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How does the number of observations of spacecraft position and velocity affect the orbit determination accuracy? Suppose in one case we have 1000 instances of position and velocity measurements with a sampling of 1 second and in other case we have 2000 instances of position and velocity measurements with sampling of 500 milliseconds. Why would the accuracy of the orbit be more in the second case?

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There are several aspects playing a role here: first of all, every measurement has a certain random inaccuracy. Ask ten people to measure the size of an object with the best precision possible and you'll likely get ten slightly different answers. The average of those measurements is usually a much better estimate for the size than any individual measurement. Thus, with twice as many measurements you know the position and velocity of the space craft better. If the error of those measurements is purely random, you expect the total error to be sqrt(2) better.

Second, if you have many measurements close together in time, you need to extrapolate a lot to get the full orbit of your spacecraft. Extrapolation is always a source of errors, as a very small variation in the starting conditions can cause tremendous differences in a later state. You specified measurements 16 minutes apart, which is a significant time of a full orbit in LEO, but not so much in GEO. Hence, a LEO orbit can be determined rather well within these 16 minutes, but for a satellite close to GEO (or a probe even further out in space) you might want to spread measurements further.

Lastly, measurement of the position of a satellite can't be done equally well in the three spatial directions. Distance to the satellite and its speed away or towards us can be measured very accurately using signal latencies and the Doppler effect. The other two dimensions are not as easy, e.g. you need to know very precisely where your telescope points, and due to the large distances even minute errors in the pointing angle sum up to large uncertainties in the position and subsequently the speed measurements. At different points in time, the relative movement of the satellite is in different directions. This means these inaccuracies can cancel out to some extent.

In summary: Having more measurements helps by reducing the error due to measuring inaccuracies. Having them spread over a larger time helps by knowing more information on very different points of the orbit and helps by reducing the amount of extrapolation to be done.

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  • $\begingroup$ @asfdex Thanks for the explanation! Can you kindly elaborate on the aspect that measurement position can't be done equally well in the three spatial directions. $\endgroup$ – Soumajit May 14 '16 at 19:50
  • $\begingroup$ This is due to different measurement principles: Distance is measured as travel time of a radio signal with very high precision (see GPS). The measurement of the other two dimensions (translating to up/down left/right in the sky) boils down to accurately point your instrument at the satellite. Any tiny error in the angle adds up to a large error in position due to the huge distance to the object. Triangulation from different observers gives better results, but still not as good as the distance measurement. $\endgroup$ – asdfex May 14 '16 at 20:01
  • $\begingroup$ @afdex Thanks once again. In that case, even the velocity in the other 2 components would be less accurate than the line of sight component ? $\endgroup$ – Soumajit May 14 '16 at 20:09
  • $\begingroup$ Exactly. The difference is even worse - velocity in radial direction is the thing we can measure with the best accuracy, even down to millimeters per second. $\endgroup$ – asdfex May 14 '16 at 20:51

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