Instructor Notes

Require SI units for impact energy and explicit ppm conversion for transit depth. Exoplanet interpretation should emphasize detection bias, not just formation theory.

Problem 1. Impact Energy and Hazard Scaling

Worked solution: Radius is 40 m. Volume \(V=4\pi r^3/3=4\pi(40)^3/3=2.68\times10^5\,\mathrm{m^3}\). Mass \(m=\rho V=3000\times2.68\times10^5=8.04\times10^8\) kg. Speed \(v=2.0\times10^4\,\mathrm{m\ s^{-1}}\). Energy \(E=0.5(8.04\times10^8)(2.0\times10^4)^2=1.61\times10^{17}\) J. TNT equivalent \(=1.61\times10^{17}/4.184\times10^{15}=38.5\) Mt. If speed doubles, energy increases by \(2^2=4\).

Criterion	Points	Notes
Mass calculation	7	Sphere volume and density.
Kinetic energy	7	Speed in m/s.
TNT conversion	5	About 39 Mt.
Speed scaling	6	Factor of 4 with explanation.

Common errors: using diameter as radius, leaving speed in km/s inside SI energy, or saying doubled speed doubles energy.

Problem 2. Comet Orbit and Activity Window

Worked solution: Semimajor axis \(a=(q+Q)/2=(1.2+35)/2=18.1\) AU. Period \(P=a^{3/2}=18.1^{1.5}=77\) yr. Activity increases near perihelion because sunlight heats surface ices, causing sublimation, coma formation, jets, and dust release. Compared with an asteroid, a comet nucleus is typically more ice-rich, lower density/porosity, and activity-prone. Ambiguity: dormant comets can resemble dark asteroids, and some asteroids show activity after impacts or volatile release.

Criterion	Points	Notes
Semimajor axis	5	Average perihelion and aphelion.
Period	7	About 77 yr.
Activity explanation	6	Sublimation near Sun.
Comet/asteroid comparison	7	Include ambiguity.

Common errors: using aphelion as semimajor axis or assuming all asteroids are dry and all comets active.

Problem 3. Transit Depths and Planet Radii

Worked solution: Jupiter-size transit: \(\delta=(0.100)^2=0.010\), or 1.0% = 10,000 ppm. Earth-size transit: \(\delta=(0.00916)^2=8.39\times10^{-5}\), or 83.9 ppm. The Jupiter-size planet is easier to detect because the signal is far larger than 200 ppm single-transit precision. Transit geometry favors close-in planets with higher transit probability, and instrumental bias favors large planets, quiet stars, short periods, and repeated transits with high signal-to-noise.

Criterion	Points	Notes
Jupiter depth	5	1%, 10000 ppm.
Earth depth	6	About 84 ppm.
Detection comparison	5	Relate to 200 ppm.
Biases	9	One geometric, one instrumental.

Common errors: using radius ratio without squaring or confusing ppm with percent.

Problem 4. Solar-System Expectations versus Exoplanet Architecture

Worked solution: Convert days to years: 3.0 d = 0.00821 yr, 8.5 d = 0.0233 yr, 31 d = 0.0849 yr. Then \(a=P^{2/3}\): b = 0.0407 AU, c = 0.0816 AU, d = 0.193 AU. All three are much closer than Mercury and larger than Earth, unlike the solar system's small inner rocky planets and distant giants. This architecture is easy for transit surveys because close-in planets transit more often and with higher probability. Close-in sub-Neptunes may form farther out and migrate inward, form in compact disks, or accrete/lose atmospheres after formation.

Criterion	Points	Notes
Period conversions	5	Days to years.
Semimajor axes	8	Use \(P^{2/3}\).
Architecture comparison	5	Solar-system context.
Detection bias and hypothesis	7	Both required.

Common errors: using days directly as years, or interpreting the discovered sample as unbiased.

OpenStax Companion Practice Notes

Chapter 13 entries support small bodies, comets, asteroids, and impact risk; Chapter 14 entries support solar-system formation and comparison with other planetary systems. Verify numbering against the adopted PDF before release.