Calculation of an Orbit from Three Observations

by Liona Fan-Chiang

1  Introduction

By the time Karl Friedrich Gauss rose from local genius to the front lines of world fame in late 1801, the campaign of terror to eliminate creativity had already settled thick, clouding the potential historic minds of Europe. The terror came in the form of liberalism, sophistry, and empiricism. The dead world of Newtonian mechanics had, only three years earlier, been resurrected by the mathematical mortician, Laplace, with his Mechanique Celeste, while one of the most prolific and only remaining explicit defenders of Cusa, Kepler, and Leibniz, against Newtonianism1, Abraham Kästner, had passed away just one year prior.
The astronomical community reeled when their efforts were put to shame by the then unknown young lad, who not only accurately calculated the orbit of the elusive planet, Ceres, from 3 degrees arc of observations, but then was able to correct the original observational error! Everyone was eager to learn of the superior tactics Gauss must have used. Only after significant pressure from an honest scientist, Heinrich Olbers, does Gauss finally respond with a letter which outlines a complete solution to the problem, finally followed by a much more extensive book in 1809. However, after dissecting and digesting both these publications, the question still remains: What did Gauss do that was different than the attempts of others, such that he could solve the problem no one else could? Even Olbers points out that the equation designated `the most important part of the entire method' in his letter, looks just like one of Laplace's equations. Is the difference in the minute detail? Or, did Gauss just not publish what he actually did?
The present report, as a smaller part of a project to discover Gauss' thought process, will provide a spotlight on the second half of Gauss' determination, as outlined in his first report to Olbers, namely the task of calculating the orbit of the planet when the geocentric distances are known, in order to attempt to illuminate the process of Gauss' mind through the cracks of public appearance. The letter will euphemistically be called the Summary Overview throughout.

2  The Summary Overview
The Second Point Approximate Determination of the Elements2

The primary result of the entire first half of the Summary Overview appeared to be the ability to determine the geocentric distances of three positions of the elusive planet, derived from the fundamental principle of gravitation in the context of the unique geometry of the infinitesimal. The final equation, which was the "most important part of the entire method," expresses the geocentric distance in a knowable way, involving only the three corresponding observations given in geocentric longitude and latitude, the time intervals between the observations, the semi-axis major of Earth's orbit, and the heliocentric distance of the Earth at those three moments of observation.
However, the nominal task at the outset was, not to find our own distance to the planet at any moment, or even three moments, but rather, to find the elements of the entire orbit as seen from the Sun! Yet, the only heliocentric relation retained in the conclusion of the first half, leading directly into determining the elements of the sought orbit, is that of the distance of the Earth from the Sun. Has Gauss led us off track? Or, has he only cast the observer in the role of mediator between the harmonic properties of the Solar System in relation to the Sun, and their lawful projection onto the sphere of creative investigation?
To begin following Gauss as far as he is willing to lead, stop pointing at the mysterious planet, and turn your extended finger toward the Sun, as he is the main character of this scene.
Gauss begins the second point:
We leave out entirely the middle observation for the time t¢, and use instead the distances d and d", which are approximately determined in the preceding point. It is clear, that from this henceforth the heliocentric longitude [l], latitude [b] and distance [r] can be derived, and hence, the longitude of W [the ascending node] and the inclination [i] of the orbit and the longitude in the orbit [v].
Thus, the first task is to transport all the prior geocentric relations to the Sun, since only the sun can directly see the longitude of the ascending node as well as the inclination.
Given the curtate distance, d, (i.e. the distance of Earth from the planet's perpendicular projection onto the ecliptic) of the two outer positions, can the above mentioned heliocentric properties be found? Gauss characteristically says, "it is clear." A diagram may help to make this proposal at least more transparent.
Sun, Earth and Palner relationships
a and b are the geocentric longitude and latitude as observed from the Earth. R, the distance of the Sun from Earth, can be calculated from the elements of the Earth, as given by Kepler in his New Astronomy[LINK?]. d is the curtate geocentric distance to the position on the ecliptic to which the actual position of the planet is projected perpendicularly, as beautifully arrived at in the first half of the Summary Overview.
The above quantities, the heliocentric distance to the planet (r) and the heliocentric longitude and latitude, (l and b respectively), can be found for both of the the outer positions of the planet geometrically. Afterward, the remaining quantities to be found are the longitude of the ascending node (W), the inclination (i) of the orbit, and the longitude in the orbit (v), all angular quantities as seen from the Sun. Therefore, the concept of distance can be left aside for the moment, as we turn here to the celestial sphere.
positions on the celestial sphere
b, b", and l, l" are the heliocentric latitudes and longitudes, respectively, of the two positions in consideration. v is an angle measured at the Sun from the vernal equinox to the actual position of the planet in orbit, which can be found by the relation cos(v-W)=cosb cos(l-W).
There are several ways to arrive at the above sought magnitudes, all of which, require an ability to navigate on the sphere. Therefore, if you are not already intimately familiar with spherical trigonometry, take an interlude and familiarize yourself with this great companion.3
Notice that a result of the above geometrical exercise, is that we have already obtained two of the six necessary elements of the orbit. Moreover, these two alone are the two required to specify the orientation of the unknown orbit with respect to that of the Earth's. However, if the entire determination were simply a matter of geometry, then Gauss would have stopped here, and moved on to write about or conduct another of his plethora of profound discoveries.
There's the Rub
Following the ability to determine the orientation of the plane of the sought orbit, is the much more challenging task of seeking the characteristics which uniquely define the harmonic characteristics of the orbit itself. Recall that Kepler had used only two primary measures. The first he revealed in his `little book' the Mysterium Cosmographicum, namely, that the approximate proportions of distances between the orbits, could be determined by a nesting of the five platonic solids. The second was elaborated in his Harmony of the World, in which Kepler proves that when considering motions of planets, each orbit could only be understood as integral proportions of the entire Solar System, but this time proportions which could only be understood only by the ear.
In Gauss' notation, the size of the orbits are given in proportion of mean distance of the plants orbit to the Sun (a) with respect to the mean distance of the Earth from the Sun, which is often replaced by the more useful semi-parameter (p=(b2)/a, where b is the semi-axis minor). The eccentricity defining the proportion of extreme motion is denoted by e, and the position at which those extreme motions occur, designated by the longitude of the apsides. Gauss chose to use the longitude of the perihelion, i.e. the point of quickest motion for the practical reason that most celestial objects of interest in the late 18th century were comets, many of which had no aphelion!
Thus the problem still remains to determine the remaining elements, namely a, e, p and the epoch
from the two longitudes in the orbit ¼v v"
the distances from the Sun ¼r r"
and the corresponding times ¼t t"
Since the relations of these magnitudes to the given ones are transcendental, we must again rely on the indirect method
Summary Overview
First, in order to gain some insight into why Gauss asserts that the sought relations are transcendental, let us dive in and try to calculate a set of elements. We begin with a useful tool, which is recognized as the shadow of those physical principles examined by Kepler; a shadow which is often misconstrued, or even misleadingly presented as one of "Kepler's Laws": the geometrical ellipse, whose mathematical expression comes in the form of

= 1

(1 - e cos(v - p))

Here are four approaches to this problem:

Case 1: Given two radii and the parameter, find the elements e and p.


= 1

(1 - e cos(v - p))

= 1

(1 - e cos(v'' - p))

give two positions of the same ellipse.

With some rearranging, namely moving all known quantities onto the right side, we obtain

1 - p

= e cos(v - p))
1 - p

= e cos(v'' - p))

How are e and p to be isolated? For this, we apply what may appear to be a very familiar construction for those familiar with Gauss.

double angle formula

The equation below the diagram expresses a geometric relation of e cos(v - p) to the difference in angle between the two observations.

If both sides are divided by e cos(v - p):
e sin (v - p)

e cos (v - p)
tan (v - p))

= cos(v'' -v) e cos(v - p) - e cos(v'' - p)

e sin (v'' -v)

cos(v'' -v) (1 - p

) - (1- p


e sin (v'' -v)

The last step was arrived by applying the original relation
1 -

e cos(v - p))
from above.
The above will give p in terms of known quantities, after which, e can be easily found.
Case 2: Given two radii and p, find p and e.
Again we have two positions on the same ellipse Only, this time, p, the size of the ellipse, is unknown. What is known, however, is that regardless of the value of the parameter, it is the same for both positions of the same ellipse. Therefore, p =r (1 - e cos(v-p))=r" (1-e cos(v" - p)), from which e can easily be isolated. Once e is isolated, p can be found by p = r (1 - e cos(v - p)).
Case 3: Given two radii and the eccentricity.
A case very similar to the previous, and therefore will be left to the reader to pursue.

Case 4: If we used all three of the geocentric distances calculated in the first half of the Summary Overview, then three heliocentric distances are had, from which three equations expressing three instances of the same ellipse can be posed.

1 - p

= e cos(v - p))
1 - p

= e cos(v'- p))
1 - p

= e cos(v'' - p))


One could again algebraically manipulate the three equations below, until three separate expressions for p, e and p, exclusively in terms of r, r¢, r" and v, v¢, v", are obtained.

Gauss concludes with the expression


p =

4 sin 1

(v"-v¢) sin 1

(v"-v) sin 1

(v¢-v) · rr¢r"

r"r¢ sin(v"-v¢) - r"r sin(v"-v) + r¢r sin(v¢ - v)

by which the other two elements can be found by applying the same steps as in case 1. Note that the denominator consists of those triangles formed from pairs of orbital positions and the sun. The sum of the three forms the triangle formed by the three positions of the planet at the respective times t, t¢, t".

Finally, once the elements are had from either of the four methods mentioned, the accuracy of the orbit must be confirmed by comparison with the original observations. Therefore, for the purposes of comparison, either a geocentric observation, or a time interval must be calculated from the hypothesized elements. We will choose time, since the time interval between two positions requires a simple calculation of the difference between their corresponding mean anomalies, namely m" - m = E" - e sin E" - (E - esinE),4 where the eccentric anomalies E and E" can be had geometrically by relation to the known true anomalies.

Mean anomoly

The mean distance a, as mentioned earlier, can be had from the parameter, p = b2/a = a (1 - e2).

Once the difference in mean anomaly is acquired, the corresponding time interval can be had directly

m" - m

= time elapsed

entire period
For an example of an exact calculation of the time from the elements from the actual elements of Ceres, see [link to Merv's Ped].
If the elements, and consequently the geocentric distances from which they were derived, are correct, the calculated time interval between the observations should coincide, thus confirming a correct calculated orbit.
However, the case which presently faces us coincides with none of the four described above. We have neither p, e nor p, which eliminates the first three cases. On the other hand, the fourth case seems plausible. Gauss however, disregards this method immediately. Why?
In the language of infinitesimal dynamics as developed earlier, the expression of the parameter, namely


p =

4 sin 1

(v"-v¢) sin 1

(v"-v) sin 1


r"r¢ sin(v"-v¢) - r"r sin(v"-v) + r¢r sin(v¢-v)
contains a third order expression above and below and therefore results in calculations which have a finite error.
Hence it is readily inferred, that if one or more of the quantities r, r¢, r", v, v¢, v" are affected by errors never so slight, a very great error may thence arise in the determination of p; on which account, this manner of obtaining the dimensions of the orbit can never admit of great accuracy, except the three heliocentric places are distance from each other by considerable intervals.
Instead, Gauss offers a fifth case:
Since it is possible to determine the whole orbit by two radii vectors given in magnitude and position together with one element of the orbit, the time also in which the heavenly body moves from one radius vector to another, may be determined, if we either neglect the mass of the body, or regard it as known: ...Hence, inversely, it is apparent that two radii vectors given in magnitude and position, together with the time in which the heavenly body describes the intermediate space, determine the orbit. But this problem, to be considered the most important in the theory of the motions of the heavenly bodies, is not so easily solved, since the expression of the time in terms of the elements is transcendental...-Theoria Motus § 84(emphasis added)5
Can we simply reverse the process above as Gauss describes? That is, can the elements p, e, p be derived from the time interval and two corresponding heliocentric distances? Are there any steps that can't be reversed?
Indeed, the last step in the previous calculation, thus our first step in this proposed inverse problem, cannot be directly inverted. As Kepler emphasized, although the direct relationship between the mean anomaly, and the eccentric anomaly could be stated, since the mean anomaly is a single magnitude which is a combined result of two incommensurable magnitudes, the latter two cannot be directly unraveled from first. It seems we are curbed before taking even one step.
Here is Gauss' proposal
Now it is evident that if the approximate value of any one of the quantities p, e, p, is known, the two remaining ones can be determined from them, and afterward, by the method explained in the first section, the time corresponding to the motion from the first place to the second. If this proves to be equal to the given time t, the assumed value of p, e, or p, is the true one and the orbit is found; but if not, the calculation repeated with another value differing a little from the first, will show how great a change in the value of the time corresponds to a small change in the values of p, e, and p... §85
In other words even though the none of elements are known, we begin by taking p, e or p as given, i.e. use an initial guess, and proceed just as before! Calculate the corresponding time interval according to the elements found, compare it to the real time interval. Then, adjust the element and repeat the calculation, until an acceptable degree of accuracy is obtained. Therefore, even though the inverse problem could not be solved directly, Gauss effectively inverted the problem, thereby creating a situation where the calculator must instead solve a series of direct ones!
The above process will, after adequate iterations, "leave nothing to be desired." However, the case remains: what if one begins with an orbit wholly unknown? How is the first approximation to be made?
For this, Gauss offers the most interesting method.
The advantage of this third method consists in the fact that a very approximate value for p can be found directly, if the arc v"-v is not too large. Namely, the sector between the two radii vectors is
g¢= a3/2Öp

(m"-m)= 1

-Summary Overview
Here, g is the area traversed, a, the semi-axis major , and m, m" and M, M", the mean longitudes of the planet and Earth respectively.

Again we encounter, as we have several times throughout this report, the seed Kepler had planted in his New Astronomy, which expresses a harmony that exists even within the organizing principle of one planet's orbit. As with every higher ordering principle, the expression appears in the form of a paradox: motion along the path of the planet at every moment is governed by the portion of area swept out with respect to the whole area encompassed by the orbit. Thus, although the whole may be thought of as consisting of its parts, the parts in turn, can only be known by its relation to the whole! This great paradox, as many of the paradoxes Kepler leaves us with, has served as sustenance to many creative minds who have shaped civilization, such as Gottfried Leibniz in his creation of the calculus. It is mathematically expressed as

traversed space

total ellipse area
= mean time traversed

total period
= mean motion

or equivalently
traversed space= total area

·mean motion
where the total area is b/a·pa2 = pab = pa3/2Öp where a and b is the semi-axis major and minor respectively and p = b2/a.
mean anomaly
This gives
g = pa3/2Öp

(m" - m) = a3/2(m"- m)

just as seen above.
Following this is another shadow, which is presented as a consequence of Kepler's subsequent investigation of the Solar System as a whole. The individual orbits are organized as an entity, but in addition the whole of each orbit is tuned with the that of all other orbits. Again, expressed mathematically, this ordering is expressed as,

= æ


where T1 and T2 signify the period of any two planets, and a1, a2, their respective mean distances.
As a consequence, since the time spent by the planet while is was being observed is the same time as that spent on Earth when observing the planet,

= m" - m

, and t

= M"-M

thus t = a3/2(m" - m) = A3/2(M" - M)
and finally
g¢= 1

A3/2(M" - M)Öp

Which gives the second part of the equation which Gauss proffers as foundation.

Since all magnitudes with respect to Earth's orbit are known, if the sector swept by the planet in its own orbit is found, p is known and thereby e and p, by same the calculation as outlined earlier in case 1. However, as mentioned earlier, it is precisely the calculation of this sector which presents the paradox of the reciprocal relation of the whole and the part. The area of a circle, which has uniform radius is easy enough to calculate, however, in an ellipse sector, the distance of the planet to the Sun is always changing at a changing rate!

equal area equal time
Gauss begins by proposing "a method by which the value of p is obtained with such accuracy that for small values of (v¢-v) it will require no further correction." This method is his application of the approximation method of COTES.
The area of a circle, given the angle traversed and the heliocentric distance to the planet, the area of a sector is pr2 ·(v"- v)/(2p)=(r2(v" - v))/2. Since the distance from the focus to the ellipse varies continuously, the sector would be =1/2 òvv"r(w)2 dw. That is, to obtain the area of the sector, you must add an infinite amount of areas which depends on a length that changes at every moment! Kepler spared his calculator by only adding area intervals of 1o. But, of course, to obtain the change in radius, and therefore the radii r(w) in the interval between our two observations, we require precisely the elements of which we now seek! Therefore, we turn to an application of COTES' simple approximation method, which uses only the two heliocentric distances calculated earlier to approximate the sector area.
The first approximation involves using two circles, one at radius r" and the other at r, both over the stretch of (v"-v)/2
Each half has an area of area p / r2 = v"-v / 2, resulting in a total approximated sector area of v"-v/2 (r2+r"2). Accordingly, the approximated parameter becomes


= (v" - v)(r2 + r"2)

A3/2(M" - M)
You can see for yourself for what combinations of positions in the orbit, and for how large of an angle, this first approximation is suitable. For the case of Ceres, the observations used were very near quadrature, and therefore suited to this method of approximation.
A successive approximation uses a division of 3 parts, taking a sixth of the angle between the two observations with a radius r, 2/3 at a radius r*, located at (v" - v)/2, and the remaining sixth with radius r". However, how is the middle distance r* to be determined? A linear approximation, i.e. (r + r")/2, would, in some places, be a worse approximation than the first! In fact, any good approximation for the middle distance would actually have to take into account the specific part of the ellipse the interval in consideration is, as well the approxiamte curvature of the ellipse. It seems as if we need to previously know the ellipse which we are trying to calculate in order to approximate the area of the same unknown ellipse! In order to remedy this paradox, clever Gauss actually does use an ellipse. Only since we don't have the ellipse necessary, he uses the next best, namely the ellipse which corresponds to p0 as calculated above!
As you can see, for a small angle, the above approximation will be very close.
first approximationn of ellipse area by cotes approximation
A further approximation can be carried out in which the angle is further divided into 8 parts, however, Gauss is satisfied with the first two.
Gauss continues on to present yet another method, but only several years later, in his 1809 Theoria Motus, in which he at last publicly addresses the true transcendental nature of this problem by replacing the COTES approximation with a brief presentation of an application of the hypergeometric series [LINK]. Although, Gauss would seem to be most satisfied with the latter method, the author of this report does not claim that either one, or any of these methods, are the first or the actual method which Gauss used in his original calculation of the orbit of Ceres.
Houston We Have an Orbit
Once the elements W, i, e, p, and p are found, we at last have a unique conic section, which is specified in orientation with respect to the orbit of the Earth.
orbital elements labeled
Is there anything missing from this picture? ¼ THERE'S NO PLANET!
The last step for this set of elements is to designate where the planet is at some point in time. This should be an easy task since we already know where it is in three places.
mean anamoly equation
After this, the first approximation of an orbit is obtained. But the job is not done! Is the orbit correct? Does it coincide with all observations, or even just the center one that was left out of this second part? Gauss initially offered four sets of elements to the December 1801 issue of the Monatliche Correspondenz [LINK].

3  Conclusion

Although the process Gauss describes may seem like a straight forward approximation of an orbit, the question which hounds the investigator of Gauss' mind remains: What was unique about the process which Gauss followed?
For an exercise, glance through a few chapters of Laplace's five volume text book the Mechanique Celeste6 or even any modern day astronomy or physics textbook. Immediately you will be introduced to certain `fundamental' and `self-evident' quantities, such as force, acceleration, mass, momentum, and energy, as well as `basic' equations such as the `inverse square law.' Before empty space begins to creep in, look back at the entire process by which Gauss calculated an orbit from three observations. How many times does Gauss apply the above mentioned `fundamental' measures? Although Newton's name appears as `the great Newton' in the prelude to his Gauss' text, how many times do Newton's Laws, or even any of the quantities which are used in Newton's Laws, i.e. force, mass, acceleration, etc., used? None! In fact, the only quantity he does mention in this regard is mass, as was excerpted above, and even then, it was only to make the point that he was going to ignore it!
In this context, we begin to see that, just as with his other writings, Gauss' apparently formal descriptions of his determination of the orbit of Ceres and other asteroids are no innocent `objective' reference material. Rather, Gauss demonstrated that what the Newtonians asserted was not only unfounded and unnecessary, it was not even adequate in solving what appeared to a practical problem: the ability to calculate an orbit from a few observations. Moreover, since his entire method involved only those principles proved by Kepler's creative investigation of the human mind, which is intentionally excluded by empiricists such as Newton, Gauss demonstrated that in fact the Newtonians had made no advancement at all beyond Kepler!
Therefore, the real question is: why would modern classrooms teach Newtonian mechanics and not Kepler's harmonies? Why would a modern civilization tolerate Newtonian mechanics at all, except as a lesson in what not to teach?


1Newtonianism-a disease which fills minds with empty space. Common symptoms include uncontrollable babble about hard balls, and a fanatical defense of free trade.
2Refer to the a pedagogical[LINK] on how to calculate a geocentric position, given a scheduled time of observation, for an interactive overview of planetary elements.
3Sphereical Trigonometry has been extensively treated by the Larouche Youth Movement through the course of investigation of Kepler's Harmony of the World [LINK TO VINNIE PED], and Gauss' Pentagramma Myrificum [BEN'S PED]. There is also an elemetry treatment by Abraham Kästner [LINK TO TRANSLATION]
4For a treatment of the derivation of this equation, see the New Astronomy[LINK TO CH60] website as well as a pedagogy[LINK TO MEAN ANOM PAGE] on calculating the positions of Mars
5Theory of the Motion of the Heavenly Bodies Moving About in Conic Sections. By Karl Friedrich Gauss, Translated by Charles Henry Davis 1857 Little, Brown and Company
6All five volumes of Laplace's Celestial Mechanics have been translated into English by American Nathaniel Bowditch, one of the leading American scientist of his time, including extensive commentary by Bowditch.

File translated from TEX by TTH, version 3.80.
On 15 Mar 2008, 21:29.