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A number of gas giant planets orbiting other stars at distances less than 1 A.U. have been discovered. Because of their proximity to their parent stars, and their compositional similarity to Jupiter, they have been labeled “Hot Jupiters”.
- The orbital radius of one of these planets is 0.06 A.U. with average orbital speed 600 km/sec. What is the length of this planet’s year in Earth (solar) days?
- Estimate the mass, M, of its parent star in terms of the mass of the sun (M) using Newton’s first form of Kepler’s 3rd Law.
- Calculate the star’s luminosity, L, in terms of the luminosity of the sun (L☉), Note: (LL=MM4where L ~ 4 × 1026 W ).
- The radius of this planet is 1.5 times the radius of Jupiter. Assuming its equilibrium temperature is the temperature at which the planet radiates as much energy as it receives from its star, estimate the temperature of the planet. The value of the planet’s albedo is 0.8. (NOTE: The intensity of the star’s radiant power at a distance d from the star is L4d2. Also assume a value of the Stefan-Boltzmann constant σ ~ 6 × 10-8 W/m2K4 and use kB = 1.4 × 10-23 m2kg/sK for the Boltzmann constant). How does this temperature compare to the boiling point of water (373 K)? (DON’T BE WORRIED IF YOU GET A HUGE TEMPERATURE VALUE)
- Using your result from part d, estimate the mass of the planet if observations indicate the presence of water vapor (H2O) in the planet’s atmosphere. (HINT: Assume the water vapor’s thermal energy does not prevent it from remaining gravitationally bound to the planet.)
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