Vis-viva equation v² = GM(2 - r − 1 - a) — derivation: Foundations

Before you can read the derivation on the parent page (topic note), every letter in the vis-viva equation must mean something to you as a picture, not just a symbol. This page builds them one at a time, in an order where each depends only on the ones before it. We won't even write the full equation until every letter in it has been earned.

0. The cast of characters (what we're even talking about)

Two things are involved:

A big body — a planet or star — that barely moves. We call its mass $M$ (capital M for "massive").
A small body — a satellite, moon, or spacecraft — that whips around the big one. Its mass is $m$ (little m).

Figure — Vis-viva equation v² = GM(2 - r − 1 - a) — derivation

The small body traces a closed loop. That loop, and the point it loops around, are the stage for everything else.

Hold on to $m$ — in §6 you'll watch it cancel out of every equation, which is why the final formula never mentions the satellite's mass at all.

1. $r$ — where you are right now

Look at the arrow in the figure below: $r$ is a rubber band from the planet's center to the satellite. As the satellite moves, the rubber band stretches and shrinks — so $r$ changes every second.

Picture: the length of the line joining the two dots.
Why the topic needs it: vis-viva answers "how fast am I going right here?" — and "here" is exactly what $r$ measures.

2. $v$ — how fast you're going

Picture: the length of the little velocity arrow riding along with the satellite, pointing where it's headed.
Why the topic needs it: $v$ is the whole output of vis-viva. Everything else exists to pin down this one number.

Notice the equation we're building solves for $v^{2}$ , the square of the speed, never $v$ itself. That's a hint that this equation comes from energy (energy carries a $v^{2}$ ), which we meet in §6.

3. $G$ and $GM$ — the strength of gravity

Before any energy formula, we need the number that says how strong gravity is. It shows up in every step from here on, so we earn it now.

Picture: one single "gravity strength" dial for the whole planet.
Why the topic needs it: $GM$ is the only planet-property that appears in vis-viva; it scales everything. Astronomers measure it directly as one lump because $G$ alone is known poorly and $M$ of a planet is hard to weigh — but $GM$ falls straight out of watching orbits.

So the symbol $GM$ you'll meet in every formula below simply means "gravity strength."

4. The ellipse — the shape of a bound orbit

Orbits aren't circles; they're ellipses — gently squashed circles. To describe one, we need three ideas: the focus, the semi-major axis, and the eccentricity.

Picture: half of the longest diameter of the oval (the mint line in the figure).
Why the topic needs it: the whole punchline of the derivation is that the orbit's energy depends only on $a$ — not its shape, not where you are. So $a$ is the single "size knob" of the whole loop.

We build the story on the bound ellipse ( $0 \leq e < 1$ ) because it's the one you can draw — but §8 shows the very same vis-viva formula stretches to the escape and fly-by cases too. That's why $e$ matters even though it never appears in the final equation.

5. Perigee and apogee — the closest and farthest points

Since $r$ changes as you orbit, there is a smallest $r$ and a largest $r$ .

The one fact you must carry forward:

Picture: the two tips of the long axis in the figure — one hugging the planet, one flung far away.
Why the topic needs it: at these two points the velocity is purely sideways (no in/out motion), which makes energy and angular momentum easy to write down. They are the derivation's "easy chairs."

6. Energy — the quantity that never changes (and where $m$ disappears)

This is the engine of the whole topic. The total energy of the orbiting body has two flavours:

$E = kinetic \frac{1}{2} m v^{2} - potential \frac{GM m}{r}$

Both pieces carry the satellite's mass $m$ . Watch what happens when we divide the whole thing by $m$ to get energy per kilogram:

$ε = \frac{E}{m} = \frac{1}{2} v^{2} - \frac{GM}{r}$

The $m$ cancels everywhere — a 1-tonne satellite and a 1-gram bolt on the same orbit have the same $ε$ . That's exactly why we work with these "specific" (per-unit-mass) quantities: they describe the orbit, not the object.

The magic fact (proven on the parent page) is that the sum of these two is frozen:

Picture: a marble rolling in a smooth bowl — it speeds up going down, slows down going up, but the total (motion + height) is always the same because nothing steals energy.
Why the topic needs it: "the same everywhere" is what lets you compute $v$ at any $r$ once you know the constant. That's literally vis-viva.

This is the payoff of your prerequisite Conservation of Energy.

7. Angular momentum — the "spin" that never changes

There's a second frozen quantity, used only to solve the two apside equations together. First, two speed labels we'll need:

Picture: a figure skater pulling their arms in — as $r$ shrinks, $v$ must grow so the product $r \times v$ stays fixed. That's why perigee is fast and apogee is slow.
Why the topic needs it: two conserved quantities (energy and angular momentum) give two equations, which pin down the two unknown speeds $v_{p}$ and $v_{a}$ . See Conservation of Angular Momentum.

8. The key collapse — energy depends only on $a$

When you feed the two conserved quantities of §6 and §7 into the two apside equations and grind the algebra (the parent page does every line), everything shape-related cancels and you're left with one astonishingly simple result:

Now set the two forms of $ε$ equal: $\frac{1}{2} v^{2} - \frac{GM}{r} = - \frac{GM}{2 a}$ Multiply by 2 and rearrange — now every letter is earned, so we may finally write the equation: $v^{2} = GM (\frac{2}{r} - \frac{1}{a})$

The picture of the equation, at last

$how fast v^{2} = gravity strength GM where you are \frac{2}{r} - size of loop \frac{1}{a}$

$\frac{2}{r}$ grows huge when you dive close (small $r$ ) → fast.
$\frac{1}{a}$ is a fixed toll set once by the orbit's size.
Subtracting them, times gravity strength, gives your speed-squared.

The edge cases (same formula, all four regimes)

Prerequisite map

Equipment checklist

Test yourself — cover the right side and say each aloud before revealing.

What does

r

measure, and from where?

The instantaneous distance from the center of the big body to the orbiting body — it changes every moment.

What does

v

represent, and why does the equation use

v^{2}

The current speed;

v^{2}

appears because vis-viva comes from energy, and kinetic energy carries a

\frac{1}{2} v^{2}

What is

G

, and what is

GM

G

is the universal gravitational constant;

GM

bundles it with the planet's mass into one "gravity strength" number, measured directly from orbits.

Where does the big mass

M

sit on the ellipse?

At one focus, never at the center.

Define the semi-major axis

a

in one sentence.

Half the length of the longest line across the ellipse — the orbit's overall size.

What do the four ranges of

e

mean?

e = 0

circle,

0 < e < 1

ellipse,

e = 1

parabola (escape,

a \to \infty

e > 1

hyperbola (fly-by,

a < 0

What do

v_{p}

and

v_{a}

denote?

The orbiting body's speed at perigee (fastest) and at apogee (slowest), respectively.

Why do we work with "per unit mass" (specific) quantities?

Dividing energy and angular momentum by

m

makes the satellite's mass cancel, so the numbers describe the orbit, not the object.

What is special about perigee and apogee for the derivation?

There the velocity is purely sideways, making energy and angular momentum easy to write; also

r_{p} + r_{a} = 2 a

State the constant specific energy in terms of

a

ε = - GM / (2 a)

— the key collapse; energy depends only on orbit size.

How does vis-viva handle escape and hyperbolic paths?

a \to \infty

gives

v^{2} = 2 GM / r

(escape);

a < 0

gives

v^{2} > 2 GM / r

(excess speed) — same formula, different sign of

a

Connections

Conservation of Energy — the frozen $ε$ that powers the derivation.
Conservation of Angular Momentum — links $v_{p}$ and $v_{a}$ via $h$ .
Orbital Elements — formal definitions of $a$ , $e$ , $r_{p}$ , $r_{a}$ .
Kepler's Laws — the geometry of ellipses and periods.
Escape Velocity — the $a \to \infty$ limit of these same ideas.
Hohmann Transfer Orbit — where you actually use all this.