Foundations — Orbit shape from eccentricity — circle (e=0), ellipse (0 - e - 1), parabola (e=1), hyperbola (e - 1)

This page builds every symbol the parent note leans on, starting from a smart-12-year-old's toolkit and nothing more. Read top to bottom — each block only uses things defined above it.

1. Distance, angle, and the two masses

Together $(r,\theta )$ are polar coordinates — a "how far, which way" address. This is why the whole topic is written as $r(\theta )$: give me the direction, I'll tell you the distance.

2. The focus — the special point the orbit is drawn around

This matters because — once we have built the orbit equation later on — the distance $r$ will always be measured from the focus, not from the geometric centre. The Sun is at the focus, off to one side of the ellipse — never in its middle.

3. Slicing a cone — where all four shapes come from

4. Two things geometry gives us: $a$, $c$, and $p$

Before energy enters, the shape of an ellipse needs three lengths.

We now have $p$ purely from geometry. Later, once angular momentum and gravity's strength are defined (Sections 6 and 8), the very same $p$ will re-appear as $p=h^2/\mu$ — the physics value of the same width. That double life of $p$ is exactly what ties the orbit's shape to the body's motion, but we cannot write that formula honestly until those symbols exist.

5. The trig tool: cosine, and why it (not sine) appears

Now that we have $p$, $e$, and $\cos \theta$, we can finally read the orbit equation the parent states: $r(\theta )=\frac{p}{1+e\cos \theta }$ Every symbol in it has now been earned. (At $\theta =90^{\circ }$, $\cos 90^{\circ }=0$, so $r=p$ — which is precisely why $p$ is the "width at the focus" we defined above.)

This single fact drives the entire "read off the denominator" argument in the parent: the denominator shrinks as $\theta$ grows toward $180^{\circ }$ (because $\cos$ falls toward $-1$), and $r$ grows.

6. Rates of change: the dot, and specific angular momentum $h$

See the connection Angular Momentum in Orbits: $h$ is what fixes the width $p=h^2/\mu$ (the $\mu$ in it is defined in Section 8).

7. Energy: $U$, $\epsilon$, $E$, and the sign that decides everything

Figure s04 draws exactly this. The curved chalk line is the potential well $U=-GMm/r$: steep and deep near the centre, flattening to the dotted "rim" (energy $=0$) far out. A body's total energy is a horizontal level line — because energy is conserved, it stays flat as the body moves. Read the figure this way: the blue level sits below the rim ($\epsilon <0$) so the body can never reach the right edge → it is trapped → ellipse. The pink dashed level sits exactly on the rim ($\epsilon =0$) → it just barely reaches infinity → parabola. The yellow level sits above the rim ($\epsilon >0$) → it reaches infinity with height to spare → hyperbola.

See Specific Orbital Energy and Escape Velocity (the $\epsilon =0$ boundary), and Vis-viva Equation (which turns $\epsilon$ into a speed at any $r$).

8. Gravity's constants: $G$ and $\mu$

Why bundle $G$ and $M$? Because they only ever appear together in orbit maths, so carrying one symbol $\mu$ is cleaner. It appears in $p=h^2/\mu$ (now every symbol in it — $p$, $h$, $\mu$ — is defined) and in $e=\sqrt{1+2\epsilon h^2/{\mu }^2}$.

9. Putting the master relation in plain words

Read the sign of $\epsilon$ straight off:

• $\epsilon <0\Rightarrow$ inside-the-square-root is less than 1 $\Rightarrow e<1\Rightarrow$ ellipse/circle.
• $\epsilon =0\Rightarrow e=\sqrt{1}=1\Rightarrow$ parabola.
• $\epsilon >0\Rightarrow e>1\Rightarrow$ hyperbola.

That single formula is the whole parent topic compressed. Everything else is decoration.

Prerequisite map

The diagram below shows the dependency order of this page: geometry symbols (left) build the orbit equation, motion symbols (right) build the eccentricity–energy relation, and both streams merge into the four-shape classification that is the parent topic. Read an arrow as "is needed before". It is the reading order made visible — nothing downstream uses a symbol that is not upstream of it.

All streams feed the parent topic.

Equipment checklist

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