# Determining distance from semi-major axis and eccentricity

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I am trying to obtain the distance covered by an object in orbit around the earth within a specified amount of time past its perigee passage. The object is in an elliptical orbit with eccentricity 0.2 and has a semi-major axis of 9600km. Where would I look to find the objects position 90 mins after it passes its perigee? Thanks for any advice

Supposing that the mass of the object is negligible compared with the mass of the Earth, you can derive the orbital period $T$ from the 3rd Keplero's law:

$frac{T^2}{a^3} = frac{4pi^2}{G(m_E + m_b)} approx frac{4pi^2}{Gm_E},$

where $a$ is the semi-major. With $T$, for each time istant you also know the mean anomaly $M$, given by (suppose $t = 0$ at perigee):

$M(t) = frac{2pi}{T}t$.

Solving numerically the Keplero's equation for the eccentric anomaly $E$ (where $e$ is the eccentricity)

$M = E - esin E$

and then use the following equation to derive the true anonaly $u$, which is the angle between the direction of periapsis and the current position of the body, as seen from the Earth:

$cos u = frac{cos E - e}{1 - ecos E}$ and $sin u = frac{sqrt{1-e^2}sin E}{1 - ecos E}$.

The distance from the Earth is just given by the orbit equation

$r = frac{a(1-e^2)}{1 + ecos u}$.

If I'm not wrong with calculation, it should be:

$T = 9364 s = 2.6 h.$

$M(90min) = 207.60°$

$E(90min) = 203.11°$

$u(90min) = 198.95°$

$r = 11'366 km$

To get the covered distance you should calculate the line integral of the orbit equation.

@Dario_Panarello I think what you're saying is correct for a non-rotating geocentric observer, but the earth's radius is fairly large compared to the orbit, so I don't think the geocentric approximation works well.

I don't have an answer, but I think the solution looks something like this:

where the light blue circle is the Earth, the small blue dot is the geocenter, the black dot in the blue circle is the center of the ellipse, the black ellipse is the orbit of the satellite, and the two black dots on the ellipse are the perigee and final positions of the satellite respectively.

Even allowing for the Earth's rotation in the 90 minute timefram, I'm not sure anyone on Earth could see the satellite both at perigee and at its final location.

I'm working on a more complete answer at https://github.com/barrycarter/bcapps/blob/master/STACK/bc-solve-astronomy-13635.m

## Determining distance from semi-major axis and eccentricity - Astronomy

The semi-minor axis, b, is half of the shortest diameter of an ellipse. Together with the semi-major axis, a, and eccentricity, e, it forms a set of related values that completely describe the shape of an ellipse:

In cartesian coordinates (x,y), an ellipse is the solution of:

or in polar coordinates (r,θ):

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## Semi-major axis

In geometry, the major axis of an ellipse is the longest diameter: a line (line segment) that runs through the center and both foci, with ends at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse essentially, it is the radius of an orbit at the orbit's two most distant points. For the special case of a circle, the semi-major axis is the radius. One can think of the semi-major axis as an ellipse's long radius.
The length of the semi-major axis a of an ellipse is related to the semi-minor axis' length b through the eccentricity e and the semi-latus rectum , as follows:

The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex (turning point) of the hyperbola.
A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping fixed. Thus and tend to infinity, a faster than b.

## Ellipse: Eccentricity

A circle can be described as an ellipse that has a distance from the center to the foci equal to 0. The greater the distance between the center and the foci determine the ovalness of the ellipse. Thus the term eccentricity is used to refer to the ovalness of an ellipse.

If an ellipse is close to circular it has an eccentricity close to zero. If an ellipse has an eccentricity close to one it has a high degree of ovalness.

Figure 1 shows a picture of two ellipses one of which is nearly circular with an eccentricity close to zero and the other with a higher degree of eccentricity.

The formal definition of eccentricity is:

ECCENTRICITY OF AN ELLIPSE:

The eccentricity (e) of an ellipse is the ratio of the distance from the center to the foci (c) and the distance from the center to the vertices (a).

As the distance between the center and the foci (c) approaches zero, the ratio of c a approaches zero and the shape approaches a circle. A circle has eccentricity equal to zero.

As the distance between the center and the foci (c) approaches the distance between the center and the vertices (a), the ratio of c a approaches one. An ellipse with a high degree of ovalness has an eccentricity approaching one.

Let's use this concept in some examples:

Step 1: Determine the values for the distance between the center and the foci (c) and the distance between the center and the vertices (a).

Length of a: The given equation for the ellipse is written in standard form. Since the major axis is 2a and the smaller minor axis is 2b, then a 2 > b 2 , therefore a 2 = 16.

Length of c: To find c the equation c 2 = a 2 + b 2 can be used but the value of b must be determined. From our discussion above, b 2 = 9. Find b and solve for c.

c 2 = 4 2 − 3 2 → c 2 = 7 → c = 7

Step 2: Substitute the values for c and a into the equation for eccentricity.

## Determining distance from semi-major axis and eccentricity - Astronomy

Support for the Heliocentric Model

Although Copernicus set down the basic principles of the heliocentric model, it was regarded as simply an alternative way of thinking about the universe, without any certainty that the Earth really moved. Two later scientists, Galileo and Kepler, gave several strong arguments in favor of the heliocentric model.

Galileo gave observational evidence:

 Moons of Jupiter: gave clear evidence of smaller objects circling larger objects (although no one knew why--see also) Phases of Venus: gave clear evidence that Venus circles the Sun Sunspots on the Sun: gave clear evidence that heaven is not "perfect" Craters and Mountains on the Moon : gave clear evidence that the Moon is another "world"
 Kepler's Three Laws (qualitative version) First Law: Planets travel in elliptical orbits with the Sun at one focus Second Law: Planets move more slowly in their orbits when far from the Sun than when close to the Sun Third Law: Planets with larger orbits move more slowly than planets with smaller orbits.
• Sep 18 - Mercury Greatest Eastern Elongation (26 Degrees)
• Oct 29 - Mercury At Its Greatest Western Elongation (18 Degrees)
• Jan 12 - Mercury At Its Greatest Eastern Elongation (19 Degrees)
• Feb 21 - Mercury at Greatest Western Elongation (27 Degrees)
• May 04 - Mercury Greatest Eastern Elongation (20 Degrees)
• Jun 21 - Mercury at Greatest Western Elongation (22 Degrees)

By measuring Mercury's greatest elongation from many places along Earth's orbit, any variation in distance of Mercury from the Sun can be determined.

The situation for the outer planets is harder, but can be done. Kepler did it by observing the outer planet at pairs of times separated by one sidereal rotation of the planet. Here is an outline of how this is done, for one pair of observations of Mars:

 Determining distance of outer planet from Sun: Take two measurements of the elongation (angle from the Sun) of Mars, one sidereal period (687 days) apart. Earth is at location E' at the time of the first observation, goes once around its orbit and arrives back at location E (almost completing two orbits) after Mars has gone around once. The figure below shows the situation at a larger scale, with the angles labeled. The two elongation angles are e and e ', and we also know the angle n, which is just the number of degrees less than two full orbits that the Earth makes in 687 days. You should be able to show that n = 42.89 degrees. Since triangle D SEE' is isoceles, we can determine a , (you should be able to show that it is 68.56 degrees) and hence the length EE' (use the Law of Sines to show that it is 0.73 AU). Subtract a from e and e ', which allows us to solve for triangle D EPE'. Finally, using the Law of Cosines for triangle D SPE', we can determine the distance r.

• Semi-major axis a = half of the long axis of ellipse
• Semi-minor axis b = half of the short axis of ellipse
• Eccentricity e = distance of focus F from center, in units of a . The eccentricity ranges from 0 (a circle) to 1 (a parabola).
• Sum of distances of a point on the ellipse from the two foci ( r and r ') is a constant: r + r ' = 2 a .
• As Kepler found, planets have an orbit that is an ellipse with the Sun at one focus. In the above drawings, the Sun would be at focus F. There would be nothing at all at focus F '.
• When the planet is at position A on the ellipse (closest to the Sun), it is at perihelion .
• When the planet is at position A' in its orbit (farthest from the Sun), it is at aphelion

Careful measurements (by Tycho Brahe) were used by Kepler to measure the orbits of the planets and show that The Earth Moves! Kepler's Laws prove quantitatively that the true situation is given by a heliocentric model in which the planets revolve around the Sun. Kepler showed that the planets move in ellipses. We learned the important terms for ellipse characteristics: focus, semi-major axis, semi-minor axis, eccentricity, and for elliptical orbits, the terms perihelion and aphelion. We examined the characteristics and mathematical formulas for conic sections (ellipse, parabola, and hyperbola).

Kepler thought that there was a clockwork universe of crystal spheres, arranged harmonically (in certain ratios, related to regular geometric solids), but we will see next time that Newton was able to put it all on a firm physical foundation with his Law of Universal Gravitation.

## Calculate flight path angle given semi-major axis, eccentricity and distance from the focal point

One method of calculating the angle involves using the ellipse reflection law. Light from one focus reflects off the ellipse into the other focus.

Thus in the picture below (by the author), the radial vector from the focus $F_1$ is reflected at $P$ onto the second focus $F_2$ , forming a triangle whose third side is the line between the foci.

Your flight angle $psi$ is the angle of incidence between the radial vector and the dashed line which us perpendicular to the (tangential) flight path, and also the angle of reflection towards the second focus. Thus the angle in the triangle at $P$ measures $2psi$ .

We now apply the Law of Cosines to this triangle:

In a circular orbit you have $epsilon=0$ and $r=alpha$ , forcing the cosine to $1$ as expected. For an elliptical orbit when you're on the minor axis ( $r=alpha$ ) you get a formula for the maximal flight angle:

## Determining distance from semi-major axis and eccentricity - Astronomy

for any ellipse:
focus (plural: foci) : points of symmetry inside the ellipse
major axis : the long axis of the ellipse
minor axis: the short axis of the ellipse
semi-major axis (a): half of the length of the major axis
eccentricity: a measure of the departure from circularity

assuming an elliptical orbit with the primary body at one focus:
periapse: the closest point on the ellipse to the focus occupied by the primary
apoapse: the most distant point on the ellipse from the focus occupied by the primary
inclination: the angle between the elliptical orbit and a reference plane

the equation for an ellipse is: r = a(1-e 2 )/[1 + e cos(theta)]
r = distance from orbiting body to the primary focus
a = semi-major axis
e = eccentricity
theta = angular position in orbit (theta = 0 defined at periapse)

What are the Earth's closest and furthest orbital distances from the Sun (assume you know a, e)?

closest: at perihelion (periapse around the sun), theta = 0
thus, rperi = a(1-e) = 1 AU * (1 - 0.017) = 0.983 AU
furtherst: at aphelion (apoapse around the sun), theta = 180
thus, raph = 1(1+e) = 1 AU * (1 + 0.017) = 1.017 AU

Given that the amount of light received (I) from the Sun changes with the square of the distance from the Sun, what is the relative change in solar insolation (amount of sunlight received) from perihelion to aphelion?

I_perihelion / I_aphelion = (1.017 / 0.983) 2 = 1.070

so the Earth receives

7% more sunlight at perihelion (about January 6) than at aphelion (about July 6).

## Average Orbital Distance

If a planet is orbiting the Sun with a semi-major axis, a, and orbital eccentricity, e, it is often stated that the average distance of the planet from the Sun is simply a. This is only true for circular orbits (e = 0) where the planet maintains a constant distance from the Sun, and that distance is a.

Let’s imagine a hypothetical planet much like the Earth that has a perfectly circular orbit around the Sun with a = 1.0 AU and e = 0. It is easy to see in this case that at all times, the planet will be exactly 1.0 AU from the Sun.

If, however, the planet orbits the Sun in an elliptical orbit at a = 1 AU and e > 0, we find that the planet orbits more slowly when it is farther from Sun than when it is nearer the Sun. So, you’d expect to see the time-averaged average distance to be greater than 1.0 AU. This is indeed the case.

The Earth’s current osculating orbital elements give us:

a = 0.999998 and e = 0.016694

Earth’s average distance from the Sun is thus:

Mercury, the innermost planet, has the most eccentric orbit of all the major planets:

Your answer is meaningless unless if you specify what UNITS it is in.

Under the assumption that you meant km, your answer is clearly wrong.

Your answer is meaningless unless if you specify what UNITS it is in.

Under the assumption that you meant km, your answer is clearly wrong.

Can anyone help me with this?

Can anyone help me with this?

Sure. Can you show us the steps you used to calculate the perihelion distance?

First you need to compute 1-e. Then you need to multiply the result of that by a (the semi-major axis of the orbit).

Here's a hint: in this example, e is a very small number, agreed? Therefore, 1-e should be fairly close to 1. If that's true, then a*(1-e) should be fairly close to a. The answer you get should be close to the length of the semi-major axis. In other words, the closest earth-sun distance does not deviate very much from the average earth-sun distance. The fact that the earth-sun distance doesn't change very much as it goes around its orbit suggests that the orbit does not not deviate too much from circularity. In other words, it is not very elliptical (remember that e = 0 would be a perfect circle, so e very small means close to circular).

Calculate the Earth-Sun distance during perihelion (at Earth’s closest approach) The Earth’s orbit has a semi-major axis of a = 1.496×108 km and
eccentricity of e = 0.017. Is Earth’s orbit far from circular? Explain.

The formula to be used is: rP = a(1 − e)

I went (1 - e) which would be 1 - 0.017 Correct?

Then I just went a x 0.017 = 2.74

Calculate the Earth-Sun distance during perihelion (at Earth’s closest approach) The Earth’s orbit has a semi-major axis of a = 1.496×108 km and
eccentricity of e = 0.017. Is Earth’s orbit far from circular? Explain.

The formula to be used is: rP = a(1 − e)

I went (1 - e) which would be 1 - 0.017 Correct?

Then I just went a x 0.017 = 2.74

Yes, (1 - e) = (1 - 0.017) = 0.983. That is correct.

It's the next part that doesn't make any sense. The formula is a*(1-e), but for some reason you have written down a*e. I don't understand why. Not only that, but the answer doesn't make sense either. Please get into the habit of including units in all of your calculation steps. It is crucial.

## Ellipses in physics

### Elliptical reflectors and acoustics

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves created by that disturbance, after being reflected by the walls, will converge simultaneously to a single point — the second focus. This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.

Similarly, if a light source is placed at one focus of an elliptic mirror, all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolate spheroid), this property will hold for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear fluorescent lamp along a line of the paper such mirrors are used in some document scanners.

Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. The effect is even more evident under a vaulted roof shaped as a section of a prolate spheroid. Such a room is called a whisper chamber. The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the U.S. Capitol (where John Quincy Adams is said to have used this property for eavesdropping on political matters), at an exhibit on sound at the Museum of Science and Industry in Chicago, in front of the University of Illinois at Urbana-Champaign Foellinger Auditorium, and also at a side chamber of the Palace of Charles V, in the Alhambra.

### Planetary orbits

In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.

More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus.

Keplerian elliptical orbits are the result of any radially-directed attraction force whose strength is inversely proportional to distance. Thus, in principle, the motion of two oppositely-charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation and quantum effects which become significant when the particles are moving at high speed.)

### Harmonic oscillators

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions, or of a mass attached to a fixed point by a perfectly elastic spring. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.

### Phase visualization

In electronics, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope. If the display is an ellipse, rather than a straight line, the two signals are out of phase.

### Elliptical gears

Two gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, will turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt. Such elliptical gears may be used in mechanical equipment to vary the torque or angular speed during each turn of the driving axle.

### Optics

In a material that is optically anisotropic (birefringent), the refractive index depends on the direction of the light. The dependency can be described by an index ellipsoid. (If the material is optically isotropic, this ellipsoid is a sphere.)

Hamburg: Friedrich Perthes and I.H. Besser, 1809.

First edition of the work which, “ with the Disquisitiones [Arithmeticae, 1801], established his reputation as a mathematical and scientific genius of the first order” (DSB). “The Theoria motus will always be classed among those great works the appearance of which forms an epoch in the history of the science to which they refer. The processes detailed in it are no less remarkable for originality and completeness than for the concise and elegant form in which the author has exhibited them. Indeed, it may be considered as the textbook from which have been chiefly derived those powerful and refined methods of investigation which characterize German astronomy and its representatives of the nineteenth century, Bessel, Hansen, Struve, Encke, and Gerling … It was forty years before the methods of the Theoria motus became the common possession of all astronomers” (Dunnington, Carl Friedrich Gauss : Titan of Science (2004), p. 91). “In this work Gauss systematically developed the method of orbit calculation from three observations he had devised in 1801 to locate the planetoid Ceres, the earliest discovered of the ‘asteroids,’ which had been spotted and lost by G. Piazzi in January 1801. Gauss predicted where the planetoid would be found next, using improved numerical methods based on least squares, and a more accurate orbit theory based on ellipse rather than the usual circular approximation. Gauss’s calculations, completed in 1801, enabled the astronomer Heinrich W. M. Olbers to find Ceres in the predicted position, a remarkable feat that cemented Gauss’ reputation as mathematical and scientific genius. Gauss found the reputation his astronomical work gained for him so attractive that he decide upon a career in astronomy, becoming director of the Göttingen Observatory in 1807” (Norman). As well as providing a tool for astronomers, Gauss’s method of orbit computation also offered a way of reducing inaccuracy of calculations arising from measurement error – 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). The French mathematician Adrien-Marie Legendre had published the method of least squares, though without justification, in 1805 in his Nouvelles méthodes pour la détermination des orbites des comète, but Gauss states in the present work that he had the method since 1795.

“On 1 January 1801, Giuseppe Piazzi in Palermo discovered a comet or planet in the constellation of Taurus, detectable only telescopically. He observed it through 11 February, when illness interrupted his observations. He informed three astronomers of his discovery, and in May sent his detailed observations to J.J. Lalande in Paris, asking that publication be postponed.

“Since the 1770s two astronomers, J.E. Bode of Berlin and Franz Xaver von Zach (1754– 1832) of Gotha, had entertained the notion of a missing planet between Mars and Jupiter. A numerical series due to J.D. Titius, publicized by Bode in 1772, gave approximate mean solar distances of the known planets, but predicted a planet in this ‘gap’. It received surprising corroboration in 1781 with the discovery of Uranus, a planet whose nearly circular orbit had a radius close to the next term after Saturn in the series. In autumn 1800 Zach and other German astronomers formed a society to promote systematic search for the missing planet.

“In spring 1801 the question arose: might Piazzi’s ‘comet’ be the quarry sought? It must be re-discovered! From June onward, Zach’s monthly reports in a periodical which he published, the Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde (hereafter, ‘MC’) gave an ongoing account of the search.

“The July issue reported the efforts of J.C. Burckhardt, in Paris, to put an orbit to Piazzi’s observations. Parabolic orbits, Burckhardt found, were unsatisfactory circular orbits could accommodate more of the data. He proposed an approximate elliptical orbit, but in agreement with P.S. Laplace (1749–1827), held that an accurate orbit determination would require more observations.

“Through late summer and autumn, cloudy weather prevented a systematic search. In the September issue, Zach published Piazzi’s revised observations. Gauss, a subscriber to the MC, set about determining an orbit.

“The November issue of the MC contained a review of Piazzi’s memoir on his discovery. Finding parabolic trajectories hopeless, he had derived two circular orbits with radii 2.7067 and 2.68626 astronomical units. From the second of these Zach computed an ephemeris for November and December. Piazzi named the planet Ceres Ferdinandea, thus honoring Sicily’s ruler.

“Zach now received Gauss’s results, and to them devoted his entire report in the December issue. Gauss had computed four different elliptical orbits, each based on a different trio of observations the four sets of elements were in near agreement with each other and with the 19 observations Piazzi had considered undoubtful. Gauss put the planet in January 1801 about a quadrant past aphelion and assigned it a considerably higher eccentricity than had Burckhardt, so that in December 1801 the planet would be 6° or 7° farther east than any of the other proposed orbits implied. He gave positions for Ceres at 6-day intervals from 25 November to 31 December.

“The weather continued unpropitious. As Zach reported in the January 1802 issue of the MC, in the early morning hours of 7–8 December he clocked a star very close to Gauss’s prediction for Ceres, but bad weather on the following nights prevented verification.

“As he reported in the February 1802 issue, early on 1 January Zach discovered the planet some 6°east of its December position, and through January he followed its motion, which agreed closely with Gauss’s orbital elements. Wilhelm Olbers (1758–1840) also re-discovered the planet, reporting the fact to the newspapers, where Gauss read about it. Gauss’s ellipse, exclaimed Zach, was astonishingly exact. ‘Without the ingenious efforts and calculations of Dr. Gauss, we should probably not have found Ceres again the greater and more beautiful part of the achievement belongs to him’” (Landmark Writings in Western Mathematics 1640-1940, pp. 317-8).

According to Kepler, the orbit of a celestial body is a conic section with focus at the centre of the Sun. To specify its orbit five parameters, or elements, are required, namely: two parameters determining the position of the plane of the body’s orbit relative to the Earth’s orbit the relative scale of the orbit the eccentricity of the orbit or perihelial distance, the shortest distance from the orbit to the centre of the Sun and the relative ‘tilt’ of the main axis of the orbit. In addition to these five parameters, a single time when the object was at a particular point in the orbit is needed, so that its location at a given time can be computed. Gauss had a total of 22 observations made by Piazzi over 41 days. The data from these observations consisted of a specific moment in time together with two angles defining the direction in which the object had been seen relative to an astronomical system of reference defined by the sphere of fixed stars. In principle, each of these observations defined a line in space, starting from the location of Piazzi’s location at the moment of observation and directed along the direction defined by the two angles. Gauss had to make corrections for various effects such as the rotation of the Earth’s axis, the motion of the Earth’s orbit around the sun, and possible errors in Piazzi’s observations or in their transcription. Gauss began by determining a rough approximation to the unknown orbit, and he then refined it to a higher degree of precision. Gauss initially used only three of Piazzi’s 22 observations, those from January 1, January 21, and February 11. The observations showed an apparent retrograde motion from January 1 to January 11, around which time Ceres reversed to a forward motion. Gauss chose one of the unknown distances, the one corresponding to the intermediate position of the there observations, as the target of his efforts. After obtaining that important value, he determined the distances of the first and third observations, and from those the corresponding spatial positions of Ceres. From the spatial positions Gauss calculated a first approximation of the elements of the orbit. Using this approximate orbital calculation, he could then revise the initial calculation of the distances to obtain a more precise orbit, and so on, until all the values in the calculation became coherent with each other and with the three selected observations. Subsequent refinements in his calculation adjusted the initial parameters to fit all of Piazzi’s observations more smoothly.

Gauss sent a manuscript summarizing his methods in a letter to Olbers dated 6 August, 1802, just seven months after the discovery of Ceres it was published only seven years later in the September 1809 issue of MC. By this time Gauss had so refined his methods of orbit calculation that he writes in the preface to Theoria motus that “Scarcely any trace of resemblance remains between the method by which the orbit of Ceres was first computed and the form given in this work.”

“In 1809, the bookseller Perthes of Hamburg published Gauss’s Theoria motus corporum coelestium in sectionibus conicis solem ambientium. The book … contains the sum of Gauss’s work in theoretical astronomy but it does not always describe the actual methods Gauss used in his research. Like Disquisitiones arithmeticae, Theoria motus was published in Latin Gauss had written it in German but had to translate it because Perthes thought it would sell better. The subject matter of Theoria motus is the determination of the elliptic and hyperbolic orbits of planets and comets from a minimum of observations and without any superfluous or unfounded assumption … Theoria motus is systematic to the point of being pedantic it consists of two books, one with preliminary material and one with the solution of the general problem. The work is the first rigorous account of Gauss’s methods for calculating the orbits of celestial bodies, directly deduced from Kepler’s laws. Up to Gauss’s time, astronomers used ad hoc methods which varied from case to case, despite the fact that the theoretical foundations had been clear for more than 100 years. Gauss’s essential contribution consisted in a combination of thorough theoretical knowledge, the unusual algebraic facility with which he handled the considerable complications which occur in a direct development of these equations, and his practical astronomical experience” (Bühler, Gauss).

“It was Gauss in his Theoria motus who first connected probability theory to the method of least squares … The Theoria motus, which was written to explain how to calculate planetary positions, came into being because the methods available to the astronomers of the eighteenth century were not adequate to determine the orbit of the planet Ceres … Against this background it was natural for Gauss to concern himself with the problem of how to use redundant observations. It seems clear that the more observations available the more accurately will the orbit be known. Gauss said on the subject: ‘But in such a case, if it is proposed to aim at the greatest precision, we shall take care to collect and employ the greatest possible number of accurate places. Then, of course, more data will exist than are required: but all these data will be liable to errors, however small, so that it will generally be impossible to satisfy all perfectly. Now as no reason exists, why, from among those data, we should consider any six as absolutely exact, but since we must assume, rather, upon the principles of probability, that greater or less errors are equally possible in all, promiscuously since, moreover, generally speaking, small errors oftener occur than large ones it is evident, that an orbit which, while it satisfies precisely six data, deviates more or less from the others, must be regarded as less consistent with the principles of the calculus of probabilities, than one which, at the same time that it differs a little from those six data, presents so much the better an agreement with the rest. The investigation of an orbit having, strictly speaking, the maximum probability, will depend upon a knowledge of the law according to which the probability of errors decreases as the errors increase in magnitude: but that depends upon so many vague and doubtful considerations — physiological included — which cannot be subjected to calculation, that it is scarcely, and indeed less than scarcely, possible to assign properly a law of this kind, in any case of practical astronomy. Nevertheless, an investigation of the connection between this law and the most probable orbit, which we will undertake in its utmost generality, is not to be regarded as by any means a barren speculation’ (Goldstine, A History of Numerical Analysis from the 16th through the 19th Century , pp. 212-3).

In Section 186 of the present work, “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).

Dibner 114n Norman 879 Sparrow, Milestones of Science 81 PMM 257n. 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).

Large 4to (296 x 236 mm), pp. [i-iii], iv-xi, [1], [1], 2-227, [1, errata], 1-20 (tables) and one engraved plate (occasional minor stains). Contemporary calf-backed marbled boards.

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