"Planets orbit about the sun in elliptical orbits where the sun is positioned at one of the focal points of the ellipse."
As mentioned previously, it was readily apparent to the astronomers of the past through observing the planets that none of them were found to travel at uniform speed when viewed from the Earth, and that even retrograde motion itself could not fully account for this on its own; hence the continued use of epicycles by Copernicus.
And yet even with their presence, a persistent margin of error of up to 5 degrees still remained when the formulations of his system were set against actual observations. Thus, it was all too evident that the movements of the planets did not conform too well to the mathematics of the epicycles that were superimposed upon them; a fact that could not be ignored.
To Kepler indeed, it was something that represented a real challenge. Initially focusing upon the planet Mars, Kepler conducted an extensive examination of its orbit with the aim of developing a far better understanding of its characteristic activity than recognised under the Copernican model. To aid him in this, Kepler actually made use of the observational data gathered by his contemporary, Tycho Brahe, whom he had once worked with for a time .
It was following his careful examination of Mars that Kepler was eventually able to discover a series of causal principles found to govern the motion of this planet, the reality of which all but eliminated the errors present under the Copernican epicyclical system. Not only this, Kepler's work also led him to uncover the unique underlying geometry of all orbital bodies. Combined, these discoveries showed up the epicycles for what they really were: nothing but a mathematical fiction. Indeed, the strongly held belief among Kepler's contemporaries that the planets travelled in perfectly circular orbits, as intuitively they were considered divine, was also overturned.
The actual breakthrough that Kepler made was the realisation that Mars travelled not in a circular orbit, or even a combination of circular orbits, but rather in an elliptical orbit about the sun; and that the non-uniform motion of the planet itself was intimately connected to this type of geometrical figure.
Thus, under the Copernican system, the combined use of two circles, one large and one small for specifying its orbit, was discarded by Kepler, who replaced them with a single ellipse representative of its total orbital path; an ellipse that deviated just slightly from an exact circle. Furthermore, under such an orbit, Kepler determined that the sun was not positioned at the exact centre of the ellipse but rather at one of its two focal points, the other being empty.
Indeed, it was this very feature of the orbit that determined the characteristic pattern of the non-uniform motion which the planet engaged in. The application of these discoveries to the orbit of Mars overcame the errors inherent to the epicyclical system, reducing them to almost nothing. Moreover, these principles were found to apply to all of the known planets and not just Mars. Each appeared to possess an ellipse uniquely characteristic of its total orbit, with the sun located at one of the focal points of the ellipse.
With these discoveries, Kepler was able then to break out of the restricted geometry of the circle and of uniform motion, to one that was inclusive of elliptical forms and of fundamentally non-uniform motion. Understanding the essential nature then of an ellipse shape and of how it differs from that of a simple circle is an absolute requirement if one is to fully comprehend the principles of planetary motion discovered by Kepler.
When considering any shapes in nature that are bounded by curvature, it is critical to understand that such shapes are generated through the activity of rotation - this being the very underlying principle governing their formation. So it is with both the circle and the ellipse. In comparing these shapes it should be noted that the former is to the latter, as a square is to a rectangle. The circle is but a special case of an ellipse.
Indeed, both shapes are generated from exactly the same essential components, namely, two centre points and a double radius. In the case of an ellipse it is quite easy to see the truth of this statement due to the fact that the two centre points are distinctly separate, as indeed are each of the two radii.
It is not so readily apparent though in the case of the circle, for upon visual inspection of this shape, the two centres appear merged, as do the two radii, giving the impression that it possesses only one of each of these two components, though two are indeed present. The truth of this may be grasped when the generation of each shape through an act of rotation is considered.
The formation of a circle begins from a point, mathematically considered to be ‘a certain something possessed of the property of zero extension’; in essence a singularity. From the point, two lines, which by contrast do indeed possess extension – at least or rather at most – in one specific dimension, emerge to produce a double radius of a set length.
The lines are then rotated one full cycle of 360 degrees upon a 2-dimensional plane of existence, the result of which sweeps out an area bounded by curvature, defining the form of the circle itself. Though it may seem that only one point and only one line (radius) are required to produce the circle, two of each is necessary, as detailed:
Diagram 1: On the left, it may be seen that both centre-points c1 & c2 are merged, simultaneously existing in the same location at the exact centre of the shape about which the rotation occurs. Also, two lines of extension, each representative of the radius of the circle, r1 & r2, are also merged and thus appear as one. The image left shows the partially swept out circle; the right, the completed form of a circle fully swept out. Strange at it may seem, because the shape is produced from the actions of two lines, the area is actually swept out twice, though this is not obvious.
An ellipse shape has the general appearance of a somewhat squashed circle. It is of course rigorously constructed though using exactly the same components as those underlying the circle. In the case of the ellipse however, the two centre points about which each of the two radii rotate, are separated by a discreet distance. This gives rise to a certain added level of complexity in the formation of this shape that is not present in the case of the circle.
Specifically, the rotational action of each radius about its centre point, ultimately generating the elliptical form, involves a non-uniform fluctuation in the length of each radius, even though the total length of both taken together remains constant throughout. This is detailed in the diagram below:
Diagram 2: On the left, the diagram details the positions of each of the two centre points C1 and C2, each being separated by a discreet distance. From an initial position wherein each radius is of the same length, as shown above, there is a clockwise rotation of both about their respective centres. The fundamental aim is to attain the greatest possible extension of each, so that the maximum area permissible is covered in order to generate the shape. The actual perimeter of the ellipse is ‘drawn’ from the point of contact of each radius as they rotate. As the process unfolds, there is a transfer of length from each radius to the other. For one half of the formation of the shape r1 increases its length whilst r2 decreases. In the second half, the process is reversed with r1 decreasing its length as r2 increases. However, the total length of both radii when added together remains constant throughout the entire process. The complete formation of the ellipse is detailed on the right, and as may be perceived from a careful consideration of the rotational movement that generates the shape, the total area of the form is swept out twice.
Though both the circle and the ellipse possess very similar properties, certain essential differences do exist between the two forms due to the added complexity of the latter. Firstly, it may be noted that a circle is symmetrical, infinitely so when cut through the centre at any angle. An ellipse however is symmetrical in two directions only, a result of the fact that the centre points are separated, whereas in the circle they are merged. By convention the centre points of both shapes are known as focal points, for they are the points about which rotational action occurs. The essential component parts of each form are detailed: Diagram 3 (a), left.
Components of the Circle:
Diameter (or Major Axis) = any straight-line cut through the form passing through the exact centre of the shape.
E.g. X1 – X2 or Y1 – Y2
Radius (or Semi-Major Axis) = Z (centre of circle) to any point on the perimeter.
The distance between focal points is always equal to 0 in the case of a circle.
Diagram 3 (b)
Components of the Ellipse:
Major Axis = X1 – X2
Minor Axis = Y1 – Y2
Semi-Minor Axis = Z to either point Y1 or Y2
Semi-Major Axis = Z to either point X1 or X2, and also either focal point f1 or f2 to either point Y1 or Y2. E.g. f1 – Y1 as detailed
Focal point to Centre = Z to either focal point, f1 or f2
Notable Properties of the Ellipse:
1) A straight line drawn from both focal points to any point (P) on the perimeter of the ellipse is always equal to 2 times the semi-major axis of the ellipse, i.e. the major axis (as detailed in above diagram).
2) Eccentricity. All elliptical figures (and thus elliptical orbits) possess a unique curvature, the measure of which is expressed mathematically as a number between 0 to 1 denoting the eccentricity of the shape. The eccentricity value itself is actually calculated by dividing the distance from either focal point to the exact centre of the figure by the semi-major axis component of the form.
onsequently, an eccentricity value of 0 always denotes a perfect circle as the focal points of this particular shape are merged upon the exact centre of the form itself. The distance between either of them to the centre is therefore always 0, and dividing 0 by any given radius must always produce an answer of 0. By contrast, in the case of an ellipse, a given distance however small is always present between the exact centre of the form and either focal point.
Thus, the eccentricity is always greater than 0. As a rule then the greater the deviation of the figure from a perfect circular form, the closer the eccentricity is to a value of 1, which itself denotes the condition of a completely ‘squashed’ figure where the focal points become merged with the perimeter of the ellipse.
NB: One may note that all of the planets within the solar system possess elliptical orbits, deviating only slightly from perfectly circular orbits, and thus their eccentricity values are all close to 0.
In order to determine the location of the focal points within an ellipse a simple formula is used. The formula in question is a basic rearrangement of what is commonly known as Pythagoras’ theorem, relating the various sides of a right-angled triangle. As shown in Diagram 3(b) above, because an ellipse has a two-way symmetry, the shortest distance from the exact centre to the perimeter is at 90 degrees to the longest distance from the centre to the perimeter. The focal points are always located on the latter line. As a result, to determine the distance between either focal point to the centre, the following formula is used:
This formula essentially relates the sides of the triangle shown above, whose three corners are designated by f1, Z and Y1.
a = f 1 to Y1 = Semi-major axis
b = Z to Y1 = Semi-minor axis
c = f 1 to Z = Distance from exact centre of ellipse to focal point
When a planet reaches the point in its orbit wherein the semi-minor axis touches the perimeter of its ellipse, e.g. at either points Y1 or Y2 (pictured), at this given moment, the distance between the planet and the sun is the arithmetical mean of the entire orbit, which as stated previously, is the semi major axis of a planet’s orbit.
Set against it, are the two extreme values of approach to the sun. The point of closest approach is known as the Perihelion, whilst the point of farthest approach is called the Aphelion. In the example ellipse shown in Diagram 3(b), were the sun positioned at f1, the other focal point being empty, then a planet would be at perihelion at point X1, and at aphelion at point X2. Of elliptical orbits per se, mathematically, the closest approach, farthest approach, and the mean distance to the sun (semi-major axis) are related by the following formula:
 P. Moore (General Editor), The Astronomy Encyclopaedia (1987). Mitchell Beazley. p.222 Ibid