Exoplanet Demographics: Occurrence Rates and the Mass-Radius Relation

Learning Goals

1. Define planet occurrence rate and explain how survey completeness corrections convert a raw detection count into a true occurrence rate.
2. Describe the observed exoplanet mass-radius relation and identify the physical transition between rocky and gas-dominated planets.
3. Interpret the observed "radius valley" in the Kepler planet-radius distribution as evidence for atmospheric mass loss.

Reading anchor: 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.

Why This Matters

With detection methods (Lectures 01-03) established, this lecture asks the population-level question every subsequent topic in this course depends on: how common are different kinds of planets, and what does the shape of the mass-radius relation reveal about planetary composition before any detailed interior model (Lecture 06) is built?

Opening Phenomenon

The Kepler mission’s multi-year transit survey of over 150,000 stars found that small planets (Earth- to Neptune-sized) are dramatically more common than Jupiter-sized planets, and revealed an unexpected near-absence of planets between about 1.5 and 2.0 Earth radii -- the "radius valley" -- a demographic gap that does not appear in any single-planet detection but only emerges from population-level statistics across thousands of systems.

First question: what here is directly observed or a published/live-verified measurement, and what follows only once a physical law is derived and applied?

Evidence We Need to Explain

• Kepler-era occurrence-rate studies (Batalha et al. 2013 and many follow-ups) correct raw detection counts for each star’s specific transit probability (Lecture 02, depends on a/Rstar) and detection sensitivity (depends on the star’s brightness and noise properties) before reporting a corrected occurrence rate, because a raw, uncorrected count would badly underrepresent small, long-period planets that are intrinsically harder to detect.
• The observed mass-radius relation for confirmed exoplanets shows a clear break: below roughly 1.5-2 Earth radii, radius increases only slowly with mass (consistent with a rocky, roughly constant-density composition), while above this break, radius increases much more steeply with only a small mass increase (consistent with a growing hydrogen/helium or ice-rich envelope) -- a transition visible directly in the data before any interior model is invoked.
• The radius valley, first robustly measured in the California-Kepler Survey (Fulton et al. 2017), separates two distinct sub-populations (smaller, likely bare-rock super-Earths and larger, likely volatile-rich sub-Neptunes) and is now widely interpreted as evidence that many sub-Neptune-sized planets lose their primordial hydrogen/helium envelopes via photoevaporation or core-powered mass loss, shrinking them across the valley over their first ~1 Gyr.

Derivation and Model

• A survey’s raw detection count for a given planet size and period bin must be divided by that bin’s combined transit probability and detection completeness (both computable from the survey’s known noise properties and target star sample) to recover a true, bias-corrected occurrence rate -- exactly the correction Lecture 03 flagged as necessary before making any demographic claim.
• This course’s own real solar-system and exoplanet dataset (Lecture 04’s worked mass-radius comparison) already shows the rocky/gas-giant dichotomy directly: Earth, Venus, and Mars cluster on a steep, roughly constant-density mass-radius track, while Jupiter, HD 209458 b, and 51 Pegasi b cluster on a nearly flat track where a 300-fold mass range corresponds to less than a factor of 2 in radius, because gas giants are supported by electron-degeneracy pressure at high mass rather than ordinary compressibility.
• The radius valley’s location and depth are predicted, with reasonable quantitative success, by photoevaporation models in which intense high-energy (X-ray/UV) irradiation from a young, active host star strips a close-in planet’s hydrogen/helium envelope over roughly its first hundred million to one billion years, converting some sub-Neptunes into bare, rocky super-Earths and leaving a relative dearth of planets at the transitional radius.

Quantitative Tool

Worked Example

1. This course’s own mass-radius sample (Lecture 04’s figure): Earth (M=1.0 Mearth, R=1.0 Rearth, rho=5514 kg/m^3), TRAPPIST-1e (M=0.772 Mearth, R=0.920 Rearth, rho=5466 kg/m^3) both sit near Earth’s density, while HD 209458 b (M=217 Mearth, R=14.9 Rearth, rho=360 kg/m^3) has a density lower than water, a direct numeric illustration of the rocky/gas-giant density dichotomy.
2. Density ratio: Earth is 15× denser than HD 209458 b despite HD 209458 b having 217× Earth’s mass -- only possible because a gas giant’s radius is set by pressure-supported hydrogen/helium, not by the same rocky-material compressibility that sets Earth’s radius.
3. TRAPPIST-1 system radius spread: the seven planets range from 0.755 to 1.129 Earth radii, entirely below the radius-valley transition, consistent with all seven being bare or thin-atmosphere rocky worlds rather than sub-Neptunes -- a real system-level illustration of the small-planet side of the demographic picture this lecture describes statistically.

Common Misconception

Treating an early-discovered, RV/transit-biased sample (dominated by hot Jupiters, since 51 Pegasi b in 1995) as evidence that hot Jupiters are the most common type of planet. Occurrence-rate studies correcting for detection bias instead find that small, close-in planets (super-Earths and sub-Neptunes) are far more numerous than hot Jupiters around Sun-like stars; the earliest discoveries were the easiest to detect, not the most common to exist.

Active Learning Segment

Given that a rocky planet’s density is roughly constant with mass (radius grows slowly with mass) while a gas giant’s radius is nearly independent of mass at high mass, sketch qualitatively why the mass-radius relation must have an inflection point somewhere between super-Earth and Saturn-mass planets, and connect this inflection to the radius valley’s location.

Lab Connection

Lab 02 builds a log-log mass-radius diagram from this course’s own real planet sample (solar system, TRAPPIST-1, 51 Peg b, HD 209458 b) and classifies each planet by its inferred bulk composition regime.

Synthesis

Correcting raw detection counts for each method’s selection effects reveals that small, close-in rocky and volatile-rich planets vastly outnumber hot Jupiters, and the mass-radius relation’s rocky/gas-giant dichotomy -- together with the radius valley’s photoevaporation-driven gap -- sets the stage for Lecture 06’s detailed interior-structure inference and this course’s later habitability discussion.

References

• 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.
• Course dataset and derivations used in this lecture’s worked example: materials/ASTR350/data/ and materials/ASTR350/src/generate_astr350_content.py.