7.1 Kepler's Laws of Planetary Motion

Kepler's laws describe how planets move around the Sun in elliptical orbits. They explain why planets speed up when closer to the Sun and slow down when farther away, and they connect orbital size to orbital period. Understanding these laws is foundational for everything from predicting planetary positions to planning satellite launches and interplanetary missions.

Kepler's Laws of Planetary Motion

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Kepler's laws of planetary motion

Kepler's First Law: The Law of Ellipses

Every planet orbits the Sun in an elliptical path, with the Sun at one focus of the ellipse. An ellipse has two focal points, but only one is occupied by the Sun; the other focus is empty.

The shape of an orbit is described by its eccentricity ($e$):

• $e = 0$: a perfectly circular orbit
• $0 < e < 1$: an elliptical orbit. The closer $e$ is to 1, the more elongated the ellipse. Earth's orbit is nearly circular ($e \approx 0.017$), while Mercury's is noticeably more elongated ($e \approx 0.206$).
• $e = 1$: a parabolic trajectory (an object with exactly escape energy)
• $e > 1$: a hyperbolic trajectory, followed by some comets that pass through the solar system only once

Kepler's Second Law: The Law of Equal Areas

A line segment drawn from the Sun to a planet sweeps out equal areas during equal time intervals. The consequence: a planet moves faster when it's closer to the Sun and slower when it's farther away.

This happens because angular momentum is conserved throughout the orbit. No external torque acts on the Sun-planet system (gravity points radially inward, producing zero torque about the Sun), so the planet must speed up at closer distances to keep sweeping the same area per unit time. Halley's Comet is a dramatic example: it whips around the Sun at perihelion but crawls through the outer solar system near aphelion.

Kepler's Third Law: The Law of Periods

The square of a planet's orbital period ($T$) is directly proportional to the cube of the semi-major axis ($a$) of its orbit:

$T^2 = k \cdot a^3$

The constant $k$ depends on the mass of the central body. For planets orbiting the Sun, $k$ is the same for all of them, which means you can compare any two planets: a planet farther from the Sun takes longer to orbit, and the relationship is precise. Jupiter ($a \approx 5.2 \text{ AU}$) has an orbital period of about 11.9 years, while Saturn ($a \approx 9.5 \text{ AU}$) takes about 29.5 years.

Calculations with Kepler's laws

Using Kepler's Third Law quantitatively

Newton showed that Kepler's third law follows from the law of gravitation. The full form is:

$T^2 = \frac{4\pi^2}{G(M + m)} \cdot a^3$

where $G$ is the gravitational constant, $M$ is the mass of the central body, and $m$ is the mass of the orbiting body. In most cases $m \ll M$, so you can simplify to:

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

For an Earth-orbiting satellite like the ISS, you'd use $M = M_E = 5.97 \times 10^{24} \text{ kg}$. Given the orbital radius, you can solve for the period, or vice versa.

The vis-viva equation

This equation relates an orbiting body's speed ($v$) to its current distance from the central body ($r$) and the semi-major axis ($a$):

$v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right)$

It's especially useful for finding the velocity at specific points in an orbit. At perigee (closest approach), $r$ is smallest, so $v$ is largest. At apogee (farthest point), $r$ is largest, so $v$ is smallest. For a circular orbit, $r = a$ everywhere, and the equation simplifies to $v = \sqrt{GM/r}$.

Orbital energy

The total mechanical energy of an orbiting body is the sum of its kinetic and gravitational potential energy:

$E = \frac{1}{2}mv^2 - \frac{GMm}{r} = -\frac{GMm}{2a}$

Notice that the total energy depends only on $a$, not on where the body currently is in its orbit. This confirms that total energy is conserved throughout the orbit. A bound orbit always has negative total energy; if $E \geq 0$, the object escapes.

Orbital position and planetary dynamics

Perihelion is the point in an orbit closest to the Sun (or perigee for orbits around Earth). At perihelion:

• Orbital velocity is at its maximum
• Gravitational potential energy is at its lowest (most negative)
• Kinetic energy is at its highest
• Earth reaches perihelion around January 3 each year, at about 147.1 million km from the Sun

Aphelion is the point farthest from the Sun (or apogee for Earth orbits). At aphelion:

• Orbital velocity is at its minimum
• Gravitational potential energy is at its highest (least negative)
• Kinetic energy is at its lowest
• Earth reaches aphelion around July 4, at about 152.1 million km from the Sun

The trade-off between kinetic and potential energy at these points is a direct consequence of energy conservation. As a planet falls inward toward the Sun, it loses potential energy and gains kinetic energy. As it moves outward, the reverse happens.

Why velocity varies (connecting the laws):

1. When a planet is closer to the Sun, it moves faster to sweep out equal areas in equal time (Kepler's second law).
2. When a planet is farther from the Sun, it moves slower for the same reason.
3. This variation maintains conservation of angular momentum: $L = mvr\sin\theta$ stays constant, so as $r$ decreases, $v$ must increase.

Pluto's orbit ($e \approx 0.25$) shows this effect clearly: its distance from the Sun ranges from about 30 AU to 49 AU, producing a significant difference in orbital speed between perihelion and aphelion.