Sensitivity of calculated orbital elements to observational errors These days, we have some very precise ways of making measurements, but I'm sure it wasn't so in Kepler's day. So I am wondering how astronomers of that time could make such accurate determinations of planetary orbits, given the likely limitations of the available instrumentation. To put the question another way: How does each added (or lost) digit of precision in the measurement affect the accuracy of the calculated orbital elements? Is there significant compounding of errors due to the nature of the calculations, or is it essentially one-for-one? How much accuracy (to how many digits of precision) must the "three observations" be in order to determine the orbital elements to a given precision? How does that precision degrade over time? If, for example, we wanted to prepare an almanac for Neptune that will be accurate to say, one second of arc in our sky after 100 years?
Kepler used data from catalogs made by Tycho Brahe which had a varying level of precision and accuracy down to about 0.5 arcminutes for a measurement. However, since Kepler thought all systems should operate in the same way, he could use a larger sample verify his laws. Once you know what the relation should be, you can then map the observations to what you would expect. For an individual case, the precision of measuring orbital parameters depeneds on how much of the orbit you have actually observed.