Stellar Stability - Hydrostatic Equilibrium and Equation of State · USAPhO 2024 #A2

Physics · modern-physics

Problem

A star in hydrostatic equilibrium has inward gravitational forces balanced by pressure gradients. Though the material in a star is not simply an ideal gas, in many cases its pressure PP and density ρ\rho are simply related by P=KργP = K\rho^\gamma for constants KK and γ\gamma. Throughout this problem, assume the star is spherically symmetric, its mass is conserved, and relativistic effects can be neglected. a. A thin shell of the star at radius r0r_0 has density ρ0\rho_0 and thickness Δr\Delta r, and experiences an inward gravitational field of magnitude g0g_0. i. What is the pressure difference ΔP\Delta P across the shell in equilibrium? ii. Suppose the entire star expands uniformly by a factor 1+x1+x, so that the shell now has radius r=r0(1+x)r = r_0(1+x). In terms of ΔP0\Delta P_0, xx, and γ\gamma, what is the new pressure difference across it? iii. By considering the forces on the shell, write an expression for d2r/dt2d^2r/dt^2 valid when xx is small, in terms of g0g_0, γ\gamma, and xx. For what values of γ\gamma will the star be stable? Next, we consider some simple models of stars, where γ\gamma can be computed. b. In a giant star, the pressure is P=Pgas+PradP = P_{\text{gas}} + P_{\text{rad}}, where PgasP_{\text{gas}} is due to the gas, which obeys the ideal gas law, and Prad∝T4P_{\text{rad}} \propto T^4 is due to blackbody radiation. In the star's "radiation zone", PradP_{\text{rad}} is much larger than PgasP_{\text{gas}}, but the two have a constant ratio. In this case, what is the value of γ\gamma? c. A white dwarf is composed of electrons and nuclei. The electrons provide the outward pressure, while the nuclei cancel the electrons' charge, and are responsible for most of the mass density. Consider a region of a white dwarf where the number density of electrons is nen_e. i. The electrons obey the Heisenberg uncertainty principle, ΔpΔx≳ℏ\Delta p \Delta x \gtrsim \hbar, where Δx\Delta x is the spacing between them, and Δp\Delta p is the typical momentum that quantum mechanics implies they must have. Find a rough estimate for Δp\Delta p in terms of nen_e and ℏ\hbar. ii. Using this result, find γ\gamma for a white dwarf. iii. A white dwarf has total mass MM, radius RR, and a relatively uniform density of order ρ∼M/R3\rho \sim M/R^3. The radius is related to the mass by R∝MnR \propto M^n for a constant nn. Find the value of nn.

Browse the problem library