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how do you calculate a planets orbit?

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how do you calculate a planets orbit?

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HallsofIvy

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Theoretically, if you know the position and velocity vectors at a given time, you can solve the differential equation

d^2r/dt^2= -GmMr/|r|^3 where r is the position vector (assuming the sun is at the origin) and |r| is the length of that vector- the distance from the sun to the planet. This is assuming Newtonian gravitation and ignoring the gravitational pull of other planets. That's a non-linear differential equation and can be very difficult.

More simply, you can use the fact that a planetary orbit is an ellipse with the sun at one focus. If you know the position of the planet and it's velocity vector at a given time, you plug that information into the general formula (the velocity vector will be tangent to the ellipse) and determine the equation of the ellipse.

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hypnagogue

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Uhh, is it one of the Lagrange points?Originally posted by hypnagogue

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selfAdjoint

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Then people like me and my fellow students would try to use Celestial Mechanics to compute the orbit. We worked in pairs and calculated (pre computer) for weeks, and to make a long story short, we got the vertex of the orbit inside the surface of the Sun! We got A's in the course anyway.

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HallsofIvy

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Chemicalsuperfreak wrote:Originally posted by hypnagogue

Is there any kind of physical/mechanical meaning whatsoever behind the other, non-sun focus? As far as I know there isn't, but I always wondered.

No, Lagrange points of an orbit are ON the orbit at certain angles before and behind the planet.Uhh, is it one of the Lagrange points?

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enigma

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There are five L points. Only two are stable - the two which HoI mentioned, L4 & L5. If you're looking down from above the system, they lie approximately 60 degrees before and behind the larger mass. The other three lie in a straight line with the two masses. L1 is located between the two masses where the gravity from both is the same. The other two are located on one side or the other. Those are orbitting the combined mass with a period equal to the orbital period of the masses. They are just like geocentric satellites of the Earth only they orbit the combined Earth/Moon mass.

To Hypnagogue's question, no there is no physical significance to the other focus. The elliptical shape simply falls out of the equations and laws derived by Kepler and Newton.

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eg. at each time interval, calculate the resultant gravitational force vector on each object (more fun if you have lots of objects, but ok with just one around the sun, I suppose). Calculate resultant acceleration. Calculate the changed velocity with v = v + at, and finally find the new position x= x+vt. All simple equations, without calculus. Then do it again and again until you plot the orbit.

Note that is a pseudo-fudge known as the leapfrog method. It happens to work in cases like these - other methods don't conserve energy in some cases.

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enigma

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I've got one of those coded up in Matlab if anyone's interested!

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By using Gauss' method,adapted for computers.All we need are 3 observations which to specify the right ascension and declination of the planet.The method is based on Kepler's laws which can be derived from Newtonian mechanics showing that the motion of a planet is entirely dependent of its orbit,there is no need to know its mass,velocity and so on.The bad thing is that it implies a lot of calculations.I've tried once to calculate such a trajectory using only a simple calculator...but I renounced rapidly,a computer would be very useful.

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is there anybody who can send me an m.file which outputs velocity and position vectors when given time and kepler's elements(eccentricity,angular momentum,anomaly etc) and outputs at another time kepler's elements?

in addition it should make the inverse process,i mean when i give position and velocity vectors at a time,it should output kepler's elements and at anoter time velocity and position vectors?

it would be very precious for me now,like a gift from heaven..i can give presents for the one who helps me,like an excellent ambigram

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DaveC426913

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I've always wondered this myself. Anyone have any insight?

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