Suppose that a planet was reported to be orbiting the sun-like star Iota Horolog

Question

Suppose that a planet was reported to be orbiting the sun-like star Iota Horologii with a period of 350.0 days. Find the radius of the planets orbit, assuming that Iota Horologii has twice the mass as the Sun. (This planet is presumably similar to Jupiter, but it may have large, rocky moons that enjoy a pleasant climate. Use 2.00 x 10^30 kg for the mass of the Sun.) To what radius would a star of mass 2.05 x 10^30 kg have to be contracted for its escape speed to equal the speed of light? (Black holes have escape speeds greater than the speed of light; hence we see no light from them.) A satellite orbits the Earth In an elliptical orbit. At perigee its distance from the center of the Earth is 22,500 km and its speed is 4230 m/s. At apogee its distance from the center of the Earth is 24100 km and its speed is 3980 m/s. Using this information, calculate the mass of the Earth.

Explanation / Answer

2.

Use equation,

T2 =(42/GM)r3

Plugging values,

350*24*606*60 = [(4*3.142)/(6.67*10-11*2*1030)]*r3        => r= 1.0*109 m

3. Use equation,

v= sqrt[2GM/R]

Plugging values,

3*108 = sqrt[(2*6.67*10-11*2.05*1030)/R]         => R=3038.56 m

4.

By Newton’s second law,

mv^2/r = GMm/r^2

M= v^2r/G

At perigee,

Mp= (4230^2*22500000)/(6.67*10^-11) = 1.2*10^10 kg = 6.036*10^24kg

At apogee,

Ma= (3980^2*24100000/(6.67*10^-11) = 1.2*10^10 kg = 5.72*10^24kg

Mave= (Mp+Ma)/2 = (6.036*10^24+5.72*10^24)/2= 5.88*10^24kg

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