Question

what would be the period of a planet whose orbit has a semimajor axis of 4 au?

Answer 1

we need 3rd Kepler's law to solve this problem. According to this law,

$$t1^2/t2^2 = a1^3/a2^3$$

where $a$ is semimajor axis and $t$ is period. So we find

$$t2 = t1(a2/a1)^3(3/2)$$

knowing that for Earth $t1 = 1$ year, $a1 = 1$ au, we can see

$$t2 = 1^*(4/1)^3 = 1^*8 = 8 \text{ years}$$

answer is 8 years