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What is the minimum information I need to be able to draw the shape of an orbit of a body that orbits the sun? With this information how can I figure out the shape of its orbit?

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If you want just the "shape" you need one number: the eccentricty.

The shape of an ellipse is determined by the eccentricty. It is a measure of how "uncircular" an ellipse is. Conventionally this is written as $e$

If the ellipse is centred at the origin, and its long axis ends at (1,0), then the focus of the ellipse is at (e,0)

A very simple geogebra construction that does just this: https://www.geogebra.org/classic/ydp9cthr

If you want the "shape" and the "size" of the ellipse, you need a second number, and conventionally this is half the length of the long axis of the ellipse, this is written as $a$.

If the half length of the short axis is $b$, then these are related by the formula

$$e^2 = \frac{a^2-b^2}{a^2}$$

So if you have two, you can work out the third, and plot the ellipse at a given size using parametric equations

$$\begin{align} x &= a \cos(t)\\ y &= b \sin(t)\\ \end{align}$$

Now, this draws the ellipse in 2d space and oriented along the x-axis. If you want the ellipse drawn in 3d space and at an arbitrary alignment, you need three more pieces of information, usually these are the "argument of periapsis" (the angle of long axis of the eclipse) "inclination" (the angle of the plane containing the eclipse to the x-y plane) and the "longitude of the ascending node" (the bearing of the point at which the object crosses the x-y plane)

A sixth number is required to define the position of the planet on the ellipse at a given time. (usually given as the "mean anomaly" or "true anomaly")

These six numbers are called the orbital elements

All this assumes Kepler's laws. Real orbits are perturbed by the gravity of other planets (and slightly deviate from Kepler due to relativity). The real paths of the planets aren't actually periodic, but Kepler's laws are a very good model.

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  • $\begingroup$ "The shape of an ellipse is determined by the eccentricty. It is a measure of how "uncircular" an ellipse is. Conventionally this is written as e" The eccentricity first of all measure how eccentric the orbit is. Difference of its shape from ellipse much less dramatic. $\endgroup$
    – Leos Ondra
    Mar 29 at 20:54
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    $\begingroup$ I don't quite understand your point. "eccentricity is measure of how eccentric". Well that is true... but not very useful to someone who doesn't know what "eccentric" means. This is why I use the bad English "uncircular". This whole answer (until the last paragraph) assumes Kepler's laws, so the orbits are exactly ellipses. I also give the exact position of the foci in an ellipse with sma=1. There is assumed to be no difference from ellipse. $\endgroup$
    – James K
    Mar 29 at 21:09
  • $\begingroup$ I don't quite understand your point. "eccentricity is measure of how eccentric". I mean how far is the central body (Sun) from the center of the orbit (which is even for Mars in ordinary diagrams just circle). $\endgroup$
    – Leos Ondra
    Mar 30 at 10:10
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    $\begingroup$ I see, well I think I've already addressed that in the third paragraph. If the centre is at 0,0, then the focus is at (0,e) for an ellipse with a = 1 (and proportionally for larger ellipses.) $\endgroup$
    – James K
    Mar 30 at 10:23

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