DYNAMICS · NEWTONIAN GRAVITATION
Gravitation and Orbits
Explore circular and elliptical orbits, escape trajectories and impacts on a surface. Explore universal gravitation, Kepler’s three laws, and conservation of energy and angular momentum.
Physical Parameters
Orbit and Gravitational Force
The x and y axes use the same distance scale: orbital shapes and distances are not distorted. The satellite marker and vector arrows are schematic.
Synchronized Graphs
Select a point on a graph or enter a specific time. The curves distinguish free orbital motion from an inelastic surface impact; energy and angular momentum are conserved during free gravitational motion.
Distance from Centre vs Time
Speed Magnitude
Kinetic, Potential and Mechanical Energy
Swept Area (Kepler’s Second Law)
Numerical Residuals During Free Flight
Relative scale ×10⁻⁹. Red: δE/(K₀+|U₀|); green: δh/h₀. The gray band and dashed lines mark ±10⁻⁸. The test does not apply after impact.
Numerical Conservation-Law Audit
Maximum relative energy error: —. Maximum relative angular-momentum error: —.
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The display and numerical verification threshold is 10⁻⁸. Residuals are computed only during free gravitational motion; they do not represent physical losses of energy or angular momentum.
Data Table
| t (s) | Phase | x (m) | y (m) | r (m) | vₓ (m/s) | vᵧ (m/s) | K (J) | U (J) | E (J) | h (m²/s) | A (m²) | Residual (J) |
|---|
Comma-separated CSV with decimal points; all columns use SI units.
Model Assumptions
Newtonian gravitation of a spherical, stationary central body, modeled as a point mass outside its surface. The satellite is a test mass (m ≪ M), and the motion of the central body is neglected. No friction, atmospheric drag, thrust or other perturbations are included.
Propagation uses theuniversal-variable Kepler equation, solved numerically. In the ideal two-body model, integrals of motion are conserved to numerical precision.
If the orbit intersects the surface, the satellite stops at the first inelastic impact and does not pass through the central body; the kinetic-energy loss is reported separately. Kepler’s laws apply to the free-flight segment, not the constrained motion after impact.
Escape trajectories are open: they have no orbital period. The period law applies only to bound orbits. Local escape speed neglects air resistance and the planet’s rotation.
FROM MODEL TO RESULTS
Step-by-Step Calculations
General equations, SI substitutions, dimensional checks and verification of Kepler’s laws.
Equations are shown separately and left-aligned. MathJax with MathML fallback. Rounding affects display only.
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