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Premium experiment • orbital state vectors

PHYSICS • MECHANICS

Orbital Mechanics Lab

Change initial altitude and tangential speed, then propagate an Earth-like two-body orbit while tracking conserved quantities and conic geometry.

● TWO-BODY GRAVITYOrbital Mechanics
Eccentricity
Altitude
Specific energy
Period

Reasoning inspector

live model state
Speed—
Circular speed—
Escape speed—
v / circular—
Specific energy—
Angular momentum—
Eccentricity—
Periapsis altitude—
Apoapsis altitude—
Period—
Trajectory class—

Read an orbit from its state

Position and velocity determine the orbit. The simulator converts the evolving state vector into energy, angular momentum, eccentricity and turning-point distances.

LEARN WITH THE SIMULATION

Understand Orbital Mechanics Lab by experimenting.

START HERE

Orbital mechanics is a state-vector problem: position and velocity together determine the conic trajectory. This laboratory starts a spacecraft tangentially at a chosen Earth-like altitude, propagates the ideal two-body equations numerically, and continuously derives specific orbital energy, angular momentum, eccentricity, periapsis, apoapsis and orbital period. The premium inspector also compares instantaneous speed with local circular and escape speeds so a learner can predict the orbit before pressing Play.

BUILD THE IDEA

What you should understand first

  • Use local circular speed and escape speed as reference velocities for classifying a tangential injection.
  • Connect the sign of specific orbital energy with bound and unbound trajectories.
  • Explain how eccentricity, periapsis and apoapsis follow from the same position,velocity state.
  • Recognize why a tangential injection below circular speed makes the starting point an apoapsis, while a faster-but-bound injection makes it a periapsis.

TRY IT YOURSELF

Why does changing speed at one fixed altitude change both the shape and the turning points of the entire orbit?

  1. Start with speed factor 1.00 and verify that eccentricity is near zero and periapsis and apoapsis nearly coincide.
  2. Lower the speed factor while keeping altitude fixed; predict whether the spacecraft is currently at periapsis or apoapsis before reading the inspector.
  3. Raise the factor above 1 but keep it below the escape threshold and compare the new eccentricity and apoapsis.
  4. Move toward a factor of √2 and watch specific orbital energy approach zero while apoapsis grows without bound.
Read the visual:

At a fixed starting radius with purely tangential velocity, circular speed is the boundary between starting at apoapsis and starting at periapsis. The escape boundary occurs when the speed reaches √2 times local circular speed, where specific orbital energy reaches zero and the bound orbital period ceases to exist.

WHY IT WORKS 01

Circular speed and escape speed are two different energy landmarks

Local circular speed is √(μ/r). At exactly that speed, kinetic and gravitational terms combine to give the energy of a circular orbit and eccentricity zero. Escape speed is √(2μ/r), larger by √2. At escape speed the specific orbital energy reaches zero; below it the conic is bound, while above it the ideal two-body trajectory is unbound.

WHY IT WORKS 02

A tangential injection begins at an apsis

The initial radial velocity is zero, so the spacecraft begins at a radial turning point of the osculating conic. If its tangential speed is below circular, gravity bends the trajectory inward and the starting point is the high point, apoapsis. If speed is above circular but still below escape, the starting point is periapsis and the spacecraft climbs to a larger apoapsis.

WHY IT WORKS 03

For this special setup, speed factor predicts eccentricity directly

Because the initial velocity is purely tangential, substituting v = f·vc into the energy and angular-momentum relations gives e = |f² − 1| for the conic generated at that instant. This makes the target-eccentricity control a reasoning exercise rather than trial and error: choose whether the target should come from a slower or faster-than-circular injection, then estimate the needed speed factor.

DO NOT MIX THESE UP

Common misconception

Orbiting objects are not beyond gravity; gravity supplies the centripetal acceleration continuously. A larger tangential speed at one radius does not simply create a larger circular orbit: unless the speed is exactly the circular value for that radius, the state belongs to an ellipse or an escape conic. The displayed trajectory also assumes an ideal spherical central body and ignores atmosphere, oblateness, third-body gravity and propulsion after injection.

PRACTICE CHALLENGE

Can you hit a target eccentricity and explain whether the initial point is periapsis or apoapsis?

  • Set a target eccentricity in the premium controls.
  • Estimate a speed factor using e ≈ |f² − 1| for this tangential initial condition.
  • Adjust the speed factor until the inspector reaches the target tolerance.
  • Use the speed-versus-circular comparison to state whether the starting point is periapsis, apoapsis or nearly circular.
Reveal the reasoning

For the tangential start used here, e = |f² − 1|, where f is speed divided by local circular speed. A target e therefore has two bound possibilities when available: f = √(1−e) gives a slower injection beginning at apoapsis, while f = √(1+e) gives a faster injection beginning at periapsis. Escape occurs as f approaches √2.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • Motion is planar and governed by Newtonian two-body gravity around a fixed central mass.
  • The spacecraft mass is negligible compared with the central body, so the gravitational parameter stays constant.
  • No atmosphere, thrust, oblateness, third-body gravity or relativistic corrections act on the trajectory.

Model limitations

  • Very close approaches and long integrations can accumulate numerical error despite the symplectic-style velocity-Verlet step.
  • The Earth-like gravitational parameter and radius are pedagogical defaults rather than a complete mission-analysis environment.

CONNECT IT

Where this matters

  • Satellite insertion and transfer-orbit reasoning
  • Spacecraft periapsis and apoapsis planning
  • Escape-speed and mission-energy analysis
  • Teaching conserved quantities in central-force motion

WORDS THAT MATTER

Key vocabulary

State vector
Position and velocity at one instant; together they determine the osculating two-body orbit.
Specific orbital energy
Orbital mechanical energy per unit mass, ε = v²/2 − μ/r.
Circular speed
Tangential speed √(μ/r) required for a circular two-body orbit at radius r.
Escape speed
Speed √(2μ/r) at which specific orbital energy reaches zero at radius r.
Eccentricity
Dimensionless conic-shape parameter: zero for a circle, between zero and one for an ellipse, one for a parabola and greater than one for a hyperbola.
Apsis
Radial turning point of an orbit; periapsis is closest and apoapsis is farthest for a bound ellipse.