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 Homework Statement:

1. Mars has a moon, Phobos. orbiting mars in circular orbit with T= 7h39min and r=9400 km.
Use Keplers laws to determine the mass of mars.
2. Calculate the mass of Mars from the orbital velocity, Newtons laws and gravitation
3. Given a star with M = 3*Mass of sun and R = 2.5*R_sun and T=11000K determine the thermal time scale of the planet if the total energy radiated during this time is 3/10 *GM^2/R
4. For the same star determine the nuclear time scale, given that the available nuclear energy is 0.1*0.007*M*c**2
 Relevant Equations:

T^2 = 4*pi^2*a^3/(G*M1+M2)
d=2*pi*r
v=d/t
v_r=sqrt(G(M1+M2)*(2/r1/a))
L=4*pi*r^2*T^4*σ
T_th = E_g/L
T_n = E_n/L
1. Keplers third law (and the asumption that M1+M2 ~ M1) gives that
M_Mars = 4*Pi^2*a^3/(G*T^2)
With numerical values inserted
Mmars = 4*3.14^2*(9400*1000+3396.97*1000)^3/((6.67*10^11*(7*60*60+39*60)^2)
2. Phobos needs 7h39 minutes to complete a circle, this gives a speed of 2*pi*(9400*1000+3396.97*1000)/(7*60*60+39*60) = 2912 m/s
The orbit is circular so the semimajor axis has the same value as radius. This gives the equation
2912 = sqrt(G*M_mars*(1/r))
3) T_th = (3/10*((3*1.981*10^30)^2*6.671*10^11)/(2.5*6.95508*10^8))/(4*3.14*(2.5*6.95508*10^8)^2*11000^4*5.67*10^8)
4) T_n = 0.1*0.007*3*1.981*10^3*(3*10^8)^2/(4*3.14*(2.5*6.95508*10^8)^2*11000^4*5.67*10^8)
M_Mars = 4*Pi^2*a^3/(G*T^2)
With numerical values inserted
Mmars = 4*3.14^2*(9400*1000+3396.97*1000)^3/((6.67*10^11*(7*60*60+39*60)^2)
2. Phobos needs 7h39 minutes to complete a circle, this gives a speed of 2*pi*(9400*1000+3396.97*1000)/(7*60*60+39*60) = 2912 m/s
The orbit is circular so the semimajor axis has the same value as radius. This gives the equation
2912 = sqrt(G*M_mars*(1/r))
3) T_th = (3/10*((3*1.981*10^30)^2*6.671*10^11)/(2.5*6.95508*10^8))/(4*3.14*(2.5*6.95508*10^8)^2*11000^4*5.67*10^8)
4) T_n = 0.1*0.007*3*1.981*10^3*(3*10^8)^2/(4*3.14*(2.5*6.95508*10^8)^2*11000^4*5.67*10^8)
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