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## Homework Statement

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 10

^{8}km; Jupiter orbit mean radius = 7.80 x 10

^{8}km; Jupiter mean radius = 69000 km]

## Homework Equations

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 (R

_{Earth}+ R

_{Sun}/ 1 U.A. ).

the Stefan-Boltzmann law: L

_{star}=4πR

_{star}

^{2}σ T

^{4}

## 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: L

_{1}=4πR

_{Sun}

^{2}σ T

^{4}, and the luminosity during the eclipse as:

L

_{2}=4π(R

_{Sun}

^{2}- R

_{Earth}

^{2}) σ T

^{4}.

Then, doing the ratio between these two luminosities, the magnitude difference can be found, and it results in a Δm of about 10

^{-4}.

**Questions:**

Is this part right (before I proceed to the other points)?

In particular, I'm not sure about the equation for L

_{2}: is the radiative surface equal to that I have reported?

Thanks in advance.