Specific orbital energy ε = −GM - 2a: Foundations

Before you can trust that boxed formula in the parent note, every letter inside it has to mean something you can picture. This page builds them one at a time, from nothing. Never do we use a symbol before it earns its place.

0. The stage: a small body looping a big one

Figure — Specific orbital energy ε = −GM - 2a

Look at the figure: the yellow blob is the central body $M$ . The blue curve is the orbit — the path the small body traces. The two special points, closest (red) and farthest (green), are where all the interesting energy exchange peaks. We will name every quantity on this picture below.

1. Distance $r$ — how far out the satellite is

Picture: in the figure, $r$ is the length of the arrow from the yellow centre out to the moving dot. As the satellite loops, this arrow grows and shrinks — long at the far point, short at the near point.

Why the topic needs it: gravity's pull depends on how far apart the bodies are, so almost every formula asks "how far?" — and the answer is $r$ .

2. Speed $v$ — how fast it moves

Picture: imagine a tiny arrow glued to the moving dot, pointing where it's headed. Its length is $v$ . In the figure that arrow is long near the red point (fast) and short near the green point (slow).

Why the topic needs it: motion carries energy, and the amount depends on speed. No $v$ , no kinetic energy.

3. The gravitational constant $G$ and central mass $M$

Picture: think of $G$ as the "exchange rate" turning masses-and-distances into a pull. $M$ is how much stuff is doing the pulling.

Why the topic needs it: they always appear glued together as the product $GM$ , which is the only gravity-strength number the orbit cares about.

4. Kinetic energy per mass — the "motion money"

Picture: the longer the velocity arrow (Section 2), the more of this "motion money." At the red near-point the arrow is longest, so this term peaks there.

Why divide by $m$ ? So the satellite's own mass drops out — a feather and a boulder in the same orbit share the same budget. That is the whole point of the word specific (= per unit mass).

Why $v^{2}$ and not $v$ ? Doubling speed quadruples the stored motion energy — that is a measured fact of physics, captured by the square.

5. Gravitational potential energy per mass — the "height money"

This is the subtle one, so we build it slowly.

Where do we call the potential energy zero? We choose it to be zero infinitely far away, because that is the one place gravity has finished pulling. Every closer position then has less energy than that — hence a negative number.

Picture: the figure shows $- GM / r$ as a hill (well) that plunges as you approach the planet. The satellite rolls in this well: far out it sits near the top (near zero); close in it sits deep in the pit (very negative). Follow the blue dot — as $r$ shrinks, it slides down the curve.

Why the topic needs it: this is the "height money" half of the energy budget. Combined with motion money it makes the total that never changes.

Why the shape $- 1/ r$ and not something else? Gravity weakens as the square of distance ( $\propto 1/ r^{2}$ ). Adding up (integrating) that pull as you move outward produces a $1/ r$ dependence in the energy — the exact reason a $1/ r$ shape appears here rather than a straight line.

6. Total energy per mass — the budget that never moves

Picture: the figure stacks the two contributions along the orbit. As the satellite dives inward (red), the height bar drops (more negative) but the motion bar rises by the exact same amount — the total line stays flat. That flat line is $ε$ .

Why "conserved"? Gravity is a conservative force: the work it does depends only on where you start and end, never the path taken. That guarantees motion-money and height-money only ever trade, never leak away.

7. The semi-major axis $a$ — the "size of the loop"

Everything above lives at a single instant. The final symbol describes the orbit as a whole.

Picture: in the figure the near point (perigee, distance $r_{p}$ ) and far point (apogee, distance $r_{a}$ ) sit at opposite ends. The full span across is $r_{p} + r_{a}$ , and $a = \frac{r _{p} + r _{a}}{2} .$

Why the topic needs it: the headline result is that the entire energy budget depends on only this size: $ε = - \frac{GM}{2 a} .$ Bigger loop $\Rightarrow$ larger $a$ $\Rightarrow$ $ε$ closer to zero $\Rightarrow$ "more loosely bound."

8. Angular momentum per mass $h$ (used in the derivation)

Picture: at perigee and apogee the velocity arrow points straight across the radius arrow — no in/out component. That is exactly where the clean product $r \times v$ holds.

Why the topic needs it: the parent's derivation equates energy at two points and needs a second equation linking the two speeds. Because a central pull exerts zero twist (torque), $h$ is conserved — that is the second equation.

See Conservation of angular momentum for the full story of why swirl is conserved.

Prerequisite map

This map shows the flow: raw ingredients ( $G$ , $M$ , $r$ , $v$ ) build the two energy halves, which sum to $ε$ ; the derivation (using $h$ ) then trades $r$ and $v$ for the pure size $a$ , giving $- \frac{GM}{2 a}$ , which unlocks the Vis-viva equation and the orbit-type rule via Conic sections in orbits.

Equipment checklist

Test yourself — cover the right side and answer out loud.

What does

r

measure, and from where?

Distance of the small body from the centre of the big body, in metres.

What is

v

The speed (length of the velocity arrow) of the orbiting body, in m/s.

Why do

G

and

M

always appear as the single product

GM

They never act separately in orbital formulas;

GM

is the one gravity-strength number the orbit cares about.

Write kinetic energy per unit mass and give its units.

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

, units

m^{2} / s^{2} = J/kg

Why is gravitational PE per mass negative?

We set it to zero at infinity; every closer position is downhill from there, hence negative.

Write PE per mass as a formula.

- \frac{GM}{r}

Why is the total

ε = \frac{v ^{2}}{2} - \frac{GM}{r}

constant along the orbit?

Gravity is conservative, so motion-energy and height-energy only trade — the sum is conserved.

What does "specific" mean and why do we bother?

Per unit mass; dividing by

m

makes the result independent of the satellite's own mass.

Define

a

in terms of

r_{p}

and

r_{a}

a = \frac{r _{p} + r _{a}}{2}

, half the major axis of the ellipse.

Why is

h = r_{p} v_{p} = r_{a} v_{a}

only at perigee and apogee?

There the velocity is perpendicular to the radius, so

r v_{⊥}

simplifies to

r v

Bigger orbit — does

ε

get more or less negative?

Less negative (closer to zero), since

ε = - \frac{GM}{2 a}

shrinks in size as

a

grows.