Gravitation and Kepler's Laws

The Geometry of Bound Orbits

Imagine a planet moving around the Sun. In the previous lecture, we learned that any central force produces a planar orbit, and the shape of that orbit is determined by the total energy and angular momentum of the system. For the gravitational force, which weakens with the square of the distance, the mathematics simplifies beautifully. If the total energy of the system is negative, meaning the planet is "bound" to the Sun and cannot escape to infinity, the resulting path is not a circle, but an ellipse. Crucially, the Sun does not sit at the center of this ellipse; it sits at one of the two foci. This is the geometric essence of Kepler's First Law. The "eccentricity" of the orbit, a number between zero and one, tells us how stretched out this ellipse is. A perfectly circular orbit has an eccentricity of zero, while a highly elongated orbit has an eccentricity close to one. This geometric result is not an assumption; it is a direct consequence of the inverse-square nature of gravity.

From Binet's Equation to the Conic Section

We begin with the Binet orbit equation derived in D-029. For a gravitational force \( F(r) = -\dfrac{GMm}{r^2} \), substituting this into the general Binet form yields a specific differential equation for \( u(\theta) = 1/r(\theta) \). The equation becomes linear and has a well-known general solution. By applying the boundary conditions of the orbit, we find that the solution is a cosine function, which geometrically represents a conic section.

The general solution to this differential equation is \( u(\theta) = A \cos(\theta - \theta_0) + \frac{GM\mu}{L^2} \). By choosing our coordinate system such that the periapsis (closest approach) occurs at \( \theta = 0 \), we can rewrite this in the standard form of a conic section. The parameter \( p \), known as the semi-latus rectum, is determined by the angular momentum \( L \) and the reduced mass \( \mu \). The eccentricity \( e \) is fixed by the total energy \( E \) together with \( L \). For bound orbits where \( E < 0 \), the eccentricity \( e \) is strictly less than 1, confirming the elliptical shape.

Here, \( p = \dfrac{L^2}{\mu GM} \). This equation is the master key for orbital shape. It tells us that the distance \( r \) varies periodically with the angle \( \theta \), reaching a minimum at \( \theta = 0 \) and a maximum at \( \theta = \pi \).

Areal Velocity and the Second Law

Kepler's Second Law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. In the language of mechanics, this is a direct statement of the conservation of angular momentum. Since the gravitational force is central, the torque is zero, and the angular momentum \( L \) is constant. The rate at which area is swept out, \( \dfrac{dA}{dt} \), is directly proportional to this constant angular momentum.

This result is profound because it implies that the planet moves faster when it is closer to the Sun (smaller \( r \)) and slower when it is farther away (larger \( r \)), such that the product \( r^2 \dot{\theta} \) remains constant. This is not an empirical rule but a necessary consequence of the rotational symmetry of the gravitational potential.

The Period and the Third Law

Kepler's Third Law relates the orbital period \( T \) to the size of the orbit. For an elliptical orbit, the "size" is characterized by the semi-major axis \( a \). By integrating the areal velocity over one full period, we can derive the relationship between \( T \) and \( a \). The area of an ellipse is \( \pi a b \), where \( b \) is the semi-minor axis. Using the geometric relation \( b = a\sqrt{1-e^2} \) and the definition of the semi-latus rectum \( p = a(1-e^2) \), we can express the total area in terms of \( a \) and \( e \). Combining this with the constant areal velocity yields the final result.

In the case where the central mass \( M \) is much larger than the orbiting mass \( m \) (such as the Sun and Earth), the term \( M+m \) is approximately \( M \). This shows that the square of the period is proportional to the cube of the semi-major axis. This law allows astronomers to determine the mass of a central body simply by observing the period and distance of a satellite.