Consider a hypothetical star of radius R with density rho that is constant, i.e.

Question

Consider a hypothetical star of radius R with density rho that is constant, i.e. independent of radius. The star is composed of a classical, nonrelativistic, ideal gas of fully ionized Hydrogen. a. Solve the equations of stellar structure for the pressure profile, P(r), with boundary condition P(R) = 0. Answer: P(r) = {2pi/3)Gp^2(R^2 - r^2). b. Find the temperature profile. T(r). c. Assume that the nuclear energy production rate depends on temperature as c ~ T^1. (This is the approximate dependence of the rate for the p-p chain at the temperature in the core of the Sun.) At what radius does e decrease to 0.1 of its central value, and what fraction of the star's volume is included within this radius?

Explanation / Answer

considering the hypothetical star, constant density rho, radius R

a. at radius r consider a small layer of thickness dr

then dP = rho*g*dr

where g = Gm/r^2

where m = rho*4*pi*r^3/3

so dP = 4*pi*rho^2*G*r*dr/3

integrating from r = R to r = r, P = 0 to P = P

P = 2*pi*rho^2*G(R^2 - r^2)/3

b. using ideal gas equation

P = rho*R'*T

here R' = specific gas constant of the gas in the star

so, R' = R / 0.002 = 4124 J / kg K [ where R is universal gas constant]

so, T = P/4124*rho = 2*pi*rho*G(R^2 - r^2)/3*4124 = pi*rho*G(R^2 - r^2)/6186

c. given e = k*[pi*rho*G(R^2 - r^2)/6186]^4 [ where k is a constant]

so at r = 0

e = k*[pi*rho*G(R^2)/6186]^4

so, for r

0.1*k*[pi*rho*G(R^2)/6186]^4 = k*[pi*rho*G(R^2 - r^2)/6186]^4

0.1[(R^2)]^4 = [(R^2 - r^2)]^4

0.5623R^2 = R^2 - r^2

r = 0.66155R

fraction of stars volume in this radius = r^3/R^3 = 0.2895

in percent, 28.953 percent of stars volume is enclosed in this radius

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.