Orbit
An orbit is the curved path one object follows around another under gravity, such as Earth around the Sun or a satellite around Earth.
Definition
An orbit is the path an object follows as it travels around another object in space, held there by gravity, as Earth orbits the Sun and the Moon orbits Earth. Orbits are ellipses, most planetary ones nearly circular, and they arise from the balance between an object's forward motion (inertia) and gravity pulling it inward: without gravity it would fly off straight, and without forward motion it would fall straight in. Newton's law of gravitation combined with his second law gives an ellipse with the central mass at one focus; for a circular orbit of radius $r$ around mass $M$, gravity supplies the centripetal force, so $v = \sqrt{GM/r}$ and $T = 2\pi\sqrt{r^3/GM}$. Total orbital energy is $E = -GMm/2a$ for semi-major axis $a$, unbound paths with $E \geq 0$ are parabolic or hyperbolic, and an orbit is fully described by six orbital elements including semi-major axis, eccentricity, and inclination.
Example
The International Space Station orbits about $400$ kilometers up and circles Earth in about $90$ minutes, so its astronauts see $16$ sunrises a day. Earth moves at about $30\ \text{km/s}$, covering roughly $940$ million km in $365.25$ days, and a low-Earth-orbit satellite must travel about $7.7\ \text{km/s}$, which is why rockets are needed. For a geostationary satellite, setting $T = 86{,}164\ \text{s}$ and solving $r = (GMT^2/4\pi^2)^{1/3}$ with $GM_{\text{Earth}} = 3.986 \times 10^{14}\ \text{m}^3/\text{s}^2$ gives $r \approx 42{,}200\ \text{km}$, about $35{,}800\ \text{km}$ above the surface.
Key Insight
An orbit is a fall that keeps missing: a satellite falls toward Earth the whole time but moves sideways so fast that the ground curves away beneath it. Closer orbits are faster, so Mercury races around the Sun at $47\ \text{km/s}$ while Neptune crawls at $5.4\ \text{km/s}$. Raising an orbit requires speeding up, yet the higher orbit is slower, because the extra energy goes into gravitational potential energy, a counterintuitive result every spacecraft navigator must handle.