Gravity · Kepler orbits

Orbital Launchpad

Launch a circle, an ellipse or an escape trajectory. Orbit the scene and replay every instant. Compare the numerical path with an analytic conic, then inspect equal-time swept areas and the energy record.

Initial conic $0$

Recorded radius $0$

Specific energy $0$

Initial eccentricity $0$

Predicted periapsis $0$

Energy error / initial potential scale $0$

SET THE INITIAL CONDITIONS

Give gravity a starting point

Recorded pathOnly the portion reached by the replay clock.Analytic conicA dashed prediction from the launch state, separate from the integrated record.Velocity and attractionMint shows

\mathbf v

; gold shows attraction. Lengths are compressed.Equal-time swept sectorsTwo recorded half-unit intervals; compare their areas.Gravitational parameter

\mu

1

Initial radius

r_0

1.5

Speed / local circular speed

v_0/v_c

0.8

Launch angle from outward radial axis

\alpha

90

Recorded time span

T

16

Normalized units. Planar Newtonian test particle around a fixed point mass. Sliders apply on launch. Recording stops inside radius 0.08; this is a numerical boundary, not a modeled impact. Body sizes and arrows are enlarged; relativity and other bodies are omitted.

One recorded trajectory drives the scene, readouts and CSV. Physical quantities use normalized units; body sizes are enlarged.

Recorded time

t=0.00

Solid paths are integrated samples; the dashed conic is calculated independently from the initial position and velocity. The coloured sectors start at launch and halfway through the record. Both span half a normalized time unit, shortened equally for a brief record.

r_{\mathrm p}=\frac{h^2}{\mu(1+e)},\qquad \frac{\mathrm d A}{\mathrm d t}=\frac{h}{2}

Time, distance and mass use normalized units. The plane and rings are coordinate guides. Display size does not represent a physical radius. Energy errors are divided by the magnitude of the initial potential energy; they remain meaningful at zero total energy. Replay never recomputes the trajectory.

Add a third body →NASA: gravity and orbital motion

Start with the essentials

Focus question: From position and velocity alone, how can we tell whether a body returns or escapes?
One-sentence intuition: Specific energy decides whether the orbit is bound; angular momentum sets its shape and periapsis.

Core mathematical model

Central gravity

\ddot{\mathbf r}=-\frac{GM}{r^3}\mathbf r

Acceleration always points inward and its magnitude falls as $1/r^2$ .

Specific mechanical energy

\varepsilon=\frac{v^2}{2}-\frac{GM}{r}

$\varepsilon<0$ is a bound ellipse, $\varepsilon=0$ is the parabolic boundary, and $\varepsilon>0$ escapes hyperbolically.

Common difficulties

Gravity does not guarantee return

Typical misconceptionA body must return because gravity never disappears.

Better mental modelGravity weakens with distance. Positive total energy allows escape even while speed decreases.

Circular motion still accelerates

Typical misconceptionConstant speed means zero acceleration.

Better mental modelVelocity direction changes continuously, requiring inward centripetal acceleration.

Run the experiment

01
Identify a circular orbit
Choose Circular and compare velocity with gravity.
What to observe: The arrows are nearly perpendicular and radius stays stable.
02
Change the energy
Hold launch radius fixed, increase speed and launch a new record each time.
What to observe: The ellipse lengthens toward the escape boundary.
03
Cross the escape threshold
Choose Escape and inspect the classification, predicted periapsis and conservation errors.
What to observe: Specific energy becomes positive and the predicted curve opens.

Orbital Launchpad

Does the orbit keep its promises?

Orbits: read conic sections through energy