From Epicycles to Ellipses without Kepler's Second Law

In my last blog, I derived the equation for a conic section under the condition of constant angular velocity, and not constant angular momentum which is the basis for Kepler's second law.

From the time of Ptolemy, it was well known that the path of the center of the epicycle was circular about the eccentric, but at uniform velocity about another point called the equant which is located on the opposite side of the eccentric at the same distance of the earth. In other words, the Ptolemic system used equants to separate uniform circular motion from the center, or eccentric, of the circular deferent.

The center of the epicycle is at a constant distance, R, from the eccentric, but the distance from the equant to the center of the epicycle, r, is variable. The angles that the eccentric and equant make with respect to a line passing through the earth, eccentric, and equant, are alpha and beta, respectively. This is shown in the figure which has been taken from mathpages, Eccentrics, Deferents, Epicycles, and Equants.

From simple trigonometry

R sin alpha = r sin beta

and

R cos alpha- E= r cos beta.

Squaring each of the two and summing result in

R^2-E^2- 2Er cos beta - r^2 =0.

Transforming to u=1/r, and neglecting the square of the ratio, E/R, give

u=R^(-1){1+e cos beta),

where e=E/R, is the eccentricity because E<<R. The semi-latus rectum, R=w^2/F, where w is the constant angular velocity and F is the constant force that is related to the inverse square law through the conserved energy,

U=(1/2r^2){w^2+(dr/dt)^2}=v^2/2r^2,

i.e.,

dU/dr=-F/r^2,

where

F=v^2/r-d^2/dt^2,

is constant. Transforming from the relative acceleration, d^2/dr/dt^2, to a coordinate oriented one, a, the force becomes

F=(dr/dt)^2/r- a = (v sin theta)^2/r- a = h^2/r^3 -a,

which is the negative of Newton's expression, where h= r X v, the angular momentum per unit mass.

The velocity, v, can have either of its components made constant: w= v cos theta, the angular velocity is constant, or dr/dt= v sin theta, the angular angular momentum per unit mass is constant.