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Here is the scenario. Two ships are orbiting an uninhabited planet (no sensors on the ground and no other satellites in orbit). At time zero they are far away (let's say 50,000 Km apart) following different high orbits and they have no information about each other.

Now one ship wants to compute the orbital parameters (position and velocity vector) of the other (e.g., because it wants to fire a kinetic weapon to it or it wants to plot an intercept course) by using either an active (radar or laser) or passive (telescope) sensor. For simplicity, we can assume a dish of 10m and a ship cross section of 100m.

For how long does it need to observe the target and how precise can the position and velocity measures be?

For instance, I know that to dock to the ISS ships often rely on GPS information to get an accurate relative position.... so I guess radar along would not be very precise. Am I wrong?

I read many "stealth in space" posts that explain how hiding in space is not an option (even though many assume array of sensors and not a single ship-to-ship scenario). But here I wonder more about the precision of the trajectory measurement and the time required for it more than just the fact that locating the target is possible.

I also read answers on how we can determine the position of a far planet with a telescope by measuring the parallax over time.. but again it was not clear to me how long the time needs to be and how precise is the estimation (for a planet it might not matter too much but if you want to aim at a 100m ship you need to be quite precise).

Finally, I found some formulas on the maximum distance a radar can cover, but not much on how precise is the measurement at that distance.

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    $\begingroup$ Depends on the orbit. Minutes to tens of minutes is we are limited to passive observation, seconds if an active technology like space Lidar is available. $\endgroup$ – Alexander Sep 21 '20 at 21:07
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    $\begingroup$ If all they want is to plot an intercept course they don't need centimeter accuracy. All they need is an estimation of the position of the other ship to the nearest kilometer, and rely on fine tuning their orbit once they are close enough; that's how spaceships going to the ISS do it. (And docking to the ISS is done using on-board sensors. Automatic docking of spacecraft has been available and in use since the late 1960s and early 1970s, long before GPS came to be.) $\endgroup$ – AlexP Sep 21 '20 at 22:39
  • $\begingroup$ I agree with AlexP. However, it really depends on whether they are in a static orbit, or if they are constantly doing maneuvers. If they are trying to evade by constantly changing their trajectory, it will be very difficult. $\endgroup$ – cowlinator Sep 21 '20 at 23:31
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There is no "Minimum to know it", the more you sample, the more accurate your orbital approximation will be.

If everything was perfect:

On a whiteboard, if your only input is a sensor that give precise distance and bearing measurements, you will need 3 measurements, the further apart they are in space, the better. To fit an ellipse to points, you need a minimum of 4 points, however in this case the centre of the earth can provide the 4th point.

enter image description here
Example fitting a ellipse in 2D to 3 fixed points and a fixed centre (FreeCAD sketch tool).

A simple thought experiment for this is to consider an ellipse as the intersection of a plane a cone:
enter image description here
By fixing the cone, and any one or 2 points on the plane in space, the rest of the plane can "Swing". Only by providing that 3rd point will the system have zero degrees of freedom

If your only input is a sensor that gives precise distance, bearing, and velocity, and the orbital system is simple enough that only 1 body is providing the orbit acceleration, you will only need 1 measurement.

How to do it from first principles:

However you are unlikely to have such overpowered sensors, so this is why the answer is "lots, as many as possible". For an example, check out the history of the orbital equation of the Asteroid Apophis. It shows thousands of oversations being made, slowly changing its trajectory from "1 in 45 chance of impacting Earth" to "1 in 20 billion" back to "1 in 5560 chance of impacting earth" back to "9-in-a-million chance"

The process of converting into an orbital equation is basically:

  • With a list of your observations of position (in 3D Cartesian space):
  • Now in the 2D space of your plane:
    • Project all your observed points onto the plane
    • Project the earth centre onto your plane. Make this the origin.
    • Project / rotate this plane such that Z = 0, allowing us to work in 2D.
    • Now convert all your 2D Cartesian coordinates to polar form, so we have an angle and a radius at each point.
    • Do a least squares regression to fit Anomaly and Distance vs time
    • The higher your $r^2$ value, the more precise your fit.
    • Keep sampling until your $r^2$ is high enough.

What you'd probably do for real

Follow Gauss's method.

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    $\begingroup$ This is exactly my problem. I already have a numerical simulation to compute the orbit given a set of points and project the position of the vehicle in the future. However, as you said, it all depends on the precision of the measurements.. and this was the nature of my question. In a hard-science setting, how precise are optical/radar/lidar measurements of an object 50K Kms away moving at a speed of (let's say) 5KM/s $\endgroup$ – Baltoh Sep 22 '20 at 7:04

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