Kepler's First Law

On this page we develop the elliptic theory used by Kepler in support of his First Law: planets move on ellipses with the Sun at one focus. We introduce central coordinates, which are polar coordinates from the center of the ellipse, solar coordinates, which are polar coordinates from the lower focus of the ellipse (the location of the sun), and Kepler's angle, which is a most useful auxiliary parameter in the context of solar coordinates. All this comprises Kepler's First Law.

Central coordinates, Fig. 04

If now we introduce standard polar coordinates centered at C, say (r, alpha). Then we have an additional representation for the cartesian coordinates of R = (X, Y),
X = r*cos(alpha)
Y = r*sin(alpha).
These expressions are valid for an arbitrary point (X, Y) in the plane. But if we restrict the point to the ellipse, the radial distance, r, varies between a maximum of a to a minumum of b, with the position of R along the ellipse. We may then express the position R = (X, Y) in the parametric form,
X = a*cos(alpha)
Y = b*sin(alpha)
in which a and b are constants, the semi-major and semi-minor axes of the ellipse, and there is only one variable, alpha. As alpha varies from 0 to 2*PI, the point R sweeps out the full ellipse. In our current context, a = 1, and b < 1.

Solar coordinates, Fig. 03

These are the polar coordinates of a point R on the ellipse, relative to the sun, S. Let rho denote the radial distance from S to R, and nu the angle subtended at S, measured counterclockwise from Q to R. We may express the cartesian coordinates of the point R, as:
X = rho*cos(nu) - e
Y = rho*sin(nu),
where X is height above C, and Y horizontal distance to the left, as shown in the figure.

Note: Let RHO denote the solar distance in arbitrary units. Given only RHO-max and RHO-min for a planet, we may determine:

A = RHO-av = (RHO-max + RHO-min)/2
C = RHO-max - A = A - RHO-min
B = sqrt(A2 - C2)
b = B/A
rho = RHO/A
c = C/A = e = (RHO-max - RHO-min)/(RHO-max + RHO-min)

Kepler's angle, Fig. 05

As above, construct a horizontal line segment through the point R. Let V denote its intersection with the line CQ, and W denote its intersection with the unit circle. Let beta denote the angle at C from Q to W. We now apply trigonometry to the triangle CVW, finding,
CV = cos(beta)
VW = sin(beta)
as CW=1. Similarly, looking at the triangle RSV, we find,

[1] SV = rho*cos(nu)
= e + CV
= e + cos(beta)

[2] RV = rho*sin(nu)
= b*sin(beta)

as we have established above that RV = b*WV. Squaring each of these expressions and adding, we may find (see below)

[3] rho ~= 1 + e*cos(beta)

Kepler first law

Now we may collect the various expressions above:
[1] rho*cos(nu) = e + cos(beta)
[2] rho*sin(nu) = b * sin(beta)
[3] rho ~= 1 + e*cos(beta)
All this, taken exactly from Katz, comprises Kepler's first law. For Kepler found that observational data conformed to these equations [3].

Derivation of [3]

Square both sides of [1] and [2] and add, getting:

rho2 = e2 + 2*e*cos(beta) + cos2(beta) + b2*sin2(beta)

Now substitute b ~= 1 - e2/2 in the last term on the right, obtaining,

rho2 ~= e2 + 2*e*cos(beta) + 1 - e2*sin2(beta) + e4*sin2(beta)/4

We now ignore the term in e4 in rho2 and take the square root of both sides and we are done. Note that the error in [3] is of the order of e4.

Rev'd 31 dec 2001 by ralph abraham