Orbits: The Four Conic Sections

Source

Lecture: L35 — Kepler orbits from first principles
Simulation: Top-down 2D orbit + cone-section diagram (Guided-Inquiry)
Concept: Eccentricity e picks the conic (circle / ellipse / parabola / hyperbola). Specific energy ε is derived from e at fixed angular momentum — sign(ε) tracks bound vs unbound. The right panel shows how each orbit is a plane cut through a cone.
Units: Natural units with GM = 1. Energies are specific (per unit mass), ε = E/m. Angular momentum held fixed via p = L²/GM.
Serves: Deriving the possible motions of planets and comets from first principles — the four conic sections.
Inquiry: Visit all four conics; find the parabolic edge where ε = 0; see that ε > 0 never closes.
Misconception: “A planet's path is just an ellipse” / “the shape is arbitrary.” In fact e fixes the conic and sign(ε) fixes bound vs unbound. (For V_eff intuition, see L36.)

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept, one sentence	e sets the conic; sign(ε) (derived) sets bound/unbound.
A2	Fewest objects	2D orbit left; cone diagram right highlighting the active cut.
C1	Numbers earn their place	Only e (chosen) and ε (derived) are shown.
E1	Animation honesty	Symplectic Verlet integrator; a single physical parameter (e at fixed L) drives everything.
H2	Structural misconception	The "E>0 can be bound" idea is refuted by overlaying a ghost "closed" path against the actual escape.
G4	Boundaries are sharp	e = 0 and e = 1 snap so students can land on the exact edge cases.

Orbits

Teach me Hide Text

Speed

Orbit · top-down

Sun

body

Sun at a focus · equal x/y scale

Conic sections

drag to rotate

Guided Inquiry

1 · The apparatus

Let’s meet the setup. A single body moves in the Sun's inverse-square gravity, seen from above. One number — the eccentricity e — sets the shape of its path. The angular momentum is held fixed, so the specific energy ε follows from e.

The right-hand panel draws the same curve as a plane cut through a double cone. Press ▶ Play whenever you’re ready to set the body moving.

2 · Where the Sun sits

Right now the path is a circle, and gravity pulls the body toward the Sun.

But what about a path that’s not circular? Where do you think the Sun sits then?

3 · Giving the body more energy

Now let’s give the body more and more energy, keeping its angular momentum unchanged. What do you think happens to its path?

4 · The edge

Let’s zoom in on that turning point. The eccentricity is raised until the path first fails to close. At that exact edge, what do you think the specific energy ε is?

5 · A closed path above the edge

Here’s a trickier one. Is there any eccentricity that gives a closed path while ε > 0? What’s your guess?

6 · Either side of the edge

Now it’s your turn to explore. The presets are available, so set e = 0.99, then e = 1.01.

How does the cutting plane on the cone differ between the two? And how far does the body travel before it turns back — or doesn’t? Take your time comparing the two.

Listen

Controls

▾

Eccentricity e 0.00

Orbit type by eccentricity

Circle — e = 0
Ellipse — 0 < e < 1
Parabola — e = 1
Hyperbola — e > 1

coniccircle

Specific energy ε−0.500

sign(ε)negative

Presets

Specific energy (ε = E/m)

Semi-major axis (bound)

Family (fixed L)

Conic classification

Natural units: GM = 1. The trajectory family here is parameterised by a single knob e with angular momentum L held fixed (semi-latus rectum p = L²/GM = p₀). Then r_peri = p/(1+e) and ε = GM(e²−1)/(2p) — continuous through the parabolic edge e = 1.

The Physics

Orbits and conic sections · what this sim shows, the physics behind it, and every control

At a glance

A single body moves under the Sun's inverse-square gravity, and one slider, the eccentricity e, sets the shape of its path. The sim shows every shape that one force allows, and how the body's energy is tied to that shape.

What you'll learn

Where the Sun sits relative to the path, for each kind of curve.
How the body's energy depends on e when its angular momentum is held fixed.
What the body's energy tells you about its path.
Why the same curves appear as slices through a double cone.

Each run starts here. The slider changes r and v together; their product, the angular momentum per kilogram, never changes.

What's on screen

Sun (amber), fixed at the origin of the orbit panel.
Body and trail (teal), with a dashed line from the Sun. The faint teal outline is the exact curve the body should follow.
Conic pill (top of the orbit panel): the name of the current curve.
Conic-section panel (right): a double cone and a cutting plane, coloured by curve type: red circle, green ellipse, blue parabola, orange hyperbola.
Readouts (sidebar): the conic, the specific energy ε, and its sign.

See itLaunch the body with e = 0. Watch for: the trail closing into a circle over the faint outline after one lap, about 11.5 s at 1×.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Eccentricity e	The shape of the path. Each move restarts the body at its closest point. The slider snaps onto 0 and 1 when you come within 0.02.	0–2.2 opens at 0
Readouts	The conic name, the specific energy ε (energy per kilogram), and the sign of ε.	panel
Presets	Jump to circle (0), ellipse (0.6), parabola (1), hyperbola (1.5), or to e = 0.99 and e = 1.01, which the slider's snap skips. Each one starts the body moving.	6 buttons
Conic-section panel	Drag to rotate the double cone and see the cut from any side.	drag
▶ Play / ⏸ Pause	Sets the body moving, or freezes it.	top bar
Speed	Plays the motion faster or slower on screen. The path stays the same.	0.25×–4×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Fixed: units with GM = 1 (1 time unit = 1 s at 1×), and p = 1.5, which holds the angular momentum per kilogram at √1.5.

The Sun sits at a focus

The Sun pulls the body straight toward itself with strength GM/r². Newton showed that this inverse-square pull allows one family of paths, the conic sections, all given by one equation:

The orbit equation. θ is measured from the closest point; p is held fixed in this sim.

The Sun sits at a focus of the curve, not at its centre. Only for the circle (e = 0) are the two the same point.
The closest approach is r = p/(1 + e). Because p is fixed, every path also passes through r = p directly beside the Sun (the two dots below).
Angular momentum is conserved, so the body moves fastest at its closest point and slowest far away (Kepler's second law).

The four presets drawn from the equation with p = 1.5.

See itLaunch an ellipse with e = 0.6. Watch for: the Sun well off the middle of the outline, and the body slowing as it climbs away from it.

Energy decides if it returns

The specific energy ε, the energy per kilogram, stays the same along the whole path:

The sim's ε readout is this number.

ε < 0: the body can never reach r → ∞, because there ½v² would have to be negative. The path closes: circle or ellipse.
ε = 0: the body just escapes, its speed falling toward zero far away. This is the parabola, e = 1.
ε > 0: it escapes with speed to spare, v∞ = √(2ε). This is the hyperbola, and no path with ε > 0 ever closes.

The conic pill and the sign(ε) readout always agree.

Key idea. The edge is ε = 0 exactly. Here e = 0.99 and e = 1.01 differ by only 1% in launch speed, yet the first body comes back (after about 4100 s at 1×) and the second never does.

See itLaunch just inside the edge, e = 0.99. Watch for: the faint outline closing far to the left, and sign(ε) reading negative.

See itLaunch just outside, e = 1.01. Watch for: the faint outline shrinking to two short arms that never meet, and sign(ε) reading positive.

How e sets the energy

With the angular momentum fixed, raising e launches the body closer to the Sun and faster, keeping r · v the same. That fixes the energy for each e:

The family the slider moves through. In the sim's units, ε = (e² − 1)/3.

This slider moves the sim too.

ε grows with e², not with e. From e = 0 to 0.5 it rises only from −0.333 to −0.250; from 0.5 to 1 it climbs the rest of the way to 0.
Only the sign of ε decides the type of path. Its size sets how big a closed orbit is, or how fast an escaping body leaves.

ε against e over the slider's range.

See itSweep e from 0 to 2.2. Watch for: ε creeping up at first, then faster, and its sign turning at the moment the pill reads parabola.

Conic sections are slices of a cone

The four curves are called conic sections because a flat plane cutting a double cone traces exactly these shapes. The right-hand panel shows the cut that matches the orbit.

A level plane gives a circle. Tilted less steeply than the cone's side, it gives an ellipse.
Parallel to the side, the cut never closes: a parabola, the edge case again.
Steeper than the side, the plane cuts both halves of the cone, giving the two branches of a hyperbola. An orbit uses only the branch that curves around the Sun.

β: tilt of the plane from the cone's base. β_gen: slope of the cone's side, 54° in the panel.

The panel always shows the right type of curve. Its tilt follows e only roughly: it is exact at e = 0 and e = 1, while at e = 0.6 the drawn cut has e ≈ 0.66.

See itSet e = 1.5. Watch for: on the right, the orange cut running through both halves of the cone; on the left, the body swinging once around the Sun and heading away.

Where it matters

The Solar System. Planets move on near-circles: Earth e = 0.017, Mercury 0.21. Halley's comet, e = 0.967, is bound but only just; it swings out beyond Neptune and returns every 76 years.
Interstellar visitors. 'Oumuamua (2017, e ≈ 1.20) and Borisov (2019, e ≈ 3.4) came in on hyperbolas: one pass, then gone for good.
Escape speed. Setting ε = 0 gives v = √(2GM/r). From Earth's surface that is 11.2 km/s; launch slower and you stay bound to Earth.

Measured eccentricities, drawn from 0 to about 1.3. Borisov (e ≈ 3.4) is off the end.

Go deeper: size, period, and why orbits close

For bound orbits, ε alone sets the size a, and a alone sets the period (Kepler's third law).

In the sim, the circle (a = 1.5) takes 11.5 s per lap at 1×, and the e = 0.6 ellipse (a = 2.34) takes 22.5 s.
Bound orbits that close on themselves every lap are special. Of all power-law central forces, only the inverse-square pull and a spring's linear pull do this (Bertrand's theorem). Small extra forces make real orbits slowly turn, or precess: general relativity alone turns Mercury's by 43″ per century.

Summary

In one line. Under inverse-square gravity every path is a conic with the Sun at a focus, and the sign of the energy picks which kind: ε < 0 closes, ε = 0 is the parabola, ε > 0 escapes.

Key points

All four curves solve r = p/(1 + e cos θ), with the Sun at a focus.
At fixed angular momentum, ε = GM(e² − 1)/2p, so e = 1 is exactly ε = 0.
No path with ε > 0 ever closes.

Common mix-ups

"The Sun is at the centre of the orbit." It is at a focus; only for a circle are the two the same.
"More energy just makes a bigger ellipse." Only while ε < 0. At ε = 0 the path opens, and it never closes again.
"e = 1 is just a very long ellipse." It is the boundary: e = 0.99 comes back, e = 1.01 never does.

Things to try on your own

Press Ellipse and time one lap at 1×. Compare it with T = 2π√(a³), a = 2.34.
Press e = 0.99, then e = 1.01, and compare the cutting plane on the cone panel.