Suppose the Sun suddenly quadrupled its mass,while Earth remained at the same distance from it.What is the ratio of Earth's orbital period after the change to that before it?
A) 1:16
B) 1:4
C) 1:2
D) 1:1

Question

Quizplus · Accepted Answer

According to Kepler's third law, the square of the period of a planet's orbit is proportional to the cube of its semi-major axis (T^2 ∝ a^3). Since Earth's semi-major axis (average distance from the Sun) remains the same, its period is only determined by the Sun's mass. When the Sun quadruples its mass, the gravitational force between Earth and Sun increases, causing the orbital velocity of Earth to increase. According to the conservation of angular momentum, this increased velocity will cause Earth to move in a larger orbit with a longer orbital period.

The period, T, of a planet's orbit can be determined by the relation:

T^2 ∝ a^3 / M
where M is the mass of the central body (the Sun in this case).

Since the mass of the Sun quadruples, the new period, T', can be found by solving:

(T')^2 ∝ a^3 / (4M)
or
(T')^2 = T^2 * (4M / M) = 16 T^2

Taking the square root:

T' / T = 4^(1/2) / 2 = 2^(3/2) / 2 = 1.41

Therefore, the ratio of Earth's orbital period after the change to that before it is approximately 1:2.

Suppose the Sun Suddenly Quadrupled Its Mass,while Earth Remained at the Same