COMP 521 · orbital geometry
From time to a place in orbit
An orbit is a path followed under gravity. An anomaly is an angle used to describe orbital position. Three anomalies describe the same instant: a clock angle , a construction angle , and a physical direction .
PeriastronThe closest point to the star.
ApastronThe farthest point from the star.
Period The time for one complete orbit: the planet’s “year.”
Time → clock
Fraction of a period
Kepler → construction
From center C to Q
Position → direction
From star S to planet
Why the speed changes — Kepler’s second law. The line from star to planet sweeps equal areas in equal times. During a short time step near periastron, the line is shorter: a larger sideways step sweeps the same area. Near apastron, the line is longer and a smaller step suffices. That is why equal-time dots are farther apart near the star and closer together farther away.
Read the geometry and units
Ellipse and focus: an ellipse is a closed oval defined by two fixed points, its foci. Distances from any point on the oval to those two points add to the same total. The star is at one focus, labeled ; is the oval’s center.
Eccentricity : measures how elongated the ellipse is. Zero gives a circle; larger values below one give more elongated orbits. In a circle, all positions are the same distance from the star. We keep the same starting direction to compare the angles.
Auxiliary circle: a helper circle centered at . Its point maps vertically to the planet on the ellipse. It is a construction, not another orbiting body.
Period and angle units: means a quarter of the orbital time, not a quarter of the distance traveled. A full turn is radians or .
Orbit in space: the orbital plane contains the path. The sky plane is an imaginary image plane perpendicular to the viewing direction. Inclination is the tilt between those planes. The ascending node is where the planet crosses the sky plane toward the observer. Orientation is the angle from that node direction to closest approach, measured within the orbital plane.
Predict → Play → Explain
Equal-time dots: where should they cluster? Turn them on, then play one orbit. Explain how equal swept areas allow different distances in the same time.
Circular limit: predict what happens to when . Keep the time fixed and test your prediction.
Three angles: at the film snapshot, less than 10% of the period has passed. Explain why . Which angle measures direction from the star?
Orientation: change while holding time fixed. Does change? Explain why orientation and position along the orbit are different.
● Yellow dots mark equal intervals of . Model: only the gravity between the star and planet is included. The bright point is the planet; the glow marks the star at a focus. Equations use radians. The geometry and units guide defines the diagram’s terms.