Apparatus and Setup
This lab computes HD 209458 b's atmospheric scale height and predicted transmission-spectroscopy signal amplitude (Lecture 07), then its Jeans escape parameter and naive mass-loss timescale (Lecture 08), comparing both to the real, published Hubble Space Telescope measurements.
Materials and Data
| Quantity | Value |
|---|---|
| Mass | 0.682 MJup |
| Radius | 1.359 RJup |
| Surface gravity (computed) | 9.57 m/s² |
| Exosphere temperature (real, HST-measured) | 10000 K |
| Real exosphere extent | 3.1 planetary radii |
| Real mass-loss rate | 3.0e+08 kg/s |
Procedure
- Compute HD 209458 b's atmospheric scale height H at its exosphere temperature, for atomic hydrogen (mu=1 amu).
- Estimate the transit-depth signal amplitude for N=5 scale heights of absorbing atmosphere above the bulk continuum radius, using Lecture 07's formula, and compare to the real sodium-detection signal's approximate size.
- Compute HD 209458 b's Jeans escape parameter at the exosphere temperature and state whether the planet is in the classical Jeans-escape regime or the hydrodynamic blow-off regime.
- Compute the naive mass-loss timescale (current mass divided by the real, published mass-loss rate) and state whether the planet is in danger of complete evaporation within a typical few-Gyr system age at its current rate.
Analysis and Uncertainty
State explicitly that the naive mass-loss timescale assumes a constant mass-loss rate, which is very likely an overestimate of the timescale (an underestimate of total historical loss), because the host star's X-ray/UV activity -- and therefore the escape rate -- was almost certainly much higher earlier in the system's history (Lecture 08).
Deliverables
- Scale height, transit-signal estimate, Jeans parameter, and mass-loss timescale, all with full derivations.
- A one-paragraph statement of which escape regime HD 209458 b is in and why this matters for interpreting its real, observed extended exosphere.
Assessment Criteria
- Correct scale-height calculation with correct units (K_B, T, mu in kg, g).
- Correct Jeans-parameter calculation and correctly identified escape regime.
- Explicit statement of the constant-rate assumption's limitation.
References and Provenance
- 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.
- HD 209458 b atmospheric/exosphere data: live-verified this session (Wikipedia, "HD 209458 b," 2026-09-24, citing Vidal-Madjar et al. 2003, Charbonneau et al. 2002).