A Shallow Transit Announced as a Rocky Planet
A team observes a star with radius 0.80 R_sun and mass 0.80 M_sun. The transit light curve shows a depth of 1.2% and a period of 12.3 days. The transit has a V-shaped bottom rather than a flat one. Radial-velocity follow-up gives a semi-amplitude K = 8.0 m/s. Photometric noise is 0.05%, but the star shows 0.1% variability at the same 12.3-day period. The team announces a 'rocky planet' because the transit is shallow, and they use the depth alone to estimate a radius. No independent radius measurement or adaptive-optics imaging is available. The host star is 120 pc away, and a nearby faint companion lies 2 arcseconds away.
1. Calculate the planet's radius, minimum mass, and bulk density from the supplied data. Is the 'rocky planet' claim supported?
Read the worked answer
The transit depth gives the radius ratio:
[\delta=\left(\frac{R_p}{R_*}\right)^2=0.012,\qquad \frac{R_p}{R_*}=0.1095.]
With (R_*=0.80,R_\odot=5.57\times10^5\ \text{km}), (R_p\approx6.1\times10^4\ \text{km}\approx0.85,R_J).
For the minimum mass, use
[M_p\sin i\approx\frac{K}{28.4\ \text{m/s}}\left(\frac{M_*}{M_\odot}\right)^{2/3}\left(\frac{P}{1\ \text{yr}}\right)^{1/3}M_J.]
Here (P=12.3/365.25=0.0337\ \text{yr}), so the factors are 0.862 and 0.323. Then (M_p\sin i\approx(8.0/28.4)(0.862)(0.323)M_J\approx0.078,M_J\approx25,M_\oplus).
The density relative to Jupiter is
[\rho/\rho_J=\frac{0.078}{(0.85)^3}\approx0.13,]
so (\rho\approx0.13\times1.33\ \text{g/cm}^3\approx0.17\ \text{g/cm}^3). That is far below rocky densities near 5 g/cm³. The planet is a low-density gas giant or sub-Saturn, not rocky. A genuinely rocky planet at 0.85 R_J would have a mass of hundreds of Earth masses and a radial-velocity signal far larger than 8 m/s. The 'shallow' transit is shallow relative to this small star, but 0.85 R_J is still a large planet.
2. How do the V-shaped transit and the 2-arcsecond companion affect the radius interpretation? What follow-up observations would you require?
Read the worked answer
A V-shaped transit is a warning sign. It can mean a grazing transit, where the planet crosses near the limb and the chord is short, so the depth does not cleanly measure (R_p/R_*); it can also indicate a blended eclipsing binary or a companion contaminating the light curve. The 2-arcsecond companion is especially important because, at 120 pc, it may fall within the photometric aperture. If the companion contributes unocculted light, the observed depth is diluted and the true transit depth is larger than 1.2%, making the planet larger, not smaller. Required follow-up: high-resolution imaging or speckle imaging to test whether the companion is bound and how much light it contributes, centroid analysis during and outside transit, and a careful transit fit allowing for grazing geometry. Without those, the radius is an upper or lower limit, not a secure value. The density conclusion is robust in the sense that dilution would only increase the radius and lower the density further.
3. The star varies by 0.1% at the same 12.3-day period as the transit. How should the team treat this variability, and does it change the composition conclusion?
Read the worked answer
The 0.1% variation is about 8% of the 1.2% transit depth, so it is not negligible. If the team measures the transit depth against a baseline taken during a different part of the stellar cycle, the depth can be biased by up to roughly 0.1%, which is about 8% of the depth. Since radius scales as the square root of depth, that is about a 4% radius error; density scales as (R^{-3}), so about a 12% density error. The team should model the stellar variability simultaneously with the transit, use out-of-transit baselines close in time, and check whether the 12.3-day signal is rotation or binarity rather than a planet. The variability does not rescue the rocky interpretation: even a 4% larger radius makes the planet larger and the density lower. The composition conclusion is driven by the combination of a large radius and a small minimum mass, not by fine details of the depth.