How far from the sun should a planet be placed so that you celebrate your birthday twice as often? Between which planets would this hypothetical...

Hello!

To find the desirable distance from the Sun, we should use Kepler's Third law (in the simplified form, when the mass of a planet is negligible with respect to the mass of the Sun):

`P^2 / a^3 = P_2^2 / a_2^3.`

Here `P` is the current orbital period, `P_2 = P/2` is the desired period, `a` is the current distance between Earth and the Sun and `a_2` is the new distance to find.

From  `P^2 / a^3 = (P/2)^2 / a_2^3`  we easily obtain that  `a_2 = a / root(3)(4).`

Now we have to use table data about the distances between the Sun and the planets. For Earth, it is about 150 millions km and the new distance will be about 94 millions km. This orbit would be between Mercury (46-70 millions km) and Venus (107-109 millions km).