Kepler's Equation: A Listing of Some Derivatives

This post serves as a compilation of some derivatives for Kepler's Equation of Elliptical Motion. Nothing fancy here. Just posting some basic characteristics of the motion that might be useful later.

For a start, here are the M and E curves plotted on the same graph:

From Kepler's Equation of Elliptical Motion, the derivative of E with respect to (w.r.t.) t is found:

For the model I've been using throughout this work, ω = 1, so this derivative can be simplified further. In fact, I'm going to examine derivatives w.r.t. M instead of time. RETURNING TO A GENERIC FORM IN WHICH ω IS NOT 1, WE HAVE TO REMEMBER TO PUT ω BACK IN.

Here is the derivative of E w.r.t. M:

Here is a graph of dE/dM:

Here is the second derivative of E w.r.t. M:

Here is the third derivative of E w.r.t. M:

And here is the fourth derivative of E w.r.t. M:

These derivatives may be useful in creating a Taylor polynomial and/or creating interpolating functions over certain intervals.

One comment I would like to make immediately is that, in using the quotient rule to perform the differentiation, the term (1 - e cos(E)) always appears in the numerator before simplification--it's not present in just the denominator. Simplifying the numerator may reveal some insight into the motion; however, if the derivatives are going to be used in a Taylor polynomial, and computed numerically, it may be a better idea to leave the (1 - e cos(E)) term as is. After all, this term has to be computed for the denominator anyhow. The better course of action may be to compute the term (1 - e cos(E)) once, save its value in a variable, and re-use it as needed. Just a thought. It appears in all the derivatives that follow too.

Some features to note:
(i) When the motion crosses the y-axis (i.e., E = π/2 rad),
dE/dt = ω (i.e., dE/dt is parallel to M; i.e., dE/dM = 1);
(ii) dE/dM is a maximum at t = 0, at which time it takes the value 1/(1 - e);
(iii) dE/dM is a minimum when M = E = π, at which time it takes the value 1/(1 + e).
(iv) from the second derivative of E w.r.t. M, the point of inflection for the E curve occurs when sin(E) is 0.

Carrying on, here are the cos(M) and cos(E) curves plotted on the same graph:

The red line is cos(E) and the blue line is cos(M).

The derivative of cos(E) w.r.t. M is found as

Here are plots of the first derivatives of the cosine curves on the same graph:

The second derivative of cos(E) w.r.t. M is

Here are plots of the second derivatives of the cosine curves on the same graph:

Note that the second derivative of cos(E) takes a zero value
when cos(E) = e; in other words, the point of inflection occurs when the motion is at the semi-latus rectum (M = 0.180371160753 rad).
Also note that when elliptical motion reduces to circular motion,
(i.e., e = 0), the point of inflection occurs at the usual times:
every time cos(M) = 0 (i.e., M = π /2, 3 π /2, etc.).

Now repeat the process for the sine curves. Here are the sin(M) and sin(E) curves plotted on the same graph:

The red line is sin(E) and the blue line is sin(M).

The derivative of sin(E) w.r.t. M is

Here are plots of the first derivatives of the sine curves on the same graph:

The second derivative of sin(E) w.r.t. M is

Note that this derivative is similar to the second derivative of E w.r.t. M; it's just missing the scaling factor e.

Here are plots of the second derivatives of the sine curves on the same graph:

Note that the second derivative of sin(E) takes a zero value
when sin(E) = 0; in other words, points of inflection occur each time M = 0, π, 2π, etc.
Again note that when elliptical motion reduces to circular motion,
(i.e., e = 0), the point of inflection occurs at the usual times:
every time sin(M) = 0 (i.e., M = 0, π , 2 π , etc.).