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.