Kepler's Law

Daniel123 · 2007-09-14 06:37:59

Hello.

For physics homework, my teacher has asked me to graphically prove Kepler's law.. i.e by plotting a graph and finding the equation of it.

The problem is I left the lesson early, and do not know what Kepler's law is... wikipedia shows there are three laws, the most simple of which is P² is directly proprtional to a³. I have understood P to mean the orbital period of a planet, and a to mean the semi-major axis of the planet's orbital ellipse. However, I do not know what units this proportionality applies to and therefore how to go about starting the question.

Can anyone help?

Thanks.

JaneFairfax · 2007-09-14 07:05:36

The simplest law to prove is actually the second law: A line joining a planet and the sun sweeps out equal areas during equal intervals of time. In other words, the area swept out is proportional to the time taken by the planet to sweep out the area.

This can be easily proven in the hypothetical case of a planet moving in a circular orbit of radius R with constant angular velocity ω and the Sun in the centre. Suppose an area A is swept out in time t. Then A = ½R[sup]2[/sup]θ, where θ is the angular displacement (in radians). But θ = ωt. Hence A = ½R[sup]2[/sup]ωt; in other words, A ∝ t.

I doubt if, at your level, you really have to prove the second law for the case of an ellipse, or the third law at all.

Last edited by JaneFairfax (2007-09-14 07:13:35)

Identity · 2007-09-14 07:18:19

I actually got this out of my Maths book, but I think it is pretty accurate

Last edited by Identity (2007-09-14 07:20:53)

Daniel123 · 2007-09-14 07:57:15

Hmm... once showing calculations for the value of the constant, how would I go about proving it graphically?

Identity · 2007-09-14 16:22:26

I suppose you could show how you came to that conclusion. You could graph R against T, observe that it isn't a straight line, try R^2 against T, then R^3 against T, eventually graphing R^3 against T^2, which does appear as a straight line, and check using linear regression (i use my calculator for that)

However I believe that if you can draw a diagram instead of graphing, then Kepler's first and second laws are the easiest to display, as Jane said.
Exhibit A.
Law 1. The sun is at one foci of the ellipse (if you don't know the terminology look it up) of orbiting planets. This is true for all planets. You could easily draw a diagram representing this.
Exhibit B.
Law 2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time. This allows us to see the acceleration of planets as they swing around the sun and decceleration as they reach the furthest of their orbit. Also easily demonstrated. Here you can see A2 sweeps out a much larger area than A1, despite not being much longer, so you can tell the object is slowing at A2.

There is a fantastic one-hour documentary on Kepler, which is great if you want to see it (although I must admit it digresses at times). It's from Cosmos (1980)
http://www.tv-links.co.uk/video/1/2542/3718/24637/38543

Last edited by Identity (2007-09-14 16:23:50)

Daniel123 · 2007-09-15 03:18:43

Well my teacher specifically asked us toprove it graphically.. not to teach us Kepler's Law, but to teach us graphical analysis. Unfortunately, I have totally forgotten how to do linear regression (it's been a while since I did stats). Can someone refresh my memory?

Last edited by Daniel123 (2007-09-15 03:38:18)

Daniel123 · 2007-09-15 03:39:44

In fact, would I have to use linear regression? Couldn't I just use y=mx+c... where m should be equal to 1962889 and c equal to 0? I doubt it has to be that accurate...

Daniel123 · 2007-09-15 05:13:45

Ok finished... thanks Jane and Identity

George,Y · 2007-09-15 05:45:45

No, it has nothing to do with regression-it has only something to do with geometry and calculus. Trust me.

Math Is Fun Forum

#1 2007-09-14 06:37:59

Kepler's Law

#2 2007-09-14 07:05:36

Re: Kepler's Law

#3 2007-09-14 07:18:19

Re: Kepler's Law

#4 2007-09-14 07:57:15

Re: Kepler's Law

#5 2007-09-14 16:22:26

Re: Kepler's Law

#6 2007-09-15 03:18:43

Re: Kepler's Law

#7 2007-09-15 03:39:44

Re: Kepler's Law

#8 2007-09-15 05:13:45

Re: Kepler's Law

#9 2007-09-15 05:45:45

Re: Kepler's Law

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