Questions: Exoplanet Detection and Orbital Parameters

Question 1

A planet detected by the radial velocity method produces a signal with M sin i = 2.0 Jupiter masses. What can be concluded about the planet's true mass?

• A. The true mass is exactly 2.0 Jupiter masses, since M sin i gives the actual mass
• B. The true mass is at least 2.0 Jupiter masses — it could be higher depending on orbital inclination
• C. The true mass is at most 2.0 Jupiter masses — sin i ≤ 1 means M sin i overestimates mass
• D. The true mass cannot be determined at all without direct imaging

Answer: (B)

Radial velocity measures only the component of stellar wobble along our line of sight. The true wobble amplitude is larger by a factor of 1/sin i. Since sin i ≤ 1, dividing by sin i gives a value ≥ M sin i. The minimum mass occurs when i = 90° (edge-on orbit, sin i = 1) — if the orbit is more face-on, the true mass is larger. We can only say the planet is at least 2.0 MJ; it could be a brown dwarf or even a star if the orbital plane happens to be nearly face-on.

Question 2

A star shows both periodic radial velocity variations AND transits by the same planet. What does combining these two detections reveal that neither method alone can provide?

• A. The planet's atmospheric composition — transit spectroscopy combined with RV mass constrains chemistry
• B. The planet's true mass and mean density — transit resolves the sin i ambiguity, and density distinguishes rocky from gaseous worlds
• C. The planet's surface temperature — the mass-radius combination constrains internal heat sources
• D. The orbital eccentricity — transits reveal the shape of the orbit that RV alone cannot

Answer: (B)

The key contribution of combining methods is resolving the sin i ambiguity. A transit occurs only if the orbit is nearly edge-on (i ≈ 90°), so sin i ≈ 1, and M sin i ≈ true mass. The transit also gives the planet's radius from the depth of the brightness dip. With both true mass and radius, you can calculate mean density (mass/volume) — the single most diagnostic quantity for distinguishing rocky terrestrial planets from gas giants with similar masses.

Question 3

The transit method measures a planet's mass directly from the depth of the brightness dip when the planet crosses its star.

• T. True
• F. False

Answer: False

The transit depth measures the fraction of starlight blocked by the planet — proportional to the ratio of their cross-sectional areas, not to mass. This gives the planet's radius (relative to the stellar radius), not its mass. Mass information comes from the radial velocity method (stellar wobble amplitude) or transit timing variations (gravitational perturbations). It's a common misconception that transit depth encodes mass; it encodes size.

Question 4

The radial velocity method is more sensitive to massive planets in short-period orbits than to Earth-mass planets in wide orbits.

• T. True
• F. False

Answer: True

Radial velocity amplitude scales with planet mass — a more massive planet pulls the star harder, producing a larger Doppler shift. Short orbital periods mean more orbits observed per year, and the star's orbital velocity around the barycenter is higher for close-in planets. This creates a strong selection bias: RV surveys detect 'hot Jupiters' (massive, close-in) far more easily than Earth-analogs (low mass, wide orbit). Understanding these biases is essential for interpreting exoplanet statistics — the observed population is not a random sample of what exists.

Question 5

Why does the radial velocity method yield only a minimum mass for a planet, and how does combining it with the transit method resolve this ambiguity?

Model answer: Radial velocity measures the component of the star's orbital velocity along our line of sight. If the planet's orbital plane is tilted relative to our line of sight (inclination i < 90°), we only see a fraction sin i of the full stellar wobble. The measured quantity is M sin i, not M. The true mass M = (M sin i) / sin i is larger whenever i < 90°. A transit can only occur when the orbital plane is nearly edge-on (i ≈ 90°), so if the same planet both transits and produces an RV signal, sin i ≈ 1 and the minimum mass is very close to the true mass — resolving the ambiguity.

This is the power of multi-method astronomy: each method's limitation is another method's strength. Radial velocity is degenerate in inclination; transits require a specific inclination. When both are observed, the degeneracy breaks and the physical parameters become fully determined. Adding the transit radius to the RV mass then enables density estimates, which would otherwise be impossible.