Transit Depth and the Density Trap
During an outreach event, students used a 6-inch telescope to record a transit of a planet across a star. They measured a transit depth of 1.1% and a period of 3.5 days. The host star has radius 0.90 R_Sun and mass 0.85 M_Sun. A teammate announces, 'The depth is about 1%, so this is an Earth-sized planet.' Another notes that the star's light curve also shows a slight dimming outside transit and that the telescope's field contains a fainter companion 4 arcseconds away. The students have no radial-velocity data. They want to submit a short note claiming discovery of a habitable-zone terrestrial planet. Their transit-depth uncertainty is ±0.1%, and the companion is about 3 magnitudes fainter.
1. Calculate the planet radius implied by the transit depth and compare it with Earth and Jupiter.
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Transit depth D ≈ (R_p/R_s)^2, so R_p/R_s = √0.011 ≈ 0.105. With R_s = 0.90 × 6.96 × 10^5 km = 6.26 × 10^5 km, R_p ≈ 0.105 × 6.26 × 10^5 km ≈ 6.6 × 10^4 km, or about 66,000 km. Earth's radius is 6,371 km, so R_p ≈ 10.3 R_E. Jupiter's radius is 71,492 km, so R_p ≈ 0.92 R_J. The ±0.1% depth uncertainty gives R_p/R_s between √0.010 = 0.100 and √0.012 = 0.110, or about 62,600–68,600 km, roughly 9.8–10.8 R_E. The planet is therefore near Jupiter's size, not Earth-sized. The teammate's inference confused a 1% depth with a 1% radius; depth scales as radius squared.
2. Evaluate the claim that this is a habitable-zone terrestrial planet.
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The claim is not supported. A radius near 0.9 R_J makes the object a gas giant or at least a low-density sub-Jupiter, not a terrestrial planet. The 3.5-day period places it extremely close to a 0.85 M_Sun star. Using Kepler's third law, a ≈ (M P²)^(1/3) with P = 3.5/365.25 ≈ 0.0096 yr gives a ≈ (0.85 × 9.2 × 10^-5)^(1/3) ≈ 0.043 AU. That is far inside the habitable zone and implies intense irradiation and likely a hot, inflated atmosphere. No radial-velocity data are available, so mass and density are unknown, but the radius alone rules out an Earth-like rocky composition under standard models unless the transit is diluted by a blended companion. The students should not claim habitability.
3. What false-positive scenarios must the students rule out before claiming a planet discovery?
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They must rule out a blended eclipsing binary or a background eclipsing system diluted by the target star's light. A fainter companion 4 arcseconds away and 3 magnitudes fainter could produce a shallow transit-like signal if it is an eclipsing binary; the slight out-of-transit dimming also raises this concern. They need high-resolution imaging or adaptive optics, centroid-motion checks during transit, and a comparison of transit depth in different bandpasses, since a true planet transit should be nearly achromatic. They should also look for secondary eclipses and odd-even transit differences. Radial velocity would provide an independent mass and reject a stellar companion. Without these checks, the 1.1% depth is only a candidate signal, not a confirmed planet.