Questions: Exoplanet Mass-Radius Relations and Interior Composition

Question 1

A newly discovered planet has 5 Earth masses but a radius of 2.5 Earth radii — much larger than expected for a rocky body of that mass. What does this imply about its composition?

• A. It must have a large iron core, which inflates the radius at high mass
• B. It contains substantial lower-density material, such as water ice or a hydrogen-helium envelope
• C. The measurement must be wrong — more massive planets are always denser due to gravitational compression
• D. It is a rocky super-Earth; the super-Earth range spans 2–5 Earth masses regardless of density

Answer: (B)

Planets that plot above the rocky sequence on the mass-radius diagram are less dense than pure rock and must therefore contain lighter material. More iron would make the planet denser, not less (option A is backwards). Option C confuses the compression effect — rocky planets do get denser with mass, but a planet above the rocky sequence is an outlier in the direction of low density, not high. Option D confuses the mass range label with compositional inference.

Question 2

The Fulton gap (radius gap) is a deficit of planets between about 1.5 and 2.0 Earth radii. What process best explains this gap?

• A. Planets cannot form at these sizes due to orbital resonance effects in protoplanetary disks
• B. The rocky planet sequence predicts no stable configurations at these radii
• C. Planets with thin hydrogen-helium envelopes lose them to stellar radiation, shrinking to bare rocky cores; those with thick enough envelopes remain puffy mini-Neptunes
• D. Water worlds at these sizes evaporate their oceans, collapsing to smaller radii over billions of years

Answer: (C)

The gap results from atmospheric mass loss: photoevaporation and core-powered mass loss strip thin hydrogen envelopes from planets close to their host stars. Planets that started with modest envelopes lose them and shrink below ~1.5 R⊕; those with thick envelopes retain them and stay above ~2.0 R⊕. The gap is not a formation artifact (option A) or a theoretical prediction from the rocky sequence (option B). Water loss (option D) is a separate process and does not produce the sharp radius gap observed.

Question 3

Two exoplanets have identical masses and identical radii, giving them the same bulk density. They should therefore have the same interior composition.

• T. True
• F. False

Answer: False

This is the degeneracy problem in mass-radius interpretation. Different mixtures of materials — for example, a water-rich planet versus a rocky planet with a thin gas envelope — can produce the same bulk density. Mass and radius constrain average density, but density alone cannot distinguish between multiple compositional models. Breaking this degeneracy requires additional data, typically atmospheric spectroscopy to determine whether a hydrogen-rich envelope or a water-dominated atmosphere is present.

Question 4

Determining an exoplanet's bulk density requires both its radius (from transit observations) and its mass (from radial velocity or transit timing variations).

• T. True
• F. False

Answer: True

Transit observations measure the planet's radius from the fractional dimming of starlight. Radial velocity measurements measure the planet's mass from the gravitational wobble it induces in the star. Density = mass / volume, so both are required. Neither alone is sufficient: a transit gives radius but not mass, and a radial velocity signal gives mass but not radius. When both measurements are available for the same planet, bulk density can be calculated, enabling compositional inferences.

Question 5

Explain the degeneracy problem in exoplanet interior modeling. Why can't bulk density alone uniquely determine a planet's composition?

Model answer: Bulk density is a single number — the average of all the materials inside the planet — and multiple combinations of materials can yield the same average. For instance, a planet made mostly of water ice and a planet made of rock plus a thin hydrogen atmosphere can have the same total mass, the same radius, and therefore the same density. Without knowing which layers are present and in what proportions, the interior is underdetermined. Breaking the degeneracy requires additional constraints, such as atmospheric composition from spectroscopy, which can reveal whether a hydrogen envelope or a water-vapor-dominated atmosphere is present.

The degeneracy problem is what makes the mass-radius diagram necessary but not sufficient for compositional inference. It also motivates combining transit spectroscopy (atmospheric chemistry) with mass-radius measurements. The degeneracy is worst in the super-Earth to mini-Neptune range (~1.5–4 R⊕), where rocky, water-rich, and gas-enveloped planets can overlap substantially.