Lesson 23 Energy and Eccentricity

According to Newton's laws, all objects in a gravitational field trace out conic sections. The precise shape of an orbit depends on the interplay between energy and eccentricity.

Conic sections have mathematical and grammatical properties:

ellipse - ellipses
parabola - parable
hyperbola - hyperbole

Asteroids follow these conic sections and display how nature obeys mathematics, particularly in the universe.

Newton interpreted Kepler's three laws and developed the equation that models the Earth's orbit:

L^2/DM/(1+ecos(theta)) = r (closely related to the equation of an ellipse)

D = G M M(o)

One of the underlying mathematical properties of the universe is that planets move in ellipses determined by angular momentum, mass of a planet, the mass of the sun, and the eccentricity of the orbit. Not every orbit must be elliptic, circular etc.

The exact shape of an orbit is determined by energy. For example, when a planet falls close to the sun, the potential energy is low, but the planet speeds up so kinetic energy is high. Potential energy is high when a planet is far away from the sun, and kinetic energy is low. Since space is a vacuum, total energy does not change and this is the key to the shape of the orbit.


All the orbits of the planets in the universe are paths of constant energy. Pluto (discovered by Percival Lowell) is responsible for irregularities in Neptune and Uranus's orbit, and its orbit intersects that of Neptune. Halley's comet also has an orbit that is extremely eccentric and elliptical.


The total energy of the orbit is the sum of kinetic and potential energy.

Potential Energy:

U = -D/r

r = L^2/DM/(1 + ecos(theta))

U = D^2M/L^2 (-1-ecos(theta))

Kinetic Energy:

K = (1/2) m v ^2

K = MD^2/L^2((1/2) + ecos(theta) + (1/2)e^2)

E = U+K is constnt because the (cosine) terms cancel in the addition of potential energy and kinetic energy.

E = D^2M/L^2 (1/2) (e^2-1) - connection between energy and eccentricity.

The shapes of orbits depend on energy. For a given angular momentum, the energy determines the eccentricity, and the eccentricity determines the orbit.

Ellipse - if the total energy is negative, and the eccentricity is less than 1, then the orbit is an ellipse. Postive kinetic energy is too small to overcome negative potential energy so the body cannot escape a solar system.

Parabola - if the total energy is zero, and the eccentrcity is exactly 1, then the orbit is a parabola. If a body started with 0 kinetic energy and fell from infinity it would whiponce around the sun and return to infinity. This is very unlikely, however possible.

Hyperbola - if the total energy is positive and the eccentricity is greater than 1, then the orbit is a hyperbola. If an object was projected towards the sun from a great distance, the positive kinetic energy would overcome the negative potential energy. Some comets have this orbit.

Circle - if the total energy has a special value, the eccentricity 0 and the orvit is a circle. This is the lowest energy a planet can have for a given angular momentum. The rings of Saturn display this.