Orbits
Falling Forever
Falling and Missing
Newton drew a picture that settles the whole subject in one move. Put a cannon on a very tall mountain and fire it horizontally. The ball arcs out and lands. Fire it faster and it lands further away. Faster still, and further still, and here is the trick – Earth’s surface is curved, so as the ball falls, the ground is also falling away beneath it.
Get the speed right and the ball falls at exactly the rate the surface curves away. It never gets any closer to the ground. It also never stops falling. That is an orbit. Not a state of floating, not a place where gravity has released its grip, but a permanent, unending fall that keeps missing.
The idea deserves its reputation. Before it, the heavens and the Earth were separate domains with separate rules. Apples fell and planets circled, and the two were not thought to be the same kind of event. Newton’s picture said they are the same event at different speeds. The Moon is not held up by anything. The Moon is falling, and has been for four and a half billion years, and keeps missing.
Why Ellipses and Not Circles
A circular orbit needs the speed to be exactly right. Anything else and the shape stretches into an ellipse, which is what real orbits almost always are. The circle is not the normal case with ellipses as exceptions. The circle is the one special value in a continuous range.
Kepler worked this out from Tycho Brahe’s observations decades before anyone could explain why. He found three rules. Orbits are ellipses with the massive body at one focus rather than the center. A planet moves faster when it is closer and slower when it is further, in a very precise way. And the further out a planet is, the longer its year, on a fixed schedule that holds across the entire solar system.
The second rule is the most beautiful and the least famous. Draw a line from the planet to the star. As the planet moves, that line sweeps out area. Kepler found that it sweeps equal areas in equal times, always. Near the star the planet races through a short fat wedge; far away it crawls through a long thin one, and the two slivers have identical area.
Newton later showed this is not a fact about gravity at all. It is conservation of angular momentum, the same rule that speeds up a skater who pulls their arms in. A planet swinging closer to the star is a skater pulling in. It has to speed up, and the sweep rate is exactly the quantity that stays fixed.
Astronauts Are Not Weightless
The space station orbits about four hundred kilometers up. At that height Earth’s gravity is still roughly ninety percent as strong as it is at the surface. An astronaut floating past the window has very nearly their full Earth weight pulling on them. So why do they float?
Because everything around them is falling at the same rate. The station is falling. The astronaut is falling. The camera, the water droplets, the loose spoon – all falling together, all missing Earth together. There is no relative motion between them, so nothing presses against anything, and the sensation of weight vanishes. What we call weightlessness was never the pull switching off. It is the absence of anything interrupting the fall.
You feel your weight right now because the floor is pushing up on you, refusing to let you fall. Remove the floor and the sensation goes with it. This is why an aircraft flying a particular arc can give passengers half a minute of the same experience without leaving the atmosphere, and it is the observation Einstein built general relativity on.
How Fast Is Fast Enough to Leave
Fire the cannonball hard enough and it never comes back. Escape velocity from Earth’s surface is about eleven kilometers per second, which is roughly forty thousand kilometers per hour. Below that, the path bends back. Above it, the object coasts away and keeps coasting, slowing forever but never quite stopping.
One point here is often garbled. Escape velocity is not a speed limit you must break to leave. It is the speed you would need if you got one single shove and then had no engine at all. A rocket with a working engine can leave at walking pace, given enough fuel, because it keeps pushing the whole way up. Escape velocity is about ballistics, not about permission.
The number depends only on mass and radius, which makes it a good measure of how deep a gravitational well is. It is about two point four kilometers per second from the Moon, sixty from Jupiter, six hundred and eighteen from the Sun’s surface. Push that logic far enough and you reach an object whose escape velocity exceeds the speed of light. The real story there needs relativity rather than this arithmetic.
The Places That Hold Still
When two bodies orbit each other, there are five particular spots where a third small object can sit and keep pace with both, staying in the same relative position indefinitely. They are called Lagrange points, and they are one of the more useful pieces of orbital mechanics ever worked out. The animation below gives the smaller body far more mass than Earth really has, so that the three points strung along the line have room to be labeled; the geometry is otherwise honest.
Three of the five sit in a line through both bodies and are unstable, like a ball balanced on a dome. Something parked there will drift off and needs occasional nudges to stay. The other two lead and trail the smaller body by sixty degrees, and those are genuinely stable – objects fall into them and stay. Jupiter has accumulated over a million asteroids in its two stable points, herded there over billions of years.
The James Webb Space Telescope sits at the unstable point directly opposite the Sun from Earth, about one and a half million kilometers out. From there Earth, Sun, and the Moon all stay in the same direction, so a single shield blocks all three at once and the telescope’s mirrors can sit at forty degrees above absolute zero. It costs a small burn every three weeks to stay there, and that fuel budget is what ultimately limits the mission’s life.
Tides, and the Moon Quietly Leaving
Gravity weakens with distance, so the Moon pulls harder on the near side of Earth than on the far side. That difference stretches the planet slightly along the Earth-Moon line, and the oceans, being fluid, respond most visibly. Two bulges form, one facing the Moon and one directly opposite, and Earth rotates through both of them each day. That is why most coasts get two high tides rather than one.
Then something subtle happens. Earth spins faster than the Moon orbits, so friction drags the bulges slightly ahead of the Moon’s position. That misaligned mass pulls the Moon forward along its path, feeding it energy and lifting it into a wider orbit. Meanwhile the Moon pulls back on the bulges, braking Earth’s rotation.
Both effects are measured. Laser reflectors left on the lunar surface by the Apollo missions show the Moon receding by about three point eight centimeters a year. Earth’s day is lengthening by a little under two milliseconds per century, which is small but relentless – fossil coral growth bands indicate that four hundred million years ago the year held about four hundred days. Angular momentum is not being lost. It is being transferred from Earth’s spin to the Moon’s orbit, and the total is unchanged.
The Moon has already been through the far end of this process. Earth raises a bulge in the Moon’s rock just as the Moon raises one in Earth’s oceans, and Earth’s pull on that bulge slowed the Moon’s spin until it matched the orbit. That is why we only ever see one face of it. The same braking is running on Earth right now, just far from finished.
The Crack That Showed at Mercury
Newtonian orbits are astonishingly accurate. They are also not quite right, and the first crack showed up in the nineteenth century at the planet closest to the Sun.
Mercury’s ellipse slowly rotates, so the point of closest approach creeps around over time. Most of that creep is caused by the tug of the other planets and was fully accounted for. A small residue was not – about forty three seconds of arc per century, a tiny amount that stubbornly refused to go away. Astronomers hunted for an unseen inner planet to explain it. There was none. The explanation was that space near a massive body is curved, and general relativity predicted the leftover exactly.
There is a second limit, and it is not about relativity. Two bodies orbiting each other have an exact solution, written down once and good forever. Three bodies do not. The general three-body problem has no such formula, and worse, the motion can be genuinely chaotic – arbitrarily small differences in the starting conditions grow into completely different futures. The solar system is stable over human timescales and not provably stable over its remaining lifetime. That is a statement about prediction rather than about determinism.
One Rule, an Entire Solar System
Orbits get an extraordinary amount out of very little. One rule about how mass attracts mass, plus the laws of motion, and out falls the shape of the solar system, the timing of the tides, the parking spot for a telescope, and the reason your phone knows where you are. No extra machinery. No separate celestial physics.
And the central image stays useful long after the mathematics has been replaced. Whatever gravity turns out to be underneath, everything in orbit is still doing the same thing Newton described – falling, continuously, and missing.
