Could someone explain to me why Kepler didnt need to know the mass of the earth and sun when he came up with his laws

Kepler was using the ratios of a planet's orbital radius to Earth's and of a planet's orbital period to Earth's. When you set up a problem saying that Jupiter orbits at five times the distance of Earth, so its orbital period is 11 times that of Earth, the masses and such cancel out of the equation and you don't have to worry about them.

We need to remember that Kepler had no dynamic theory of what was going on here. This was an empirical kinematic exercise on the special case of planets orbiting the Sun.

I don't know whether or not he did a similar exercise on Jupiter's moons. Such an exercise would yield a similar equation relating the orbital period and radius, but with a different proportionality constant.

The general case would be of the form P² = KR³, with Jupiter's K being about 1,000 times that of the Sun. As a result of Newton's work we now know that K = 1/M, where M is the mass of the primary object.

Somewhat more pedantically, the value of the constant in Kepler's third law scales with the inverse of (Msun + Mplanet) for each planet, which is always within a range of 0.1%. An additional small effect comes from the fact that the masses of planets interior to the one considered also ad to the net force on its orbit in a time-averaged sense. This is a huge effect for distributed systems such as galaxies, but again doesn't matter at the 0.1% level in the Solar System.

Kepler did find that the Galilean satellites of Jupiter followed the same pattern with a different constant, using data Galileo set him. (I suspect Galileo was frustrated by the same thing that frustrates some of my students when I set them this exercise - they have similar brightnesses, and we usually see their orbits from nearly their common plane, so its a real chore to keep track of which one is which unless you have frequent position measurements).

Because no one had calculated them prior to that time.

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I neglected to say that my simplified account is only valid if the planet's mass is vanishingly small. You are right.

And is there also, presumably, an effect from the exterior planets that has to be subtracted?

I guess there would be - in fact, isn't this the way some of the outer planets were discovered, through their gravitational effect on previously known planets?

Although it may not be exactly analogous. If a planet is always closer to the sun than you are, then it must always be pulling you at least a little in the direction of the sun. But if a planet is farther away, sometimes it will be pulling you towards the sun (when it is on the other side), and sometimes it will be pulling you away (when it is on the same side and behind you).

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I wonder if Kepler contributed to Galileo's relatively accurate Medician moon ephemeris? [The hunger for an accurate way to determine time for navigation gave him considerable interest in this. He built a head-mounted contraption to help steady the view of Jupiter off a ship's deck, but it didn't help much. Spain was a good prospect but dumped the idea for various reasons.]

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I think you are mistaken here. If Jupiter is behind the Sun, the Earth will gravitate toward Jupiter and thus toward the Sun, but the Sun will gravitate toward Jupiter even more strongly because it is closer, thus tending to pull away from the Earth. The effect will be a subtraction in either position. Remember, two objects of different masses accelerate toward a third body at the same rate if at the same distance. The fact that the third body, Jupiter in this case, is less massive than the Sun does not change this.

When Jupiter is at quadrature, the gravitational acceleration vectors of Earth and Sun are of equal magnitude, but are angled inward. This would give the illusion of strengthening the Earth-Sun gravitational attraction.

I think the average effect over a complete revolution is a subtraction, but I am not absolutely sure.

It sounds like you are describing a tide. Tidal forces are very small, though, relative to that under discussion.

Yes, this is the same gravitational gradient phenomenon that is involved at a smaller scale with tides.

Arg, I hate it when that happens

Yes, I was considering the sun to be so massive that its movement would be unaffected by the other planets, but that's not appropriate if we are considering the affect of one planet on another

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A spherical shell outside an orbit has no effect whatsoever on an orbit. This is because of Gauss theorem. Or else, an area and therefore the mass of a spherical shell is proportional to the square of distance, and since gravitational attraction is proportional to inverse square, the opposite portions of a spherical shell cancel exactly.

But an uniform ring does have a gravitational effect inside. The length and mass are proportional to distance, so a nearer part of the ring will exert stronger attraction.

Does the effect of a ring differ from that of a planet orbiting the same path?

I'm a little confused by this. What exactly is a uniform ring? I think you must be referring to something that is different than the picture in my head right now.

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Absolutely. But the difference seems so obvious to me, that I'm not sure that we are talking about the same thing.

If you gave more details of your ring, compared to a planet, maybe that'll answer both my questions and Arnold Layne's

Compare the gravitational effects, to a planet orbiting inside, of
1) a ring with mass uniformly distributed along its circumference
2) a point planet with the same total mass as 1), orbiting on the same path as the ring did
3) a point planet with the exact same mass and orbit as in 2), but orbiting in the opposite direction.

How different are the effects?

Didnt Kepler base his laws on observation?

Orbital resonances would imply that there is a difference, but I dont know a proof off the top of my head.

Yes. Tycho's data was accurate and respected enough in Kepler's view to discover the ellipitical orbit as the solution. This also ended the idea of uniform motion around the orbit. This probably got him going on finding the relationship between period and distance, which he did discover.

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