Midterm 2 Solutions

Released solutions for Midterm 2

Author

Dr. Anna Rosen

These brief solutions are meant to help you check both your answers and your reasoning. Module 2 emphasizes how observable quantities (parallax, brightness, spectral lines, Doppler shifts) translate into inferred stellar properties (distances, luminosities, temperatures, masses, and eventually a star’s life history).

Answer Key

Q Ans Q Ans Q Ans Q Ans Q Ans
1 A 6 D 11 A 16 A 21 D
2 D 7 A 12 D 17 D 22 C
3 B 8 B 13 B 18 B 23 C
4 B 9 D 14 C 19 C 24 A
5 C 10 C 15 D 20 A 25 B

Brief Solutions

  1. A. Hot, ionized hydrogen near a young O-type star is an emission nebula (an H II region). The O star’s UV photons ionize the surrounding gas, which then re-emits at characteristic wavelengths.
  2. D. A 1 \(M_\odot\) star like the Sun is too low-mass to produce a core-collapse supernova or a neutron star/black hole. It ends as a planetary nebula leaving behind a white dwarf.
  3. B. Only nuclear fusion can sustain the Sun’s luminosity for billions of years. Hydrogen fusing to helium converts a tiny fraction of mass to energy via \(E = mc^2\). Chemical fuels would last \(\sim 10^4\) years, not \(10^9\).
  4. B. Inverse-square law: \(F \propto 1/d^2\). At three times the distance, brightness is \(1/3^2 = 1/9\).
  5. C. Once core hydrogen runs out, the inert helium core contracts and heats up, hydrogen-shell burning ignites just outside the core, and the outer envelope expands and cools — the star becomes a red giant.
  6. D. Stefan-Boltzmann at fixed radius: \(L \propto T^4\). Doubling \(T\) multiplies \(L\) by \(2^4 = 16\).
  7. A. Only stars more massive than \(\sim 8\,M_\odot\) build inert iron cores and undergo core collapse. A 25 \(M_\odot\) O-type star is the natural Type II progenitor.
  8. B. Kepler’s third law in solar units: \(M_1 + M_2 = a^3/P^2 = 2^3/2^2 = 2\,M_\odot\).
  9. D. Wien’s law: \(\lambda_\mathrm{peak} \propto 1/T\). Halving \(T\) (6000 K → 3000 K) doubles \(\lambda_\mathrm{peak}\).
  10. C. \(R_s = 2GM/c^2 \propto M\). A 10 \(M_\odot\) black hole has a Schwarzschild radius about ten times the Sun’s value, \(\sim 30\) km.
  11. A. Interstellar dust scatters blue light more strongly than red, and absorbs/dims the through-light. A background star seen through dust appears both dimmer and redder — interstellar reddening.
  12. D. Mass-luminosity relation: \(L \propto M^{3.5}\). For \(M = 4\,M_\odot\), \(L \approx 4^{3.5} = 4^3 \cdot 4^{0.5} = 64 \cdot 2 = 128\,L_\odot\).
  13. B. Direct chain: parallax → distance; apparent brightness measured at the telescope; combining with the inverse-square law gives the intrinsic luminosity. Mass and temperature alone cannot give luminosity directly.
  14. C. Stefan-Boltzmann: \(L = 4\pi R^2 \sigma T^4\). Even at low surface temperature, an enormous radius makes a red giant extremely luminous.
  15. D. Mass-lifetime: \(t_\mathrm{MS} \propto M^{-2.5}\). For \(M = 4\,M_\odot\), \(t_\mathrm{MS} \approx 10^{10}/4^{2.5} = 10^{10}/32 \approx 3 \times 10^8\) yr ≈ 0.3 Gyr.
  16. A. Bulk motion of hot plasma rising and cooler plasma sinking is convection, which dominates energy transport in the Sun’s outer convective zone.
  17. D. When a star leaves the main sequence, its envelope expands and cools while its luminosity rises — it moves into the upper-right (cool, luminous) region of the H-R diagram.
  18. B. Same apparent brightness, but Star B is 10× farther (parallax is 10× smaller). Inverse-square: 10× farther means 100× more luminous to look the same.
  19. C. The Chandrasekhar limit (\(\approx 1.4\,M_\odot\)) is the maximum mass an electron-degenerate white dwarf can support. Above it, electron degeneracy pressure cannot resist gravity.
  20. A. Newton’s law of gravity depends only on mass and distance. Replacing the Sun with a 1 \(M_\odot\) black hole at the same location leaves Earth’s orbit unchanged — the gravitational pull at 1 AU is identical.
  21. D. The main sequence is a snapshot of stars of different masses simultaneously fusing hydrogen in their cores — not an evolutionary track that a single star follows.
  22. C. Iron sits at the peak of the binding-energy-per-nucleon curve, so fusing iron absorbs energy rather than releasing it. The core can no longer support itself against gravity, and collapse begins.
  23. C. Elements heavier than iron are forged in the extreme temperatures and densities of supernova explosions (and neutron-star mergers), through rapid neutron capture (the r-process).
  24. A. O-type stars are very massive and short-lived (\(\sim\) Myr). If a cluster has G-type main-sequence stars but no surviving O stars, the cluster must be old enough that all its O stars have already evolved off the main sequence.
  25. B. Neutron-star mergers produce both a chirp of gravitational waves during inspiral and a brilliant electromagnetic counterpart from the colliding matter (kilonova) — first observed together as GW170817 in 2017.