# Planet orbit

## Main Question or Discussion Point

how do you calculate a planets orbit?

HallsofIvy
Homework Helper
With a great deal of work!

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.

hypnagogue
Staff Emeritus
Gold Member
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.

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.
Uhh, is it one of the Lagrange points?

Staff Emeritus
Gold Member
Dearly Missed
Back in college I spent most of a semester trying to calculate the approximate orbit of a comet (we used a parabola) from a "Harvard card". I don't know if they have them anymore. Astronomers who spotted a new comet would submit the right ascension and declination of their observation to Harvard, which acted as a clearinghouse. Whaen it had three different observations on the same comet it would publish a postcard with the three data points.

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.

HallsofIvy
Homework Helper
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.
Chemicalsuperfreak wrote:
Uhh, is it one of the Lagrange points?
No, Lagrange points of an orbit are ON the orbit at certain angles before and behind the planet.

enigma
Staff Emeritus
Gold Member
Lagrange points only apply if you have a three body problem. Earth/moon/satellite is one such situation.

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.

FZ+
With a computer, it's pretty easy. Turn the differential into a difference equation, set a small increment to each calculation, and then run the simulation. With a small enough increment, you can get pretty good accuracies.

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.

enigma
Staff Emeritus
Gold Member I've got one of those coded up in Matlab if anyone's interested!

how do you calculate a planets orbit?

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.

kepler's elements

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 DaveC426913
Gold Member
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.
I've always wondered this myself. Anyone have any insight?