What are the units for a in Kepler's 3rd law?

[Calpalned]

Homework Statement

Kepler's 3rd law can be written as $p^2=a^3$
If p, the period, is given in years, what are the units for a?

Homework Equations

n/a

The Attempt at a Solution

The answer is AU. Is there a proof for this or is this merely a definition? Thank you.

 

[Daeho Ro]

Let's consider the Earth.

If we choose the "year" for period, then the value of $a$ also have to be $1$ with some unit and the only thing is $1 \text{AU}.$ You may think that this is the definition for $\text{AU}.$

BTW, be careful about dimensions of each side. $(\text{Year})^2 \neq (\text{AU})^3,$ and so usually we write the law as $p^2 \propto a^3.$

 

[S.G. Janssens]

Please pardon my ignorance, but I would feel that, strictly speaking, we cannot know the units of $a$ as long as we do not know the units of the proportionality constant $C$ in the relation $\frac{a^3}{p^2} = C$? And vice versa, of course.

 

[Daeho Ro]

Please pardon my ignorance, but I would feel that, strictly speaking, we cannot know the units of $a$ as long as we do not know the units of the proportionality constant $C$ in the relation $\frac{a^3}{p^2} = C$? And vice versa, of course.

Otherwise, you may fix first the units of $a$ and $p$, then you can get the constant $C$ with some unit.