Orbits aren't circles. Once you accept that, the next question is — what are they, exactly? Two numbers tell you everything you need to know about an orbit's shape and size.
Newton showed that a body moving under an inverse-square gravitational force traces out a conic section — a circle, ellipse, parabola, or hyperbola, depending on its energy. We'll skip his derivation (it leans on calculus we haven't introduced) and just take the result as given. For any bound orbit — anything that doesn't fly off to infinity — the path is an ellipse, with the central body sitting at one of the two foci.
That last detail is easy to miss but worth pausing on. The central body is not at the center of the ellipse. It's offset, sitting at one focus. The other focus is empty — a mathematical landmark with no physical object there.
An ellipse is fully specified by its semi-major axis $a$ (half the long diameter, which sets the size) and its eccentricity $e$ (a dimensionless number between 0 and 1 that sets the shape). From those two, everything else follows:
The semi-minor axis $b$ — half the short diameter — comes from a Pythagorean-flavored relation:
$$ b = a\sqrt{1 - e^2} $$The distance from the center of the ellipse to either focus is $c = ae$. So when $e = 0$, the foci collapse onto the center and the ellipse becomes a circle. As $e$ approaches 1, the foci slide outward toward the ends, and the ellipse stretches into something long and cigar-shaped.
The closest point on the orbit to the central body is called periapsis, and the farthest is apoapsis. (For Earth orbits, you'll see perigee and apogee; for solar orbits, perihelion and aphelion. Same idea, different prefixes.) Both distances are easy to read off from $a$ and $e$:
$$ r_{\text{peri}} = a(1 - e) \qquad r_{\text{apo}} = a(1 + e) $$Notice that they average to $a$. The semi-major axis is the mean distance, in this specific sense.
Drag the sliders below to see how $a$ and $e$ shape an orbit. The orange disk is the central body, sitting at the right-hand focus; the small gray dot marks the empty focus. The dashed lines mark periapsis and apoapsis distances.
The presets above are revealing. Earth's orbit, often drawn as a dramatic ellipse in textbooks, is so close to circular that you can barely tell it apart from a circle by eye — its eccentricity is less than 0.02. Most planets are like this. Mercury and Pluto are the standout exceptions among the major bodies, with eccentricities around 0.2.
Comets are different. Halley's Comet has $e \approx 0.967$, which means its orbit is so elongated that at aphelion it's nearly 60 times farther from the Sun than at perihelion. That's why we only see it for a few months every 76 years — it spends almost all of its time far out in the dim, slow part of its orbit.
An eccentricity of exactly 1 is the boundary case: a parabolic trajectory, just barely unbound. Anything past 1 is a hyperbola — an open curve that comes in once, swings around, and leaves forever. Voyager 1's exit trajectory from the solar system is hyperbolic.
Two numbers — $a$ and $e$ — fully describe the size and shape of an orbit in its own plane. But that's not enough to pin down a real orbit in three-dimensional space. We still need to specify how the orbit is tilted, where the long axis points, and where the orbiting body is along the path at any given moment. Those are the rest of the six orbital elements, and they're the subject of the next module.