Accuracy of Laplace Method for determining orbital elements

Accuracy of Laplace Method for determining orbital elements

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Recently I had to implement Laplace method and apply it to 3 observations of Mars (10 days in between two observations). The results were pretty good, with discrepancies with real data well below 5% in most of the orbital elements.

I supposed that Laplace method was very accurate, so I tried to apply it to an asteroid (CASLEO, 5387). The 3 observations were also 10-days spaced but now the results widely diverge from the real orbital elements.

I expected it to be less accurate than in the Mars case since the observations covered less fraction of the orbit, but some of the elements were quite far from expected.

So my question is: What is the accuracy of the Laplace Method? Is it really that sensitive to the distance?

In principal, Laplace's method (and Gauss's method) should be very accurate for determining orbits.

Of course "garbage in garbage out" applies, and if the observations were inaccurate the results may also be inaccurate. However, you should not get great discrepancies. I would consider careful checking of your calculation, and perhaps comparing your results with those of an independent orbital determiner, to check for a mistake in the maths.

Accuracy of Laplace Method for determining orbital elements - Astronomy

Laplace's method is a standard for the calculation of a preliminary orbit. Certain modifications enhance its efficacy: reduce the observations, if necessary, by use of the L1 criterion use a polynomial, whose order is determined by impersonal criteria, to calculate the first and second derivatives of observational quantities combine the separate equations, one to determine the heliocentric distance of the object and the other its geocentric distance, into one polynomial equation for the heliocentric distance, whose roots are found by a standard algorithm use recursion to calculate the f and g series. At least one differential correction is recommended to increase the accuracy of the computed orbital elements. Difficult problems, lack of convergence of the differential corrections, for example, can be handled by total least squares or ridge regression. The method is first applied to calculate a preliminary orbit of Comet P/ 1846 D1 (de Vico) from 59 observations made during five days in 1995 and then to a more difficult object, the Amor type minor planet 1982 DV (3288 Seleucus).

Accuracy of Laplace Method for determining orbital elements - Astronomy

Laplace’s method is a standard for the calculation of a preliminary orbit. Certain modifications, briefly summarized, enhance its efficacy. At least one differential correction is recommended, and sometimes becomes essential, to increase the accuracy of the computed orbital elements. Difficult problems, lack of convergence of the differential corrections, for example, can be handled by total least squares or ridge regression. The differential corrections represent more than just getting better agreement with the observations, but a means by which a satisfactory orbit can be calculated. The method is applied to three examples of differing difficulty: to calculate a preliminary orbit of Comet 122/P de Vico from 59 observations made during five days in 1995 a more difficult calculation of a possible new object with a poor distribution of observations Herget’s method fails for this example and finally a really difficult object, the Amor type minor planet 1982 DV (3288 Seleucus). For this last object use of L 1 regression becomes essential to calculate a preliminary orbit. For this orbit Laplace’s method compares favorably with Gauss’s.

A simple procedure to extend the Gauss method of determining orbital parameters from three to N points

A simple procedure is developed to determine orbital elements of an object orbiting in a central force field which contribute more than three independent celestial positions. By manipulation of formal three point Gauss method of orbit determination, an initial set of heliocentric state vectors r i and (dot>_) is calculated. Then using the fact that the object follows the path that keep the constants of motion unchanged, I derive conserved quantities by applying simple linear regression method on state vectors r i and (dot>_) . The best orbital plane is fixed by applying an iterative procedure which minimize the variation in magnitude of angular momentum of the orbit. Same procedure is used to fix shape and orientation of the orbit in the plane by minimizing variation in total energy and Laplace Runge Lenz vector. The method is tested using simulated data for a hypothetical planet rotating around the sun.

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Orbital Elements-Identification of method

I'm looking at a book that has a method for calculating the six orbital elements. I'm attaching the relevant pages. My knowledge about orbits is limited to knowing the meaning of the six elements. What I'd like to know is where did these equations come from, and is the author right when he says, "e from (87) and (88)" at the bottom of page 39. I suspect he meant 86 and 87.

I've heard of Gauss' and LaPlace's methods for determining orbits. Is the approach given one of those? He notes Herget as a source at the top of page 40. I have access to it, but is not a simple matter to browse through to find the approach on page 40.

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H. Debehogne R.R. Mourao G. Vieira (1984) ArticleTitle ‘Positions of comet Bowell (1982 b), asteroid 1982 DV and 51 Nemausa, March 1982, with the GPO, ESO-CHILE’ Acta Astronomica. 34 129–134 Occurrence Handle 1984AcA. 34..129D

H. Eichhorn (1990) ‘Choosing the correct model for the adjustment of observations’ C. Jaschek F. Murtagh (Eds) Errors, Bias and Uncertainties in Astronomy Cambridge University press Cambridge 133–152

P. Herget (1948) The Computation of Orbits Edwards Bros. Ann Arbor, MI

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N.J. Higham (2002) Accuracy and Stability of Numerical Algorithms EditionNumber 2 SIAM Philadelphia

P.S. Laplace (1966) Celestial Mechanics NumberInSeries Vol 1 Chelsea New York

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H.C. Plummer (1960) An Introductory Treatise on Dynamical Astronomy Dover New York 76–77

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Orbit determination has a long history, beginning with the prehistoric discovery of the planets and subsequent attempts to predict their motions. Johannes Kepler used Tycho Brahe's careful observations of Mars to deduce the elliptical shape of its orbit and its orientation in space, deriving his three laws of planetary motion in the process.

The mathematical methods for orbit determination originated with the publication in 1687 of the first edition of Newton's Principia, which gave a method for finding the orbit of a body following a parabolic path from three observations. [1] This was used by Edmund Halley to establish the orbits of various comets, including that which bears his name. Newton's method of successive approximation was formalised into an analytic method by Euler in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits by Lambert in 1761–1777.

Another milestone in orbit determination was Carl Friedrich Gauss' assistance in the "recovery" of the dwarf planet Ceres in 1801. Gauss's method was able to use just three observations (in the form of celestial coordinates) to find the six orbital elements that completely describe an orbit. The theory of orbit determination has subsequently been developed to the point where today it is applied in GPS receivers as well as the tracking and cataloguing of newly observed minor planets.

In order to determine the unknown orbit of a body, some observations of its motion with time are required. In early modern astronomy, the only available observational data for celestial objects were the right ascension and declination, obtained by observing the body as it moved in its observation arc, relative to the fixed stars, using an optical telescope. This corresponds to knowing the object's relative direction in space, measured from the observer, but without knowledge of the distance of the object, i.e. the resultant measurement contains only direction information, like a unit vector.

With radar, relative distance measurements (by timing of the radar echo) and relative velocity measurements (by measuring the Doppler effect of the radar echo) are possible using radio telescopes. However, the returned signal strength from radar decreases rapidly, as the inverse fourth power of the range to the object. This generally limits radar observations to objects relatively near the Earth, such as artificial satellites and Near-Earth objects. Larger apertures permit tracking of transponders on interplanetary spacecraft throughout the solar system, and radar astronomy of natural bodies.

Various space agencies and commercial providers operate tracking networks to provide these observations. See Category:Deep Space Network for a partial listing. Space-based tracking of satellites is also regularly performed. See List of radio telescopes#Space-based and Space Network.

Orbit determination must take into account that the apparent celestial motion of the body is influenced by the observer's own motion. For instance, an observer on Earth tracking an asteroid must take into account the motion of the Earth around the Sun, the rotation of the Earth, and the observer's local latitude and longitude, as these affect the apparent position of the body.

A key observation is that (to a close approximation) all objects move in orbits that are conic sections, with the attracting body (such as the Sun or the Earth) in the prime focus, and that the orbit lies in a fixed plane. Vectors drawn from the attracting body to the body at different points in time will all lie in the orbital plane.

If the position and velocity relative to the observer are available (as is the case with radar observations), these observational data can be adjusted by the known position and velocity of the observer relative to the attracting body at the times of observation. This yields the position and velocity with respect to the attracting body. If two such observations are available, along with the time difference between them, the orbit can be determined using Lambert's method, invented in the 18th century. See Lambert's problem for details.

Even if no distance information is available, an orbit can still be determined if three or more observations of the body's right ascension and declination have been made. Gauss's method, made famous in his 1801 "recovery" of the first lost minor planet, Ceres, has been subsequently polished.

One use is in the determination of asteroid masses via the dynamic method. In this procedure Gauss's method is used twice, both before and after a close interaction between two asteroids. After both orbits have been determined the mass of one or both of the asteroids can be worked out. [ citation needed ]


The Laplace method is a typical method of orbit computation for short arcs in the model of two-body problem. If the angular measurement is as accurate as 10 −4 – 10 −5 (2″ – 20″ ), then the oblateness of the Earth should be considered in the computation. In this paper, the theory and procedure of an improved Laplace method is given together with numerical examples. Besides orbit computation with multiple measurements, the method discussed in this paper is an efficient way to improve the accuracy of orbit computation, in general. It is the more useful the larger the oblateness.

Accuracy of Laplace Method for determining orbital elements - Astronomy

Qbasic programs
by Neil H. Jacoby,Jr.

Laplace.bas Program
LP3SCG.bas Program

These programs, which apply the principles of astrodynamics, compute a heliocentric orbit of any object in our Solar System given three observations of its right ascension,alpha, and its declination,delta and the times of these observations in Julian days. Each of these observations must be nearly equally spaced in time as much as possible and also the solar coordinants must be at the same time the observations are made. These solar coordinants are obtained from the Astronomical Almanac which is published by the U.S. Naval Observatory.

The solar coordinates can also be obtained by using a solar coordinate generator (applying the principles of astrodynamics),which the author developed and incorporated into his present Laplace program using three observations. This means that instead of inputting into the program the corresponding solar coordinates for each of the three observations, the user needs only to input the three Julian dates (in days for each observation),which cuts the inputting time almost in half. The author is naming this program LP3SCG, which is discussed in the Laplace method section of this report. The user is most welcome to download the listing of this program, LP3SCG, by clicking on LP3SCG.bas Program. However after the user downloads this program, it is not an executable program. In order to make LP3SCG.bas executable, the user must type the downloaded program LP3SCG into his computer using the Q-basic language. The author tested this program, LP3SCG, on a number of planetary orbits in our Solar System and the accuracy was found to be just as good as the original Laplace method when the solar coordinates (as obtained from The Astronomical Almanac) were inputted into the Laplace program without the solar coordinate generator.

In all these orbit determination programs the observed object's right ascension,alpha,and its declination,delta are all based on the earth's equator system of coordinates as well as the solar coordinates,X,Y and Z whereas the orientation angles i,Node and Omega of the orbit of the observed object are all based on the earth's ecliptic system of coordinates. The user of all these orbit determination programs should always remember that due to the earth's precession, the equator and equinox change through time. This earth's precession is known as the earth's general precession in longitude along the earth's ecliptic and its rate is 1.397 degrees per Julian century. Therefore the object's right ascension and declination referred to a particular equator and equinox of date must agree with the solar coordinates X,Y,Z referred to the same equator and equinox of that date.

If the orbit of the object is geocentric, then alpha,delta,X,Y,Z, and the orientation angles i,Node,and omega are all based on the earth's equator system of coordinates. Of course, the x-axis always points toward the vernal equinox in both the equator and ecliptic coordinate systems. In this case, the observation times are then measured in minutes. The same principles of astrodynamics apply in this case.

Returning to the heliocentric case, the times of each of these observations must be specified and the units are in days. The output of this program are the orbital elements which are a,the semi major axis, e , the eccentricity of the orbit and the orientation angles, which are ,i, the inclination of the orbit to the ecliptic plane,the ascending node OMEGA, and the argument of perihelion, omega. The final element, T, which is the time of the last epoch passage is not computed in this program but will be added later. The time interval between each of the three observations must be small enough to permit use of the f and g series so as to minimize truncation error but large enough for accurate orbit determination.

Next the input, which this program asks the user, is an smart initial guess for r, which is the distance in astronomical units (a.u.'s) from the sun to the observed object. The final value for r is determined from Newton's method of iteration where delta(r) is a correction to r in the Laplace method. This iteration continues until the ABS(delta(r)) is less than or equal to eps, where eps is the tolerance or limit of convergence. Usually eps is less than or equal to 0.000001 a.u., which the user must specify. On this same line of input the user must also specify eps2 which is the tolerance (in degrees) for the representation of the observations of alpha and delta. Usually eps2 is specified to be equal to .00001 of a degree of the actual alpha and delta. eps2 is used in the Leuschner differential correction portion of this program using three observations and the one for five observations.

Finally, the program asks the user to specify the maximum number of iterations,i.e. imax, and mu, where mu is defined as the ratio of the mass of the sun plus the observed object to the mass of the sun. Since the mass of the observed object, say an asteroid or comet, is negligible compared to mass of the sun, then mu can assumed to be equal to 1.0. In practice it is best to set imax = 10.

An smart guess for r can be found either graphically or as follows. First, most of the objects of concern to us in our Solar System are the asteroids which lie between the orbits of Mars and Jupiter, where r ranges between 1.52 a.u.'s and 5.23 a.u.'s from the sun approximately. Second, as long as the eccentricity of the observed asteroid's orbit is not too large, then r can be approximated by knowing the asteroid's synodic period, which can be observed or approximated from the observations. The synodic period is defined as the time interval between two successive conjunctions of the observed asteroid relative to the same stellar background as observed from earth. From knowing this synodic period, the observed asteroid's sidereal period can be computed from the following formula. First,let E = 1 year,sidereal period of the earth. Then let Psy = observed synodic period of asteroid,in years. Therefore the observed asteroid's sidereal period,P,is found from P = Psy/(Psy-1),where P is in years. Then knowing the observed asteroid's sidereal period, r can then be approximated by applying Kepler's Third Law which is P^2 = a^3, where P is in years and a is the semi- major axis in a.u.'s.

This program uses the Laplace method of preliminary orbit determination and the Leuschner differential correction method. These methods use the f and g series, (one of the basic concepts in astrodynamics), which is carried up to and including the sixth time derivative. The final output of all these orbit determination programs are the elements of the preliminary orbit and the final elements after the Leuschner differential correction is completed. If the three or the five observations are good, then convergence in the Leuschner differential corrections, should achieve the desired tolerance within say 0.0001 of a degree of the actual observed alpha and delta and within ten iterations.

If the declination of the observed object is negative, the degrees, minutes, and seconds are all have to be entered as negative numbers.

This program works the same way (applying the basic principles of astrodynamics) as the three observation-Laplace, except five observations instead of three must be inputted as well as the solar coordinates which are obtained at the same time the observations are made. As before, these solar coordinants are obtained from the Astronomical Almanac.
It was found that five observations instead of the usual three, that overall greater accuracy was obtained in determining the orbital elements, i.e., the semi major axis, the eccentricity, the inclination, the ascending node, and omega, the argument of perihelion, for low to moderate eccentricity orbits. In addition, this five-observation program outputs T, the time of last perihelion passage. As before, the orientation angles, the inclination i, angle of the ascending node, called node, and omega, the argument of perihelion, are all based on the earth's ecliptic coordinate system.

A very important finding so far, in this research on the use of the Laplace and Leuschner methods of orbit determination (in astrodhynamics), is that each of the three or five observations should be chosen such that the object's right ascension and declination be nearly equally spaced as much as possible in alpha and delta as well as their observation times for the best accuracy. It was found that if any two of the three or five observations of alpha and delta are spaced too close together, poor determination of the orbital elements resulted and no convergence in the differential corrections. This unfortunate situation can occur if any two or more of the observations is at or very near a point when the observed object is starting or ending retrograde motion as observed from earth. Therefore it is best to choose the observations during the time when the observed object is totally in either direct or retrograde motion and not at the beginning or end of retrograde motion. Also is was found that the most accurate heliocentric orbit determinaton resulted when the time interval between each succesive observation was 15 to 25 days for the asteroids between the orbits of Mars and Jupiter. For heliocentric objects whose orbits are beyond Jupiter's orbit this time interval should range between 25 to 50 days.

Another very important finding in this research was that the most accurate determination of the orbital elements occurred when the middle observation of the three and five observations was at or very near the object's opposition point,i.e.,its elongation angle was 180 degrees. (The elongation angle is defined as the angle measured from the direction to the sun to the direction to the object as observed from earth.) This was the finding for low to low-moderate eccentricity heliocentric orbits. Also at or near the opposition point the object's right ascension and declination change most rapidly, where the object's motion is nearly perpendicular to the line of sight. In the case of high eccentricity orbits, such a comet's orbit, the three or five observations should be chosen such that the comet's change in right ascension and declination are maximum. This maximum change may not necessarily be at or near the opposition point however.

I conducted further research in determining the elements of the orbits of comets using three and five observations of right ascension and declination and the important result of this research was that using three observations instead of five gave a far more accurate determination of the elements of the comet's orbit. The reason for this is that there was not only rapid convergence within ten iterations in the differential corrections to the desired tolerance, but also a very accurate prediction of the comet's right ascension and declination at some future time. This prediction accuracy was found to be within 0.01 of degree of the actual comet's right ascension and declination in almost all cases.

On the other hand, when five observations of right ascension and declination were used, poor results were obtained because there was no convergence in the differential corrections no matter how many iterations were allowed by the user. Apparently, the additional two observations corrupted the final results because of the very high eccentricity of the comet's orbit. These unfortunate results occurred in almost all cases. Therefore it is strongly advised that the user use three observations and not five in determining the elements of very high eccentricity orbits of comets.

The orbits of Comet Hale-Bopp and Comet S4 Linear were used in this research because their orbits have a very high eccentricity,i.e., 0.9 < e < 1.0.

Finally in cases of orbits ranging from low to high eccentricity heliocentric orbits, all of the three or five observations of the object should have an elongation angle greater than 120 degrees east of the sun and greater that 120 degrees west of the sun for the best visibility of the object being observed.

This program computes the heliocentric object's right ascension ,alpha, and declination, delta, given the object's orbital elements, the semi-major access a, the eccentricty e, the inclination i,the angle of the ascending node, called node, the argument of perihelion, and the mean anomaly, M, at the time of the reference observation. The angles i, Node, and omega all expressed in degrees. If the 3 observation program is used, the reference observation is the second observation, and if the 5 observation program is used, the reference observation is the 3rd observation. The desired time of the future observation must be specified and the solar coordinants at this desired time, must also be specified. As before, the solar coordinants are obtained from the Astronomical Almanac. The desired time of the future observation is measured from the time of the second observation if 3 observations are used and if 5 observations are used, this desired time is measured from the time of the 3rd observation. As before, the time is measured in Julian days.
The user of this program, should remember, that the orientation angles, i, node, and omega, are based on the ecliptic system of coordinants and the right ascension and declination are based on the equator system of coordinates.

Kepler.bas Program
This program solves Kepler's equation when the Mean Anomaly,M, and eccentricity, e, are given and one needs to solve for the eccentric anomaly,E. E, is solved by applying Newton's method of iteration. In most cases, M and e are given and not the eccentric anomaly,E.

Kepler's Equation, M=E-e*sinE, is also used in the closed form of the f and g series. Kepler's Equation is useful when one is given the orbital elements and one needs to predict the object's, right ascension and declination at some given future or past time.

When running this program, the program will ask the user to input the eccentricity,e,and the mean anomaly M in radians, and the specified limit of convergence or tolerance,eps. Usually eps is 0.000001 of a radian. If e is between 0 and 1 the user will be directed to the elliptical part of the program, and if e is greater than 1 the user will be directed to the hyperbolic part of the program.

The case of parabolic orbits, where e = 1 and the semi-major axis a is infinity, is not presented here because these types of orbits exist only in theory and not in reality.

In all these heliocentric orbit determination programs, which I developed, it should be noted that these programs involve only Keplerian (two-body) orbits and that no perturbations are included. The orbital elements and the ephemeris of a heliocentric object, which are given in the Astronomical Almanac, are determined by including the perturbations of the nearby planets in our solar system. Therefore the object's orbit is not Keplerian. Even with this Keplerian approximation, however, it was found that as long as the observations are good, the results of all these orbit determination and alpha,delta position-prediction programs ranged from good to excellent. However if a highly accurate ephemeris of an object is required, the perturbations of the planets on this object must be included.

First edition, second issue, but the first to contain the Supplement, of the invention of the method of least squares, “the automobile of modern statistical analysis” and the origin of “the most famous priority dispute in the history of statistics” (Stigler). “ In 1805 Legendre published the work by which he is chiefly known in the history of statistics, Nouvelles méthodes pour la détermination des orbites des comètes. At eighty pages this work made a slim book, but it gained a fifty-five page supplement (and a reprinted title page [i.e., the offered issue]) in January of 1806 . For stark clarity of exposition the presentation is unsurpassed it must be counted as one of the clearest and most elegant introductions of a new statistical method in the history of statistics. In fact, statisticians in the succeeding century and three-quarters have found so little to improve upon that … the explanation of the method could almost be from an elementary text of the present” (ibid.). “ The great advances in mathematical astronomy made during the early years of the nineteenth century were due in no small part to the development of the method of least squares. The same method is the foundation for the calculus of errors of observation now occupying a place of great importance in the scientific study of social, economic, biological, and psychological problems. Gauss says in his work on the Theory of Motions of the Heavenly Bodies (1809) that he had made use of this principle since 1795 but that it was first published by Legendre. The first statement of the method appeared as an appendix entitled “Sur la méthode des moindres quarrés” in Legendre’s Nouvelles méthodes pour la détermination des orbites des comètes , Paris 1805” (Wolberg, ‘The Method of Least Squares,’ in Designing Quantitative Experiments, 2010). This is a rare book on the market, ABPC/RBH listing just four copies of either issue since 1941.

“[By] 1800 the principle of combining observational equations had evolved, through work of [Tobias] Mayer and [Pierre-Simon] Laplace, to produce a convenient ad hoc procedure for quite general situations. We have also seen how the idea of starting with a mathematical criterion had led, in work of [Roger] Boscovich and Laplace, to an elegant solution suitable for simple linear relationships involving only two unknowns. The first of these approaches developed through problems in astronomy the second was (at least in these early years) exclusively employed in connection with attempts to determine the figure of the earth. These two lines came together in the work of a man who, like Laplace, was an excellent mathematician working on problems in both arenas − Adrien Marie Legendre.

“Legendre came to deal with empirical problems in astronomy and geodesy at a time when [methods] had been developed separately in the two fields. It was also a time when a half-century's successful use of these methods had seen a change in the view scientists took of them − from Euler’s early belief that combination of observations made under different conditions would be detrimental to the later view of Laplace that such combination was essential to the comparison of theory and experience. Legendre brought a fresh view to these problems and it was Legendre, and not Laplace, who took the next important step.

“Legendre did not hit upon the idea of least squares in his first exposure to observational data. From 1792 on he was associated with the French commission charged with measuring the length of a meridian quadrant (the distance from the equator to the North Pole) through Paris. One of the major projects initiated by the National Convention in the early years after the French Revolution had been the decision in 1792 to change the ancient system of measurement by introducing the metric system as a new order, toppling existing standards of measurement in an action symbolic of the French Revolution itself. The basis of the new system was to be the meter, defined to be 1/10,000,000 of a meridian quadrant. It remained for French science to come up with a new determination of the length of this arc. In keeping with the nationalism that inspired the enterprise, the determination was to be based only on new measurements made on French lands. To this end an arc of nearly 10', extending from Montjouy (near Barcelona) in the south to Dunkirk in the north, was measured in 1795. By 1799 the complex task of reducing the multitude of angular measurements to arc lengths had been completed by J. B. J. Delambre and P. F. A. Mechain … In early 1799, Delambre published an extensive discussion of the theoretical results underlying the reduction of the raw data on this arc. This volume is prefaced by a short memoir by Legendre that is dated 9 Nivose, an VII (30 December 1798) and indicates that Legendre did not have the method of least squares at that time …

“The occasion for Legendre’s reconsideration of observational equations, and for the appearance of the method of least squares, was the preparation in 1805 of a memoir on the determination of cometary orbits. The memoir is a scant seventy-one pages (excluding the appendix) and, aside from a few brief remarks at the end of the preface, the method of least squares makes no appearance before page 64. Even this first mention of least squares seems to be an afterthought because, after presenting an arbitrary solution to five linear equations in four unknowns (one that assumed that two equations held exactly and two of the unknowns were zero), Legendre wrote that the resulting errors were of a size “quite tolerable in the theory of comets. But it is possible to reduce them further by seeking the minimum of the sum of the squares of the quantities E', E", E"'” (Nouvelles méthodes, p. 64). He then reworked the solution in line with this principle. It seems plausible that Legendre hit on the method of least squares while his memoir was in the later stages of preparation, a guess that is consistent with the fact that the method is not employed earlier in the memoir, despite several opportunities. It is clear, however, that Legendre immediately realized the method’s potential and that it was not merely applications to the orbits of comets he had in mind. On pages 68 and 69 he explained the method in more detail (with the word minimum making five italicized appearances, an emphasis reflecting his apparent excitement), and the memoir is followed on pages 72-80 by the elegant appendix. The example that concludes the appendix reveals Legendre’s depth of understanding of his method (notwithstanding the lack of a formal probabilistic framework). It also suggests that it was because Legendre saw these problems of the orbits of comets as similar to those he had encountered in geodesy that he was inspired to introduce his principle and was able to abstract it from the particular problem he faced. Indeed, the example he chose to discuss was not just given as an illustration, it was a serious return to what must have been the most expensive set of data in France − the 1795 measurements of the French meridian arc from Montjouy to Dunkirk” (Stigler).

The appendix is dated 6 March 1805. Legendre states the principle of his method of “distributing the errors among the equations” on pp. 72-3: “Of all the principles that can be proposed for this purpose, I think there is none more general, more exact, or easier to apply, than that which we have used in this work it consists of making the sum of the squares of the errors a minimum. By this method, a kind of equilibrium is established among the errors which, since it prevents the extremes from dominating, is appropriate for revealing the state of the system which most nearly approaches the truth.”

“The clarity of the exposition no doubt contributed to the fact that the method met with almost immediate success. Before the year 1805 was over it had appeared in another book, Puissant’s Traite de geodesie and in August of the following year it was presented to a German audience by von Lindenau in von Zach’s astronomical journal, Monatliche Correspondenz. Ten years after Legendre's 1805 appendix, the method of least squares was a standard tool in astronomy and geodesy in France, Italy, and Prussia. By 1825 the same was true in England. The rapid geographic diffusion of the method and its quick acceptance in these two fields, almost to the exclusion of other methods, is a success story that has few parallels in the history of scientific method” (ibid.).

An unintended consequence of Legendre’s publication was a protracted priority dispute with Carl Friedrich Gauss, who claimed in 1809 that he had been using the method since 1795. In Theoria motus corporum coelestium (1809, Section 186), “Gauss writes: “Our principle, which we have made use of since the year 1795, has lately been published by Legendre in the work Nouvelles méthodes pour la détermination des orbites des comètes, Paris, 1806, where several other properties of this principle have been explained, which, for the sake of brevity, we here omit.”

“The Theoria motus was originally written in German and completed in the autumn of 1806. In July 1806 Gauss had for some weeks at his disposal a copy of Legendre’s book before it was sent to Olbers for reviewing. It was not until 1807 that Gauss finally found a publisher, who, however, required that the manuscript should be translated into Latin. Printing began in 1807 and the book was published in 1809. Gauss had thus ample time to elaborate on the formulation of the relation of his version of the method of least squares to that of Legendre, if he had wished so.

“Gauss’s use of the expression “our principle” naturally angered Legendre who expressed his feelings in a letter to Gauss dated May 31, 1809. The original is in the Gauss archives at Gottingen it contains the following statement:

“It was with pleasure that I saw that in the course of your meditations you had hit on the same method which I had called Méthode des moindres quarrés in my memoir on comets. The idea for this method did not call for an effort of genius however, when I observe how imperfect and full of difficulties were the methods which had been employed previously with the same end in view, especially that of M. La Place, which you are justified in attacking, I confess to you that I do attach some value to this little find. I will therefore not conceal from you, Sir, that I felt some regret to see that in citing my memoir p. 221 you say principium nostrum quojam inde ab anno 1795 usi sumus etc. There is no discovery that one cannot claim for oneself by saying that one had found the same thing some years previously but if one does not supply the evidence by citing the place where one has published it, this assertion becomes pointless and serves only to do a disservice to the true author of the discovery.”

“It therefore became important for Gauss to get his claim of having used the method of least squares since 1795 corroborated. He wrote to Olbers in 1809 asking whether Olbers still remembered their discussions in 1803 and 1804 when Gauss had explained the method to him. In 1812 he again wrote to Olbers saying “Perhaps you will find an opportunity sometime, to attest publicly that I already stated the essential ideas to you at our first personal meeting in 1803.” In an 1816 paper Olbers attested that he remembered being told the basic principle in 1803.

In 1811 Laplace brought the matter of priority before Gauss, who answered that “I have used the method of least squares since the year 1795 and I find in my papers, that the month of June 1798 is the time when I reconciled it with the principle of the calculus of probabilities.” [In his Théorie analytique des probabilités (1812)] Laplace writes that Legendre was the first to publish the method, but that we owe Gauss the justice to observe that he had the same idea several years before, that he had used it regularly, and that he had communicated it to several astronomers” (Hald, pp. 394-5).

“The heat of the dispute never reached that of the Newton − Leibniz controversy, but it reached dramatic levels nonetheless. Legendre appended a semi-anonymous attack on Gauss to the 1820 version of his Nouvelles méthodes pour la détermination des orbites des comètes, and Gauss solicited reluctant testimony from friends that he had told them of the method before 1805. A recent study of this and further evidence suggests that, although Gauss may well have been telling the truth about his prior use of the method, he was unsuccessful in whatever attempts he made to communicate it before 1805. In addition, there is no indication that he saw its great general potential before he learned of Legendre’s work. Legendre’s 1805 appendix, on the other hand, although it fell far short of Gauss's work in development, was a dramatic and clear proclamation of a general method by a man who had no doubt about its importance” (Stigler).

Nouvelles méthodes was first issued in 1805, and reissued in January 1806 with a reset title-page and a 55-page supplement. In the supplement Legendre makes some improvements to the methods he had introduced for determining the orbital elements of comets. When these methods had been applied to the observations by Alexis Bouvard of a comet that appeared in 1805 (now called 2P/Encke), Legendre had found that the elements were “not sufficiently exact”. He traced the problem to a coefficient in one of his equations which, if it happened to be very small (as it was for the observations relating to Bouvard’s comet), could lead to large errors in the calculation of the orbital elements. In the supplement he introduced a modification of his method which avoided this problem. In the second part of the supplement he applied his new method to Bouvard’s observations of a second comet that had appeared in 1805. Bouvard is best known for his prediction of the existence of a planet beyond Uranus, but he died before he could complete his investigations and the discovery of Neptune was made by Adams and Le Verrier.

“Legendre (born 18 September 1752, died 10 January 1833) was a mathematician of great breadth and originality. He was three years Laplace’s junior and succeeded Laplace successively as professor of mathematics at the École Militaire and the École Normale. Legendre’s best-known mathematical work was on elliptic integrals (he pioneered this area forty years before Abel and Jacobi), number theory (he discovered the law of quadratic reciprocity), and geometry (his Éléments de géométrie was among the most successful of such texts of the nineteenth century). In addition, he wrote important memoirs on the theory of gravitational attraction. He was a member of two French commissions, one that in 1787 geodetically joined the observatories at Paris and Greenwich and one that in 1795 measured the meridian arc from Barcelona to Dunkirk, the arc upon which the length of the meter was based. It is at the nexus of these latter works in theoretical and practical astronomy and geodesy that the method of least squares appeared” (Stigler).

Hald, A History of Mathematical Statistics from 1750 to 1930, 1998 Stigler, A History of Statistics, 1986 (see pp. 12-15, 55-61 & 145-6).

4to (266 x 212 mm), pp. [2] viii, 80 and 1 engraved plate 1-55 (supplement), uncut in original pink plain wrappers, very light sporring to a few leaves, overall a very fine and untouched copy.

Preliminary Orbit Determination – 2-Body Problem

When we are dealing with two bodies, we can determine their orbital paths exactly using a certain set of orbital elements. If you want to learn more about these orbital elements you can find my post on them here and how to use them to determine an orbital path here. If we have a certain set of orbital elements, assuming Keplerian motion, we can tell where the satellite was, is, and will be for all time. But what if we don’t have those orbital elements? That’s where orbit determination comes in.

First, let’s think of some cases where we wouldn’t have direct knowledge of the orbital elements. One case would be if our rocket had a malfunction and placed our satellite in the wrong orbit. Another would be if we were trying to track another countries spy satellite. A historical need for orbit determination, and one that let many of the earliest mathematical advances, was trying to determine the orbits of the other planets in our solar system.

Orbit Determination is a Big Topic

Now that we know there are many uses, both for past and present problems, is there one correct way to do orbit determination? No.

There are hundreds of different orbit determination schemes. They differ based on the necessary inputs, the constraints of the problem, and the solving accuracy they provide. I could spend the next year just wring posts on orbit determination. Comparing and contrasting the different methods, deriving some, leaving others as vague ideas, and still have topics to talk about. It’s a large topic and the rest of this post will only scratch the surface of three extremely basic methods.

The three methods I’m highlighting are from section 3.7 of Battin’s Introduction to Astrodynamics.

Orbits from 3 Co-planar Positions

The first of the three methods requires us to have 3 co-planar position measurements in polar coordinates. That means we have r1,θ1,r2,θ2,r3, and θ3. We return to our equation of orbit we found in a prior post which is.

By appealing to angle difference identities we can transform our equation of orbit to the following

Let’s now introduce two variables P and Q that are defined as

Which lets us linearize our equation of orbit with

Through some algebraic manipulation we get the following

We have 3 unknowns in the above equation, p,P, and Q. This is why we need 3 different measurements. 3 equations, 3 unknowns, I leave determining the values for p,P, and Q up to a suitable method of your choosing. Once you have these three determining the eccentricity and ω are trivial. Hint think of using trig identities and come out with the following.

Orbit Determination with 3 Position Vectors (Gibb’s Method)

Before we began with 3 co-planar positions. But what if we are not in the same orbital plane as the object we are observing? This method is like the last method, just with vector algebra. Here, we have 3 position vectors, r1,r2, and r3. Because we are dealing with Keplerian orbits, those 3 measurements can be assumed to be to co-planar points. First calculate alpha and beta

We can now find the semi-latus rectum, p, using the following equation

Now in order to find the eccentricity lets take the cross product of n and e

Because we also know that e is normal to n we have the following relationship

Combining these last two relations we get

This is known as Gibb’s method and is often used as an initial orbit determination method. Once you know the rough orbital elements you can narrow down your solution space and use this as your initial estimate in more powerful iterative orbit determination algorithms.

Approximate Orbit from Three Position Fixes

While the above two methods will give us the exact orbital elements if we have perfect inputs, they both fall victim to increased numerical errors durring calculation if the angles between the measurements are small. Because we’ve only been looking at 3 position vectors, we have been using geometry to determine the orbital elements. If we begin to also record the times at which we take our measurements, we can also use our understandings of the dynamics of the system to improve our estimates of the system.

Now lets say we have measurement vectors r1,r2,r3 and the time of respective measurement t1,t2,and t3. These 6 pieces of information can be related by a fifth order power series expansion.

Take it’s derivative to get

one more derivative where A is acceleration

Note that we had 6 pieces of information and we now have 6 unknown coefficients from a0 to a5.

Let’s also define a set of time intervals just to make our equations a bit tidier

Now, I’m going to give you the next step and it might not be obvious where we are going with this so take a moment to stop and think about what it looks like we’re doing. replace t in the power series successively by -τ3,0, and τ1. You should get the following 7 equations.

Note: It reminded me of taking a finite difference. Namely a central difference. We’re finding the velocity at the second point by using the time it took to go from the first point to the third point and incorporating the system dynamics.

If we solve this system of equations for v2 we get the following

Now, this looks promising. We get v2 and by combining that with r2 we can pull out the orbital elements in the classical way. Unfortunately, this solution happens to have a problem. It doesn’t work when τ1 = τ3. Try plugging those in and see what you get! This mea ns that we cant use this method with regularly spaced measurements. Additionally because we live in the real world of numerical accuracy, any measurements that are nearly regularly spaced will suffer from large computational error. I haven’t done a sensitivity analysis but I would expect this error to grow exponentially as we get closer to regularly spaced measurements.

Luckily for us, Samuel Herrick determined a work around. This article is already quire long so I will just present it here and leave the derivation to the determined reader (The skeleton of the derivation is presented on Page 135 of Battin’s Intro to Astrodynamics if you get stuck).

Again once we have v2 we can use it with r2 to determine the orbital elements using formulas we derived here.

How we get these Measurements?

Radar stations on the ground obtaining the position vectors of a satellite in orbit

Much like how this post is a very basic introduction to orbit determination, I will only briefly be covering the most common way of how we get these measurements from Earth-based observatories. We can ping the satellite with an electromagnetic wave and measure how long it takes to get there and come back. We can estimate the velocity of the signal in the earth’s atmosphere and the vacuum of space so we can figure out the range rather well. If we have two ground stations that receive the same message, we can determine the satellites relative location by looking at the difference in the ping’s time of flight. When done with a laser this is called laser range-finding and when done with radio waves it’s called radar.

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