Foundations — Kepler's first law — orbits are conic sections

Before you can read the parent note Kepler's first law, you need to genuinely own every symbol it uses. This page builds each one from nothing — plain words, then a picture, then why the topic can't live without it.

1. Position: where is the planet? ($r$ and $\theta$)

Imagine you are standing on the Sun, watching a planet swing around you. To say exactly where it is, you need two numbers:

• How far away it is — call this $r$, the distance from the Sun to the planet.
• What direction it lies in — call this $\theta$, the angle you have to turn to point at it.

Why the topic needs this. The whole goal of Kepler's first law is a formula for the shape of the orbit. A shape is a rule that says "for each direction $\theta$, the planet sits at this distance $r$." That is exactly what $r(\theta )$ means — read it as "$r$ as a function of $\theta$," i.e. give me an angle, I'll give you the distance. Ordinary left–right / up–down $(x,y)$ coordinates would work too, but because gravity pulls straight along the line to the Sun, using distance-and-angle makes the mathematics dramatically cleaner.

2. The dots: rates of change ($\dot{\theta }$, $\dot{r}$, $\ddot{r}$)

The planet moves, so $r$ and $\theta$ are not frozen — they change with time $t$. Physicists write a small dot over a symbol to mean "how fast this is changing per second."

Picture $\dot{r}$ as the speed of a bead sliding along the spoke from Sun to planet; $\dot{\theta }$ as the speed of that spoke sweeping around like a clock hand.

3. The two masses and $\mu =GM$

Why bundle them into $\mu$? In every orbit equation the combination $GM$ appears as a single lump — the mass of the small planet $m$ cancels out (a feather and a cannonball fall the same way). So we give the lump one name, $\mu$, to keep the algebra tidy. See Two-body problem and reduced mass for the fine print on when $m$ can be ignored.

4. The inverse-square pull and how it becomes an acceleration

This is the engine of the whole story. Gravity's strength is not constant — it fades with distance, and specifically it fades as one-over-distance-squared.

Why the topic needs this exact law. As the parent note stresses, only the $1/r^2$ shape produces clean, closed, non-repeating orbits (see Inverse-square law and Bertrand's theorem). A slightly different power — say $1/r^3$ — would make orbits spiral or precess instead of tracing a fixed ellipse. The magic of Kepler's first law is a gift of this specific power law.

5. The little hat: direction ($\hat{r}$)

So "$-\frac{\mu }{r^2}\hat{r}$" reads: a pull of strength $\mu /r^2$, aimed opposite to $\hat{r}$ — i.e. inward, back toward the Sun. The minus sign flips "outward" into "inward."

Why the topic needs it. Force is a vector — it has both a size and a direction. $\hat{r}$ is how we bolt a direction onto the size $\mu /r^2$. Without it we could only say how hard, not which way.

6. Central force and angular momentum ($h=r^2\dot{\theta }$)

Because gravity always points exactly along $\hat{r}$ — straight at the Sun — it is called a central force. This single fact hands us a treasure: a quantity that never changes.

Why the topic needs it. $h$ being constant is the lever the derivation uses to erase time and get pure geometry $r(\theta )$. It also secretly is Kepler's second law, Kepler's second law — equal areas in equal times: constant $h$ means the spoke sweeps equal areas in equal times.

7. The radial equation of motion (what the force actually drives)

Once you have the inward acceleration $\mu /r^2$ and the constant $h$, Newton's second law in the in–out (radial) direction becomes a single equation:

Why this exists. In polar coordinates the true radial acceleration is not just $\ddot{r}$; swinging around adds an outward term $r{\dot{\theta }}^2$ (the same effect that throws you outward on a merry-go-round). Setting the net radial acceleration equal to the inward pull gives the equation above. This is the single relation the parent note solves — using the constant $h$ to erase time — to arrive at the orbit shape $r(\theta )$. You do not need to solve it here; you just need to recognise it when the flowchart below points at it.

8. The conic family ($p$, $e$, and $r=\frac{p}{1+e\cos \theta }$)

A conic section is any curve you get by slicing a cone with a flat plane: tilt the plane different ways and out come circles, ellipses, parabolas, hyperbolas. Full geometry lives in Conic sections — geometry of ellipse, parabola, hyperbola.

Reading the formula, all cases (with $\theta =0$ at perihelion):

• $\theta =0$ (perihelion, closest): $\cos 0=1$, denominator biggest, $r$ smallest $=\frac{p}{1+e}$.
• $\theta =90^{\circ }$: $\cos =0$, $r=p$ exactly.
• $\theta =180^{\circ }$ (aphelion, farthest): $\cos =-1$, denominator smallest, $r$ largest $=\frac{p}{1-e}$ (only finite if $e<1$).
• If $e\ge 1$, then at some angle $1+e\cos \theta =0$, making $r\to \infty$ — the curve runs off to infinity and never returns. That is precisely an escape trajectory.

Why the topic needs it. This is the target: proving orbits obey this formula is Kepler's first law.

9. Energy per unit mass ($\mathcal{E}$) and speed ($v$)

The parent note ties it together with $e=\sqrt{1+\frac{2\mathcal{E}{h}^2}{{\mu }^2}}$: notice $\mathcal{E}<0$ shrinks the square root below $1$ (ellipse), $\mathcal{E}=0$ gives exactly $1$ (parabola), $\mathcal{E}>0$ pushes above $1$ (hyperbola). Sign of energy → type of conic. Perfect harmony.

How it all fits together

Read it top-down: coordinates and masses feed the force; dividing the force by the planet's mass gives the acceleration $\mu /r^2$; the force being central freezes $h$; acceleration plus $h$ give the radial equation of motion; solving it yields the shape $r(\theta )$, which matches a conic; energy then tells you which conic.

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