An interesting relationship had been discovered by mathematicians much earlier, that if you have any three points in a coordinate plane, there is always a simple curve that can be drawn through those three points and that curve was always a parabola. For this reason, this is sometimes called the Parabolic Rule. In advanced math and Physics, is more often called Simpson's Rule, after the mathematician who discovered it.
We can show how this works for a slightly simplified case where the distance along the X-axis between the first-and-second and second-and-third points is the same. For unequal distances, the proof still works but it is a lot more complicated mathematically.
So we could say that our three points have X coordinates of -x, 0, and +x. There is a Y-coordinate dimension for each of these, which we can call y1, y2, and y3.
We can take the generic form of the parabola equation, y = a * x2 + b * x + c.
This then means that our three Y-coordinates are:
y1 = a * (-x)2 + b * (-x) + c
y2 = c
y3 = a * (+x)2 + b * x + c
It turns out that it is a simple Calculus problem to confirm this and to establish the values of a, b and c. This then gives the (approximate) shape of the orbit. If you have been paying attention, you should have noticed an apparent problem here! Asteroids orbit in ELLIPSES and not following a PARABOLIC path! You get a Gold Star for noticing that! But it turns out not to be a serious problem in any real discovery. An astronomer generally makes his position determinations at about the same time on consecutive days, or within a few days. It turns out that the curvature of the orbit during such a short time period is virtually identical for an ellipse or a parabola. By using the Parabolic Rule to determine an APPROXIMATE orbit (within a few days of discovery), the chances of others finding and confirming it a week or two later is greatly improved. Once plenty of accurate positions are later measured, a more accurate (and far more mathematically difficult) orbit determination can be made.
In any case, fairly quickly, we can determine an approximate orbit. Determining c is easy, just the observed Y-coordinate of the middle observation. b is nearly as easy, just half the difference between the Y-coordinate of the first and third observation. a is then determined, and it really is the term that has the square-factor for the curvature of the orbit.
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One effect of this is that a truly critical observation measurement is the most precise possible actual spacing of the positions of the object in the sky. It turns out that the DIFFERENCE noticed between the first-second and second-third observations gives a good estimate for the a we needed above, which defines the curvature of the orbit. The difference between the first and third observations gives us b, which is essentially an estimate of a straight line indicating the current portion of the orbit. (c is not directly determinable, but is actually unimportant at that time, being the actual distance the object happened to be from us at the time of the observations. Once a and b are determined, an approximate orbit shape is already known. The b value and some creative applications of Kepler's Laws can give us a rough idea of where the object is going around the Sun as compared to where we are, and we can then establish what the c factor is.)
However, this is still not actually accurate! The observed motion toward one side or the other MIGHT mean an actual motion in that direction, but far more likely is that the asteroid is actually also moving toward or away from us too! Kepler's Laws helps here, too. When we calculate the apparent sideways velocity (by first assuming that it IS straight sideways) we can calculate a velocity that it must be moving in its orbit around the Sun. But from the other data (above) we were able to estimate the actual distance the asteroid is from the Sun. Kepler then lets us calculate the Period to go all the way around the Sun, and so we can calculate how fast the asteroid MUST BE moving, in order to comply with Kepler's Laws. That actual velocity is generally higher than what we calculate by assuming a straight lateral motion. With simple Trigonometry (the sine, or actually arc-sine) we can use those two numbers to tell us what angle the asteroid is moving toward or away from us (we do not know which!)
Note that, piece by piece, we develop information, all based on just three accurate observations! We are able to estimate the X and Y position in the Solar System, the X and Y velocities, and even the accelerations that are curving that path. Combined with Kepler and a decently reliable assumption, we can finally estimate the path, and therefore the orbit.
When people did all this in the early 1800s, all calculation was with paper and pencil, and many days of complex calculations were involved to get really good estimates of the orbit, especially the part where an actual ellipse was to be calculated! Now, computers whiz through all of this in a fraction of a second!
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago