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Jul 2002
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Based on observations made by eye alone, with minimal instrumentation, what information would you need to calculate an orbit, using fairly elementary calculus and/or differential equations, just like Newton did it? Show me.

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Oct 2001
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It ain't easy!

Let's assume it's something nice and regular, like an asteroid in an elliptical orbit.

At minimum, you need two good sightings and the exact time. The more sightings, the better (obviously.)

In theory, mathematically, this is enough to establish the orbit. (K.F. Gauss pulled it off!)

From here on, though, I can't describe it neatly, as it starts getting into some really gnarly equations...

Silas

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ON 2002-11-18 19:44, SILAS WROTE: To? HUb'
It ain't easy!

My guess, {& I can only guess}
GET "Fortran"
in spacific DE200
once you find out
that YOU CANNOT get EiThOR
pay M\$ for the new Realease of the New Reallease
Boo # 8 4 & 6

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A telescope is useful for getting positions that can be much more precise than positions obtained with telescope-less observations, but that's about it.

An orbit is determined by the values of is parameters at some time, and there are always six of these values: three position and three velocity components or some appropriate combination of these.

So you will need six values and a time. Positions and extra times can count as values.

If you measure an object's position against the stars, you will get two coordinates: right ascension and declination, or some equivalent. So you will need only three observations to completely determine an object's orbit.

If you use a sextant, however, you will only get elevations, and you will need at least 4 observations.

You also must make your observations over a long-enough range of time so that the object's gravitational acceleration will be noticeable in one's observations. This is only a few minutes for a low-earth-orbit satellite, but it can be weeks or months for an interplanetary object.

But one obvious application, a visible comet, will have its aphelion much farther away than its perihelion, making it OK to approximate its orbit with a parabola. This makes only 5 parameters necessary, meaning that one only needs two observations.

Once one has one's data, there are various techniques for finding initial estimates of orbit parameters, and then for improving those estimtes. I don't know offhand where to look, though a celestial-mechanics textbook will likely have some details on the subject.

5. A Few Words About Initial Orbit Determination

The equations of Newtonian mechanics are examples of second-order differential equations. In order to solve such an equation in one variable two numbers are required in order to completely specify the solution. Either the value of the variable and its first derivative may be specified at one time, in what is called an "initial value problem", or values of the variable may be given at two different times, in what is called a "boundary value problem". In an elementary calculus course, an example of the initial value problem would be: "A ball is thrown straight up in the air from ground level at 30 meters per second, where is it after 3 seconds?". An example of the boundary value problem would be: "A ball is thrown straight upwards from ground level and 2 seconds later it is at a height of 10 meters. With what velocity was it thrown upwards?" Boundary value problems are very difficult and do not always have unique solutions. Initial value problems have unique solutions (usually).

Newtonian mechanics works fairly well in the Solar System, and because there are three spatial dimensions we must specify six constants in order to determine the solution. There are infinitely many choices for these six numbers. We can specify the position and velocity vectors at one time (initial value), or position vectors at two times (boundary value), or we may specify some set of orbital elements, such as semimajor axis, eccentricity, inclination, longitude of node, argument of pericenter, and time of pericenter passage, for example.

When we are observing an unknown asteroid or comet for the first time, there is an important complication--we have no idea of the distance to the object, all we have are two numbers per observation (right ascension and declination, or longitude and latitude), so we need at least three observations, not two.

Much ingenuity has been spent on developing methods of orbit determination that use just three observations and the two main methods bear the names of Laplace and Gauss. The method of Laplace uses the observations to estimate the position and velocity of the object at one time, usually the middle observation and is thus an application of the initial value problem. The method of Gauss uses the observations to estimate two positions (usually the first and the third) and is thus an application of the boundary value problem.

But there is one huge complication lurking under the surface, and that is the quality of the observations. All observations are subject to error, but the traditional three-observation methods of preliminary orbit determination fit an orbit to the observations exactly. Usually it works reasonably well and will just have to do until more observations are available. Sometimes the results can be disastrous.

Laurence Taff in his book Celestial Mechanics (New York, Wiley, 1985) gives the case history of the discovery of 2060 Chiron, as told by the first few IAU circulars. I also have these circulars, and here is the story:
• Circular 3129 gives three points, one of them not at the same precision as the other two, but no orbit.
• Circular 3130 replaces one of the points and adds three others for a total of six, and describes the orbit as "extremely indeterminate". They give an orbit that is almost circular with a period of 66.1 years.
• Circular 3134 adds two more points, declares that a near-circular orbit is viable, but a high-eccentricity orbit is not.
• Circular 3140 adds four more points and is silent about the orbit.
• Circular 3143 adds three more points and is also silent about the orbit.
• Circular 3145 adds two historical observations from 1969 and gives an orbit with a period of 50.70 years and eccentricity 0.37860. (Not very circular anymore, is it?) It took 15 data points spanning 37 days and 2 historical data points to arrive at this orbit.
• Circular 3147 adds five historical observations from 1941, 1943, 1952, and 1976.
• Circular 3151 adds two historical observations from 1895 and 1976, and a correction to the 1941 observations. An orbit is given that satisfies the observations to within 2 arcsec except for the 1943 point. (Planetary perturbations are taken into account.) The orbit has a period of 50.68 years and eccentricity 0.378623.
• Circulars 3156 and 3215 correct previous historical observations, and add further historical and current data points, but no more is said about the orbit, which seems to have settled down into reasonably definitive values.

Taff declares that no orbit was given in circular 3129, and only says that it would have been an embarrassment. Using the three data points and software from Montenbruck and Pfleger's book it is easy to see why: a calculation using the method of Gauss gives an eccentricity of nearly 8! (I don't believe any comet has been observed with an eccentricity greater than 1.1, and perhaps not even that high.) Taff does not say whether the method of Laplace would have done any better here.

I am currently researching a method of applying a statistical method to the problem of preliminary orbit determination. I have not yet programmed the method I am developing, but I can say that the data in those circulars will be my case study. My method uses the data from three or more observations and uses them all symmetrically in the sense that none of the observations is satisfied exactly. (The method of Herget satisfies two observations exactly and lets the others float.) My method may not give the "perfect" orbit, but it will come with error estimates so that you will have some idea of how bad it is! [img]/phpBB/images/smiles/icon_smile.gif[/img]

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