Drafting office · Sheet 03

Field guide: how curved paths keep their shape

Each section takes one idea, one drawing and one plain-language rule. Read them in order, or jump to the glossary at the end.

A path bends toward its source

Drop a stone and it falls straight toward the ground. Throw it sideways and its path curves. Throw it harder and the curve flattens, and the stone lands farther away. The bending has one cause: every mass pulls on every other mass, and that pull weakens with distance.

Isaac Newton stated the pull as an inverse-square law. Double the distance and the pull falls to a quarter. Triple the distance and it falls to a ninth. The rule is short, and it is enough to produce the orbit curves in this guide.

First law: an orbit is an ellipse with the sun at one focus

A closed orbit is an ellipse. An ellipse has two focus points, and the sun sits at one of them. The other focus is empty. A circle is the special case in which both foci meet at the center.

Fig. 03. An ellipse with semi-major axis a and semi-minor axis b. The focus F lies a distance c from the center O, and the sun sits at F.

The eccentricity, written e, is c divided by a. At e equal to zero the ellipse is a circle. As e grows, the ellipse flattens, and at values close to one it becomes a long, narrow loop.

Second law: equal areas in equal times

Draw a line from the sun to the body. As the body moves, that line sweeps across the orbit. Kepler found that the line sweeps out equal areas in equal times. The body therefore moves fastest near the sun, where the orbit is closest, and slowest far away from it.

Fig. 04. Two wedges swept in equal times. Wedge A1 covers the stretch near the closest point, and wedge A2 covers the stretch far from it. Their areas are equal.

Third law: the period follows the size

The square of the period grows in step with the cube of the semi-major axis. Double the size of an orbit and its period grows by a factor of about 2.8. Quadruple the size and the period grows by a factor of eight.

In the orbit lab, the constant is set so that a semi-major axis of one unit has a period of one model year. The period of every other body follows from that single rule.

Glossary

Each term is used on the simulator page or in one of the diagrams above.

Apoapsis: The point of an orbit farthest from the sun.
Eccentricity: How far an ellipse departs from a circle. Zero is a circle, and values close to one are long and narrow.
Focus: One of two fixed points inside an ellipse. For a planetary orbit, the sun sits at one focus.
Mean anomaly: The angle a body would sweep if it moved at a steady rate around the orbit. Kepler's equation converts it into the angle that fixes the body's position on the ellipse.
Period: The time for one complete orbit.
Periapsis: The point of an orbit closest to the sun.
Semi-major axis: Half the longest diameter of an ellipse. It sets the size of the orbit.
Semi-minor axis: Half the shortest diameter of an ellipse.

Next, try it

Open the orbit lab and switch on trails. Compare how long each body takes to close its loop, then check the periods against the third law above.