ASTR 120 Problem Set 06

Lectures 11-12: Small bodies and exoplanet systems

Due and Scope

Due: [insert date]. Submit calculations, plotted or tabulated data where requested, and short interpretations.

Learning objectives: use kinetic-energy scaling for impact hazards; distinguish asteroids, comets, and meteorites by orbit and composition; infer basic exoplanet properties from transit data; identify detection biases.

Instructions: Show equations and units. Label synthetic data clearly. Revisions may be resubmitted with a change note.

Problems

1. Impact Energy and Hazard Scaling 25 points

A stony asteroid has diameter 80 m, density \(3000\,\mathrm{kg\ m^{-3}}\), and impact speed \(20\,\mathrm{km\ s^{-1}}\). Treat it as a sphere and use \(E=\frac{1}{2}mv^2\).

  1. Compute its mass.
  2. Compute its kinetic energy in joules.
  3. Convert to megatons of TNT using \(1\,\mathrm{Mt}=4.184\times10^{15}\) J.
  4. If speed doubles while diameter and density stay fixed, by what factor does energy change?

Deliverables: mass, energy, TNT equivalent, and scaling explanation.

2. Comet Orbit and Activity Window 25 points

A comet has perihelion 1.2 AU and aphelion 35 AU.

  1. Compute the semimajor axis.
  2. Use \(P^2=a^3\) to estimate the orbital period in years.
  3. Explain why activity increases near perihelion.
  4. State two ways a comet nucleus differs from an asteroid of similar size, and one observational ambiguity.

Deliverables: orbit calculation and composition/activity interpretation.

3. Transit Depths and Planet Radii 25 points

Use \(\delta=(R_p/R_*)^2\). Assume a Sun-like star with \(R_*=1.00R_\odot\), \(R_J=0.100R_\odot\), and \(R_E=0.00916R_\odot\).

  1. Compute the transit depth for a Jupiter-size planet.
  2. Compute the transit depth for an Earth-size planet in parts per million.
  3. A transit survey has single-transit precision of 200 ppm. Which planet is easier to detect and why?
  4. Explain one geometric and one instrumental bias of the transit method.

Deliverables: depth calculations and detection-bias paragraph.

4. Solar-System Expectations versus Exoplanet Architecture 25 points

A discovered system has three planets: Planet b, period 3.0 d and radius \(1.8R_E\); Planet c, period 8.5 d and radius \(2.7R_E\); Planet d, period 31 d and radius \(3.5R_E\). The star is Sun-like.

  1. Use \(a=P^{2/3}\) with \(P\) in years to estimate each semimajor axis in AU.
  2. Compare the architecture with the solar system's inner rocky planets and outer giants.
  3. Explain why this system might have been easier to discover than a true solar-system analog.
  4. State one formation or migration hypothesis that could account for close-in sub-Neptunes.

Deliverables: semimajor-axis table and comparative interpretation.

OpenStax Companion Practice

Recommended for additional practice: OpenStax Astronomy 2e, Chapter 13, Review Questions 1-14; Thought Questions 16-18; Figuring for Yourself 22-29; Chapter 14, Review Questions 5-9; Thought Questions 17 and 19; Figuring for Yourself 21-23. Verify numbering against the adopted PDF before LMS release.

References and Data Sources