Interactive two-body physics
Why does gravity, not density, hold the Solar System together?
Drag mass, density, distance, and velocity to bend a real orbit — then flip on a rival "density-based" force law and watch it fall apart against 400 years of astronomical measurement.
01 · Simulator
Bend an orbit in real time
The Sun sits fixed at the origin. Set a starting distance and a tangential velocity, then watch Newton's inverse-square law decide the shape. Toggle the model to see what happens if attraction depended on density instead of mass.
Under Gravity-only, this trajectory is completely unaffected by mass or density — only distance and velocity matter. That's Galileo's equivalence principle. Switch to Density-based and move the density slider away from Earth's value (5514 kg/m³) to see the predicted path peel away from reality.
02 · The two models
One law depends only on mass. The other quietly depends on density too.
Gravity-only model
F = G·M☉·m / r²
Newton's law: the force on an orbiting body depends on the mass of the Sun and the distance r — never on the properties of the orbiting body itself. Divide by the orbiting mass m to get acceleration and m cancels out entirely:
a = G·M☉ / r²
This is why a spacecraft, a boulder, and a planet all follow the same path at the same distance and speed — mass and density are irrelevant to the trajectory.
Density-based model
a = G·M☉ · (ρbody / ρEarth) / r²
This strawman model — echoing the pre-Newtonian intuition that "denser objects feel gravity more" — swaps the orbiting body's density into the force law, calibrated to agree with reality only at Earth's density (5514 kg/m³).
Because planetary density varies far less than mass (Jupiter is 318× Earth's mass but only 0.24× its density), this model badly under- or overshoots the true pull for anything that isn't rock-Earth-density — as the data below shows.
03 · Force-law data
Predicted vs. observed gravitational pull, Mercury to Neptune
For each planet we take its real orbital velocity and distance and derive the observed centripetal acceleration (v²/r) — this is what actually keeps it in orbit. Then we compare two predictions against that observation.
| Planet | Distance | Density | Observed accel. | Gravity-model prediction | Gravity error | Density-model prediction | Density error |
|---|
Data: JPL Solar System Dynamics — Planetary Physical Parameters, Orbital Mechanics & Astrodynamics reference tables. Observed acceleration derived from measured mean orbital velocity and semi-major axis (a = v²/r).
04 · Moon trajectories
The same failure shows up around Jupiter
Calibrate the density-based model on Io — the innermost, densest Galilean moon — so it matches reality there. Then test it on Europa, Ganymede, and Callisto, whose densities drop steadily with distance from Jupiter.
| Moon | Distance | Density | Observed velocity | Gravity-model prediction | Gravity error | Density-model prediction | Density error |
|---|
Data: JPL Planetary Satellite Physical Parameters, NASA Galilean Moons of Jupiter fact sheet. Observed velocity from mean orbital radius and period (v = 2πr/T); density-model calibrated to match Io exactly, then applied unchanged to the other three moons.
The pattern, in one line
Gravity's prediction tracks every planet and moon in this dataset to within a fraction of a percent — because it correctly ignores the orbiting body's composition. The density-based model only agrees at its calibration point and drifts by 25–90% everywhere else, because density is the wrong variable. Mass — and only mass — determines gravitational attraction.