Chapter 25 Quiz — Nuclear Physics of Neutron Stars
Instructions: Select the best answer for each question. Answers are provided at the end.
Q1. The average density of a typical neutron star ($1.4\,M_\odot$, $R \approx 12$ km) is approximately:
(a) $10^{10}$ g/cm$^3$ — comparable to a white dwarf (b) $10^{12}$ g/cm$^3$ — about 100 times white dwarf density (c) $10^{14}$--$10^{15}$ g/cm$^3$ — comparable to nuclear saturation density (d) $10^{18}$ g/cm$^3$ — comparable to a black hole singularity
Q2. The equation of state (EOS) of neutron star matter is the relationship between:
(a) Temperature and luminosity (b) Pressure and energy density (or baryon density) (c) Mass and radius (d) Magnetic field and rotation period
Q3. The Tolman-Oppenheimer-Volkoff (TOV) equation differs from the Newtonian hydrostatic equilibrium equation because:
(a) It includes quantum corrections from the Pauli exclusion principle (b) It accounts for the nuclear force between neutrons (c) It includes general-relativistic corrections where pressure contributes to gravity and spacetime curvature modifies the force law (d) It includes the effects of rotation and magnetic fields
Q4. All three general-relativistic correction terms in the TOV equation act in the same direction. Specifically, compared to Newtonian gravity, they make gravity:
(a) Weaker, allowing higher maximum masses (b) Stronger, reducing the maximum mass for a given EOS (c) Unchanged — the corrections cancel out (d) Sometimes stronger, sometimes weaker, depending on the EOS
Q5. The observation that neutron stars exist with masses of $\sim 2\,M_\odot$ (e.g., PSR J0740+6620) implies that:
(a) The EOS must be sufficiently stiff (high pressure at high density) to support this mass (b) General relativity breaks down at neutron star densities (c) Neutron stars are not made of nuclear matter (d) The Chandrasekhar mass limit must be wrong
Q6. Neutron star matter in the outer core is approximately 95% neutrons and 5% protons. The reason for this extreme neutron richness is:
(a) The strong force preferentially attracts neutrons over protons (b) Beta equilibrium: the balance between neutron decay and electron capture, combined with the high electron Fermi energy (c) All the protons have been converted to neutrons during the supernova (d) The Coulomb force repels all protons to the surface
Q7. The "neutron drip" transition in the neutron star crust occurs at $\rho \approx 4 \times 10^{11}$ g/cm$^3$. Below this density (in the outer crust), the matter consists of:
(a) Free neutrons only (b) A uniform mixture of neutrons, protons, and electrons (c) Neutron-rich nuclei arranged in a crystalline lattice, immersed in a degenerate electron gas (d) Quark matter
Q8. The "nuclear pasta" phases at the base of the neutron star crust (spaghetti, lasagna, etc.) arise from:
(a) The competition between Coulomb energy and nuclear surface energy at high density (b) The effects of strong magnetic fields on nuclear matter (c) Quantum tunneling between different nuclear configurations (d) The Pauli exclusion principle applied to protons
Q9. GW170817 — the first detected neutron star merger — constrained the neutron star EOS through a measurement of:
(a) The maximum mass of neutron stars (b) The temperature of the kilonova ejecta (c) The tidal deformability — how much the stars are deformed by each other's gravitational field during the inspiral (d) The magnetic field strength of the merger remnant
Q10. NASA's NICER instrument on the International Space Station constrains neutron star properties by:
(a) Directly imaging the neutron star surface with X-ray optics (b) Measuring gravitational waves from neutron star oscillations (c) Modeling X-ray pulse profiles from hot spots, where gravitational light bending depends on the mass-to-radius ratio (d) Detecting neutrinos from the neutron star interior
Q11. The "hyperon puzzle" in neutron star physics refers to:
(a) The fact that hyperons have never been observed in laboratory experiments (b) The prediction that hyperons appearing at high density soften the EOS, potentially reducing the maximum mass below the observed $\sim 2\,M_\odot$ (c) The inability of QCD to predict hyperon masses (d) The observation that neutron stars contain no strange quarks
Q12. Magnetars are neutron stars with magnetic fields of $\sim 10^{14}$--$10^{15}$ T. Their primary energy source is:
(a) Nuclear fusion in the core (b) Gravitational contraction (c) Rotational energy (like ordinary pulsars) (d) The decay of the magnetic field (magnetic energy)
Q13. Pulsar glitches — sudden spin-up events — are believed to be caused by:
(a) Accretion of matter from a companion star (b) The sudden transfer of angular momentum from the superfluid neutron component to the rigid crust, when pinned vortices unpin (c) Magnetic field reconnection events (d) Thermonuclear explosions on the surface
Q14. The symmetry energy slope parameter $L$ is important for neutron star physics because:
(a) It determines the maximum mass (b) It controls the pressure of neutron-rich matter near saturation density, which correlates with the radius of a $1.4\,M_\odot$ neutron star (c) It sets the magnetic field strength (d) It determines the rotation period
Q15. Which of the following statements about neutron star cores is correct?
(a) The composition of the inner core (above $\sim 2\rho_0$) is well established as pure neutron matter (b) Lattice QCD can straightforwardly calculate the EOS at the relevant densities (c) The inner core may contain hyperons, meson condensates, or quark matter, but the actual composition is unknown — this is a frontier problem (d) X-ray observations have directly confirmed the presence of quark matter in neutron star cores
Answer Key
| Question | Answer | Brief Explanation |
|---|---|---|
| Q1 | (c) | $\bar\rho \approx 4 \times 10^{14}$ g/cm$^3 \approx 1.5\rho_0$ for a canonical $1.4\,M_\odot$ star |
| Q2 | (b) | $P(\varepsilon)$ determines all macroscopic properties via the TOV equation |
| Q3 | (c) | GR corrections: pressure gravitates, volume pressure contributes to mass, metric curvature factor |
| Q4 | (b) | All three corrections strengthen gravity, reducing $M_\text{max}$ compared to Newtonian gravity |
| Q5 | (a) | Stiff EOS = high pressure at high density = supports more mass against GR-enhanced gravity |
| Q6 | (b) | Beta equilibrium $\mu_n = \mu_p + \mu_e$ with high $\mu_e$ drives the proton fraction to $\sim 5\%$ |
| Q7 | (c) | The outer crust is a BCC lattice of neutron-rich nuclei in a relativistic electron gas |
| Q8 | (a) | Coulomb favors spread-out shapes; surface energy favors compact shapes. The competition produces non-spherical geometries at intermediate volume fractions |
| Q9 | (c) | Tidal deformability $\Lambda$ was extracted from the late inspiral gravitational wave signal |
| Q10 | (c) | Gravitational light bending of X-ray emission from hot spots encodes $M/R$ |
| Q11 | (b) | Hyperons soften the EOS, potentially making $M_\text{max} < 2\,M_\odot$ in conflict with observations |
| Q12 | (d) | Magnetic energy $E_B \sim 10^{46}$--$10^{47}$ erg exceeds rotational energy for slowly rotating magnetars |
| Q13 | (b) | Superfluid angular momentum reservoir transfers to crust when pinned vortices unpin collectively |
| Q14 | (b) | $L$ controls $P_\text{NM}(n_0) \approx n_0 L/3$, which correlates linearly with $R_{1.4}$ |
| Q15 | (c) | The inner core composition above $\sim 2\rho_0$ is genuinely unknown — one of the great open questions in nuclear physics |