Question

Callisto orbits Jupiter at an average distance of 1.88 106 km with an orbital period of 0.0457 yr. The Moon, which is one of the satellites of the Earth orbits its parent at an average distance of 3.84 105 km with an orbital period of 0.07481 yr.(a) Use the above information to find the orbital speeds of Callisto around Jupiter and of the Moon around the Earth.

vCallisto =	What is the distance covered by the satellite in one orbital period? Check units for consistency. m/s
vMoon =	m/s

(b) What is the expression for the mass M of the parent in terms of the orbital speed v of the satellite, the orbital radius R of the satellite and the gravitational constant G? (Do not substitute numerical values; use variables only.)
M =   

(c) Now use your answers from parts (a) and (b) to find the ratio of the mass of the Earth to that of Jupiter.

ME

MJ

=

Answer 1

a)

for callisto ::

radius of orbit = R = 1.88 x 10⁹ m

orbit period = T = 0.0457 yr = 0.0457 x 365 x 24 x 3600 sec = 1.44 x 10⁶ sec

distance travelled = circumference of the circular orbit = 2 R

orbital speed is given as , V_callisto = 2 R / T = 2 (3.14) (1.88 x 10⁹) / (1.44 x 10⁶) = 8198.89 m/s

for moon ::

radius of orbit = R = 3.84 x 10⁸ m

orbit period = T = 0.07481 yr = 0.07481 x 365 x 24 x 3600 sec = 2.36 x 10⁶ sec

distance travelled = circumference of the circular orbit = 2 R

orbital speed is given as , V_moon= 2 R / T = 2 (3.14) (3.84 x 10⁸ ) / (2.36 x 10⁶ ) = 1021.83 m/s

b)

let the mass of satellite "m" and mass of parent "M"

centripetal force = gravitational force of attaraction

m V²/R = GMm / R²

M = V² R / G

c)

M_jupitor = V_callisto² R_callisto / G

M_earth = V_moon² R_moon / G

M_e / M_j = V_moon² R_moon / (V_callisto² R_callisto )

M_e / M_j = (10.21.83)² (3.84 x 10⁸ ) / ((8198.89)² (1.88 x 10⁹ ))

M_e / M_j = 0.923