This page assumes you know nothing about the parent note's symbols. We build each one from the ground up, in an order where each idea leans only on the ones before it. By the end you should be able to read the sentence "M=n(t−tp) and M=E−esinE" and feel that every letter has already earned its place.
An ellipse has a centreO (the middle, where the two axes cross). But it also has two special inner points called foci (singular: focus). For a planet, the Sun sits at one focus, which we call S — not at the centre.
Two facts about e that the parent note uses constantly:
Why we need it: ae is the exact offset that later produces the esinE correction term, and b2=a2(1−e2) is the algebra key used to simplify the radius formula.
Recall Where does
b2=a2(1−e2) come from?
It is the defining relation of ellipse geometry (built from the distances to the two foci). We treat it here as a given tool; see Eccentricity and ellipse geometry (a, b, ae, b²=a²(1−e²)) for the full construction.
An angle measures "how much turn". You may know degrees (a full circle =360°). Physics here uses radians instead.
Why radians and not degrees? Every formula in this topic came from areas and arcs of circles, where the "arc = radius × angle" rule holds only in radians. A circular sector of angle θ has area 21R2θ — again only in radians. Plug degrees into M=E−esinE and you get nonsense.
A point on a circle of radius R, at angle θ measured from the rightward axis, has coordinates
x=Rcosθ,y=Rsinθ.
The identity we lean on: cos2θ+sin2θ=1 (Pythagoras on that right triangle). It is what collapses the messy radius algebra in Derivation 2 down to r=a(1−ecosE).
Now apply the squash. On the auxiliary circle (radius a) the helper point is (acosE,asinE). Squash vertically by b/a and the planet on the ellipse sits at
x=acosE,y=bsinE.
This is where the eccentric anomaly E is about to enter — as the angle feeding these cosines and sines.
These are the plainest symbols, but the topic is meaningless without them.
Perihelion itself is a place: the point of the orbit nearest the Sun (distance a(1−e)). Its opposite, farthest point, is aphelion (distance a(1+e)). We label the perihelion direction as the axis X from which all angles are counted.
Why we need them: they turn a time question into an angle question in the very next step.
Here is the "steady imaginary clock" from the core idea.
Why we need it: it is the input we always know (we know the time), and Kepler's equation's whole job is to convert this honest clock reading M into the real geometry.
For completeness (the parent uses it once): the true anomalyν (Greek "nu") is the actual angle of the planet as seen from the Sun at focus S, measured from perihelion. It is what you would literally point a telescope along. It relates to E by
tan2ν=1−e1+etan2E.
See True anomaly ν and the orbit equation r = a(1−e²)/(1+e cos ν) for its own story. For now: M (clock) → E (centre helper) → ν (real angle) is the full chain.
M=E−esinE has E both outside (E) and inside a sin. You cannot rearrange it into "E= (formula in M)". Equations like this are called transcendental; you solve them by repeated guessing that homes in on the answer (see Numerical root-finding — Newton–Raphson).
