In modules 1 and 2 we worked out the speed of a circular orbit at any altitude. The vis-viva equation extends that to any orbit — circular, elliptical, parabolic, hyperbolic — by way of one of the deepest ideas in physics: the conservation of energy.
An orbiting body has energy of two kinds. The first is kinetic energy, the energy of motion, which you've seen before:
$$ \mathrm{KE} = \frac{1}{2} m v^2 $$The second is gravitational potential energy, which is the energy a body has by virtue of where it is in a gravitational field. Lift an object higher and you've stored energy in it — release it and that energy comes back as motion as it falls. Near the Earth's surface this is the familiar $\mathrm{PE} = mgh$, where $h$ is height above some reference point.
That formula breaks down when "height" isn't well-defined and gravity isn't constant — which is exactly the situation in orbit. The honest, distance-based form of gravitational potential energy is:
$$ U = -\frac{G M m}{r} $$The minus sign is real, and it's the single most confusing feature of orbital mechanics for newcomers. It's worth three sentences.
Potential energy is always relative — you can only ever measure differences in PE, not absolute values. So we get to choose where to call PE zero. For orbits, the natural choice is to set $U = 0$ at infinity, infinitely far from the central body, where gravity has no effect. Then any finite distance $r$ is "lower in the well" than infinity, and lower means less PE. Less than zero is negative. The closer you are to the central body, the deeper in the well you sit, the more negative your PE becomes.
This is the same idea as defining "ground level" as your zero for elevation: a basement has negative elevation. Here, "infinity" is the reference instead of "ground level," and being anywhere near a planet means you're below it.
The sum of an orbiting body's kinetic and potential energy is its total mechanical energy $E$:
$$ E = \frac{1}{2} m v^2 - \frac{G M m}{r} $$The signature property of an inverse-square gravitational field is that $E$ doesn't change as the body moves along its orbit. Energy can transfer between kinetic and potential — and it does, constantly — but the total stays put. As the body falls toward the central mass, gravity pulls it faster, and $r$ shrinks; KE goes up, PE becomes more negative, and the two changes exactly cancel. As it climbs back out, the reverse happens. This is what conservation of energy means in action.
Because energy depends on the mass of the orbiting body, it's often more useful to talk about specific energy $\varepsilon$ — energy per unit mass — by dividing through by $m$:
$$ \varepsilon = \frac{1}{2} v^2 - \frac{GM}{r} $$The orbiting body's mass cancelled out, just as it did in the circular-orbit derivation. A satellite, an astronaut, and a stray bolt floating outside the ISS all have the same specific energy when they're in the same orbit.
Now for the punchline. There's a remarkable fact about elliptical orbits — which we'll prove in module 5 — that the specific energy depends only on the orbit's size, specifically its semi-major axis $a$:
$$ \varepsilon = -\frac{GM}{2a} $$Two orbits with the same semi-major axis have the same total energy, regardless of how circular or elongated they are. A nearly circular orbit at radius $a$ has the same energy as a wildly stretched ellipse with the same semi-major axis — just distributed differently between KE and PE at any given moment.
This gives us two expressions for the same $\varepsilon$, one in terms of position and speed and one in terms of orbit size. Setting them equal:
Equate the two expressions for specific energy:
$$ \frac{1}{2} v^2 - \frac{GM}{r} = -\frac{GM}{2a} $$Move the $GM/r$ to the right and multiply both sides by 2:
$$ v^2 = \frac{2 GM}{r} - \frac{GM}{a} $$Factor out $GM$:
$$ \boxed{\, v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right) \,} $$This is the vis-viva equation, and it's the single most useful formula in orbital mechanics. Tell me where you are ($r$) and what orbit you're on ($a$), and it tells you exactly how fast you're moving. The Latin name comes from vis viva, "living force" — Leibniz's seventeenth-century term for what we now call kinetic energy.
Vis-viva contains every speed result we've already derived as a special case. For a circular orbit, $r = a$ everywhere, so $2/r - 1/a = 1/r$, and $v = \sqrt{GM/r}$ — exactly the circular-orbit formula from module 2.
For a parabolic trajectory — the boundary between bound and unbound — the orbit is "infinitely long," meaning $a \to \infty$ and the $1/a$ term vanishes. That gives $v = \sqrt{2GM/r}$, the formula for escape velocity. At Earth's surface this works out to 11.2 km/s, the number that appeared as a magic threshold in the Newton's cannonball simulator.
For an elliptical orbit, vis-viva tells you the speed at any point. Plugging in $r = a(1-e)$ at periapsis or $r = a(1+e)$ at apoapsis gives the maximum and minimum speeds:
$$ v_{\text{peri}} = \sqrt{\frac{GM}{a} \cdot \frac{1+e}{1-e}}, \qquad v_{\text{apo}} = \sqrt{\frac{GM}{a} \cdot \frac{1-e}{1+e}} $$For Halley's Comet, with $e \approx 0.967$, the ratio $v_{\text{peri}}/v_{\text{apo}} = (1+e)/(1-e) \approx 60$. The comet sprints through perihelion almost sixty times faster than it crawls past aphelion — exactly the wildly varying speed you see in the Kepler's-laws animation in module 4.
The interactive below puts a body on an elliptical orbit and shows its kinetic, potential, and total energy live as it moves. Watch what happens: KE and PE constantly swap value, but the total — the blue bar — never moves. The conservation law is right there on the screen.
Specific energy (units: GM / a)
Try a few eccentricities. At $e = 0$ (a circle), KE and PE are both constant — the bars don't move at all, because the body's distance from the focus never changes. Crank the eccentricity up and the bars start to swing wildly: deep PE wells at periapsis, shallow PE at aphelion, with KE doing the inverse. The total bar — what vis-viva is really about — stays exactly where it is.
Vis-viva is the workhorse equation for the rest of the curriculum. In module 7 we'll use it to compute the velocity changes ("delta-v's") needed for a Hohmann transfer between two circular orbits. In module 8 we'll use it again to plan a trip to Mars. The fact that any orbital-velocity question can be answered by the same one-line equation is what makes orbital mechanics feel less like a sprawling subject and more like a single well-organized idea.