Foundations — Specific angular momentum h = √(GMp)

How to read this page

Below, each foundation is: plain words → the picture → why the topic needs it. They are ordered so every new symbol only uses ideas already built above it. Nothing is assumed — no symbol is used before the section that defines it.

1. A vector $\vec{r}$ — an arrow with length and direction

Look at figure s01. The black dot is the heavy central body. The red arrow is $\vec{r}$: it points at the planet, and its length is the distance $r$ (no arrow — plain $r$ means just the length, forgetting direction; we make this precise in the next section).

Why the topic needs it: the orbit is the set of all positions $\vec{r}$ the planet visits. Every later idea on this page — the velocity arrow, the spin-making cross product, the orbit's shape — is built on top of this one arrow.

2. Length $r$ and the unit arrow $\hat{r}$

Picture: take the red arrow, keep it pointing the same way, but squash it until it's exactly one unit long. That stub is $\hat{r}$. So any position splits as $\vec{r}=r\hat{r}$ = (how far) $\times$ (which way).

Why the topic needs it: the whole page distinguishes "how far" ($r$) from "which way" ($\hat{r}$). Gravity pulls with a strength set by $r$ and a direction set by $\hat{r}$; separating them now lets every later statement about "outward" versus "inward" be exact.

3. Velocity $\vec{v}$ — the arrow of motion

In figure s02, the red arrow $\vec{v}$ is drawn tangent to the orbit path — it always points along the direction of travel. It is generally not aligned with $\vec{r}$: $\vec{r}$ points outward from the centre, while $\vec{v}$ points along the curve.

Why the topic needs it: the spin amount is built from both where you are ($\vec{r}$) and where you're going ($\vec{v}$). One without the other tells you nothing about the orbit's rotation.

4. The angle $\varphi$ between $\vec{r}$ and $\vec{v}$ (flight-path angle)

Look again at figure s02: the small wedge marks $\varphi$. Here is what each size of $\varphi$ looks like physically (we describe motion in words now; the exact rate symbols come in §8):

Why the topic needs it: the angle between $\vec{r}$ and $\vec{v}$ is exactly what the cross product in §5 measures, and it is what makes the simple picture "distance $\times$ speed" work only at special points. Note that $\sin \varphi$ takes the same value at $\varphi$ and $180^{\circ }-\varphi$, so a point on the outbound half and its mirror on the inbound half share the same value of the sideways ingredient — a first hint of a conserved quantity. See Flight-path angle.

5. The cross product $\vec{r}\times \vec{v}$ — the "spin-area" machine

This is the heart of the whole topic, so we build it slowly.

Figure s03 shows why the length is that parallelogram area. Two arrows sweep out a slanted box; its area is base $\times$ height $=|\vec{a}|\times (|\vec{b}|\sin \theta )$. When the arrows point the same way ($\theta =0$) the box is flat — zero area — so $\vec{a}\times \vec{a}=0$.

Why the topic needs it: $\vec{h}=\vec{r}\times \vec{v}$ is the definition of specific angular momentum. Its length is (twice) the rate at which the position arrow sweeps out area — the link to Kepler's Second Law. Its direction is fixed, which is why the orbit stays in one plane. Because $h=rv\sin \varphi$, the simple $h=rv$ holds only at the apsides where $\varphi =90^{\circ }$ (this is the parent's "$h=rv$ everywhere" mistake).

6. $G$, $M$, and the product $GM$ (the parameter $\mu$)

Now we can finally read the gravity arrow the parent note uses: the acceleration of the planet is $\vec{a}=-\frac{GM}{r^2}\hat{r}$. Every piece is now defined — $GM$ (strength), $r^2$ (weaker with distance, length from §2), and $-\hat{r}$ (direction: inward, from §2).

Why the topic needs it: because the small mass cancels out of orbital motion (see the parent's "why divide by $m$"), the only mass that matters is the central $M$, and it always appears glued to $G$. So $GM=\mu$ is the natural single knob. It sits inside $h=\sqrt{GMp}$ and inside gravity $\vec{a}=-\frac{GM}{r^2}\hat{r}$.

7. The conic orbit and its shape symbols $p$, $e$, $a$, $\theta$

The four shape symbols, each a picture (figure s04):

Why the topic needs it: $h=\sqrt{GMp}$ links the conserved spin $h$ to the shape symbol $p$. Without $p$ (and its cousins $e,a,\theta$) the formula would have nothing geometric to say. These come out of the Two-body problem.

8. Derivatives and the dot notation $\dot{r},\ddot{r}$

Plain picture: if $r$ is your distance from the centre, $\dot{r}$ is your outward/inward speed ($\dot{r}>0$ outward, $\dot{r}<0$ inward), and $\ddot{r}$ is whether that speed is speeding up. Zero $\dot{r}$ means momentarily neither approaching nor receding — exactly the apsides, the same $\varphi =90^{\circ }$ points from §4 (now stated as a rate: $\dot{r}=0$).

Why the topic needs it: the conservation proof uses $\frac{d\vec{h}}{dt}=0$, and the orbit derivation uses $\dot{r},\ddot{r},\dot{\theta }$ in polar form. Every step in the parent note that has a dot or a $\frac{d}{dt}$ is standing on this idea.

Prerequisite map

Equipment checklist

Read each line, answer in your head, then reveal.

Connections

• Parent: h = √(GMp) — this page supplies its every symbol.
• Two-body problem — where the conic orbit equation (and $p,e,a,\theta$) is born.
• Kepler's Second Law — the cross-product area meaning of $\vec{h}$ leads straight to equal areas in equal times.
• Flight-path angle — the angle $\varphi$ built in section 4.
• Eccentricity vector — deepens the eccentricity $e$ introduced here.
• Vis-viva equation and Specific orbital energy — the energy partners to $h$.
• Hinglish version →