2.1 Kepler's laws and orbital dynamics

Kepler's laws describe how objects move in orbit: elliptical paths, varying speeds, and a precise relationship between orbital period and distance. They form the foundation of celestial mechanics and are essential for everything from predicting planetary positions to planning satellite trajectories.

Orbital dynamics builds on these laws with mathematical tools that let you calculate an object's position, velocity, and energy at any point in its orbit.

Kepler's Laws of Planetary Motion

Kepler's laws of planetary motion

First Law (Law of Ellipses): Every planet orbits the Sun along an elliptical path, with the Sun at one of the two foci. A circle is just a special case where both foci overlap. This was a major break from the old assumption of perfectly circular orbits, and it finally allowed accurate predictions of planetary positions.

Second Law (Law of Equal Areas): A line drawn from the Sun to a planet sweeps out equal areas in equal time intervals. The practical consequence: a planet moves faster when it's closer to the Sun (near perihelion) and slower when it's farther away (near aphelion). This isn't arbitrary; it's a direct result of angular momentum conservation.

Third Law (Law of Periods): The square of a planet's orbital period is proportional to the cube of its semi-major axis. This applies universally to any objects orbiting the same central body, whether they're planets, moons, or artificial satellites. It means that more distant orbits take disproportionately longer to complete.

Orbital period and semi-major axis

The Third Law in its full Newtonian form is:

$T^2 = \frac{4\pi^2}{GM}a^3$

where $T$ is the orbital period, $a$ is the semi-major axis, $G$ is the gravitational constant, and $M$ is the mass of the central body. Notice that the orbiting object's mass doesn't appear here (it cancels out, assuming $m \ll M$).

Derivation outline (for a circular orbit as a simplified case):

1. Set gravitational force equal to centripetal force: $\frac{GMm}{r^2} = \frac{mv^2}{r}$
2. Solve for orbital velocity: $v = \sqrt{\frac{GM}{r}}$
3. Express the period using circumference and velocity: $T = \frac{2\pi r}{v}$
4. Substitute the velocity expression and simplify. Replace $r$ with $a$ to generalize for elliptical orbits, yielding $T^2 = \frac{4\pi^2}{GM}a^3$

This relationship is used constantly in practice. For example, if you measure an exoplanet's orbital period from transit data, you can determine its semi-major axis (provided you know the star's mass). It also works in reverse for designing satellite orbits at specific altitudes.

Orbital Dynamics

Velocity and position in elliptical orbits

Three orbital elements define the basic geometry and the object's location within the orbit:

• Semi-major axis ($a$): half the longest diameter of the ellipse; sets the orbit's size
• Eccentricity ($e$): how elongated the ellipse is (0 = circle, closer to 1 = very stretched)
• True anomaly ($\theta$): the angle measured from periapsis to the object's current position, as seen from the focus

Position in polar coordinates (with the central body at the focus):

$r = \frac{a(1-e^2)}{1 + e\cos\theta}$

This gives the distance $r$ from the central body at any true anomaly $\theta$. You can verify that at $\theta = 0$ (periapsis), this reduces to $r_p = a(1-e)$, and at $\theta = \pi$ (apoapsis), it gives $r_a = a(1+e)$.

Velocity components at any point in the orbit:

• Radial (toward/away from focus): $v_r = \sqrt{\frac{GM}{a(1-e^2)}}\, e\sin\theta$
• Tangential (perpendicular to radial): $v_\theta = \sqrt{\frac{GM}{a(1-e^2)}}\,(1 + e\cos\theta)$

The total orbital speed is $v = \sqrt{v_r^2 + v_\theta^2}$. At periapsis, $v_r = 0$ and the speed is purely tangential (and at its maximum). The same is true at apoapsis, where the speed reaches its minimum.

Kepler's equation connects position to time:

$M = E - e\sin E$

Here $M$ is the mean anomaly (a fictitious angle that increases uniformly with time) and $E$ is the eccentric anomaly (a geometric angle related to the true anomaly). This equation is transcendental, meaning you can't solve it for $E$ in closed form. In practice, you solve it numerically (e.g., Newton's method) to find where an object is at a given time.

Effects of orbital eccentricity

Eccentricity $e$ controls how much an orbit deviates from a perfect circle. Different values correspond to fundamentally different trajectory types:

Total orbital energy for a bound orbit depends only on the semi-major axis:

$E = -\frac{GMm}{2a}$

This is constant throughout the orbit. The negative sign indicates a bound system. A key subtlety: for a given semi-major axis, the total energy is the same regardless of eccentricity. A circular orbit and a highly elliptical orbit with the same $a$ have identical total energy; what differs is how that energy shifts between kinetic and potential as the object moves.

The extreme distances in the orbit are:

• Periapsis: $r_p = a(1-e)$
• Apoapsis: $r_a = a(1+e)$

As eccentricity increases, the gap between these two distances widens, which means the velocity contrast between periapsis and apoapsis becomes more extreme. This velocity variation is a direct consequence of angular momentum conservation: since $L = mvr\sin\phi$ is constant, a smaller $r$ at periapsis requires a larger $v$, and vice versa at apoapsis.