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View Full Version : How many data points are needed to plot an orbit?

Trigonal Planar
07-07-2003, 10:50 PM
Can an orbit be fully described by only 2 sets of points? Ie, Observation 1 (X, Y, Z directions + Speed), Observation 2 (X, Y, Z directions+ Speed) or are more points needed in order to interpolate the function?

scr4
07-07-2003, 11:35 PM
You just need one point if you know position, velocity and time. If one data point isn't enough, that would imply that two objects at the same position and same velocity at the same time can be in different orbits - and that's not true.

Or you can think of it in terms of the number of parameters you need to describe an orbit. You need 2 to describe the shape of the orbit (radius and eccentricity), 3 to describe the orientation of that ellipse in space, and one to describe phase (time) for a total of 5 parameters. One set of (X,Y,Z,Vx,Vy,Vz,T) is all you need. If you just have (X,Y,Z,T) and no velocity, you need two data points.

Achernar
07-08-2003, 12:12 AM
It depends on if you know the position and mass of the sun, and it depends on what you can measure: position, velocity, and time. I'll assume you can always measure position.

If you know the position and mass of the sun, you will need either: Three pos measurements, or
One pos/vel measurement, or
Two pos/time measurements. It's possible that this will be degenerate, though, and you'll need to take a third. In general, two pos/time measurements will work.
If you have the sun's position but not mass, you need:
Three pos measurements, or
Two pos/vel measurements, or
Three pos/time measurements

If you don't have the sun's position you need:
As many as five pos measurements*, or
Two pos/vel measurements, possibly three, or
I don't know. At least three and no more than five.

Now, all of this assumes that your measurements are in the reference frame of the sun. If the sun is not stationary, it's a bit harder.

* - this one I'm not sure about. It's equivalent to how many points it takes to define a conic section in a plane. You may be able to get away with four in cases of non-degeneracy. I'm not sure.

Trigonal Planar
07-08-2003, 12:16 AM
Probably a dumb question, but what exactly do you need the time element for? If you know the exact position and velocity, those should be unique to a given point in time.

Achernar
07-08-2003, 01:09 AM
I don't understand what you're asking. First off, for closed orbits, objects go back through the same point with the same velocity every orbit. Even if it's not a closed orbit, though, yes the position and velocity define a time, but you can only get that time if you know the parameters of the orbit to begin with.

However, knowing the time T of a single observation does no good, as far as I can tell, unless you have a reference point like T = 0 is perihelion or something.

scr4
07-08-2003, 05:26 AM
Originally posted by Achernar
If you know the position and mass of the sun, you will need either: Three pos measurements
Nitpick: You need at least one time measurement also.

Trigonal Planar, you don't need any time measurements if you just want the shape of the orbit (radius, ellipticity and orientation of the orbit). But usually when people talk about determining the orbit, we mean finding enough information to predict position as a function of time. If two satellites have the same orbital radius, ellipticity, etc. but one is 5 minutes ahed of the other, the two are considered to be in different orbits.

RM Mentock
07-08-2003, 06:27 AM
Originally posted by scr4
Or you can think of it in terms of the number of parameters you need to describe an orbit. You need 2 to describe the shape of the orbit (radius and eccentricity), 3 to describe the orientation of that ellipse in space, and one to describe phase (time) for a total of 5 parameters.
Is it just me, or does that add up to 6?

Achernar
07-08-2003, 02:56 PM
Originally posted by scr4
Nitpick: You need at least one time measurement also.

Trigonal Planar, you don't need any time measurements if you just want the shape of the orbit (radius, ellipticity and orientation of the orbit). But usually when people talk about determining the orbit, we mean finding enough information to predict position as a function of time.You're right. All of my figures were if you only wanted the shape. If you want position as a function of time, then no amount of position-measruing alone will give you the answer.