§ 13.4 — Kepler's Laws of Planetary Motion

Historical Context and Empirical Discovery

Johannes Kepler, in the early 17th century, analyzed decades of precise astronomical observations by Tycho Brahe and discovered three fundamental laws describing planetary orbits. These laws, published between 1609 and 1619, preceded Newton's gravitational theory by over 50 years and provided the empirical foundation that Newton later explained through his universal law of gravitation.

Kepler's First Law: The Orbit Shape

Planets orbit the Sun in elliptical paths, with the Sun at one focus of the ellipse.

An ellipse is defined by two parameters:

• Semi-major axis (\( a \)): half the longest diameter, defining the size of the orbit
• Eccentricity (\( e \)): a dimensionless parameter ranging from 0 (perfect circle) to 1 (parabola, escape trajectory)

For any ellipse:

• Perihelion (closest approach): \( r_p = a(1 - e) \)
• Aphelion (farthest point): \( r_a = a(1 + e) \)

Earth's orbit has \( e \approx 0.017 \), making it nearly circular. Mercury's orbit is more eccentric with \( e \approx 0.206 \). Comets may have very high eccentricity, and if \( e \geq 1 \), they follow parabolic or hyperbolic escape trajectories.

Kepler's Second Law: Conservation of Angular Momentum

A line connecting a planet to the Sun sweeps out equal areas in equal times.

This is a direct consequence of conservation of angular momentum. Since gravity acts along the line joining the planet and Sun, it produces no torque about the Sun:

\[ \tau = \vec{r} \times \vec{F} = 0 \]

Therefore, angular momentum is constant:

\[ L = mvr\sin\theta = \text{constant} \]

The areal velocity (area swept per unit time) is:

\[ \frac{dA}{dt} = \frac{L}{2m} = \text{constant} \]

This explains why planets move faster near perihelion (closer to the Sun) and slower near aphelion. Earth moves fastest in early January (near perihelion) and slowest in early July (near aphelion).

Kepler's Third Law: Period and Orbital Radius

The square of a planet's orbital period is proportional to the cube of its semi-major axis.

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

or equivalently:

\[ \frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3} \]

For objects orbiting the Sun, this can be expressed in convenient units (period in years, semi-major axis in AU):

\[ T^2 = a^3 \]

where 1 AU (Astronomical Unit) = 1.496 × 10¹¹ m.

This law demonstrates why inner planets orbit faster: Mercury (0.387 AU) has a period of 0.241 years, while Neptune (30.1 AU) requires 165 years.

Derivation from Newtonian Mechanics

Kepler's laws emerge naturally from Newton's theory. For a circular orbit (special case of ellipse with \( e = 0 \)):

\[ G\frac{Mm}{r^2} = m\frac{v^2}{r} \]

The orbital velocity is \( v = \sqrt{GM/r} \). The period is:

\[ T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r^3}{GM}} \]

Squaring:

\[ T^2 = \frac{4\pi^2 r^3}{GM} \]

For ellipses, \( r \) is replaced by the semi-major axis \( a \), yielding Kepler's Third Law exactly.

A Parallel Problem: Binary Star System

Two stars with masses \( M_1 = 2.0 M_{\odot} \) and \( M_2 = 1.5 M_{\odot} \) (where \( M_{\odot} = 1.989 \times 10^{30} \, \text{kg} \) is the solar mass) orbit their common center of mass with a separation of \( d = 5.0 \times 10^{11} \, \text{m} \) (approximately 3.3 AU).

Determine: (1) the orbital period, (2) the orbital velocities of each star, and (3) the distance of each star from the center of mass.

Solution:

The center of mass condition gives:

\[ M_1 r_1 = M_2 r_2 \]

where \( r_1 \) and \( r_2 \) are distances from the center of mass. Also:

\[ r_1 + r_2 = d = 5.0 \times 10^{11} \, \text{m} \]

Solving:

\[ r_1 = \frac{M_2}{M_1 + M_2}d = \frac{1.5}{3.5}(5.0 \times 10^{11}) = 2.14 \times 10^{11} \, \text{m} \]

\[ r_2 = \frac{M_1}{M_1 + M_2}d = \frac{2.0}{3.5}(5.0 \times 10^{11}) = 2.86 \times 10^{11} \, \text{m} \]

For binary orbits, Newton's form of Kepler's Third Law applies:

\[ T^2 = \frac{4\pi^2 d^3}{G(M_1 + M_2)} \]

\[ T^2 = \frac{4\pi^2 (5.0 \times 10^{11})^3}{(6.674 \times 10^{-11})(3.5 \times 1.989 \times 10^{30})} \]

\[ T^2 = \frac{4.93 \times 10^{34}}{4.64 \times 10^{20}} = 1.06 \times 10^{14} \, \text{s}^2 \]

\[ T = 3.26 \times 10^7 \, \text{s} \approx 376 \, \text{days} \approx 1.03 \, \text{years} \]

The orbital velocities are:

\[ v_1 = \frac{2\pi r_1}{T} = \frac{2\pi(2.14 \times 10^{11})}{3.26 \times 10^7} = 4.13 \times 10^4 \, \text{m/s} \approx 41.3 \, \text{km/s} \]

\[ v_2 = \frac{2\pi r_2}{T} = \frac{2\pi(2.86 \times 10^{11})}{3.26 \times 10^7} = 5.51 \times 10^4 \, \text{m/s} \approx 55.1 \, \text{km/s} \]

Note that the more massive star moves slower (closer to the center of mass), while the less massive star orbits faster. Their momentum vectors always point in opposite directions and cancel: \( M_1 v_1 = M_2 v_2 \).

Real-World Applications and Extensions

Kepler's laws enable astronomers to determine masses of stars in binary systems, predict cometary returns, and design satellite orbits. The Hubble Space Telescope's orbit was carefully calculated using these principles. Modern exoplanet detection relies on Kepler's laws: periodic variations in a star's light indicate orbiting planets, and the period-radius relationship determines planetary masses and orbital characteristics.

In 1687, Newton proved that Kepler's laws follow as rigorous mathematical consequences of his inverse-square law, providing perhaps the greatest synthesis in the history of physics—unifying the heavens with earthly mechanics.

⇧ Back to chapter