An interactive laboratory exploring the mathematics behind planetary orbits, scale, sunlight, spaceflight and time. Change the assumptions and let the equations show you what happens.
The Solar System spans an enormous range of distances. Switch between a linear scale and a logarithmic scale to see why representing it visually is difficult. Then introduce orbital eccentricity and watch the planets move faster and slower around their orbits.
The planets themselves occupy remarkably little of the space in which they move. Most of the Solar System is empty space.
At this scale Earth would be approximately 1 cm across, but would orbit roughly 107 metres from the Sun.
The Sun would be roughly 109 mm across and the average Earth-Sun distance would be about 11.7 metres.
The average Earth-Moon distance is approximately 384,400 km. Even the nearest planetary companion is separated by a vast distance.
Neptune orbits at roughly 30 AU, around thirty times the average Earth-Sun distance.
One of the simplest and most powerful relationships in planetary astronomy is Kepler's third law.
When T is measured in Earth years and a is measured in astronomical units, the relationship becomes particularly simple.
Orbital speed depends on both the distance from the central body and the mass of that body. For an approximately circular orbit:
Real planetary orbits are ellipses rather than perfect circles. Eccentricity controls how elongated the ellipse is.
A planetary year is simply the time required for one complete orbit.
Solar radiation spreads over an expanding spherical surface. As distance increases, the energy received per unit area falls with the square of the distance.
1 AU → 100%
1.52 AU → about 43%
5.20 AU → about 3.7%
30.05 AU → about 0.11%
Light travels at approximately 299,792 km/s. The finite speed of light means that looking into space is also looking into the past.
A simplified Hohmann transfer uses an elliptical orbit to move a spacecraft from one circular planetary orbit to another. It is one of the classic applications of orbital mathematics.
This is an idealised two-body calculation. Real missions also require consideration of planetary positions, inclination, launch windows, gravity assists, atmospheric effects and spacecraft design.
Newtonian mechanics treats time as universal. Einstein's relativity shows that the rate at which time passes depends on motion and gravity.
For an object moving at velocity v relative to an observer:
Time measured by the stationary observer.
Time experienced by the traveller.
Gravity also changes the rate at which clocks run. For a non-rotating spherical body, a simplified Schwarzschild expression is:
Imagine two identical clocks. Clock A stays at Earth while Clock B is placed near another object. How much does the gravitational field change the rate?
GPS satellites experience both special-relativistic and gravitational time dilation. The effects act in opposite directions and both must be accounted for.
Change one physical assumption and see which mathematical relationships change with it.
Approximate mean orbital values used by the interactive models.
| Planet | Distance | Year | Orbital speed | Eccentricity |
|---|---|---|---|---|
| Mercury | 0.387 AU | 0.241 years | 47.4 km/s | 0.206 |
| Venus | 0.723 AU | 0.615 years | 35.0 km/s | 0.007 |
| Earth | 1.000 AU | 1.000 year | 29.8 km/s | 0.017 |
| Mars | 1.524 AU | 1.881 years | 24.1 km/s | 0.093 |
| Jupiter | 5.203 AU | 11.862 years | 13.1 km/s | 0.049 |
| Saturn | 9.537 AU | 29.457 years | 9.7 km/s | 0.057 |
| Uranus | 19.191 AU | 84.016 years | 6.8 km/s | 0.047 |
| Neptune | 30.069 AU | 164.79 years | 5.4 km/s | 0.009 |
The Solar System demonstrates how mathematics can describe systems spanning billions of kilometres.
Simple power-law relationships explain why orbital periods increase so rapidly with distance.
Gravity provides the physical mechanism behind orbital motion.
At high velocities and strong gravitational fields, even time itself becomes part of the mathematical problem.