From a description to a cause
In the previous guide you met [[keplers-laws-of-planetary-motion|Kepler's laws]]: planets trace ellipses, sweep equal areas in equal times, and obey the tidy rule that the orbital period squared tracks the orbit's size cubed. Those three laws are magnificent, but they are a description, not an explanation. Kepler found the pattern in the sky the way a careful naturalist catalogues the shapes of leaves — accurately, beautifully, and without yet knowing why the tree grows them. He could tell you exactly how Mars moves, but not one word about what was holding Mars on its path.
Isaac Newton's leap was to ask the why, and to answer it with a single, audacious idea: the same force that makes an apple fall to the ground also reaches across empty space to hold the Moon. Not a different 'celestial' physics for the heavens and an 'earthly' physics for the ground — one rule, everywhere. This is the move that turns a catalogue into a science. If one principle can both drop the apple and steer the Moon, then the sky stops being a separate magic realm and becomes a place where the same physics we test in a laboratory plays out on a colossal stage.
One law for the whole sky
Here is the law itself, in plain words. [[newtons-law-of-universal-gravitation|Newton's law of universal gravitation]] says that every mass pulls on every other mass with a force that grows with both masses and falls off with the square of the distance between them. Double either mass and you double the pull; move twice as far apart and the pull drops to a quarter, not a half. That 'square of the distance' part is the same inverse-square shape you already met for starlight spreading over a sphere, and it is not a coincidence — both spread their influence over the surface of an expanding sphere.
F = G * (m1 * m2) / r^2 F = gravitational pull between the two bodies m1 = mass of one body m2 = mass of the other r = distance between their centers G = 6.674 x 10^-11 (a tiny, universal constant)
The triumph is that all three of Kepler's laws fall out of this one equation as theorems, not as separate discoveries. Feed Newton's gravity into his laws of motion and the mathematics hands you ellipses automatically; the equal-areas rule turns out to be the conservation of angular momentum; and the period-versus-size relationship emerges with the masses now written in. Kepler had three rules he could not derive; Newton had one cause from which all three follow — and which keeps working far beyond the planets Kepler ever saw.
An orbit is just perpetual falling
Now to the most beautiful idea in this whole guide — the one that, once seen, you can never un-see. An orbit is not a state of weightless floating, and it is certainly not a place where gravity has switched off. The Moon is falling toward the Earth right now, hard, every second. The astronauts on the space station are falling too; their 'weightlessness' is simply the feeling of falling freely, the same drop-in-your-stomach sensation as the first moment off a diving board, except it never ends. Gravity is fully on. So why don't they hit the ground?
Because they are also moving sideways, fast. Newton imagined firing a cannonball from a mountaintop. Fire it gently and it arcs down and lands nearby. Fire it harder and it lands farther away, its curving fall stretched by its speed. Fire it hard enough and something marvellous happens: the ground curves away beneath it exactly as fast as the ball falls toward the ground. Now it is still falling — always falling — but it never gets any closer. It has fallen all the way around the world. That endless miss is an orbit, and the Moon is doing nothing fancier than Newton's cannonball going fast enough to keep missing the Earth forever.
Two magic numbers: orbit speed and escape speed
If an orbit is a perfect, never-ending miss, there must be a particular sideways speed that produces it — too slow and the cannonball spirals down to the ground, too fast and it climbs away. For a circular orbit just skimming a body's surface, that special speed is the circular orbital velocity. It depends only on the central body's mass and the orbit's radius, nothing about the orbiting object itself. That is why a feather and a battleship, given the same push at the same height, would circle the Earth on identical paths — gravity treats all masses alike, which is why Galileo's dropped objects all fell together.
Push faster than the circular speed and the orbit bulges from a circle into an ellipse — the same shape Kepler found, now understood as just a faster, lopsided miss with some eccentricity. Push faster still and you reach the second magic number: [[escape-velocity|escape velocity]], the speed at which the object is moving quickly enough to climb away forever and never fall back. At exactly escape speed it coasts outward, slowing but never quite stopping, finally crawling to rest only at infinite distance. To leave the Earth's surface for good takes about 11.2 kilometres per second — about 40000 kilometres per hour, which is why launching anything off a planet is so monstrously expensive.
Escape speed has a deeper meaning: it is the price tag of a body's gravity, its [[gravitational-binding-energy|gravitational binding energy]] turned into a velocity. A bigger or denser body charges a higher price. The Moon's escape speed is a gentle 2.4 km/s, which is why its weak grip let almost all its air leak away into space, leaving it airless. The Sun's is over 600 km/s. Push this idea to its breaking point — imagine an object so dense that its escape speed would exceed the speed of light — and you have stumbled onto the seed of a black hole, the very edge where, in a later rung, Newton himself must hand the baton to Einstein.
Two bodies, one dance — and the honest limits
There is one more refinement Newton hands us. We loosely say the Moon orbits the Earth, but gravity always pulls both ways with equal force: the Earth tugs the Moon, and the Moon tugs the Earth back just as hard. So neither truly sits still. Both circle their common [[center-of-mass|center of mass]] — the shared balance point, which for the Earth-Moon pair lies inside the Earth, but for two stars of similar mass sits in the empty space between them. This clean [[two-body-problem|two-body problem]] has an exact, elegant solution, and it is the engine behind one of the field's great tricks: by watching how a star wobbles, astronomers weigh unseen companions and even planets they have never directly seen.
Be honest about where the neatness ends, though. The exact solution is for two bodies only. Add a third — say the Sun, Earth, and Moon all pulling on one another — and there is no tidy formula at all; the motion can only be chased by computer, and that humbling fact opened the whole study of chaos. Newton's gravity is also not the final word. It is breathtakingly accurate for everyday orbits and got humanity to the Moon, but it is a [[newtonian-limit-of-gravity|low-speed, weak-gravity limit]] of Einstein's deeper theory. Where gravity grows fierce or speeds approach that of light — near the Sun, around a pulsar, at a black hole — Newton's numbers drift wrong, and the next guides will be where Einstein takes over.