Kepler’s Laws and Conic Sections

In the senior year at the Moscow high school no. 2, my astronomy term paper¹¹ Astronomy teacher: N. V Mart’yanova, geometry and algebra teacher: Z. M. Fotieva, calculus teacher: V. A. Senderov. was about Kepler’s laws of planetary motion. The intention was to derive the three famous empirical laws from the first principles of Newtonian gravitation. Assuming the 1st law, which stipulates that each orbit is a conic section with the center of attraction at its focus, I had no difficulty deriving the 2nd and 3rd laws (about conservation of sectorial velocity, and the relation between the periods of revolutions with the orbits’ sizes respectively). However, the derivation of the 1st law per se resisted my effort, apparently, for the lack of skill in solving ordinary differential equations.

Physics and math were taught in our school at an advanced level, and so I had no problems with differentiation of functions, and even with setting Kepler’s problem in the ODE form:

Here ${\mathbf{r}}$ denotes the radius-vector of the planet with respect to the center of attraction, and $k$ is the coefficient of proportionality in the Newton law “$F=ma$” between the acceleration and the “inverse-square radius.” Yet, solving such ODEs was beyond our syllabus.

The unfortunate deficiency was overcome in college, but the traditional solution of Kepler’s problem—by integrating the ODE in polar coordinates—left a residue of frustration. Namely, it remained unclear why, after all, the trajectories are conic sections. So, the high-school project became transformed into the question:

Where is the cone whose sections are the planetary orbits?

In this article, the answer to this question is presented among several other related achievements, including a derivation of Kepler’s 1st law for readers familiar with the cross-product of vectors and differentiation, but not necessarily integration or ODEs.

Thirty years ago, an abstract of this article was included by V. I. Arnold into his joint with V. V. Kozlov and A. I. Neishtadt survey of classical mechanics ([Arnold et al.1987]). Since then, I gave several presentations of this elementary topic: on a lunch seminar, student colloquium, math circle, in some lecture courses, but it might still be worth appearing in print.

In fact, there is no lack of accounts on Kepler’s problem, including some of elementary nature [e.g. ([Arnold1990], [Goodstein and Goodstein1996], [Milnor1983])], though in those I’ve seen, I don’t notice much overlap with ours. In any case, our treatment can hardly provide anything but one more geometric interpretation of what’s been revolving around for ages.

. The angular momentum vector is defined as the cross-product

where $m$ is the mass of the planet. Differentiating, one finds

whenever the force field is central [i.e. $m\ddot{{\mathbf{r}}}={\mathbf{F}}({\mathbf{r}})$ is proportional to the radius-vector ${\mathbf{r}}$]. Thus, in a central force field, the angular momentum vector is conserved. In particular, the direction of ${\mathbf{M}}$ is conserved, that is, the motion of the planet occurs in the plane spanned by the radius-vector ${\mathbf{r}}(0)$ and the velocity vector $\dot{{\mathbf{r}}}(0)$ at the initial moment $t=0$. Thereby Kepler’s problem in 3D is reduced to Kepler’s problem in 2D.

describes relative times the celestial body spends in various parts of the trajectory: In equal times, the radius-vector of the body sweeps equal areas. In other words, the “sectorial velocity” is constant. This is true for motion in any central force field and follows from the the conservation of the angular momentum vector ${\mathbf{M}}=m({\mathbf{r}}\times\dot{{\mathbf{r}}})$. In particular, the length $\mu:=m|{\mathbf{r}}\times\dot{{\mathbf{r}}}|$ of the angular momentum vector is conserved, and so is the sectorial velocity $|{\mathbf{r}}\times\dot{{\mathbf{r}}}|/2=\mu/2m$.

In the 3-dimensional space with coordinates $(x,y,r)$, consider the cone given by the equation

A purely geometric theorem below recasts Kepler’s 1st law as the claim that in Kepler’s 2D problem with the center of attraction located at the origin $(x,y)=(0,0)$, trajectories are the projections to the $(x,y)$-plane of plane sections of this cone.

This is one of classical definitions of ellipses, parabolas, and hyperbolas. The case of circles corresponds to $e=0$ and the directrix “located at infinity”.

Suppose that a point is moving on the surface of the cone $r^{2}=x^{2}+y^{2}$ in such a way that the projection of the point to the plane obeys the equation of motion $\ddot{{\mathbf{r}}}=-k{\mathbf{r}}/|{\mathbf{r}}|^{3}$. We represent the position of the point on the cone by its radius-vector ${\mathbf{R}}$ in space with respect to the origin shifted by distance $l$ along the axis of the cone (Fig. 2):

We choose the shift to be different for different trajectories and determined by the value of sectorial velocity: $l=\mu^{2}/km^{2}$. Since it is conserved, the same value of $l$ serves all points of one trajectory.

In particular, the direction of vector ${{\mathbf{N}}}$ is conserved, and so the trajectory lies in the section of the cone by the plane passing through the point $l{\mathbf{e}}$ and spanned by vectors ${\mathbf{R}}(0)$ and $\dot{{\mathbf{R}}}(0)$ (perpendicular to ${{\mathbf{N}}}$). We have arrived at

Note that $l$ here is non-negative. To obtain negative values of $l$ one must assume that $k<0$, i.e. that the planet is repelled from the Sun by the central anti-gravity force inversely proportional to the square of the distance (as in Coulomb’s law for electric charges of the same sign). A plane crossing the axis of the cone at a point below the plane $r=0$ can intersect the part of the cone above this plane along one of the branches of a hyperbola, namely the branch unlinked with the axis. Thus, as a bonus, we obtain:

The definition of an ellipse as a quadratic curve with the eccentricity $<1$ competes with another definition of an ellipse as the locus of points on the plane with a fixed sum of distances to two given points called foci. The following famous geometric proof of the fact that closed conic sections are ellipses was invented by French mathematician G. P. Dandelin (1794–1847).

Into a conical cup (Fig. 3), place two balls so that they, being tangent to the cone, also touch the secting plane (one from above the other from below). The claim is that the points $F$ and $G$ of tangency with the plane are the foci of the conic section. Indeed, given a point $A$ on the conic section, the segments $AF$ and $AG$ are tangent to the respective balls at the points $F$ an $G$ since the balls touch the plane at these points. The segments $AB$ and $AC$ of the generator $OA$ of the cone are also tangent to the balls, since the balls touch the cone at $B$ and $C$ respectively. Since all tangents to the same ball from the same point have the same length, one concludes that $AF=AB$, $AG=AC$, and hence $AF+AG=BC$. The latter is the distance along a generator of the cone between the parallel circles along which the cone touches the balls. This distance does not depend on the position of $A$ on the conic section.

Here we reconcile the two definitions of ellipses by locating the second focus of a Keplerian orbit.

The period $T$ of revolution can be found as the ratio of the area enclosed by the orbit to the sectorial velocity. The latter is $\mu/2m$, and the square of it $\mu^{2}/4m^{2}$ coincides with $kl/4$ (in our earlier notation). The area of an ellipse with semiaxes $a\geq b$ is equal to $\pi ab$. Combining, we find a geometric expression for the square of the period:

The farthest from and closest to the Sun positions of the planet are called the aphelium and perihelium respectively. Denote by $r_{1}$ and $r_{2}$ the respective distances to the Sun, and examine Fig. 5a showing the axial cross-section of the cone over the major axis of an elliptic orbit.

We have $al=b^{2}$: the geometric mean between two quantities is the geometric mean between their arithmetic and harmonic means. Thus, $b^{2}/l=a$, and

This is Kepler’s 3rd law: The squares of the periods are proportional to the cubes of the orbits’ major semiaxes.

The sum of the kinetic and potential energy

is conserved, as it is readily verified by differentiation.

On the other hand, at the aphelium and perihelium, where the velocity vector $\dot{{\mathbf{r}}}$ is perpendicular to ${\mathbf{r}}$, we have (from conservation of sectorial velocity) $|{\mathbf{r}}|\,|\dot{{\mathbf{r}}}|=\mu/m$. Excluding the velocities, we obtain the following quadratic equation for $r=r_{1}$ and $r=r_{2}$:

By Vieta’s theorem, we have

The first equality shows (again) that at a fixed value of sectorial velocity the secting planes hit the axis of the cone at the same point. The second equality implies that elliptic orbits have negative total energy, and that the value of $E/m$ is determined by the major semiaxis of the orbit.

The Keplerian dynamics lifted from the plane to the cone $x^{2}+y^{2}=r^{2}$ can be considered as Lagrangian dynamics on the cone itself with potential energy $-mk/r$, where however the distance $r$ to the vertex of the cone, as well as the kinetic energy, are dictated by the metric on the cone obtained from the degenerate metric $(dx)^{2}+(dy)^{2}+0(dr)^{2}$ in 3-space, induced by the projection of the space to the plane. Consider now how the dynamics changes when the metric on the cone is induced from a more general metric of the form $(dx)^{2}+(dy)^{2}+\epsilon(dr)^{2}$, Euclidean for $\epsilon>0$ and Minkovsky for $\epsilon<0$ (but ${>-}1$ so that the cone remains Euclidean). When cut along a generator, the cone can be developed isometrically to the plane. The development covers a sector which for $\epsilon\neq 0$ differs from full angle $2\pi$ by the angular defect $\alpha$ (easily computed as $\alpha=2\pi(\sqrt{1+\epsilon}-1)$. The ratio $\alpha/2\pi$ can be interpreted as the amount of the Gaussian curvature of the cone accumulated at the vertex. This curvature has the following effect on the Keplerian dynamics (see Fig. 6): the usual Keplerian orbits on the development plane jump from one edge of the cut to the other and proceed along the trajectory rotated through the angle $\alpha$. This results in the rotation of the perihelium by $\alpha$ radians per revolution—somewhat similar to the rotation of the perihelium of Mercury explained by the general theory of relativity.