How many observations does it take to determine the orbit of a TNO Several observations of a distant solar object must be taken before determining its orbit, but in the case of objects discovered beyond Neptune's orbit, just how many are required over what period of time (or what portion of arc) before a reasonably accurate orbit can be determined, sufficient, say, to be added to the Minor Planet Database or a JPL ephemeris?

I'd be interested to know to what degree of certainty orbital elements for objects beyond 30 AU can be determined when we have only observed them for (so far) short portions of their orbits. What about 60 AU, or 90 AU?

As an example, the Wiki for VNH004, a TNO that is a photo target for the New Horizons probe, states:


  The asteroid only was observed 12 times by the Mauna Kea and Las Campanas Observatory over a period of about 33.8 days between May 29th and July 2nd, 2011, so its current orbit is extremely uncertain.


How uncertain? Here are orbital elements along with +/- values:


  Heliocentric elements and errors
  Epoch . . . . . .  2455710.5000  =  2011/05/29
  Mean Anomaly: . . .       309.75557 +/-    31.234
  Argument of Peri: .     112.06593 +/-    54.010
  Long of Asc Node:  .    247.74234 +/-     3.336
  Inclination: .  . . . . . . . .   3.81106 +/-     0.396
  Eccentricity: . . . . .     0.29886369 +/-    0.8050
  Semi-Major Axis: .    37.47217640 +/-   14.2156
  Time of Perihelion: 2467404.0720 +/-    2926.4
  Perihelion: . . . . . . .   26.27310354 +/-   31.7695
  Aphelion: . . . . . . . .    48.67124926 +/-   35.3678
  Period (y) . . . . . . . . . . 229.3885 +/-    130.53  


An orbital period of over 229 years, plus or minus over 130 years. How many more observations would be required to half the margin of error?
The main problem with determining an object's orbit is we only know the position with certainty in two dimensions.  The distance to the object is largely unknown.  This accounts for the large uncertainty in the period of newly discovered TNOs.  Many possible orbits could fit the early observations, and therefore the uncertainty is large.

As time goes on, and more observations are made, more and more of these possible orbits are excluded and the orbit fit becomes more exact.  

So to answer your question, it really depends on how exact you want the answer, and how good your observations are.  The more observations you take, and the better they are, the closer your orbital fit becomes.  And more importantly, the bigger the arc of the orbit your observations cover, the more accurate the orbit becomes. It is impossible to give a specific number.

To give a practical example, Pluto takes 248 years to orbit the Sun.  It was discovered in 1930, so we've only observed it for 85 years, or about 1/3rd of one orbit.  One of the problems the New Horizons probe sent to Pluto had to overcome was that we didn't know, at launch, exactly how far away Pluto was, so therefore we didn't know exactly when New Horizons would pass the dwarf planet. That made it difficult to tell the probe where to point to take pictures at closest approach.