Initial position r₀km
Initial velocity v₀km/s
System geometry
Every body, pair edge, barycenter, and label is generated from the active state.
TIER A · DIRECT STATE
/ Teaching Tools / Gravitational Dynamics
Anything orbiting anything
No Earth-centered assumption is required. Enter the mass, absolute initial position, and absolute initial velocity of any two bodies. The model uses their relative state to classify the orbit while retaining the inertial motion of both bodies.
Two-body Cartesian initial-condition builder
Body A can move naturally with Body B, or it can be held as an explicit external constraint. Values accept ordinary scientific notation and forms such as 1.989 × 10^30.
Initial position r₀km
Initial velocity v₀km/s
Ready for any finite masses and Cartesian initial conditions. The orbit is determined by the numbers, not by either body’s name.
Extend the complete initial-value problem
The two-body builder feeds this general editor. Add a third body—or up to 64—then edit every mass, position, velocity, and external constraint directly.
Initial state vector
N =
How to edit: click any gold
MASS cell, enter a positive value—including 7.35e22, 7.35 × 10^22, or 7.35×10²²—then compile the edited model.Earth + satellite preset compiled| # | Name | Mass · kg · editable | x · km | y · km | z · km | vₓ · km/s | vᵧ · km/s | v_z · km/s | Fixed |
|---|
Build model to compile the right-hand sideDirect summation · O(N²) · N ≤ 64
Compile orbital geometry into initial conditions
Circular, elliptic, parabolic, and hyperbolic motion are not separate force laws. They are different energy–angular-momentum regimes of the same inverse-square equation. Select an ordered pair, inspect its current osculating conic, or write a new state from first-principles orbital elements.
Osculating-conic laboratory
The secondary is described relative to the primary. For a pure two-body model, these elements define the exact Kepler orbit. In an N-body model they define the instantaneous osculating orbit at the chosen epoch; all other masses perturb it after compilation.
Construction uses p = rₚ(1+e), then rotates the perifocal state by R₃(Ω)R₁(i)R₃(ω). For two free masses, the pair barycenter and total momentum are preserved exactly.
The audit reads the selected pair directly from the declared state.
Derived from r, v, μ—never inferred from the drawing
select two massesε = v²/2 − μ/r determines bound versus escape; e⃗ = (v×h)/μ − r̂ fixes conic shape; h = r×v fixes its plane. Circular requires e ≈ 0 and radial velocity ≈ 0—not a special animation preset.Generated equations of motion
The generator expands the acceleration of each declared body across every other body. A closed analytic solution is claimed only where one exists; otherwise the equation is propagated numerically and audited.
Compile the model to generate its governing equation.
The requested position solution, ri(t)
not yet compiled
Compile the model to construct the position map.
Acceleration is the differential equation, not the completed motion. This layer reports the exact position function when available, or the declared numerical interpolant when it is not.
Numerical state and interaction audit
Direct pair force
Each RHS evaluation computes all N(N−1)/2 unordered pairs. With ε = 0, acceleration is exactly G m_j Δr / |Δr|³.
Leapfrog KDK
Second-order, symmetric, time-reversible, symplectic kick–drift–kick splitting. Continuous Run advances only whole declared steps; an explicit time seek may use one final shortened step.
Yoshida fourth order
A symmetric composition of second-order symplectic maps with Yoshida coefficients. Continuous Run uses a literal fixed step and reports the last accepted h.
Dormand–Prince 5(4)
Embedded adaptive Runge–Kutta. Rejected steps do not advance time. Accurate locally, but not symplectic.
RK4
Classical fourth-order fixed-step propagation. Useful as a familiar reference, not a structure-preserving long-horizon default.
Position output r(t)
One body is integrated exactly. An unsoftened free pair—or one moving body about one externally fixed primary—uses universal Kepler variables. Every other compiled model exposes a continuous cubic-Hermite position function on accepted solver intervals.
Validity and honesty ledger
What each stage is allowed to claim
Acceleration field
Every glyph evaluates the same direct-summation acceleration law at a massless test point.
TIER A · COMPUTED FIELD
Trajectories
Solid fading paths are accepted numerical states. The dashed two-body conic, when present, is independently generated from the exact universal-variable position solution.
TIER A/B · EXACT REFERENCE + MODEL
Invariant history
Residuals are recomputed from the current state and normalized by characteristic scales. Barycenter error is measured against its expected inertial worldline—not against a falsely stationary origin.
TIER A · DIRECT AUDIT
Collision boundary
The point-mass law is singular at zero separation. Softening changes the model; it is never enabled silently.
BOUNDARY DECLARED
Not included
No post-Newtonian terms, radiation, finite radii, mergers, regularization, tree code, or close-encounter controller.
NEWTONIAN DOMAIN ONLY
Methods and primary references
- Yoshida, H. (1990). Construction of higher-order symplectic integrators. paper ↗
- Dormand, J. & Prince, P. (1980). A family of embedded Runge–Kutta formulae. paper ↗
- Hairer, Lubich & Wanner. Geometric Numerical Integration. reference ↗
- JPL Solar System Dynamics. Astrodynamic parameters and reference constants. source ↗