by RJN » Thu May 27, 2010 12:37 pm
Smallfish,
Thanks for your questions. In the wikipedia link you highlight below on Kepler's laws, it shows that really the full relation for Kepler's third law would include mass. In the "Non-Planetary Mass" section, one sees that a^3/P^2 is proportional to M+m. Since M for the Sun is so much higher than m for a planet, the planetary mass can be ignored. Next, since M is the same for all of the planets -- they all revolve around the same Sun -- the proportionality constant is the same for all of the planets and so one can just estimate that a^3 is proportional to P^2.
Next, yes, I assumed in applying Kepler's third law that the orbit is close enough to circular that "a", actually half the major axis, is close enough to all other distance measures between a planet and the Sun. Even if this is off by 20%, I felt that simplicity was more important that accuracy at that level. I apologize if you found this confusing.
- RJN
Smallfish,
Thanks for your questions. In the wikipedia link you highlight below on Kepler's laws, it shows that really the full relation for Kepler's third law would include mass. In the "Non-Planetary Mass" section, one sees that a^3/P^2 is proportional to M+m. Since M for the Sun is so much higher than m for a planet, the planetary mass can be ignored. Next, since M is the same for all of the planets -- they all revolve around the same Sun -- the proportionality constant is the same for all of the planets and so one can just estimate that a^3 is proportional to P^2.
Next, yes, I assumed in applying Kepler's third law that the orbit is close enough to circular that "a", actually half the major axis, is close enough to all other distance measures between a planet and the Sun. Even if this is off by 20%, I felt that simplicity was more important that accuracy at that level. I apologize if you found this confusing.
- RJN