Keplers 3rd law and derivations of it.
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Attached are three questions. the first relates to kepler's 3rd law and calculating the maxima and minima of acceleration of a planet around a Sun. The second asks for this equation to be combined with Newton's law of universal gravitation to find the mass of the sun and the third asks for the period of another planet with a larger radius. i am a bit rusty and can't remember how i should approach these, i have included in orange typeface my thoughts and ideas of how i should be going about answering these questions. Please can you confirm if i was heading in the right direction and show me how i should answer them? thank you
Question 1
A hypothetical star, S, has a planet, P, that moves in an elliptical orbit with period T. The minimum distance of P from S is d and the maximum distance is 1.4d. Use this equation for the magnitude of acceleration,
(eq. 1)
(The magnitude of acceleration at any time t is therefore dependent entirely on the distance between planet and sun at that instant)
To,
Q1)
Find the maxima and minima of the magnitude of the acceleration of P, in terms of d and T.
(My ideas on this question are,
I know that the magnitude of the acceleration is always directed towards the Sun, Keplers 3nd law, and I know that this acceleration only depends on the distance of the planet from the Sun. I know that the first term on the right of the above equation ( ) is the same for all planets. And that , I am just not sure what the question is asking me to do? Are they asking me to determine the actual acceleration or just find an expression that if I knew all the relevant information would enable me to find the correct values?)
Q2)
Use Newton's law of universal gravitation to find the mass of S, in terms of d, T and the gravitational constant G.
(My ideas are,
I think this question is asking me to determine a new expression that will find the mass of the Sun if I know all the relevant information. In this case I think I would need to combine equation 1 above, with,
(eq. 2)
Where x = the points of 2 particles and e is a vector of unit length pointing from 1 to 2. Do I then need to rearrange equation 2 to find an expression for then substitute this into equation 1 and rearrange?, please can you show me)
Q3)
The star S has another planet Q that moves in a circular orbit of radius 6d. Obtain the period of Q's orbit as a multiple of T.
(Can this be found from a rearrangement of Kepler's third law? I.e as and r = 6 T must be a multiple of this?
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Solution Summary
The solution examines Keplers 3rd law and the derivations of it.
Solution Preview
The correct solutions are explained in the attached pdf file.
For the formulae on Kepler's laws, you should have all that are needed in your textbook(s) or lecture notes, however, just in case, here is a page where you can look up the 3rd law of Kepler:
http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion
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Here is the plain TEX source
centerline{bf Kepler laws}
In the formula you quote at the start, $r_1$, is the bf semi-major rm axis.
Therefore
$$
r_1 = {d+1.4dover 2} = 1.2 ...
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