Orbits & Kepler's Laws
Launch a satellite and watch its orbit, equal-area sweeps and period obey Kepler.
About this simulation
Set a launch distance and speed: below escape speed the body settles into an ellipse (a circle at the right speed), above it flies away. A swept sector shows equal areas in equal times, and the period tracks the T² ∝ a³ relation as students try different orbits, with the escape-velocity threshold marked.
How to use it
- Set the Launch speed to 1.00 × v_circ for a circular orbit, then lower it to see an ellipse with the star at one focus.
- Watch the shaded sectors: each is swept in the same time, so they are long and thin near the star and short and wide far away.
- Raise the Launch speed past √2 × v_circ, about 1.42, and see the label change to Escapes.
- Change the Launch distance and Central mass and watch the orbit's size, and the period T and e at the top of the stage.
- Try the presets 'Perfect circle', 'Stretched ellipse', 'Just escapes' and 'Heavier star'.
What to notice
- A bound orbit is an ellipse with the star at a focus, and the marked centre of the ellipse is visibly away from the star for an eccentric orbit.
- Equal areas are swept in equal times, so the satellite moves fastest at its closest approach and slowest at its farthest point.
- The period depends on the semi-major axis: T² ∝ a³ for orbits around the same central mass.
- At √2 times the circular speed the total energy reaches zero and the satellite escapes instead of returning.
The formulas
- v_c = √(GM / r)
- Speed for a circular orbit of radius r around a central mass M; G is the gravitational constant.
- v_esc = √(2GM / r) = √2 · v_c
- Escape speed from a distance r from the central mass.
- T = 2π √(a³ / GM)
- Orbital period for an ellipse of semi-major axis a; this is Kepler's third law, T² ∝ a³.
Questions
What are Kepler's three laws of planetary motion?
First, every planet moves in an ellipse with the Sun at one focus. Second, the line joining the planet to the Sun sweeps out equal areas in equal intervals of time. Third, the square of the period is proportional to the cube of the semi-major axis, T² ∝ a³. The simulation shows all three: the elliptical path, the shaded equal-area sectors and how much longer a bigger orbit takes to go round.
Why does a planet move faster when it is closer to the Sun?
Gravity is a central force, so it exerts no torque about the Sun and the planet's angular momentum stays constant. Angular momentum is m × v × r for the perpendicular part of the velocity, so when r is smaller, v must be larger. This is why Kepler's second law holds. In the simulation, the shaded sectors near the star are long and narrow because the satellite covers more distance in the same time.
What is the relation between orbital velocity and escape velocity?
At the same distance r from the centre of a planet or star, the circular orbital speed is √(GM/r) and the escape speed is √(2GM/r). So the escape speed is √2, about 1.41, times the orbital speed. A satellite in a circular orbit that is given a speed boost of about 41 percent will escape. In the simulation, set the launch speed above 1.42 × v_circ to see this.
What is the shape of an orbit if the launch speed is less than the circular speed?
If a satellite is launched sideways with less than the circular speed, it cannot stay at that distance and falls inward, tracing an ellipse. The launch point becomes the farthest point of the orbit, and the closest point is on the opposite side of the star. Between the circular and escape speeds, the launch point is instead the closest point. In the simulation, try the 'Stretched ellipse' preset.
