

A. In order to find the mass stratification, m(r), we can use the equation: m(r) = (4/3)πρ(1-r²/R²)R³.

B. Here, ρ is the density at a given radius and R is the radius of the star. To find the pressure stratification, P(r), we can use the equation: P(r) = P(1-r²/R²). Here, P is the pressure at a given radius and R is the radius of the star.

C. To derive the relation between the total mass of the star, M, and R, we can integrate the mass stratification equation over the entire star and equate it to the total mass, giving us the relation: M = (4/3)πρAR³.

4. The average density of the star in units of the core density can be found by dividing the total mass by the volume of the star.

The volume of the star can be calculated using the formula for the volume of a sphere, V = (4/3)πR³. Therefore, the average density can be expressed as: average density = M / V = (3M) / (4πR³).