Exoplanet Transit: Radius, Density, and a Stellar Puzzle
A transit survey observes a star of radius 1.1 R_sun and mass 1.0 M_sun with a periodic dip of depth 1.2% and duration 3.5 hours. Radial velocity follow-up gives a semi-amplitude of 12 m/s and a period of 4.2 days. The team computes a planet radius and mass, then finds the implied density is about 0.4 g/cm³, which is low for a rocky planet. They must decide whether the object is a low-density planet, a blend with a background eclipsing binary, or a star with an unresolved companion.
1. Estimate the planet radius from the transit depth and the planet mass from the radial velocity amplitude, then compute the density.
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Transit depth ≈ (R_p/R_star)² = 0.012, so R_p/R_star ≈ 0.1095. With R_star = 1.1 R_sun ≈ 1.1 × 6.96 × 10⁸ m ≈ 7.66 × 10⁸ m, R_p ≈ 0.1095 × 7.66 × 10⁸ ≈ 8.4 × 10⁷ m, about 1.2 R_Jupiter. For mass, the RV semi-amplitude K ≈ 12 m/s with P = 4.2 days. Using K = (2πG/P)^(1/3) × M_p sin i / (M_star + M_p)^(2/3) and neglecting M_p, M_p sin i ≈ K (M_star)^(2/3) (P/2πG)^(1/3). Numerically this gives M_p sin i of order 0.5–0.7 M_Jupiter. With R_p ≈ 1.2 R_Jupiter, density is roughly 0.4–0.5 g/cm³, consistent with a gas giant, not a rocky planet. So the low density is expected for a bloated gas giant.
2. What observational tests would distinguish a genuine planet from a blend or stellar companion?
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Several tests: (1) Check the transit shape - a blend or stellar companion often produces a V-shaped or diluted transit rather than a flat-bottomed one; a flat bottom supports a planet. (2) Look for a secondary eclipse and phase-curve variations; a genuine hot Jupiter shows a small secondary dip and phase modulation. (3) Use high-resolution imaging or speckle imaging to search for a nearby star within the photometric aperture. (4) Check the RV signal for consistency: a stellar companion would give a much larger K for the same period, so 12 m/s at 4.2 days favors a planet. (5) Examine the spectral lines for a second set of lines or line-profile variations. A blend would also show a color-dependent transit depth; a planet's depth is achromatic.
3. The team worries the density is too low. Quantify what density a rocky composition would require and whether the data rule it out.
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A rocky planet of this radius (about 1.2 R_Jupiter ≈ 15 Earth radii) would have a mass of order hundreds of Earth masses and a density of several g/cm³, roughly 5–8 g/cm³ depending on composition. The measured density is about 0.4 g/cm³, an order of magnitude lower. To match a rocky density at that radius, the mass would need to be roughly 10–20 times larger than the RV-derived mass, which is ruled out by the 12 m/s RV amplitude unless the inclination is extremely small (sin i near zero), which is statistically unlikely. So the data strongly favor a low-density gas giant, not a rocky planet, and the 'too low' concern is misplaced.