Orbiting is falling
Isaac Newton imagined a cannon on a very tall mountain. Fire a ball slowly and it falls to the ground nearby; fire it faster and it lands farther away. Fire it fast enough and the ground curves away beneath it as quickly as it falls — it never lands. That is an orbit: an object continuously falling toward a planet or star while moving sideways fast enough to keep missing it. The same gravity that drops an apple holds the Moon and the planets in their paths.
Circles, ellipses and escape
- At exactly circular speed and launched sideways, the pull of gravity bends the path into a circle.
- Slower, gravity wins at first: the planet dives inward, speeds up, swings around, and returns — an ellipse with the launch point farthest away.
- Faster (but below escape speed), the planet climbs outward and slows down before falling back — a larger ellipse with the launch point closest.
- At or above escape speed (√2 ≈ 1.41 times circular speed), the path opens up and the planet never returns.
Kepler’s three laws
Johannes Kepler discovered these laws in the early 1600s from careful observations of Mars. Newton later showed that all three follow from his law of gravity.
- Orbits are ellipses, with the star at one focus (not the centre).
- Equal areas in equal times: a line from the star to the planet sweeps out equal areas in equal times, so the planet moves fastest at its closest point (perihelion) and slowest at its farthest (aphelion).
- T² ∝ a³: the square of the orbital period is proportional to the cube of the orbit’s average size (semi-major axis). Planets farther out take much longer to orbit.
Formulas
Gravity: F = G·M·m / r²Circular speed: v_c = √(GM / r) Escape speed: v_esc = √(2GM / r)Kepler’s third law: T² = 4π²·a³ / (G·M)In the units used by the simulator (AU, years and solar masses), the third law becomes simply T² = a³ / M. For Earth, a = 1 AU and M = 1, so T = 1 year.
Key terms
- Astronomical unit (AU)
- The average Earth–Sun distance, about 150 million km.
- Semi-major axis (a)
- Half the longest diameter of an ellipse — the orbit’s average distance.
- Eccentricity (e)
- How stretched an orbit is: 0 for a circle, between 0 and 1 for an ellipse, 1 or more for escape paths.
- Perihelion and aphelion
- The closest and farthest points of an orbit around the Sun.
- Bound orbit
- An orbit with negative total energy, which repeats forever.
Real-world applications
- Satellites: engineers choose orbit heights and speeds for GPS, weather and communications satellites using these same equations.
- Space missions: spacecraft travel between planets on elliptical “transfer orbits” and use gravity assists to gain speed.
- Weighing stars and planets: measuring a moon’s orbit lets astronomers calculate the mass of the planet it orbits.
- Finding exoplanets: a star wobbling in response to an orbiting planet reveals the planet’s mass and orbit.
Common misconceptions
- “There is no gravity in space.” Gravity in orbit is nearly as strong as on the ground; astronauts float because they are falling together with their spacecraft.
- “Seasons happen because Earth gets closer to the Sun.” Earth’s orbit is nearly circular (e ≈ 0.017); seasons are caused by the tilt of Earth’s axis.
- “The star is at the centre of an elliptical orbit.” It sits at one focus, off-centre.
What the model simplifies
The star is fixed in place and much heavier than the planet, and there are no other planets tugging on the orbit. The orbit is calculated step by step with a method that conserves energy well, so ellipses close neatly. Real orbits slowly change because of other planets and, for Mercury, effects described by Einstein’s general relativity.
Key takeaways
- An orbit is a balance between sideways motion and the pull of gravity.
- Circular speed is √(GM/r); escape speed is √2 times larger.
- Planets move fastest when closest to their star (equal areas in equal times).
- T² ∝ a³: larger orbits take disproportionately longer.