Earth, planets and orbits
1687
Universal gravitation
Isaac Newton
Why doesn’t the Moon fall down?
In the Principia, Newton showed that one force explains a falling apple and an orbiting Moon. A pull that weakens with the square of distance produces exactly Kepler’s ellipses. The heavens and the Earth now obeyed the same law.
First, make a guess
Roughly how fast must a cannonball be fired sideways (above the air) so it never lands?
- Last shot
- Set a speed and fire.
The cannon fires sideways from 200 km up, above nearly all the air, so there is no drag. The mountain is drawn much taller than any real one. Circular orbit here needs about 7.8 km/s; escape needs about 11.0 km/s.
How they figured it out
In January 1684, Edmond Halley, Robert Hooke and Christopher Wren argued in a London coffee house about whether a force falling off with the square of distance would produce elliptical orbits. Halley went to Cambridge to ask Newton, who said he had already calculated it: an ellipse. He couldn’t find his notes, so he redid the work. It grew into the Principia, published in 1687 at Halley’s own expense.
Newton’s key test was the Moon. In one second, an apple falls about 4.9 m. The Moon, curving around the Earth, “falls” about 1.4 mm towards us each second. The ratio is about 3,600, which is 60². And the Moon is about 60 Earth radii away. The same force, weakened by distance squared, pulls both. The Moon is falling; it just keeps missing.
The equation
Every mass pulls on every other; double the distance and the pull drops to a quarter.
See it in real life
Why astronauts float
Astronauts on the space station are not beyond gravity: up there it is still about 90% as strong. They float because they and the station are falling together, moving sideways so fast (7.66 km/s) that the ground curves away as quickly as they fall.
Where engineers use it
Spaceflight
Launch, orbit and transfer calculations start from Newton’s law of gravity.
Tides and geodesy
Tide tables and precise maps of Earth’s gravity use the same law.
Worked example: The space station’s speed
- Orbit radius km.
- km/s.
- About 92 minutes per orbit.
Where it came from
- A telescope on Jupiter 1610
Built on
Galileo’s falling bodies and his idea of inertia became Newton’s laws of motion.
- Kepler’s laws of planets 1609–1619
Led to
Newton showed that an inverse-square pull produces exactly Kepler’s ellipses and his third law.
- The invention of calculus 1665–1684
Built on
Getting from a force to an orbit needs the ideas of calculus, though Principia’s proofs are written as geometry.
What it led to
- A comet returns on schedule 1705–1758
Led to
Newton’s gravity let Halley predict a comet’s return.
- Balance points in space 1772
Led to
Lagrange solved a special case of Newton’s three-body problem.
- Finding a lost world 1801
Built on
Gauss’s orbit for Ceres is a Newtonian orbit fitted to a few observations.
- A planet found on paper 1846
Led to
Newton’s law, trusted completely, said something unseen was pulling Uranus.
- The divergence theorem 1813–1826
Led to
Gauss proved his case while studying the gravity of solid bodies, Newton’s problem.
- Galaxies spin too fast 1970
Built on
Rubin’s argument is Newton’s gravity applied to stars around a galaxy.