An Earth-like planet orbits around a Sun-like star with a circular orbit of period T=1 year. The system is very far from us. Assuming that our Earth is on orbit plane of the Earth-like planet, calculate:
i) the period of the partial occultation of the star from the planet;
ii) the resulting apparent magnitude variation of the star.
iii)Repeat the exercise for a giant planet, Jupiter-like, with an orbital period of T=4332 days, placed in the same system;
iv) Are the above mentioned phenomena observable with a ground-telescope?
[Data: Earth radius = 6370 km; Sun radius = 694000 km; 1 U.A. = 1.49 x 108 km; Jupiter orbit mean radius = 7.80 x 108 km; Jupiter mean radius = 69000 km]
This is a sketch of the eclipse: http://www.webalice.it/mizar02/figure/StarEcl.gif
So I've found the angle AOB as: α= 2 arcsin (REarth + RSun / 1 U.A. ).
the Stefan-Boltzmann law: Lstar=4πRstar2 σ T4
The Attempt at a Solution
After the angle α is found, the eclipse period can be found as: Δt = (α/2π) T = 12.9 hours.
ii) Since luminosity can be written through the Stefan-Boltzmann law, I write the luminosity of the star before the occultation as: L1=4πRSun2 σ T4, and the luminosity during the eclipse as:
L2=4π(RSun2 - REarth2) σ T4.
Then, doing the ratio between these two luminosities, the magnitude difference can be found, and it results in a Δm of about 10-4.
Is this part right (before I proceed to the other points)?
In particular, I'm not sure about the equation for L2: is the radiative surface equal to that I have reported?
Thanks in advance.