Planetary motion tackled kinematically

Orbital motion expressed in terms of the auxiliary angle

Introduction

Nowadays astronomers accept that planetary motion has to be treated dynamically, as a many-body problem, for which there is bound to be no exact solution. This situation is currently modelled in textbooks by the two-body problem, which became amenable to analysis only after the publication in 1687 of the work [1] that initiated the science of dynamics as we now know it.

Before that breakthrough, planetary motion involved merely a path (a curve), together with a measure of time, represented geometrically: that is, a strictly kinematical treatment - which by definition involves the dimensions of length and time alone, while excluding altogether the dimension of mass. This simpifies the situation to one in which a planet (regarded as a point) moves in a plane about a fixed source of motion [2]. Additionally, the motion of each individual planet will occur in isolation, entirely unaffected by any other member of the system. This will be specifically referred to as the 'one-body problem'. It is the situation we shall now examine, adjusting our terminology accordingly (in particular, replacing any mention of 'velocity' with the more appropriate term 'motion' throughout). The astronomical solution to the one-body problem consists of the two laws:

This composite solution represents what is in fact the earliest instance of a planetary orbit: it will be succinctly referred to in what follows as 'the Sun-focused ellipse'. We shall now prove that, subject to its obvious external limitations, this unique solution is of universal applicability as a self-contained piece of mathematics. Moreover the topic is of great historical significance - since the discovery of the two laws stated above actually took place during the period 1600-1630 [3], under the kinematical circumstances described above: see Kepler's Planetary Laws. Therefore it should be of interest to assess the validity of the techniques actually employed at the tine, applying the present rigorous modern methods as a standard of comparison.

Unexpectedly, this analysis is carried through in terms of the auxiliary angle, rather than the polar angle (at the Sun) that is invariably used nowadays: this came about for historical reasons [4]. As a consequence, our kinematical solution is qualitatively different from any later, dynamical one in that it is exact on the basis of geometry alone, as we shall demonstrate.

In what follows, we establish the properties of an ellipse, both as a path, in Part I, and as an orbit, in Part II; while in Part III we will derive Law III, the relationship that synthesizes the planetary system.


Part I. Geometrical properties of the ellipse with focus at the origin

(i) Determination of the radius vector



The figure shows an ellipse with its major auxiliary circle diameter CD, centre B, whose given measures will be denoted by BC = BD = a, the major semiaxis of the ellipse, and BF = b, its minor semiaxis. The focus A is constructed geometrically by drawing FM parallel to CD to cut the circle at M, and dropping a perpendicular from M to cut CD at A (thus making AM = BF). Then we set AB = BE = ae, where ae is derived from the relationship that connects the three determining constants of an ellipse (it may be referred to as 'the focus-fixing property'):

a2e2 = a2 - b2.             (1)

It is essential to appreciate here that e not only denotes the focal eccentricity (the 'ellipticity') but the polar eccentricity as well, since A is both the focus and the origin or pole of coordinates (which here coincides with the position of the Sun). Otherwise, the present treatment will be ineffective.

By considering the (evidently) congruent right-angled triangles ABF and ABM, we find AF = BM = a. This length AF is subsequently recognized as 'the mean distance', which is of great significance in Part III below.

Our derivation will be carried out exclusively in terms of the auxiliary angle

angleQBC = beta.

This will be unfamiliar to modern readers since the standard treatment is nowadays invariably based on the polar angle anglePAC = theta.

We start from what was almost certainly the earliest definition of an ellipse (because it can be derived from the plane section of a cone in three easy steps, as set out in [5]). It enables the ellipse to be regarded as a 'compressed circle', and this relation is known nowadays as 'the ratio-property of the ordinates':

PH/QH = b/a.             (2)

Now from bigdeltaQHB,

QH = asin beta ,

so from (2),

PH = (b/a).QH = b sin beta .

We will first find the radius vector AP = r in terms of beta (though it will be convenient to introduce the polar angle theta temporarily, in a subsidiary capacity). Then two geometrical equivalences can be derived from bigdeltaAPH, again shown in the figure:

PH = r sin theta = b sin beta             (3)

AH = rcos theta = a(cos beta + e).             (4)

Applying Pythagoras' theorem to bigdeltaAPH, we derive:

r2 = AP2 = PH2 + AH2

and thus,

r2 = b2sin2 beta + a2(cos beta + e)2.

Using (1),

r2 =a2(1 - e2)sin2 beta + a2(cos2 beta + 2e cos beta + e2)

= a2(sin2 beta - e2sin2 beta + cos2 beta + 2e cos beta + e2)

= a2(1 + 2e cos beta + e2cos2beta).

Hence,

r = AP = a(1 + e cos beta).             (5)

This is Law I: the equation of the elliptic path with respect to the origin at one focus. It was discovered by Kepler, in 1609, where it was of course expressed in geometrical terms:
see Kepler's Planetary Laws: Section 6.


(ii) Proof that the ellipse is 'simpler than any circle (except one)'

We consider the equation of a circle with origin at some eccentric point: as an illustration we may take the circle CQD, centre B, shown in the figure, where A is to be regarded as the origin or pole; just for our present purpose, we set AB = ae to represent the 'polar distance' alone (since the focal distance for a circle is zero). Then we use the information from (i) above to calculate the radius vector AQ of the circle:

AQ2 = AH2 + QH2

= a2(cos beta + e)2 + a2sin2 beta

= a2(1 + 2e cos beta + e2).

So,

AQ = a(1 + 2e cos beta + e2)1/2

Hence,

AQ = a(1 + e cos beta + 1/2 e2sin2 beta + ...).

Therefore it is clear that this expression for the radius vector of a circle with its origin at an eccentric point is much less simple than that for the radius vector of the ellipse with the same origin when that point is its focus, as set out in (5) just above.

Further, this argument could be generalized by carrying out a similar brief calculation to find the radius vector of any ellipse belonging to the system of conics whose origin is at the Sun (again setting polar distance AB = ae) that has CQD as its auxiliary circle and its typical point lying on QH (still defined by auxiliary angle beta). However, because each such ellipse possesses its own individual eccentricity, this would introduce a separate constant (say epsilon ) to represent the focal eccentricity of that particular ellipse, and thus produce a still more complicated expression. Since both the focal distance and the polar distance are measured from the centre B of the ellipse, it is only when these two distances coincide (aepsilon = ae), uniquely, that we obtain the simplest possible equation -- as expressed in (5). (And mathematicians will not need convincing that the simplest of all circles, having its origin at the centre B, is no more than a special case of that system of conics, with e = epsilon = 0.)

(iii) Evaluation of the transradial arc r dtheta

From the equivalences for PH set out in (3), we obtain:

sin theta/sin beta = b/r.            (6)

So, applying the formula for the radius vector from (5), we have:

sin theta = (b/a)sin beta/(1 + e cos beta).

Differentiating with respect to beta, we derive:

and using (4),

.

Hence,

dtheta/dbeta = b/r.            (7)

This identity acts as the bridging relation (inverse or direct) between the modern treatment by polar angle and the present treatment by auxiliary angle (which will only work effectively in the case of the unique Sun-focused ellipse alone).

Moreover, this purely geometrical relationship is unexpectedly of enormous significance in connection with one kinematical component of the orbit, as we shall see in Part II(i) below. Meanwhile we point out that the transradial arc is constant with respect to the auxiliary angle:

rdtheta = bdbeta.            (8)

Part II. Kinematical properties of the ellipse with focus at the origin


(i) The transradial component of motion rdtheta/dt


(This motion is known to some mathematicians as the transverse component of motion). Whatever we call it, this motion is defined to take place round the Sun instantaneously in a circle.

In 1687, Newton proved the characteristic property of orbital motion in its most general form [1]: Book I, Prop.1. (For simplicity, he had not yet introduced the consideration of mass.) This property can be formulated in various ways, all equivalent to the statement that equal areas correspond to equal times. The constant of proportionality involved (1/2h is standard usage) is expressed mathematically by the following relationship, in which r represents the radius vector measured from the source of motion at the Sun, still taken as the origin of coordinates, again with reference to the figure:

r2dtheta/dt = h.

This is the modern mathematical expression of the kinematical area-time law.

We now apply this to the special case of the Sun-focused ellipse, whose total area is πab and periodic time T, in order to evaluate its particular constant. For one complete circuit, the area-time law gives:

ab/T = h.            (9)

So in this case,

r2dtheta/dt = 2πab/T,

and hence,

rdtheta/dt = 2πab/T(1/r).            (10)

Now from the evaluation of the transradial arc in (8) above, we have:

rdtheta/dt = bdbeta/dt.            (11)

Thus for the Sun-focused ellipse alone, we deduce from (10) and (11):

dbeta/dt = 2πa/T(1/r).            (12)

We digress to consider the inverse form:

dt/dbeta = T/(2π).(r/a) = T/(2π)(1 + e cos beta), from (5).

Hence by integration,

t is proportional to b + e sin beta.

This is Law II: the time expressed in angular measure, discovered by Kepler in 1609, where he established that (when the dimensional constant is specified, for the Sun-focused ellipse alone) time is proportional to area. See Kepler's Planetary Laws: Section 7.

[Later, in 1621, Kepler demonstrated a less precise version of equation (10) -- simply that the transradial motion is proportional (inverse-linearly) to the distance. See Kepler's Planetary Laws: Section 10.]


(ii) The radial component of motion dr/dt


This motion takes place linearly in the direction of the radius vector -- towards or away from the Sun.

We return to equation (5), the formula for the radius vector:

r = a(1 + e cos beta).

Hence,

dr/dbeta = (-)ae sin beta.            (13)

This is the radial variation of the distance with respect to beta. [Kepler got no further than this: see Kepler's Planetary Laws: Section 11.]

Continuing our modern treatment, we carry out a change of variable, using (12) and (13):

,

and thus,

.            (14)

It can easily be checked that calculating the resultant of these two components (10) and (14) will produce the modern value of the 'velocity' in orbit -- but there is no reason to do so since the present treatment by components is entirely adequate -- and much simpler -- for a kinematical approach.

(iii) The radial acceleration

On the other hand, for the removal of doubt, we should confirm that this treatment is compatible with the modern dynamical approach, by determining the acceleration that corresponds to this motion (as has been said, this concept was an anachronism in Kepler's day). There are several ways of carrying this out, which unfortunately involve either sophisticated calculus or fairly heavy algebra. We start from the formula analogous to that found in textbooks of dynamics:

Radial acceleration directed towards the Sun.            (15)

Then we apply result (11) above to the first term, and, as one possibility for the second term, introduce a formula for change of variable which is found in some calculus textbooks:

.            (16)

This can be expressed in terms of r as required by differentiating (12)and (13), and also using (14), and then simplified by applying (5) and (1). Lastly by using (12), we obtain:

Radial acceleration = (2π)2a3/T2.(1/r2) towards the Sun.

Now introducing, provisionally, the quantity mu0 to represent (2π)2a3/T2,
we express the radial acceleration in the more familiar form:

Acceleration = mu0/r2 towards the Sun.

This quantity mu0 is evidently determined by the particular orbit, and is thus a (kinematical) constant associated with the individual planet. It will be interpreted further in Part III below.

We conclude that this theory is rigorously exact in kinematical terms for an individual planet, in accordance with presentday standards. Moreover, subject to precise determination of the values of all the constants involved, Kepler's own treatment was entirely satisfactory, up to the level of first order differentiation.

Part III. Corollary: the derivation of Law III for a system of planets

A geometrical lemma to Part I(i) above will enable us to evaluate AL = l, the semilatus rectum, shown in the figure (where L is the point of the ellipse lying on AM). By the original construction, AM = BF = b. Accordingly, applying the ratio-property of the ordinates, we obtain:

AL/AM = l/b = b/a.

Hence,

b2 = al.            (17)

Now we return to equation (9), which stated the area-time law (in kinematical terms) for one complete circuit:

h = 2πab/T,

and so,

h2 = (2π)2a2b2/T2

Using (17), we obtain:

h2 = (2π)2a3l/T2.

Rearranging,

a3/T2 = 1/(2π)2.h2/l.

Since h and l are constants determined by the particular orbit, we will follow Cohen [6] (who presumably chose the notation to commemorate the discoverer of this relationship) to write the result:

a3/T2 = K.            (18)

Hence we have uncovered the existence of a kinematical relationship between the square of the periodic time and the cube of the mean distance for each of the (six) planets independently, each apparently possessing its own individual value of the constant K. In 1618 the value of K was tested empirically [7], and found to be common for every pair of planets (within observational limits); it is therefore presumed to be constant for the whole planetary system -and the relationship is known as Law III. Actually, it is possible to formulate a rational basis for the above deduction, founded on geometry - and so to produce a theoretical proof of Law III (which would have been almost within the capability of the mathematicians of that time to discover [8]).

Accordingly, it is evident that the quantity mu0 provisionally defined in Part II(iii) -- there associated with an individual planet -- may now be identified as a constant that will operate to synthesize the planetary system. We will name it appropriately 'the coefficient of gravitational intensity' [9], and in correlation, we have:

mu0 = (2π)2a3/T2 = (2π)2K.


Article by A E L Davis

Notes


  1. Isaac Newton, The Mathematical Principles of Natural Philosophy (London, 1687).
  2. Nicolaus Copernicus(1473-1543): On the Revolutions of the Heavenly Spheres, Nürnberg 1543.

  3. The laws appeared in Johannes Kepler (1571-1630): New Astronomy, Heidelberg 1609. They were validated in his later work: Epitome of Copernican Astronomy, Book V, (Frankfurt ,1621).

  4. From ancient times, astronomy had involved circles whose centres were eccentrically placed with respect to the Sun - which itself did not, until the arrival of the heliocentric view, play an explicit part in planetary theory.

  5. A E L Davis, Some plane geometry from a cone ... Mathematical Gazette (forthcoming, July 2007). We demonstrate that an ellipse can be derived geometrically from a plane section of a cone in three easy steps.

  6. I. Bernard Cohen, The Birth of a New Physics (Norton, 1985: updated), p.166.

  7. Johannes Kepler, The Harmony of the World (Linz, 1619).
    In Book V, Ch.3, Prop.8 he established the value by comparing the planets in pairs.

  8. A E L Davis, "Kepler's potential proof of his Third Law" in Miscellanea Kepleriana, ed. Boockman, Di Liscia, Kothmann, (Augsburg, 2005).
    The proof involved two distinct kinematical conic-systems, and a notional orbit acted as a bridge between the two systems.

  9. The corresponding value μ in the dynamical system which is in accordance with our more sophisticated presentday knowledge depends on the relative masses as well as the actual constant of gravitation. This is explained in elementary textbooks of modern astronomy.

Further references are at this link


JOC/EFR October 2006

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