Solving Kepler's equation — Newton-Raphson iteration: Foundations

Before you can read that central equation, every letter in it must mean something visual to you. This page introduces the letters one at a time, in build-up order, and only after all of them are defined do we finally write the equation down. Read top to bottom — each symbol leans on the one before.

1. The ellipse and its two secret numbers: $a$ and $b$

An ellipse is a circle that has been stretched more in one direction than the other. Picture squashing a round balloon between two hands.

The picture: stand at the centre of the ellipse. Walk to the far end — that distance is $a$ . Turn 90° and walk to the near edge — that distance is $b$ . Because an ellipse is squashed, $b < a$ (for a circle, $b = a$ ).

Figure 1 (below) draws exactly this: a violet ellipse with the magenta arrow marking $a$ along the long axis and the orange arrow marking $b$ along the short axis, plus the centre and the off-centre focus we meet next.

Why the topic needs it: the whole orbit lives inside the box set by $a$ and $b$ . The clever trick that derives Kepler's equation is to compare the ellipse to a perfect circle of radius $a$ drawn around it — so $a$ and $b$ are the first tools we reach for.

Figure — Solving Kepler's equation — Newton-Raphson iteration

2. The focus and the offset $c$ — where the Sun sits

An ellipse has a special point called the focus. The Sun (or Earth, for a satellite) sits at one focus — not at the centre. This off-centre-ness is the source of all the difficulty.

The picture: in Figure 1 the centre is the navy dot at the geometric middle; slide toward the "fat" end by a distance $c$ (the dashed navy segment) and you reach the orange star — that is where the Sun sits. If $c = 0$ the Sun is dead centre and the orbit is a perfect circle.

Why the topic needs it: because the Sun is offset by $c$ , the body is sometimes close to it and sometimes far. Close = fast, far = slow. That uneven speed is exactly why we cannot compute position from time with simple algebra.

3. Eccentricity $e$ — the "how squashed" dial

We now package the squash into a single dimensionless number.

Rearranging gives $c = a e$ — the focal offset written using $e$ . Combined with the triangle law $c^{2} = a^{2} - b^{2}$ from the last section, this also yields the handy relation $b = a 1 - e^{2}$ (substitute $c = a e$ and solve for $b$ ).

Read the dial:

$e = 0$ → focus at centre → a perfect circle (no squash).
$e$ near $0$ (e.g. $0.017$ for Earth) → almost a circle.
$e$ near $1$ (e.g. $0.97$ for some comets) → a long, thin, cigar-shaped orbit.
$e = 1$ → the ellipse breaks open into a parabola (not an orbit anymore).

4. Angles measured in radians — the unit that makes $sin$ addable

The topic's central equation will add an angle to a sine of that angle. For that to make sense, the angle must be measured in radians, not degrees.

The picture: take a circle of radius $1$ . Lay a piece of string of length $1$ along its rim. The angle from the centre spanning that string is exactly one radian ( $\approx 57.3°$ ).

Why the topic needs it: all three anomalies we meet next are radian angles, and every $sin$ / $cos$ in the topic assumes radians.

5. The three anomalies: $M$ , $E$ , $ν$ — three ways to say "where"

"Anomaly" is an old astronomy word that just means angle telling you where the body is. There are three of them because there are three natural reference points.

Figure 2 (below) shows all three at once: the magenta ellipse (real orbit) inside its violet dashed auxiliary circle. The body is the magenta dot; its "shadow" pushed straight up onto the circle is the violet dot. The violet angle $E$ is measured at the navy centre, and the orange angle $ν$ is measured at the orange Sun — you can literally see them pointing in slightly different directions.

Why the topic needs it: the mission is time $\to$ position, i.e. $M \to E \to ν$ . $M$ is easy from time, $ν$ is easy from $E$ , but $M \to E$ is the transcendental knot the whole topic exists to untie.

6. Mass, gravity, and the parameter $μ$ — the strength of the pull

Before we can turn time into an angle, we need one number that captures how strongly the central body pulls.

The picture: think of $μ$ as a single "gravity dial" for the Sun. You never need $G$ and $M_{⋆}$ separately in orbit work — they always travel together as this one product, so astronomers bundle them into $μ$ once and for all.

Why the topic needs it: the steady rate at which the mean angle grows (defined next) depends on how strong gravity is, so $μ$ must be on the table before we can write that rate.

7. Mean motion $n$ , period $T$ , and periapsis time $t_{p}$ — turning time into $M$

The picture: if a full lap ( $2 π$ radians) takes time $T$ , then each second earns $2 π / T$ radians. That rate is $n$ . From it, $M = n (t - t_{p}),$ the elapsed radians since the body last passed closest approach at time $t_{p}$ . More in Mean Motion and Orbital Period.

The link between $a$ , $T$ and $μ$ hidden in $n = μ / a^{3} = 2 π / T$ is exactly Kepler's third law.

8. The moving tools: $sin$ , $cos$ , the derivative $f^{'}$ , and the root

The final ingredients are the mathematical machines the topic runs.

Why $sin E$ enters Kepler? In the derivation the triangle from centre-to-focus-to-circle-point has height $a sin E$ . That height is what pulls $E$ away from the steady $M$ — so the correction term ends up being $- e sin E$ .

9. Assembling the central equation and its root function

Now that $M$ , $E$ and $e$ all have pictures, we may finally write the topic's central relation:

We cannot rearrange this to get $E$ alone, so we hunt for the $E$ that makes it balance. Move everything to one side:

Why the derivative at all? Newton-Raphson uses this slope to predict where the curve will cross zero: follow the tangent line down to the axis and land near the root. Figure 3 (below) shows exactly this — the violet curve $f (E)$ , its magenta root, an orange guess, and the dashed orange tangent whose slope is $f^{'}$ sliding the guess toward the crossing. Compare this with the slower slope-free cousin in Fixed-Point Iteration for Kepler.

How it all feeds the topic

Equipment checklist

Test yourself — cover the right side and answer out loud.

What do

a

and

b

measure on an ellipse?

Half the longest (

a

) and half the shortest (

b

) diameters — the semi-major and semi-minor axes.

How are

a

b

, and

c

related?

By the right-triangle law

c^{2} = a^{2} - b^{2}

; the focus offset, minor axis, and major axis form a right triangle with

a

as hypotenuse.

Where does the Sun sit on the orbit?

At one focus, offset from the centre by

c = a e

— never at the centre.

What does the eccentricity

e

tell you?

How squashed the orbit is:

e = 0

is a circle,

e

near

1

is a long thin ellipse; it scales every correction term.

Why must every angle be in radians?

Because Kepler's equation adds a raw angle to a sine, and only in radians does an angle behave like a plain addable number.

What is the difference between

M

E

, and

ν

M

= steady fake-body angle (a proxy for time);

E

= angle to the auxiliary-circle shadow at the centre;

ν

= the real angle at the Sun.

Why can plain

arctan

give the wrong

ν

, and how do you fix it?

Because

arctan

only returns

(- \frac{π}{2}, \frac{π}{2})

and cannot tell top half from bottom; use atan2 or carry the sign of

sin E

E

and

ν

stay in the same half-plane.

What is

t_{p}

and why is it there?

The clock reading at closest approach; measuring

t - t_{p}

makes

M = 0

exactly at perihelion.

What is the gravitational parameter

μ

The product

μ = G M_{⋆}

of Newton's constant and the central mass — a single "gravity dial" that sets how fast the body must move.

How do you get

M

from a time

t

M = n (t - t_{p})

with mean motion

n = 2 π / T = μ / a^{3}

On a unit circle, what are

sin E

and

cos E

The height and the rightward reach of the point at angle

E

; each lies in

[- 1, 1]

What is a "root" of

f

, and what is

f^{'}

A root is an input making

f = 0

;

f^{'} = 1 - e cos E

is the local slope of

f

, used by Newton to aim at the root.

Ready? Head back to the main topic and every symbol should now feel like an old friend.