Question

If it takes a planet 5 years to orbit the Sun, how long (in years) will it take the planet to go all the way around our sky once?

Answer 1

An average distance from the Sun will be given by -

using a Kepler's 3^rd law, we have

p² = a³

a = (p)^2/3 (5 years)^(2/3)

a = 2.92 AU

converting AU into m :

a = 4.3683 x 10¹¹ m

How long (in years) will it take the planet to go all the way around our sky once?

we know that, T = 2r³ / G m_E

where, m_E = mass of earth = 5.972 x 10²⁴ kg

r = distance from the center of Earth = r_E + a

then, we get

T = (6.28) {[(6.371 x 10⁶ m) + (4.3683 x 10¹¹ m )]³ / [(6.67 x 10^-11 Nm²/kg²) (5.972 x 10²⁴ kg)]}

T = (6.28) [(8.33 x 10³⁴ m³) / (3.98 x 10¹⁴ m³/s²)]

T = (6.28) (1.446 x 10¹⁰ sec)

T = 9.08 x 10¹⁰ sec

converting sec into years :

T = 2877.3 years