Question

If you want to launch a probe that orbits the Sun with a period of 1.00 years, what should be its orbital radius? (Part A)

Give your answer in astronomical units.

The astronomical unit AU is defined as the mean distance from the Sun to the Earth 1AU=1.50×1011m.

What is the orbital speed of this probe? (part B)

Answer 1

If V is the orbital velocity of an object of mass m, orbiting around another object of mass M where R is the distance between the two bodies, then

the gravitational force of attraction is balanced by the centrifugal force of the circular motion (in case of planets elliptical, but assumed to be circular here) and hence they do not collide.

Force fo Gravity = Centrifugal force

GMm / R²  = m V² / R

( GM/ R)^1/2 = V

this is the orbital velocity formula.

Distance between the Sun and the probe be R_p

V_p  = ( GM /R_p )^1/2

Time period of the probe will be T_p = 2πR_p / V_p

Part A:

And for time period to be 2 years, T_p = 2 x 365 x 24 x 60 x60 seconds = 31536000 s

31536000 = 2π R_p  /  ( GM /R_p )^1/2

31536000 = 2π R_p^3/2 / ( GM )^1/2

31536000 x ( 6.67408 × 10^-11 x 1.989 x 10³⁰ )^1/2  = 2π R_p^3/2

0.5782822 x 10¹⁷ = R_p^3/2

R_p = 1.49540865 x 10¹¹ m = ( 1.49540865 x 10¹¹ ) / ( 1.5 x 10¹¹ ) AU = 0.996939 AU ~ 1.0 AU

Part B:

V_p  = ( GM /R_p )^1/2

V_p = (  6.67408 × 10^-11 x 1.989 x 10³⁰ / 1.49540865 x 10¹¹ )^1/2 = 29794.2976 m/s