Question

According to Kepler's Third Law, if the semi-major axis of an orbit increases by a factor of 4 (in other words, became 4 times larger), what would that indicate about the period of the orbit?

Using the simplified version of kepler's third law:

p²/a³

Answer 1

If the semi-major axis of an orbit increases by a factor of 4, according to Kepler's Third Law, the new orbital period will be 8 times the original period, as the period is proportional to the square root of the cube of the semi-major axis.

Step-by-step explanation:

According to Kepler's Third Law, the square of a planet's orbital period (P) is directly proportional to the cube of the semi-major axis of its orbit (a). If the semi-major axis of an orbit increases by a factor of 4, we would have to cube this factor because of the direct proportionality (a becomes a^3, so 4 becomes 4^3 or 64). Then, to find the new orbital period squared (P^2), we multiply the original period squared by this factor.

Therefore, the new orbital period squared would be 64 times the original orbital period squared. To find the new orbital period, we take the square root of this new P^2, which means the new orbital period would be 8 times the original period (since the square root of 64 is 8).