Kepler’s laws

Draft

Kepler extracted three empirical regularities from Tycho Brahe’s observations between 1609 and 1619, without a dynamical theory to explain any of them. Newton showed in 1687 that all three are consequences of an inverse-square attraction. That order matters: the laws are descriptions of a solution, and knowing which piece of the dynamics each one encodes tells you exactly when it will fail.

First law: the orbit is a conic

The orbit of a planet is an ellipse with the Sun at one focus.

The orbit equation derived from the eccentricity vector is

r = \frac{p}{1 + e\cos\nu}, \qquad p = \frac{h^2}{\mu} = a(1-e^2)

which is the polar form of a conic section with the origin at a focus. Kepler’s statement is the bound case $0 \le e < 1$ ; the same equation covers parabolic and hyperbolic trajectories, which Kepler had no reason to consider.

What this law really encodes is the conservation of the eccentricity vector, and that conservation is special to the inverse-square law. Under any other radial force law the apsidal line rotates and the orbit does not close. The observed precession of Mercury’s perihelion is exactly this: a small departure from inverse-square behaviour, resolved by general relativity. See Relativity for space systems.

Useful geometry:

Quantity	Expression
Periapsis radius	$r_p = a(1-e)$
Apoapsis radius	$r_a = a(1+e)$
Semi-major axis	$a = (r_p + r_a)/2$
Eccentricity	$e = (r_a - r_p)/(r_a + r_p)$
Semi-latus rectum	$p = a(1-e^2)$
Semi-minor axis	$b = a\sqrt{1-e^2}$

Second law: equal areas in equal times

The line joining a planet to the Sun sweeps equal areas in equal times.

The area swept in time $\mathrm{d}t$ is $\mathrm{d}A = \tfrac{1}{2}|\mathbf{r} \times \mathrm{d}\mathbf{r}|$ , so

Areal velocity

\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{1}{2}|\mathbf{r} \times \mathbf{v}| = \frac{h}{2} = \text{constant}

The second law is conservation of angular momentum and nothing else. It therefore holds for any central force, not just inverse-square, and it is the most robust of the three. A spacecraft under thrust directed purely along $\mathbf{r}$ still obeys it.

Operationally it says a satellite moves fastest at periapsis and slowest at apoapsis. A Molniya orbit spends roughly eleven of its twelve hours near apogee, which is the entire point of the design.

Third law: period and semi-major axis

The square of the orbital period is proportional to the cube of the semi-major axis.

Integrating the areal velocity over one full revolution, the swept area is the ellipse area $\pi a b = \pi a^2\sqrt{1-e^2}$ , so $T = 2\pi a^2\sqrt{1-e^2}/h$ . Substituting $h = \sqrt{\mu a(1-e^2)}$ :

Orbital period

T = 2\pi\sqrt{\frac{a^3}{\mu}}, \qquad n = \sqrt{\frac{\mu}{a^3}}

where $n$ is the mean motion in radians per second.

Eccentricity cancels. Two orbits with the same semi-major axis have the same period regardless of shape, which is what makes phasing manoeuvres work: raise or lower apogee slightly, drift for some revolutions, restore.

Kepler’s own statement omitted $\mu$ , so it was a proportionality valid for bodies orbiting the same primary. Newton’s version has the constant in it, and strictly $\mu = G(m_1 + m_2)$ , so the proportionality is only approximate when the secondary mass is not negligible.

Orbit	$a$ (km)	Period
400 km circular (ISS-like)	6 778	92.6 min
800 km circular (SSO)	7 178	100.9 min
GPS (semi-synchronous)	26 560	11 h 58 min
Geostationary	42 164	23 h 56 min
Moon around Earth	384 400	27.3 days

The GPS and GEO periods are one half and one full sidereal day (86 164.09 s), not solar days. That four-minute difference is why a geostationary satellite holds its longitude and why the GPS ground track repeats daily.

Position as a function of time

The first law gives shape, the third gives period, but neither gives position at an arbitrary time. Kepler’s second law makes $\nu(t)$ transcendental. The standard route uses three angles.

True anomaly ν: The physical angle from periapsis to the body, measured at the focus.
Eccentric anomaly E: The corresponding angle on the circumscribed circle, measured at the ellipse centre.
Mean anomaly M: A fictitious angle increasing uniformly at rate n. Not a physical direction.

Mean anomaly is trivial to propagate:

M = n(t - t_p)

The link from $M$ to $E$ is the one that resists closed-form solution:

Kepler's equation

M = E - e\sin E

and once $E$ is known, the physical angle and radius follow directly:

\tan\frac{\nu}{2} = \sqrt{\frac{1+e}{1-e}}\,\tan\frac{E}{2}, \qquad r = a(1 - e\cos E)

Solving Kepler’s equation

No closed-form solution exists in elementary functions. Newton’s method converges quadratically and is what nearly every propagator uses. The derivative is $1 - e\cos E$ , which for $e < 1$ never vanishes, so the iteration is well behaved.

The one subtlety is the starting guess. $E_0 = M$ works for moderate eccentricity but converges slowly near periapsis on a highly eccentric orbit, where the function is nearly flat; $E_0 = \pi$ is the standard fallback.

Stated as an iteration, each step is

E_{k+1} = E_k - \frac{E_k - e\sin E_k - M}{1 - e\cos E_k}

with the numerator being the residual of Kepler’s equation and the denominator its derivative.

A working implementation needs four things beyond that formula, and each one corresponds to a failure that is easy to miss.

Step	Why it is needed
Wrap $M$ into $[-\pi, \pi]$	Keeps the starting guess close to the root regardless of how many revolutions have elapsed
Choose $E_0 = M$ for $e < 0.8$ , otherwise $E_0 = \pi$	Near periapsis on a highly eccentric orbit the curve is nearly flat and $M$ is a poor guess
Stop when the correction falls below about $10^{-12}$	Testing the residual rather than the correction converges one iteration later for no benefit
Cap the iteration count	Guards against a non-converging case reaching a caller as an infinite loop rather than an error

Convergence is fast enough that the iteration count is rarely the bottleneck: for $e < 0.9$ it typically converges to machine precision in four to six iterations.

Where the laws hold and where they do not

The laws are exact for two point masses, or two spherically symmetric bodies, interacting only with each other. Nothing else.

They are a good approximation when the perturbations tabulated in Gravitation and the two-body problem are small over the time span of interest.

They fail in the following situations, all of them ordinary.

Situation	What breaks	Where it is handled
Oblate primary	First law: the node and apsidal line precess	Orbital perturbations
Atmospheric drag	Third law: $a$ decays, so the period is not constant	Orbital perturbations
Three comparable masses	All three: no closed-form solution exists	Lagrange points
Continuous low thrust	All three: energy and angular momentum both change	Electric propulsion
Strong-field relativity	First law: perihelion precesses secularly	Relativity for space systems

The practical consequence for a satellite operator is that a Keplerian propagation of a low Earth orbit is useful for hours and worthless after a few days. Two-line element sets carry a mean-element model with $J_2$ and drag folded in for exactly this reason, and even those degrade within a week.

References

Kepler, J. Astronomia Nova, 1609, and Harmonices Mundi, 1619.
Newton, I. Philosophiae Naturalis Principia Mathematica, 1687, Book I.
Colwell, P. Solving Kepler’s Equation Over Three Centuries, Willmann-Bell, 1993.
Vallado, D. A. Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press, 2022, chapter 2.