Kepler's equation

Key takeaways

Kepler's equation belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kepler's equation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kepler's equation from memory before moving on to harder problems.

Reference excerpt

In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova, and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative solution to the equation. This equation and its solution, however, first appeared in a 9th-century work by Habash al-Hasib al-Marwazi, which dealt with problems of parallax. The equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics.

Equation

Kepler's equation is

where M {\displaystyle M} is the mean anomaly, E {\displaystyle E} is the eccentric anomaly, and e {\displaystyle e} is the eccentricity. The 'eccentric anomaly' E {\displaystyle E} is useful to compute the position of a point moving in a Keplerian orbit. As for instance, if the body passes the periastron at coordinates x = a ( 1 − e ) {\displaystyle x=a(1-e)} , y = 0 {\displaystyle y=0} , at time t = t 0 {\displaystyle t=t_{0}} , then to find out the position of the body at any time, you first calculate the mean anomaly M {\displaystyle M} from the time and the mean motion n {\displaystyle n} by the formula M = n ( t − t 0 ) {\displaystyle M=n(t-t_{0})} , then solve the Kepler equation above to get E {\displaystyle E} , then get the coordinates relative to the central gravitational body from:

where a {\displaystyle a} is the semi-major axis, b {\displaystyle b} the semi-minor axis. Kepler's equation is a transcendental equation because sine is a transcendental function, and it cannot be solved for E {\displaystyle E} algebraically. Numerical analysis and series expansions are generally required to evaluate E {\displaystyle E} .

Alternate forms There are several forms of Kepler's equation. Each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits ( 0 ≤ e < 1 {\displaystyle 0\leq e<1} ). The hyperbolic Kepler equation is used for hyperbolic trajectories ( e > 1 {\displaystyle e>1} ). The radial Kepler equation is used for linear (radial) trajectories ( e = 1 {\displaystyle e=1} ). Barker's equation is used for parabolic trajectories (for which e = 1 {\displaystyle e=1} ). With the parabolic orbit, unlike the elliptical or hyperbolic orbits, it is possible to solve Barker's equation and find a closed-form expression for the position as a function of time. When e = 0 {\displaystyle e=0} , the orbit is circular. Increasing e {\displaystyle e} causes the circle to become elliptical. When e = 1 {\displaystyle e=1} , there are four possibilities:

a parabolic trajectory, a trajectory that goes back and forth along a line segment from the centre of attraction to a point at some distance away, a trajectory going in or out along an infinite ray emanating from the centre of attraction, with its speed going to zero with distance or a trajectory along a ray, but with speed not going to zero with distance. A value of e {\displaystyle e} slightly above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, and as e {\displaystyle e} goes to infinity, the orbit becomes a straight line of infinite length.

Hyperbolic Kepler equation The Hyperbolic Kepler equation is:

where H {\displaystyle H} is the hyperbolic eccentric anomaly. This equation is derived by redefining M to be the square root of −1 times the right-hand side of the elliptical equation:

M = i ( E − e sin ⁡ E ) {\displaystyle M=i\left(E-e\sin E\right)}

(in which E {\displaystyle E} is now imaginary) and then replacing E {\displaystyle E} by i H {\displaystyle iH} .

Radial Kepler equations The Radial Kepler equation for the case where the object does not have enough energy to escape is:

where t {\displaystyle t} is proportional to time and x {\displaystyle x} is proportional to the distance from the centre of attraction along the ray and attains the value 1 at the maximum distance. This equation is derived by multiplying Kepler's equation by 1/2 and setting e {\displaystyle e} to 1:

t ( x ) = 1 2 [ E − sin ⁡ E ] . {\displaystyle t(x)={\frac {1}{2}}\left[E-\sin E\right].}

and then making the substitution

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Kepler's equation

Start with the simplest possible case. Write down what Kepler's equation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kepler's equation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kepler's equation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kepler's equation

In research: Kepler's equation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kepler's equation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kepler's equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Johannes Kepler, Orbits, so understanding it makes those chapters shorter.
In everyday life: Look for Kepler's equation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Kepler's equation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kepler's equation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kepler's equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kepler's equation in simple terms?

In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova, and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative so…

Why does Kepler's equation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kepler's equation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kepler's equation.