Reproduce and extend this unit's founding detection-method calculations.
51 Pegasi b has e≈0. Using the mass-function equation, show algebraically that in the limit e→0 and Mp<<M*, K ∝ Mp sin(i) P^(-1/3) M*^(-2/3). Using this scaling relation (not the full numerical solve), estimate by what factor K would change if 51 Pegasi b's period were instead 8.46 days (double the real value) with all other parameters fixed.
A newly discovered planet orbiting a 1.11 Msun star (51 Pegasi's real mass) shows K=25.0 m/s and P=10.0 days with e≈0. Compute this planet's minimum mass in Jupiter masses.
Using only HD 209458 b's real published mass (0.682 MJup) and mean density (compute it from mass and radius, 1.359 RJup) together with the host star's radius (1.20 Rsun), compute the transit depth two independent ways: (a) directly from Rp and Rstar, and (b) by first computing Rp from the mass and density and then applying the same formula. Confirm both methods agree.
Using T ∝ P^(1/3) (from a ∝ P^(2/3) via Kepler's third law, holding Rstar fixed), estimate the transit duration a planet identical to HD 209458 b would have if its period were instead 1.0 day (roughly a quarter of the real value), and state whether this makes such a planet's transit easier or harder to schedule follow-up observations for.
Using HD 209458 b's real transit-derived radius and radial-velocity-derived true mass (both given in this problem set's reading), compute its bulk density in kg/m^3 and in g/cm^3, and state which solar-system body (if any) it is closest to in density among Mercury, Venus, Earth, Mars, Jupiter (see materials/ASTR350/data/).
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 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.
materials/ASTR350/data/ where referenced above.