Energy of the Orbit

Source

Lecture: L36 — Effective potential & energy of the orbit
Simulation: Orbit + V_eff(r), with optional θ̇(r) (Guided-Inquiry)
Concept: Radial motion is 1D motion in V_eff; bound vs unbound follows sign(E); θ̇ = l/r² ties angular speed to radius.
Units: GM = m = 1; angular momentum fixed at l = 1.
Serves: Effective potential, turning points, and how θ̇(r) builds the planar orbit.
Inquiry: Read V_eff; predict E<0 / E>0; identify peri/apo as E∩V_eff; find where θ̇ peaks.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept, one sentence	Motion in V_eff + θ̇(r) = l/r² makes the orbit.
A2	Fewest objects	Orbit left; V_eff right; θ̇ stacks under V_eff when asked for.
C1	Numbers earn their place	E slider + regime/e; turning points live on the diagram.
F3	Guided-inquiry stepper	Orient V_eff → E<0 → E>0 → turning points → θ̇.
F7	Prediction gates Play	Play is disabled until the student commits an answer on gated steps.

Effective Potential

Teach me Hide Text

Orbit — Sun + body

Sun

body

V_eff(r) — energy diagram

V_eff

θ(r) — l / r²

Guided Inquiry

1 · Orient: read V_eff

Let’s start by getting to know the curve. Where do you think V_eff approaches zero?

So the right panel shows the effective potential: the attractive −1/r plus an angular-momentum barrier that rises as r→0.

2 · Predict: E = −0.35

Now let’s give the body some energy. It’s set negative, but above the well minimum (E = −0.35). Before you press Play, what motion do you predict?

3 · Predict: E = +0.15

Now flip the sign: the energy is positive (E = +0.15). What do you think the body does this time?

4 · Observe: turning points

Let’s go back to E < 0 and look more closely. The coloured markers where E meets V_eff are perihelion and aphelion — the true radial turning points.

What do you think happens to the body at those intersections?

5 · θ̇(r): where is angular speed largest?

Check Show θ̇(r) plot (it opens for this step). One last idea: angular momentum fixes θ̇ = l/r².

So where along the orbit do you expect θ̇ to be largest?

Listen

Controls

▾

Total energy E −0.50

Show θ(r) plot

State

▾

regimebound

e0.00

Units: GM = m = 1, fixed l = 1. Then V(r) = −1/r and V_eff(r) = −1/r + l²/(2r²).

Effective potential

Energy (radial form)

Angular rate

Eccentricity (fixed l)

Circular orbit

Regime by sign of E

The Physics

Orbit energy · what this sim shows, the physics behind it, and every control

At a glance

Whether a body orbits the Sun or leaves for good depends on its total energy E. This sim reduces the orbit to one coordinate, the distance r, so its size and shape can be read off an energy diagram.

What you'll learn

What the effective potential V_eff(r) is made of.
How the energy E, drawn as a line on that curve, limits where the body can be.
How E sets the orbit's type, size and eccentricity.
How the angular speed θ̇ depends on r.

The same body in two views. In the energy diagram, the bar joins the body's dot to the E line.

What's on screen

Sun and body in the Orbit panel. The body's colour follows the sign of E: blue, amber or pink.
V_eff(r) curve (grey) and the E line (coloured like the body). The body's dot rides on the curve.
Green bar: the gap between the dot and the E line.
Coloured dots on the E line mark where it meets the curve.
State panel (regime, e, l) and the optional θ̇(r) panel.

See itRun the sim at E = −0.35. Watch for: the body's dot sliding along the V_eff curve as the body goes round the orbit.

The controls

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

Control	What it changes	Range
Total energy E	The body's energy per unit mass. It moves the E line, and with it the orbit.	−0.50 to +0.60 opens at −0.50
Show θ̇(r) plot	Opens the third panel, the angular speed against r, with a marker for the body.	on / off opens off
State	The regime (circular, bound, parabolic or unbound), the eccentricity e, and the fixed l.	panel
▶ Play / ⏸ Pause	Starts or freezes the motion.	top bar
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Fixed: GM = m = 1 and the angular momentum l = 1, so every number is in these units. Near E = 0 the slider snaps to exactly 0.

The effective potential

The orbit has a conserved angular momentum l. With l fixed, only the distance r is left to follow, and it moves like a marble rolling on the effective potential:

In the sim's units this is −1/r + 1/(2r²), with its lowest point at r = 1, where V_eff = −0.50.

The first term is gravity's potential energy. The second is the kinetic energy of the sideways motion, rewritten using l. It acts as a barrier that rises as r shrinks. Energy conservation then reads:

ṙ is the rate at which r changes. Its kinetic energy cannot be negative.

Key idea. The body can only be at radii where the E line lies on or above the curve.

See itRun the sim at E = −0.35. Watch for: the dot never rising above the E line, and the green bar shrinking to nothing where the line meets the curve.

The sign of E sets the orbit's type

E < 0. The line cuts the curve twice, so the body is confined between two radii and traces an ellipse. The lowest allowed energy is E = −1/(2l²) = −0.50, where the two radii merge and the orbit is a circle at r = 1.
E = 0. The line meets the curve once, at r = l²/2 = 0.5. The body escapes and arrives far away with almost no speed.
E > 0. Again one meeting point. The body passes it once and leaves, with speed √(2E) far from the Sun.

Three energies on one curve.

E is conserved, so the type is fixed at launch and cannot change on the way.

See itRun the sim at E = +0.15. Watch for: the E line meeting the curve at one dot only, and the body passing it once and heading out.

E sets the shape and size

With l = 1, the energy alone fixes the orbit's shape and size. For E < 0, a is the ellipse's semi-major axis (half its long diameter):

r_peri is the closest distance and r_apo the farthest.

This slider moves the sim's energy too.

Raising E from −0.50 toward 0 takes e from 0 up to 1. r_apo grows without limit, while r_peri only slides from 1 down to 0.5.
For E > 0 the same e formula gives e > 1. Then a = 1/(2E) and r_peri = a(e − 1).

See itRun E = −0.35, then E = −0.175. Watch for: the card listing e for each run, and the second orbit longer and thinner.

The barrier keeps the body out

As r → 0 the barrier l²/(2r²) grows faster than gravity's −1/r, because 1/r² beats 1/r. For any l ≠ 0 the curve climbs without limit, and no E line can reach it. The body never reaches the Sun.
At the two meeting points, perihelion (closest) and aphelion (farthest), ṙ = 0, but the body's speed is not 0. The green bar closes, yet the body keeps sweeping round the Sun, so r turns around and the orbit goes on.
With l = 1 and E ≤ 0 the closest approach is between 0.5 and 1. At E = +0.60, the top of the slider, it is 0.40.

See itRun the sim at E = +0.60. Watch for: the path bending away from the Sun at the inner dot, short of reaching it.

The angular speed follows r

θ̇ is the rate at which the angle θ around the Sun changes.

Since l is fixed, θ̇ is large close to the Sun and small far from it. The θ̇(r) panel draws this law as a curve, and the body's marker rides on it.
The same l that keeps the body off the Sun makes the closest approach the fastest part of the orbit. This is Kepler's second law: r²θ̇ = l is twice the area swept per unit time.
At E = −0.35: θ̇ ≈ 2.4 at r_peri = 0.65, against ≈ 0.20 at r_apo = 2.21. That is about 12 times faster close in.

See itRun E = −0.35 with the θ̇(r) panel open. Watch for: the marker riding the θ̇ curve, high when the body is near the Sun and low when it is far away.

Where it matters

E = 0 written in SI units: this is all "escape speed" means.

Planets all have E < 0 and travel on closed ellipses, most of them nearly circular (Earth: e ≈ 0.017).
Long-period comets have e very close to 1. They spend most of their time far out and sweep past the Sun quickly.
ʻOumuamua (2017) arrived with E > 0 and e ≈ 1.2: one pass around the Sun, then gone for good.
Spacecraft. A probe launched from Earth's surface faster than 11.2 km/s (ignoring air) has E > 0 with respect to Earth and does not come back.

Go deeper: why one coordinate is enough

Gravity points straight at the Sun, so it exerts no torque about the Sun, and l = m r²θ̇ stays constant. That is what lets the sheet follow r alone.
The −1/r potential also gives closed ellipses. Other force laws give the same one-dimensional picture, but the orbits generally do not close.

Summary

In one line. Draw V_eff(r) and put the E line on it: the body lives where the line is above the curve, and the sign of E decides whether it comes back.

Key points

V_eff = −1/r + l²/(2r²); the second term is a kinetic barrier.
E < 0 gives an ellipse, E = 0 a parabola, E > 0 a hyperbola. E cannot go below −1/(2l²).
e = √(1 + 2El²) and, for E < 0, a = −1/(2E).
θ̇ = l/r² is largest at the closest approach.

Common mix-ups

"Given time, an orbit spirals into the Sun." E and l are conserved, so the inner edge of the allowed region is a wall. Spiralling in would need drag or thrust.
"At aphelion the planet stops." Only ṙ is zero there. θ̇ = l/r² never vanishes.
"E = 0 means no motion." It is the escape threshold: the body keeps moving and reaches infinity with no speed to spare.

Things to try on your own

Set E = −0.20. From r_peri and r_apo in section 05, work out how many times faster θ̇ is at perihelion, then check it on the θ̇(r) panel.
Move E from −0.50 to −0.05 in four steps. Check each e in the State panel against √(1 + 2E).