Chapter 25 Exercises — Nuclear Physics of Neutron Stars

Reference constants and values for these exercises:

Section A: Basic Properties and Scales (Fundamental)

Problem 25.1 — Density estimates. (a) Calculate the average density of a $1.4\,M_\odot$ neutron star with radius $R = 12$ km. Express your answer in g/cm$^3$ and in units of $\rho_0$. (b) Repeat for a $2.0\,M_\odot$ neutron star with $R = 11.5$ km. (c) The central density is typically 3--5 times the average density. Estimate the range of central densities for both stars. (d) The density at the center of the Sun is $\sim 150$ g/cm$^3$. How many orders of magnitude denser is the center of a neutron star?

Problem 25.2 — Compactness and general relativity. (a) Calculate the dimensionless compactness parameter $\xi = GM/(Rc^2)$ for a $1.4\,M_\odot$, $R = 12$ km neutron star. (b) Calculate $\xi$ for the Sun ($M = M_\odot$, $R = 6.96 \times 10^5$ km). (c) Calculate $\xi$ for a white dwarf ($M = 0.6\,M_\odot$, $R = 8,000$ km). (d) For a Schwarzschild black hole, $\xi = 0.5$. What fraction of the way from a white dwarf to a black hole is a neutron star, in terms of compactness? (e) At what compactness do general-relativistic corrections become $\sim 10\%$? Is Newtonian gravity adequate for neutron stars?

Problem 25.3 — Gravitational redshift. (a) Derive the gravitational redshift factor for a photon emitted from the surface of a neutron star: $1 + z = (1 - 2GM/Rc^2)^{-1/2}$. (b) Calculate $z$ for a $1.4\,M_\odot$, $R = 12$ km star. (c) The iron K$\alpha$ line has a rest energy of 6.40 keV. What energy would this line be observed at from the neutron star surface? (d) If the redshift $z$ were measured precisely (along with the mass from binary pulsar timing), could you determine the radius? Write $R$ as a function of $M$ and $z$.

Problem 25.4 — Escape velocity. (a) Calculate the Newtonian escape velocity from the surface of a $1.4\,M_\odot$, $R = 12$ km neutron star. Express as a fraction of $c$. (b) The relativistic escape velocity is $v_{\text{esc}} = c\sqrt{2\xi/(1+2\xi)}$ where $\xi = GM/(Rc^2)$. Calculate this and compare to the Newtonian result. (c) How much kinetic energy (in MeV) would a neutron need to escape from the surface? Compare this to the neutron's rest mass energy.

Problem 25.5 — Number of baryons. (a) Estimate the total number of baryons in a $1.4\,M_\odot$ neutron star. (b) The gravitational binding energy of the star is $E_{\text{bind}} \sim 0.6 GM^2/(Rc^2) \times Mc^2 \approx 3 \times 10^{53}$ erg. What fraction of the total baryonic rest mass energy is this? (c) The gravitational mass $M$ is therefore less than the baryonic mass $M_B = N_B m_N$. Calculate the difference $M_B - M$ in solar masses.

Section B: The Equation of State (Conceptual and Quantitative)

Problem 25.6 — Beta equilibrium composition. In beta equilibrium, $\mu_n = \mu_p + \mu_e$, and charge neutrality requires $n_p = n_e$ (ignoring muons). For a free Fermi gas, $\mu_i = \sqrt{(p_{F,i} c)^2 + (m_i c^2)^2}$ and $n_i = p_{F,i}^3/(3\pi^2\hbar^3)$. (a) In the nonrelativistic limit for nucleons and the ultrarelativistic limit for electrons ($m_e c^2 \ll p_{F,e} c$), show that the proton fraction $x_p = n_p/n_B$ satisfies:

$$x_p \approx \frac{1}{8} \left(\frac{3\pi^2 n_B}{(m_N c)^3}\right)^{-1} \left(\frac{4 a_{\text{sym}} m_N}{\hbar^2}\right)^3$$

or alternatively, use the simplified estimate based on the symmetry energy: $4 S(n_0)(1 - 2x_p) \approx \mu_e \approx \hbar c (3\pi^2 n_B x_p)^{1/3}$ to find $x_p$ numerically at $n_B = n_0 = 0.16$ fm$^{-3}$ with $S = 32$ MeV. (b) Show that the proton fraction is of order 4--5%. (c) At what density does the muon threshold $\mu_e = m_\mu c^2$ get reached? (d) Explain qualitatively why the proton fraction increases with density.

Problem 25.7 — Symmetry energy and neutron star radii. The approximate relation between the symmetry energy slope $L$ and the radius of a $1.4\,M_\odot$ neutron star is $R_{1.4} \approx (9.5 + 0.045 L)$ km, where $L$ is in MeV. (a) Calculate $R_{1.4}$ for $L = 40$ MeV (soft), $L = 60$ MeV (moderate), and $L = 100$ MeV (stiff). (b) The GW170817 constraint implies $R_{1.4} \lesssim 13.5$ km. What upper bound does this place on $L$? (c) The $2\,M_\odot$ constraint requires $R_{1.4} \gtrsim 10.5$ km (approximately). What lower bound does this place on $L$? (d) The PREX-II measurement of the neutron skin of $^{208}$Pb implies $L \approx 106 \pm 37$ MeV. Is this consistent with the astrophysical constraints from parts (b) and (c)?

Problem 25.8 — Polytropic EOS. For a polytropic EOS $P = K \varepsilon^\Gamma$: (a) Show that $\Gamma = d\ln P / d\ln \varepsilon$ is the adiabatic index. (b) For an ideal nonrelativistic degenerate Fermi gas, $P \propto n_B^{5/3}$ and $\varepsilon \propto n_B$, so $P \propto \varepsilon^{5/3}$. What is $\Gamma$? (c) For an ultrarelativistic degenerate Fermi gas, $P = \varepsilon/3$. What is $\Gamma$? (d) For nuclear matter near saturation, the pressure increases steeply with density. Typical values of the effective $\Gamma$ at $\rho \sim 2\rho_0$ are $\Gamma \approx 2$--$3$. Explain qualitatively why the nuclear force produces a stiffer EOS than a free Fermi gas. (e) The causality limit requires $c_s^2 = dP/d\varepsilon \leq c^2$. For a polytrope $P = K\varepsilon^\Gamma$, what is $c_s^2$? At what density does the sound speed reach $c$ for $\Gamma = 2$ and $K = 1/(4\varepsilon_0)$, where $\varepsilon_0 = \rho_0 c^2$?

Section C: The TOV Equation and Mass-Radius Relation

Problem 25.9 — GR corrections. Consider a neutron star with $M = 1.4\,M_\odot$ and $R = 12$ km. (a) Evaluate each of the three GR correction factors in the TOV equation at the surface ($r = R$): (i) $(1 + P/\varepsilon c^2)$ — using $P_{\text{surface}} = 0$, this is trivially 1 at the surface. Evaluate instead at $r = R/2$ using the estimate $P(R/2) \approx P_c/2 \approx 18$ MeV/fm$^3$ and $\varepsilon(R/2) \approx 3\varepsilon_0 \approx 450$ MeV/fm$^3$. (ii) $(1 + 4\pi r^3 P/(mc^2))$ at $r = R/2$. (iii) $(1 - 2Gm/rc^2)^{-1}$ at $r = R/2$, assuming $m(R/2) \approx M/4$. (b) Multiply all three factors together. By what factor does GR strengthen gravity compared to Newtonian gravity at this point? (c) Repeat for a $2.0\,M_\odot$, $R = 11$ km star (more compact). Which correction factor changes the most?

Problem 25.10 — The Oppenheimer-Volkoff limit. Oppenheimer and Volkoff (1939) solved the TOV equation for a free (noninteracting) neutron Fermi gas and found $M_{\text{max}} = 0.71\,M_\odot$. (a) The free neutron gas EOS at zero temperature is $P = (3\pi^2)^{2/3} \hbar^2 n_n^{5/3}/(5 m_n)$ for nonrelativistic neutrons. Calculate $P$ at $n_n = n_0$ and convert to MeV/fm$^3$. (b) A realistic nuclear EOS gives a pressure roughly 5--10 times higher than the free gas at the same density. How does this affect $M_{\text{max}}$? (c) Use the rough scaling $M_{\text{max}} \propto P_c^{1/2}/\varepsilon_c$ to estimate the maximum mass for an EOS with 5 times the pressure of the free gas. Compare to the observed $\sim 2.1\,M_\odot$. (d) What is the physical origin of the extra pressure? (Hint: the repulsive core of the nuclear force.)

Problem 25.11 — Stability criterion. The TOV equation gives, for each central density $\varepsilon_c$, a star with mass $M(\varepsilon_c)$ and radius $R(\varepsilon_c)$. (a) Explain why the stability criterion is $dM/d\varepsilon_c > 0$: if adding matter (increasing $\varepsilon_c$) increases the mass, the star can support the extra weight, but if it decreases the mass, the star cannot. (b) On the $M$-$R$ curve, stable stars are on the branch where $M$ increases as $R$ decreases (for a given EOS). Sketch a qualitative $M(R)$ curve showing the stable branch, the maximum mass, and the unstable branch. (c) What happens to a star on the unstable branch if it is perturbed slightly toward higher central density? (d) What is the maximum mass for the free neutron gas, and why is it so much lower than for realistic EOS models?

Problem 25.12 — Mass from the M-R diagram. A neutron star EOS predicts the following mass-radius pairs (from numerical TOV integration):

(a) Plot $M$ versus $R$ and $M$ versus $\varepsilon_c$. (b) Identify the maximum mass and the corresponding central density. (c) Which entries on the table correspond to unstable configurations? (d) Is this EOS consistent with the observation of PSR J0740+6620 ($M = 2.08 \pm 0.07\,M_\odot$)? (e) Is it consistent with the NICER radius measurement for PSR J0030+0451 ($R = 12.71^{+1.14}_{-1.19}$ km at $M = 1.34\,M_\odot$)?

Section D: Neutron Star Crust and Pasta

Problem 25.13 — Neutron drip. (a) The neutron drip density $\rho_{\text{drip}} \approx 4.3 \times 10^{11}$ g/cm$^3$ is where the neutron separation energy of the equilibrium nucleus drops to zero. Express $\rho_{\text{drip}}$ in units of $\rho_0$. (b) At the neutron drip point, the equilibrium nucleus is approximately $^{118}$Kr ($Z = 36$, $N = 82$). Calculate the proton fraction $Z/A$. (c) Why is $N = 82$ not a coincidence? (Hint: magic numbers.) (d) Compare the neutron drip density to the central density. What fraction of the star's volume is at densities below neutron drip?

Problem 25.14 — Nuclear pasta energetics. The total energy density of non-uniform matter (relative to uniform matter) can be written as $e = e_{\text{surf}} + e_{\text{Coul}}$. For spherical nuclei of radius $r_N$ in a Wigner-Seitz cell of radius $R_c$ with volume fraction $u = (r_N/R_c)^3$: (a) The surface energy density is $e_{\text{surf}} = 3\sigma u / r_N$ where $\sigma \approx 1$ MeV/fm$^2$ is the surface tension. Explain this formula. (b) The Coulomb energy density is $e_{\text{Coul}} = \frac{2\pi}{5} (n_p e)^2 r_N^2 f_3(u)$ where $f_3(u) = 1 - (5/3)u^{1/3} + u$ is the lattice correction factor. Minimizing $e = e_{\text{surf}} + e_{\text{Coul}}$ with respect to $r_N$, show that at the optimum, $e_{\text{surf}} = 2 e_{\text{Coul}}$ (the virial theorem for the crust). (c) The same analysis for cylindrical rods ($d = 2$) gives $e_{\text{surf}} = e_{\text{Coul}}$, and for slabs ($d = 1$) gives $e_{\text{surf}} = (2/3) e_{\text{Coul}}$. For each geometry, calculate the optimal total energy as a function of $u$ and determine which geometry is favored at $u = 0.1$, $u = 0.3$, and $u = 0.5$.

Problem 25.15 — Crust thickness. The transition from crust to core occurs at approximately $\rho_t \approx 0.5\rho_0 = 1.35 \times 10^{14}$ g/cm$^3$. For a $1.4\,M_\odot$ neutron star with $R = 12$ km: (a) The crust occupies the region from $R_{\text{core}} < r < R$. Using the approximate relation $\Delta R_{\text{crust}} / R \approx (0.04/\xi)(P_t / \varepsilon_0 c^2)$ where $P_t$ is the pressure at the crust-core transition, estimate the crust thickness for $P_t \approx 0.5$ MeV/fm$^3$. (b) Estimate the mass of the crust using $\Delta M_{\text{crust}} / M \approx (28\pi/3)(P_t R^4)/(Mc^2)$. Express in solar masses. (c) The crust contains the same nuclear physics as the outer layers of the nuclear chart. Why might studying exotic nuclei at FRIB help us understand the neutron star crust?

Section E: Observational Constraints

Problem 25.16 — Tidal deformability. The dimensionless tidal deformability is $\Lambda = (2/3) k_2 (Rc^2/GM)^5$, where $k_2$ is the tidal Love number. (a) For a $1.4\,M_\odot$ neutron star, calculate $(Rc^2/GM)^5$ for $R = 11$ km and $R = 13$ km. How sensitive is $\Lambda$ to the radius? (b) The GW170817 constraint is $\Lambda_{1.4} \lesssim 800$. Assuming $k_2 \approx 0.1$ (a typical value), what upper bound does this place on $R_{1.4}$? (c) The approximate scaling $\Lambda \propto R^{5\text{--}6}$ means a 10% change in radius produces a $\sim 60\%$ change in $\Lambda$. Explain why gravitational wave observations are so sensitive to the EOS.

Problem 25.17 — NICER and the mass-radius plane. The NICER measurements give $M = 1.34^{+0.15}_{-0.16}\,M_\odot$, $R = 12.71^{+1.14}_{-1.19}$ km for PSR J0030+0451 and $M = 2.08 \pm 0.07\,M_\odot$, $R = 12.39^{+1.30}_{-0.98}$ km for PSR J0740+6620. (a) Plot these measurements as error ellipses on the $M$-$R$ plane. (b) Sketch the $M$-$R$ curves for a "soft" EOS ($R_{1.4} \approx 11$ km, $M_{\text{max}} \approx 2.0\,M_\odot$) and a "stiff" EOS ($R_{1.4} \approx 14$ km, $M_{\text{max}} \approx 2.5\,M_\odot$). Which is more consistent with the NICER data? (c) The fact that the two stars have similar radii despite very different masses constrains the density dependence of the EOS. Explain why.

Problem 25.18 — Magnetar energetics. A magnetar has a surface magnetic field $B \sim 10^{15}$ T and radius $R = 10$ km. (a) Estimate the total magnetic energy stored in the field: $E_B \sim B^2 R^3 / (6\mu_0)$. Express in ergs and in units of $M_\odot c^2$. (b) The giant flare from SGR 1806-20 on December 27, 2004, released $\sim 2 \times 10^{46}$ erg in 0.2 s. What fraction of the total magnetic energy was released? (c) The rotational energy of a magnetar with $P = 5$ s and $I = 10^{45}$ g cm$^2$ is $E_{\text{rot}} = 2\pi^2 I / P^2$. Calculate $E_{\text{rot}}$ and compare to $E_B$. Which energy reservoir is larger? (d) This is why magnetars are "magnetically powered" rather than "rotationally powered" like ordinary pulsars.

Section F: Computational Problems

Problem 25.19 — TOV solver. Using the provided tov_solver.py as a starting point (or writing your own): (a) Solve the TOV equation for a polytropic EOS $P = K\varepsilon^2$ with $K = 100$ km$^2$ (in geometric units $G = c = 1$). Plot the mass-radius curve. (b) Find the maximum mass and the corresponding radius and central density. (c) Repeat for $K = 50$ km$^2$ (softer) and $K = 200$ km$^2$ (stiffer). (d) Plot all three $M(R)$ curves on the same axes. Overlay the observational constraints: $M_{\text{max}} \geq 2.0\,M_\odot$, $R_{1.4} \in [11.5, 13.5]$ km. Which values of $K$ are observationally viable?

Problem 25.20 — Proton fraction in beta equilibrium. Write a Python program that, for a given baryon density $n_B$, solves the beta equilibrium condition $\mu_n(n_B, x_p) = \mu_p(n_B, x_p) + \mu_e(n_B, x_p)$ self-consistently for the proton fraction $x_p$. Use a parabolic approximation for the nucleon energies: $\varepsilon(n_B, x_p) = \varepsilon_0(n_B) + S(n_B)(1 - 2x_p)^2 n_B$ where $S(n_B) = J(n_B/n_0)^\gamma$ with $J = 32$ MeV and $\gamma = 0.6$. (a) Plot $x_p$ vs. $n_B/n_0$ for $n_B/n_0 = 0.5$ to 5. (b) At what density does the muon threshold get reached? (c) At what density does the direct Urca threshold ($x_p = 0.11$) get reached? (d) How sensitive are your results to the parameter $\gamma$?

💻 Problem 25.21 — EOS comparison. Modify the tov_solver.py to implement a piecewise polytropic EOS with two segments: a soft crust ($\Gamma = 4/3$ for $\varepsilon < 2\varepsilon_0$) and a stiffer core ($\Gamma = 2.5$ for $\varepsilon > 2\varepsilon_0$). Ensure pressure continuity at the transition density. (a) Compute the $M(R)$ curve. (b) Compare to the single polytrope with $\Gamma = 2$. (c) How does the piecewise model compare to observational constraints? (d) Vary the transition density from $1.5\varepsilon_0$ to $3\varepsilon_0$. How does $M_\text{max}$ change?

💻 Problem 25.22 — Internal structure visualization. Using the tov_solver.py structure output: (a) For a $1.4\,M_\odot$ star (moderate EOS), plot the enclosed mass $m(r)$, pressure $P(r)$, and energy density $\varepsilon(r)$ as functions of radius. (b) Identify the approximate radius at which $\varepsilon$ drops below $\varepsilon_0$ (the edge of the core). (c) At what fraction of the stellar radius is 50% of the mass enclosed? 90%? What does this tell you about the density profile? (d) Compute the gravitational redshift $z(r) = (1 - 2Gm(r)/(rc^2))^{-1/2} - 1$ as a function of radius and plot it.

💻 Problem 25.23 — Maximum mass sensitivity. Using the stiffness scan feature of tov_solver.py: (a) Plot $M_\text{max}$ as a function of $K$ for $\Gamma = 2$. (b) Determine the minimum $K$ consistent with $M_\text{max} \geq 2.0\,M_\odot$. (c) Now vary $\Gamma$ from 1.5 to 3.0 at fixed $K = 100$ km$^2$. Plot $M_\text{max}(\Gamma)$. (d) Which parameter ($K$ or $\Gamma$) has a stronger effect on $M_\text{max}$?

Section G: Synthesis and Estimation Problems

Problem 25.24 — Comparing energy scales. A neutron star contains several forms of energy. For a $1.4\,M_\odot$, $R = 12$ km star: (a) Calculate the gravitational binding energy $E_{\text{grav}} \approx 0.6 GM^2 / R$. Express in ergs, joules, and MeV. (b) Calculate the rotational kinetic energy for a rotation period of $P = 33$ ms (the Crab pulsar) using $I \approx 10^{45}$ g cm$^2$. (c) Calculate the magnetic field energy for a typical pulsar ($B = 10^{8}$ T) and a magnetar ($B = 10^{15}$ T). (d) Rank the four energy scales and discuss which processes tap into which reservoirs.

Problem 25.25 — The neutron star zoo. Classify the following observed neutron star systems by their primary energy source (rotation, accretion, magnetic field) and by the key nuclear/particle physics they probe: (a) The Crab pulsar (PSR B0531+21): $P = 33$ ms, $\dot{P} = 4.2 \times 10^{-13}$, $B \approx 4 \times 10^{8}$ T. (b) SGR 1806-20: magnetar, $P = 7.6$ s, $B \approx 2 \times 10^{11}$ T, giant flare in 2004. (c) Scorpius X-1: accreting neutron star in a low-mass X-ray binary, $L_X \sim 2 \times 10^{38}$ erg/s. (d) PSR J0737-3039 (the double pulsar): $P_A = 23$ ms, $P_B = 2.8$ s, orbital period 2.4 hours. (e) PSR J0348+0432: $M = 2.01\,M_\odot$, $P = 39$ ms, in orbit with a white dwarf. For each, identify which EOS-sensitive observable (mass, radius, moment of inertia, cooling rate, tidal deformability, glitch behavior) could in principle be measured.

Section H: Challenge Problems

Problem 25.26 — The hyperon puzzle. The $\Lambda$ hyperon ($uds$, mass 1116 MeV) appears when $\mu_n \geq m_\Lambda c^2$. (a) For a free neutron Fermi gas, at what density does the neutron chemical potential reach 1116 MeV? Express in units of $n_0$. (b) The appearance of hyperons opens a new degree of freedom, reducing the neutron Fermi pressure. Explain qualitatively why this softens the EOS. (c) Many models that include hyperons predict $M_{\text{max}} < 2.0\,M_\odot$, contradicting observations. This is the "hyperon puzzle." Propose at least two possible resolutions. (d) If the hyperon puzzle is resolved by strong three-body forces involving hyperons, what does this tell us about the nuclear force at high density?

Problem 25.27 — Twin stars. If the EOS has a strong first-order phase transition at some critical density $\varepsilon_{\text{crit}}$ (e.g., from hadronic matter to quark matter), the mass-radius curve can develop a second stable branch. (a) Sketch a $M(R)$ curve with a phase transition, showing the hadronic branch, the unstable region, and the quark matter branch. (b) Two stars on different branches can have the same mass but different radii — "twin stars." If twin stars were observed (e.g., two neutron stars of mass $1.4\,M_\odot$ but radii differing by $\sim 2$ km), what would this prove about the EOS? (c) How could NICER distinguish between twin star and single-branch scenarios?

Problem 25.28 — Sound speed and causality. The speed of sound in dense matter is $c_s = \sqrt{dP/d\varepsilon}$. Causality requires $c_s \leq c$, i.e., $dP/d\varepsilon \leq c^2$. (a) For a free ultrarelativistic gas, $P = \varepsilon/3$, so $c_s^2 = c^2/3$. This is the "conformal limit." (b) Bedaque and Steiner (2015) showed that supporting $2\,M_\odot$ requires $c_s^2/c^2 > 1/3$ at some density above $\sim 2\rho_0$. Why? (c) If $c_s^2/c^2$ must significantly exceed $1/3$, this means the matter is far from conformal (far from weakly-interacting quark matter). What does this tell us about the interactions at $2$--$5\rho_0$?

Solutions to Selected Problems

Problem 25.1 Solution:

(a) Average density of a $1.4\,M_\odot$, $R = 12$ km star:

$$\bar{\rho} = \frac{3M}{4\pi R^3} = \frac{3 \times 1.4 \times 1.989 \times 10^{33}}{4\pi \times (1.2 \times 10^6)^3} \text{ g/cm}^3$$

$$= \frac{8.354 \times 10^{33}}{7.238 \times 10^{18}} = 1.15 \times 10^{15} \text{ g/cm}^3 \approx 4.3\,\rho_0$$

Wait — let me redo this more carefully. $M = 1.4 \times 1.989 \times 10^{30}$ kg $= 2.785 \times 10^{30}$ kg. $R = 12$ km $= 1.2 \times 10^4$ m. In CGS: $M = 2.785 \times 10^{33}$ g, $R = 1.2 \times 10^6$ cm.

$$\bar{\rho} = \frac{3 \times 2.785 \times 10^{33}}{4\pi (1.2 \times 10^6)^3} = \frac{8.354 \times 10^{33}}{7.238 \times 10^{18}} = 1.15 \times 10^{15} \text{ g/cm}^3$$

Correcting: $\bar\rho = 1.15 \times 10^{15}$ g/cm$^3$, which is $1.15 \times 10^{15} / 2.7 \times 10^{14} \approx 4.3\,\rho_0$.

Note: the textbook stated a "typical" average density of $\sim 1.5\rho_0$ for the average of the entire star including the low-density crust. The more precise calculation gives $\sim 4\rho_0$ because the bulk of the star is near or above nuclear density. The factor-of-three difference arises from the density profile: the outer layers contribute significant volume but little mass.

(b) For $2.0\,M_\odot$, $R = 11.5$ km: $\bar\rho = 1.87 \times 10^{15}$ g/cm$^3 \approx 6.9\,\rho_0$.

(c) Central densities: $1.4\,M_\odot$ star: $(4\text{--}8)\rho_0 \times (4.3/4.3) \sim 4$--$8\,\rho_0$ (using factor 3--5 times average). $2.0\,M_\odot$ star: $\sim 7$--$12\,\rho_0$.

(d) Ratio: $10^{15} / 150 \approx 7 \times 10^{12}$, or about 13 orders of magnitude.

Problem 25.2 Solution (partial):

(a) $\xi_{NS} = (6.674 \times 10^{-11} \times 2.785 \times 10^{30}) / (1.2 \times 10^4 \times 9 \times 10^{16}) = 1.858 \times 10^{20} / 1.08 \times 10^{21} = 0.172$.

(b) $\xi_{\odot} = (6.674 \times 10^{-11} \times 1.989 \times 10^{30}) / (6.96 \times 10^8 \times 9 \times 10^{16}) = 1.327 \times 10^{20} / 6.264 \times 10^{25} = 2.12 \times 10^{-6}$.

(c) $\xi_{WD} = (6.674 \times 10^{-11} \times 0.6 \times 1.989 \times 10^{30}) / (8 \times 10^6 \times 9 \times 10^{16}) = 7.96 \times 10^{19} / 7.2 \times 10^{23} = 1.1 \times 10^{-4}$.

(d) In terms of compactness: WD is 0.02% of the way to a BH; NS is 34% of the way to a BH. The neutron star is roughly 1,500 times more compact than the white dwarf.

Problem 25.9 Solution (partial):

(a) At $r = R/2 = 6$ km with $P \approx 18$ MeV/fm$^3$, $\varepsilon \approx 450$ MeV/fm$^3$:

(i) $1 + P/(\varepsilon c^2)$: In natural units ($c=1$), $P/\varepsilon = 18/450 = 0.04$. Factor: 1.04.

(ii) $m(R/2) \approx M/4 = 0.35\,M_\odot$. $4\pi r^3 P / (m c^2)$: We need consistent units. In geometric units, $4\pi (6 \text{ km})^3 \times 18 \text{ MeV/fm}^3 / (0.35\,M_\odot c^2)$. Converting: $18$ MeV/fm$^3 = 2.88 \times 10^{34}$ dyn/cm$^2 = 2.88 \times 10^{33}$ Pa. $4\pi (6 \times 10^3)^3 = 2.71 \times 10^{12}$ m$^3$. Numerator: $7.81 \times 10^{45}$ J. $mc^2 = 0.35 \times 1.989 \times 10^{30} \times 9 \times 10^{16} = 6.27 \times 10^{46}$ J. Ratio: 0.12. Factor: 1.12.

(iii) $2Gm/(rc^2) = 2 \times 6.674 \times 10^{-11} \times 6.965 \times 10^{29} / (6 \times 10^3 \times 9 \times 10^{16}) = 9.294 \times 10^{19} / 5.4 \times 10^{20} = 0.172$. Factor: $(1 - 0.172)^{-1} = 1.208$.

(b) Combined factor: $1.04 \times 1.12 \times 1.208 \approx 1.41$. GR strengthens gravity by about 41% compared to Newtonian gravity at this location.