Predict → set conditions → observe → record → change one variable. Loading an example clears records and creates a reproducible paused state.
01 · Center versus focus
For any point on an ellipse, distances to the two foci have a constant sum. The star occupies one focus.
r_1+r_2=2a,\qquad c=ae,\qquad b=a\sqrt{1-e^2}
Try itLoad a=2 AU and e=0.6. Seek around the orbit, inspecting both focus lines and their distance sum.
Prediction. The distance sum is 4 AU, the focus offset is 1.2 AU and the semi-minor axis is 1.6 AU.
Compare and explain. What distance and speed interpretations would be wrong if the star were at the ellipse center?
Model limit. Star and planet sizes are enlarged relative to the orbital scale.
02 · Same semi-major axis, different shapes
Increasing eccentricity brings periapsis closer and apoapsis farther, while period remains unchanged at fixed central mass and semi-major axis.
r_\mathrm{p}=a(1-e),\qquad r_\mathrm{a}=a(1+e)
Try itAt a=2 AU, record e=0 and e=0.6. Show comparison paths and compare periods.
Prediction. Both periods are 2.828 yr. For e=0.6, periapsis is 0.8 AU and apoapsis is 3.2 AU.
Compare and explain. Why can the nearest distance not substitute for semi-major axis in the period law?
Model limit. The period is not circumference divided by an arbitrary constant speed.
03 · Equal areas in equal times
A central force conserves angular momentum, giving constant swept area per unit time.
\frac{dA}{dt}=\frac{1}{2}|\vec r\times\vec v|
Try itLoad Equal-time areas and compare 0.1T intervals at periapsis and apoapsis. In advanced settings, change the interval to 0.05T and 0.2T.
Prediction. For a=2 and e=0.6, total area is 3.2π AU²; each 0.1T sector covers about 1.005 AU².
Compare and explain. Why is the angular sweep larger near periapsis for the same area?
Model limit. Areas are calculated from orbital coordinates and time, not screen pixels.
04 · Fast periapsis, slow apoapsis
At periapsis and apoapsis, velocity is perpendicular to radius, so distance times speed can be compared directly.
\frac{v_\mathrm{p}}{v_\mathrm{a}}=\frac{1+e}{1-e}
Try itLoad e=0.5 and record t=0 and 0.5T. Inspect the maximum and minimum speeds.
Prediction. Periapsis speed is three times apoapsis speed; the distances have the reverse ratio, 1:3.
Compare and explain. Is rv constant everywhere, or is the perpendicular velocity component required?
Model limit. Angular momentum uses the component perpendicular to radius, not total speed.
05 · Test the third law graphically
For orbits around the same star, period squared is proportional to semi-major axis cubed.
T^2=\frac{4\pi^2}{GM}a^3
Try itRecord a=1, 2 and 4 AU, then open Period comparison. Add planet presets if desired.
Prediction. For M=1, periods are 1, 2.828 and 8 yr; every T²/a³ ratio is 1.
Compare and explain. What is gained by plotting T² against a³ rather than T against a?
Model limit. These are model-generated values, not astronomical observations with measurement errors.
06 · Change the central mass
A more massive star gives a shorter period for the same orbital size; the proportionality constant depends on central mass.
T\propto\frac{1}{\sqrt{M}},\qquad\frac{T^2}{a^3}=\frac{1}{M}\ \text{(AU, yr, solar mass)}
Try itRecord a=1 AU with M=1 and M=4, then inspect the period comparison plot.
Prediction. The period falls from 1 yr to 0.5 yr; T²/a³ changes from 1 to 0.25.
Compare and explain. What happens if planets around different stars are forced onto one comparison line?
Model limit. Planet mass is neglected; a massive companion requires total mass and barycentric motion.
07 · Connect speed and energy
Speed changes around an ellipse but specific mechanical energy remains constant.
v^2=GM\left(\frac{2}{r}-\frac{1}{a}\right),\qquad\varepsilon=-\frac{GM}{2a}
Try itAt a=1 AU and e=0.6, record several times. Compare distance, speed and specific energy.
Prediction. For M=1, specific energy is −2π²≈−19.739 AU²/yr² at every time.
Compare and explain. Why can the circular-orbit formula v²=GM/r not be used everywhere on an ellipse?
Model limit. Negative energy follows from zero potential at infinity; it is not a lack of energy.
08 · A circle is an ellipse
At zero eccentricity the foci coincide and distance, speed and angular speed are constant.
e=0,\qquad r=a,\qquad v=\sqrt{\frac{GM}{a}}
Try itLoad e=0, inspect the sectors and speed graph, then gradually increase eccentricity.
Prediction. At a=1 AU and M=1, speed is 2π≈6.283 AU/yr and period is 1 yr.
Compare and explain. Why do equal areas correspond to equal angles for a circle?
Model limit. This model covers bound orbits with e≤0.9. Explore escape in Newton’s cannon.