Relating Newton's universal law of gravitation with Kepler's 3rd law

Rumor

I'm trying to rewrite Newton's law - GMP² = 4(pi)²r³ as P² = a³ under these conditions: that the mass M is the sun's mass, the radius is 1 AU, and G being converted to the appropriate units.

I get stuck on this problem, and I'm hoping you guys are willing to help me out.

To begin with, to make things easier, I set this problem up as a ratio:
P²/a³ = 1 and P²/r³ = 4(pi)²/GM

and then I compare the two. In an elliptical orbit, the radius r can be equaled to a, so I get:

P²/r³ = P²/a³ = 4(pi)²/GM.

Does this look right to you guys so far? I was told that if I did it correctly, my answer should come out to 1, but I'm not getting that once I work things out. I thought that I wouldn't have to worry about the units of G in this setup, but maybe I'm wrong? I'm not sure.

To clarify, I'm using 1.98 x 10³⁰kg for M and 6.67 x 10^-11 m³/kgs² for G.

Any help would be very much appreciated.

cepheid

I think you mean that in a circular orbit, the semi-major axis, 'a' is equal to the radius, 'r'.


1. Of *course* you have to worry about the units for G. It's not a dimensionless quantity, it has units.

2. The whole thing with Kepler's 3rd Law is that it can be expressed as P² = a³ if and only if you express 'a' in AU and 'P' in years. The reason is because, using those units, the constant of proportionality between P and a, namely 4(pi)²/GM, is equal to 1. But if you don't use those units, and you use SI units instead, then this constant will not be equal to 1.