### Kepler's Constant

Recall again,

[4] nu-dot = K/rho2 (K a constant)
We now wish to evaluate this constant, which depends of course on the planetary data, and the units used to measure angle, distance, and time. At this point we will refer to planetary data for the first time, as thus far the ellipse has been arbitrary, subject only to the assumption that the eccentricity be small, so that e4 may be neglected. Among the six planets known to Kepler, the most eccentric is Mercury, with e ~= 0.2. On this page we are going to refer to (Moore/Hunt, 1983) for our data.

#### Mean motion

Suppose nu-dot were constant. Then also rho would be constant, and nu-dot would be equal to one full cycle around the ellipse, 2*PI radians, divided by one sidereal period, P.

Let us agree to measure angles in arc-minutes, and time in days (mean earth days). We let K1 denote Kepler's constant in these units.

We will argue not with rho constant, but rather with the "mean distance" which is to say, the semi-major axis, a = 1. This may be rescaled later to the mean distance in millions of kilometers. Thus, [4] becomes,

[4 mean] K1 = rho-mean2 * nu-dot-mean = 21600/P minutes per day
as there are 360*60 = 21600 minutes in a full circle.

For Mercury, P = 87.969 days. Thus, K1 = 21600/87.969 = 245.5410.

#### Table of approxiate values for K1

Continuing in this fashion for all six planets,
Planet P K1=21600/P
Mercury 87.969 245.5410
Venus 224.701 96.1277
Earth 365.256 59.1366
Mars 686.980 31.4420
Jupiter 4332.59 4.9855
Saturn 10759.20 2.0076
But this is just an intuitive argument. Here is a more precise method.

#### Planet at beta=PI/2

When the planet is at the end of the semi-minor axis, beta=+/-PI/2 radians (+/-90 degrees). Choose +PI/2. Then cos(beta)=0, sin(beta)=1, and our main equations (which are correct for all positions of the planet on the ellipse) become,
[1] X = e (at beta=PI/2)
[2] Y = b (at beta=PI/2)
[3] rho ~= 1 (at beta=PI/2)
[4] nu-dot ~= K (at beta=PI/2)
Actually, at this point, equations [3] and [4] are exact. Thus, we see that K=nu-dot not in the mean, but at this special point. We will use the values of K1 from the table, but understand that corrections might be required with computations for the more eccentric planets, Mercury and Mars.

#### Parameter nu-prime

We also will find it convenient to introduce a new parameter, nu-prime, which is identical to nu-dot except for a change of units. The variable nu-dot is an angular velocity measured in radians per mean-earth-day (2*PI radians = 360 degrees = 21600 arc minutes) while nu-prime is the same angular velocity measured in arc-minutes per mean-earth-day. Thus, equivalent to [4] above, we have,
[4] nu-prime = (21600/2*PI)*nu-dot ~= K1 (at beta=PI/2)
where K1 = (21600/2*PI)*K. Above we have a table of K1's, and thus also, mean nu-primes.
Revised 01 jan 2002 by ralph abraham