Kepler's First Law: Planetary Orbits are Ellipses

Geometry of the Ellipse

a = semi-major axis
b = semi-minor axis
e = eccentricity
F, F' = foci
r, r' = foci distance
theta = true anomaly

Orbit is defined by a and e

Position is defined by r and theta

Equation of the Ellipse

Eccentricity describes the shape of the orbit:

Orbital Eccentricities
Earth	e = 0.017	b/a = 0.99986
Venus	e = 0.007	b/a = 0.999975
Pluto	e = 0.25	b/a = 0.968
Comets	e = 0.995	b/a = 0.01

Periapse: Closest approach (Perihelion, Perigee, Periastron)

r_p = a(1-e)

Apoapse: Furthest away (Aphelion, Apogee, Apastron)

r_a = a(1+e)

Mars	Earth
e=0.093, a=1.524 AU	e=0.017, a=1.000 AU
Perihelion = 1.382 AU	Perihelion = 0.983 AU
Aphelion = 1.666 AU	Aphelion = 1.017 AU