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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/r-1/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 semi-major 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 semi-major 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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