Six numbers that pin down an orbit
In the earlier guides of this rung you met [[keplers-laws-of-planetary-motion|Kepler's laws]] — orbits are ellipses, they sweep equal areas in equal times, and the period grows with the size of the orbit — and you saw how [[newtons-law-of-universal-gravitation|Newton's law of gravity]] explains why. But a single word, 'orbit', hides a lot of detail. Two planets can both trace ellipses around the Sun and still share almost nothing: one nearly circular and close in, the other long and slanted and far out. To pin down a real orbit completely we need a small set of numbers, and astronomers call them the [[orbital-elements|orbital elements]].
Think of describing an orbit in three plain questions: how big, what shape, and how is it tilted? The size is set by the semi-major axis — half the long width of the ellipse — which also fixes the period, exactly as Kepler's third law promised. The shape is set by the [[orbital-eccentricity|eccentricity]], a single number from 0 to just under 1: zero is a perfect circle, and the closer it creeps to 1 the more stretched and cigar-like the ellipse becomes. Earth's orbit has an eccentricity of about 0.017 — so nearly circular you would not notice by eye — while a comet's can be 0.99, a thin loop that dives close to the Sun and then flees far beyond the planets.
The remaining elements answer 'how is it tilted and turned in space?' — the angle of the orbital plane against a reference plane, the direction in which the ellipse points, and where along the loop the body sits right now. None of these change the size or shape; they only orient the orbit and set the clock. Together this handful of numbers is a complete address: give them to a colleague and they can rebuild the exact path and tell you where the body will be on any future night.
Two bookkeepers that never cheat
Why does the ellipse stay an ellipse, year after year, with no engine and no maintenance? Because two quantities are conserved — they are fixed totals that the motion is never allowed to spend. The first is energy: the sum of the body's kinetic energy (energy of motion) and its gravitational potential energy (a sort of stored debt, deeper the closer you fall in) stays constant around the whole loop. The second is [[orbital-angular-momentum|angular momentum]], a measure of how much spinning-around-the-center the body carries, combining its distance, its speed, and the angle between them.
These two are not abstractions tacked on afterwards — they are exactly what makes Kepler's laws true. Conserved angular momentum is the equal-areas law in disguise: when a comet swings close to the Sun it must speed up, and when it drifts far out it must slow down, because the product of distance and sideways speed cannot change. That is why a planet races through the near part of its orbit and dawdles through the far part. Energy conservation, meanwhile, is what fixes the size of the orbit and ties it to the period. The bookkeepers do not just describe the orbit; they enforce it.
Bound, or free? The escape question
Now the most important fork in the road, and energy decides it. Gravitational potential energy is counted as negative — it is a debt that deepens as you fall closer in, and reaches zero only at infinite distance. So add up a body's total energy. If the total comes out negative, the body is in debt: it can never climb all the way out, and it is trapped on a closed ellipse forever circling its companion. This is a bound orbit, and how deep the debt runs is its [[gravitational-binding-energy|binding energy]] — literally how much energy you would have to pour in to set it free.
If instead the total energy is zero or positive, the body has enough to pay the debt in full and still have motion left over at infinity. Its path is no longer a closed ellipse but an open curve — a parabola right at zero, a hyperbola if positive — that swings once past the central body and never returns. The exact speed that brings the total to zero, the dividing line between captured and free, is the [[escape-velocity|escape velocity]]. From Earth's surface it is about 11 kilometres per second; from the Sun's surface, far deeper in its gravity well, about 618. Reach it and you are unbound; fall short and you fall back.
Why angular momentum keeps things spinning
Angular momentum has a second, subtler job: it is why so much of the universe is shaped like a spinning disk. Picture a vast, slowly turning cloud of gas that begins to collapse under its own gravity. As it shrinks, its angular momentum cannot vanish, so just as a figure skater pulling in her arms spins faster, the contracting cloud whirls ever quicker. It can fall freely along its spin axis, flattening top and bottom, but it cannot fall straight in toward the axis — the faster spin throws up a barrier. The result is a flat, rotating disk: this is the origin of planetary systems, of the accretion disk, and of the broad shape of spiral galaxies.
The same barrier explains why orbits are stable and why things do not simply plunge into the body they circle. For a planet with real sideways motion, falling closer means spinning faster, which means an ever-stiffer angular-momentum wall pushing back against the approach. So the planet does not spiral in; it swings past the closest point and climbs out again, tracing the same ellipse next time around. Angular momentum is the reason gravity, an attractive force with nothing to oppose it, nonetheless produces eternal circling instead of a single catastrophic fall.
This raises an honest puzzle. If a collapsing cloud spins ever faster, how does gas in a disk ever fall the last stretch onto the star or black hole at its center? It cannot, unless something carries its angular momentum away. Real disks solve this with friction, turbulence, and magnetic forces that hand angular momentum outward to the disk's outer parts, letting the inner gas finally drain inward. How efficiently this happens is still an area of active research — a reminder that even a textbook idea like 'disks spin' opens onto genuine open questions once you ask how the gas actually gets in.
Putting the bookkeepers to work
Here is how an astronomer actually reasons about a body whose orbit they are trying to understand — energy and angular momentum used as tools, not just facts. The procedure works for a moon, a planet, a star in a binary, or a comet swinging in from the dark.
- Add up the total energy at any one point — kinetic plus the negative gravitational potential. Its sign alone tells you the verdict: negative means bound and circling forever, zero or positive means unbound and gone.
- If it is bound, the total energy fixes the semi-major axis — the size of the orbit and therefore its period, via Kepler's third law.
- Use the angular momentum to pin down the eccentricity — how round or stretched the ellipse is, and how close the body comes at its nearest approach.
- Because both totals are conserved, you can now predict the body's speed and distance at every other point on the orbit — fast and low near perihelion, slow and high at the far end.
This is the quiet power of conservation laws. We do not have to track the body's complicated path moment by moment; we only have to know two totals it is never allowed to change, and the whole future of the orbit unrolls from them. In the next guides this same accounting scales up: it lets astronomers weigh stars in a binary, weigh whole clusters of galaxies, and finally feel for the place where Newton's tidy bookkeeping begins to fail and Einstein's gravity must take over.