The Balancing Act — Hydrostatic Equilibrium

Show explicit units, and run a sanity check on every result. A scaling answer is judged by its exponents, not its coefficient. Worked solutions are released after the homework due date.

Useful constants (CGS):

Constant	Value
$G$	$6.674 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$M_\odot$	$2.0 \times 10^{33}~\mathrm{g}$
$R_\odot$	$7.0 \times 10^{10}~\mathrm{cm}$
$\bar\rho_\odot$	$1.4~\mathrm{g\,cm^{-3}}$
$m_p$	$1.67 \times 10^{-24}~\mathrm{g}$
$k_B$	$1.38 \times 10^{-16}~\mathrm{erg\,K^{-1}}$
$\mu$	$0.6$
$P_{c,\odot}$ (scaling estimate)	$1.1 \times 10^{16}~\mathrm{dyn\,cm^{-2}}$
$1~\mathrm{atm}$	$1.013 \times 10^6~\mathrm{dyn\,cm^{-2}}$

Conceptual

Problem

⭐⭐ Pressure is not enough. A student writes: “The Sun survives because the pressure inside it is enormous.”

(a) Identify what is incomplete about this claim.
(b) State the corrected version in one sentence, using the idea of a pressure gradient.
(c) Name the observable, the model, and the inference that connect “the Sun holds its size for billions of years” to the condition $dP/dr < 0$ .

Problem

⭐⭐ Where is the squeeze hardest? Compare a shell deep in the interior ( $r \approx 0.2\,R$ ) with one near the surface ( $r \approx 0.9\,R$ ).

(a) Using $dP/dr = -\rho g$ , argue where $|dP/dr|$ is larger.
(b) How do $\rho(r)$ and $g(r)$ each contribute to that conclusion?
(c) What does this imply about where inside the star most of the central pressure is built up?

Problem

⭐⭐ Three ways to hold up a star. Stars are supported by thermal gas pressure, radiation pressure, or degeneracy pressure.

(a) State the physical origin of each pressure in one phrase.
(b) Give the temperature dependence of each ( $P_\text{gas}$ , $P_\text{rad}$ , $P_\text{deg}$ ).
(c) Name a stellar context where each one dominates.

Problem

⭐⭐ When does light hold up a star? The ratio of radiation to gas pressure scales as $P_\text{rad}/P_\text{gas} \propto T^3/\rho$ .

(a) Using the hydrostatic estimates $T_c \sim M/R$ and $\rho \sim M/R^3$ , show that $P_\text{rad}/P_\text{gas} \propto M^2$ .
(b) What does this predict about which stars become radiation-pressure-supported?
(c) Why does this make very massive stars more loosely bound and prone to instability?

Calculation

Problem

⭐⭐ How compact to double the pressure? At fixed mass, $P_c \sim G M^2/R^4$ .

(a) By what factor must the radius shrink to double the central pressure?
(b) By what factor to raise it tenfold?
(c) Comment on how sensitively $P_c$ responds to radius, and connect that sensitivity to the exponent.

Problem

⭐⭐ The crushing core of a red dwarf. A red dwarf has $M = 0.3\,M_\odot$ and $R = 0.3\,R_\odot$ .

(a) Using $P_c \sim G M^2/R^4$ , find its central pressure relative to the Sun’s.
(b) Estimate $P_c$ in $\mathrm{dyn\,cm^{-2}}$ , taking $P_{c,\odot} \sim 1.1\times10^{16}\,\mathrm{dyn\,cm^{-2}}$ .
(c) The star is far less massive than the Sun, yet its core pressure is higher. Resolve the paradox.

Problem

⭐⭐ …but not a hotter core. Take the same red dwarf ( $M = 0.3\,M_\odot$ , $R = 0.3\,R_\odot$ ).

(a) Using $T_c \sim \mu G M m_p/(k_B R) \propto M/R$ , find its core temperature relative to the Sun’s.
(b) Compare with your pressure result from the previous problem. Why does $P_c$ change so dramatically while $T_c$ barely moves?
(c) What does this say about how uniform core temperatures are across the main sequence?

Problem

⭐⭐ How fast does balance restore? A star knocked slightly out of hydrostatic equilibrium re-establishes it on the dynamical timescale $\tau_\text{dyn} \sim 1/\sqrt{G\bar\rho}$ .

(a) Compute $\tau_\text{dyn}$ for the Sun ( $\bar\rho_\odot \approx 1.4\,\mathrm{g\,cm^{-3}}$ ). Express it in minutes.
(b) The red dwarf above is $\sim 11\times$ denser than the Sun. Find its $\tau_\text{dyn}$ .
(c) Given a nuclear timescale $\tau_\text{nuc} \sim 10^{10}\,\mathrm{yr}$ , explain why we may treat the star as static when modeling its structure.

Synthesis

Problem

⭐⭐ Losing energy, getting hotter. A protostar of mass $M$ contracts quasi-statically from radius $R$ to $R/2$ while staying bound, with $U_\text{grav} \sim -G M^2/R$ .

(a) By what factor does $|U_\text{grav}|$ change, and how much gravitational energy is released?
(b) Use the virial theorem to say how that released energy splits between heating the gas and being radiated away.
(c) Explain in 2–3 sentences why this makes the core hotter as the star loses energy — the negative-heat-capacity argument.

Problem

⭐⭐⭐ Reading the Math, audited. From only a star’s measured mass and radius, derive its core temperature and then audit the derivation.

(a) Run the three moves of Reading the Math: approximate $dP/dr$ as a finite difference to obtain $P_c$ , then couple it to the ideal-gas law to obtain $T_c$ . Show each move.
(b) List every assumption the derivation made — the full audit, inherited rows included.
(c) The worked estimate of $P_c$ comes out $\sim 20\times$ below detailed solar models. Which single assumption is most responsible, and why? What would fixing it require?