Question

Suppose that a star is modelled as having a density that decreases linearly from the centre to the surface, ?(r) = ?_c (1 - r/R), where ?_c is the central density and R is the stellar radius. (i) Show that m = (4?/3) ?_c r^3 (1 - 3r/4R), where m is the mass enclosed at radius r and thus that ?_c = 3M/(? R^3) where M is the mass of the star. (ii) Show that the central pressure is P_c = 5GM^2/(4? R^4), where R is the radius of the star. Hence show that the central temperature is T_c = (5/12) (? m_H / k_B) (G M / R).

          Suppose that a star is modelled as having a density that decreases linearly from the centre to the surface,
?(r) = ?_c (1 - r/R),
where ?_c is the central density and R is the stellar radius.
(i) Show that
m = (4?/3) ?_c r^3 (1 - 3r/4R),
where m is the mass enclosed at radius r and thus that
?_c = 3M/(? R^3)
where M is the mass of the star.
(ii) Show that the central pressure is
P_c = 5GM^2/(4? R^4),
where R is the radius of the star.
Hence show that the central temperature is
T_c = (5/12) (? m_H / k_B) (G M / R).

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Suppose that a star is modelled as having a density that decreases linearly from the centre to the surface,
?(r) = ?c (1 - r/R),
where ?c is the central density and R is the stellar radius.
(i) Show that
m = (4?/3) ?c r^3 (1 - 3r/4R),
where m is the mass enclosed at radius r and thus that
?c = 3M/(? R^3)
where M is the mass of the star.
(ii) Show that the central pressure is
Pc = 5GM^2/(4? R^4),
where R is the radius of the star.
Hence show that the central temperature is
Tc = (5/12) (? mH / kB) (G M / R).

Added by Vicenta V.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Urvashi Arora

01/21/2022

Step 1

Step 1: Start with the expression for mass enclosed at radius r, which is given as \( m = \frac{4}{3} \pi \rho r^3 \). Show more…

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Suppose that a star is modelled as having a density that decreases linearly from the centre to the surface, (r) = ρ_c (1 - r/R), where ρ_c is the central density and R is the stellar radius. Show that m = (4π/3) ρ_c r^3 (1 - 3r/4R), where m is the mass enclosed at radius r and thus that ρ_c = 3M/(π R^3) where M is the mass of the star. (ii) Show that the central pressure is P_c = 5GM^2/(4π R^4), where R is the radius of the star. Hence show that the central temperature is T_c = (5/12) (μ m_H / k_B) (G M / R).

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