Gravity · Kepler orbits

Orbital Launchpad

How can one force law draw three different conic sections? The starting position and velocity fix the energy and angular momentum, and those choose the curve. Negative energy gives a bound ellipse or circle, zero gives a parabola, and positive energy gives a hyperbola that escapes. For a given energy, angular momentum sets how stretched the orbit is and how close it comes.

Launch a body into a circle, an ellipse or an escape path. Rotate the view and replay any moment. Compare the computed path with the exact conic section, then look at the areas swept in equal times and at the recorded energy.

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

Choose the launch radius, speed and angle

Recorded pathDraws only the part the replay has reached so far.Analytic conicA dashed curve predicted from the launch state, shown apart from the integrated path.Velocity and attractionMint shows

\mathbf v

; gold shows the pull of gravity. Arrow lengths are compressed.Equal-time swept sectorsCompare the areas swept in two recorded intervals of half a time unit each.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

Units are normalized. A test particle moves in a plane around a fixed point mass under Newtonian gravity. Slider changes take effect at launch. Recording stops once the radius falls below 0.08; this is a numerical cutoff, not a modeled impact. Body sizes and arrows are enlarged, and relativity and other bodies are left out.

The scene, the readouts and the CSV file all come from the same recorded trajectory. All quantities are in normalized units, and the bodies are drawn larger than scale.

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: The energy per unit mass decides whether the body stays bound; the angular momentum sets the shape of the orbit and its periapsis, the point of closest approach.

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 $\varepsilon>0$ means escape on a hyperbola.

Common difficulties

Gravity does not guarantee return

Typical misconceptionA body must return because gravity never disappears.

Better mental modelGravity weakens with distance. If the total energy is positive, the body escapes even though it keeps slowing down.

Circular motion still accelerates

Typical misconceptionConstant speed means zero acceleration.

Better mental modelThe direction of the velocity changes all the time, and that takes a centripetal acceleration pointing inward.

Run the experiment

01
Identify a circular orbit
Choose Circular and compare velocity with gravity.
What to observe: The two arrows are nearly perpendicular and the radius stays steady.
02
Change the energy
Keep the launch radius fixed, raise the speed step by step and launch a new record each time.
What to observe: The ellipse stretches longer and approaches the escape boundary.
03
Cross the escape threshold
Choose Escape and inspect the classification, predicted periapsis and conservation errors.
What to observe: The specific energy turns positive and the predicted curve no longer closes.

Orbital Launchpad

Does the orbit keep its energy and angular momentum?

Orbits: read conic sections through energy