Effective Potential

The core idea

A planet orbiting the Sun moves in two dimensions — its distance r changes and its angle θ sweeps around. Tracking both at once is awkward. The effective potential is a bookkeeping trick that hides the angular motion inside an energy term, leaving behind a problem that looks exactly like a single particle sliding back and forth on a one-dimensional hill.

The trick works because of a conservation law. Any central force — one that points along the line to the center and depends only on distance, like gravity or the Coulomb force — exerts zero torque about the center. So the angular momentum L = mr²(dθ/dt) is constant throughout the orbit. That single conserved number lets us eliminate θ entirely.

Deriving the 1D energy equation

Start from the total energy of a particle of mass m in a central potential V(r). In polar coordinates the kinetic energy splits into a radial part and an angular part:

E = ½m(dr/dt)² + ½m·r²(dθ/dt)² + V(r)

Now use angular momentum conservation, L = mr²(dθ/dt), to solve for the angular speed: dθ/dt = L/(mr²). Substitute it into the middle term:

½m·r²(dθ/dt)²  =  ½m·r²·(L/(mr²))²  =  L²/(2mr²)

The angular kinetic energy has turned into a function of r alone. Group it with the real potential:

E = ½m(dr/dt)² + V_eff(r),     V_eff(r) = V(r) + L²/(2mr²)

This is the payoff. The energy now has the form of a 1D problem: kinetic energy in r plus an "effective" potential. Everything we know about 1D motion — turning points, equilibria, oscillation — applies directly to the radial coordinate.

The centrifugal barrier

The added term L²/(2mr²) is the centrifugal barrier. It is always positive and grows without bound as r → 0. Physically, conserving L means that as the particle gets closer to the center it must spin faster, and that spinning costs rotational kinetic energy. The barrier is the energy price of approaching the center while keeping L fixed.

For any nonzero angular momentum, this barrier is infinitely tall at the origin, so the particle can never reach r = 0. That is why a planet doesn't simply fall into the Sun: it has too much sideways motion. Only a head-on, zero-angular-momentum trajectory (L = 0) has no barrier and can plunge straight in.

Reading orbits off the curve

Because the radial kinetic energy ½m(dr/dt)² can never be negative, the particle is confined to radii where V_eff(r) ≤ E. The geometry of the orbit is set by where the horizontal energy line E intersects the curve:

Energy line vs. V_eff	Radial behavior	Orbit type (gravity, V = −k/r)
E = V_eff,min (touches the bottom)	single radius, no radial motion	Circular orbit
V_eff,min < E < 0 (cuts twice)	oscillates between r_min and r_max	Bound ellipse
E = 0 (one inner crossing)	comes in, leaves to infinity	Parabolic escape
E > 0 (one inner crossing)	comes in, leaves to infinity	Hyperbolic flyby
E < V_eff,min	impossible — no real solution	(no orbit at this L)

The inner crossing is the perihelion (closest approach, r_min); the outer crossing is the aphelion (farthest, r_max). At both, the radial velocity vanishes — these are the turning points where the particle reverses its radial motion even though it keeps sweeping around in angle. The bound orbit is therefore an annulus traced over and over.

Circular orbits and stability

A circular orbit corresponds to the particle sitting at the bottom of the well, where the net radial force is zero:

dV_eff/dr = 0   →   dV/dr = L²/(mr³)   (real force balances centrifugal term)

For the Kepler potential V(r) = −k/r this gives the circular radius r_c = L²/(mk). Whether the orbit is stable depends on the curvature: if d²V_eff/dr² > 0 the bottom is a true minimum and small radial nudges oscillate around r_c; if the curvature is negative the equilibrium is a hilltop and the slightest push sends the particle spiraling away. For an attractive power-law potential V ∝ −rⁿ, stable circular orbits exist only for n > −2 — equivalently, the force law F ∝ −rⁿ⁻¹ must fall off slower than inverse-cube. That is exactly why inverse-square (n = −1, force ∝ 1/r²) gravity gives stable planetary orbits while an inverse-cube force (the n = −2 boundary) does not.

Worked numbers

System	Quantity	Value
Earth around Sun	specific angular momentum h = L/m	≈ 4.46 × 10¹⁵ m²/s
Earth around Sun	circular radius r_c = h²/(GM☉)	≈ 1.50 × 10¹¹ m (1 AU) ✓
Earth's orbit eccentricity 0.0167	r_min (perihelion)	≈ 1.471 × 10¹¹ m
Earth's orbit eccentricity 0.0167	r_max (aphelion)	≈ 1.521 × 10¹¹ m
Halley's Comet (e ≈ 0.967)	r_min / r_max	0.59 AU / 35 AU (deep, narrow well)
Hydrogen ground state (Bohr)	circular radius from V_eff min	5.29 × 10⁻¹¹ m (Bohr radius)

The same curve scales across 21 orders of magnitude in size. A near-circular orbit like Earth's sits in a shallow, nearly parabolic dip, so its turning points hug 1 AU; a comet like Halley's lives in a deep, narrow well that lets r swing by a factor of ~60.

JavaScript — effective potential and turning points

// Kepler effective potential: V_eff(r) = -k/r + L^2/(2 m r^2)
function Veff(r, k, L, m) {
  return -k / r + (L * L) / (2 * m * r * r);
}

// Bottom of the well: stable circular-orbit radius r_c = L^2 / (m k)
function circularRadius(k, L, m) {
  return (L * L) / (m * k);
}

// Minimum effective potential value (energy of the circular orbit)
function VeffMin(k, L, m) {
  const rc = circularRadius(k, L, m);
  return Veff(rc, k, L, m);          // equals -m k^2 / (2 L^2)
}

// Turning points solve V_eff(r) = E  ->  E r^2 + k r - L^2/(2m) = 0
function turningPoints(E, k, L, m) {
  // Rearranged: E r^2 + k r - L^2/(2m) = 0
  const a = E, b = k, c = -(L * L) / (2 * m);
  const disc = b * b - 4 * a * c;
  if (disc < 0) return null;          // E below V_eff min: no orbit
  const sq = Math.sqrt(disc);
  if (Math.abs(E) < 1e-30) {          // E = 0: parabola, single turning point
    return { r_min: (L * L) / (2 * m * k), r_max: Infinity };
  }
  const r1 = (-b + sq) / (2 * a);
  const r2 = (-b - sq) / (2 * a);
  const roots = [r1, r2].filter(r => r > 0).sort((p, q) => p - q);
  if (E < 0) return { r_min: roots[0], r_max: roots[1] }; // bound: two crossings
  return { r_min: roots[0], r_max: Infinity };            // E > 0: hyperbolic
}

// Classify the orbit straight off the energy line
function classify(E, k, L, m) {
  const Emin = VeffMin(k, L, m);
  if (E < Emin - 1e-12) return 'forbidden (E below V_eff min)';
  if (Math.abs(E - Emin) < 1e-12) return 'circular';
  if (E < 0) return 'bound ellipse';
  if (E === 0) return 'parabolic escape';
  return 'hyperbolic flyby';
}

// Sun-Earth, per unit mass (set m = 1, k = G M_sun)
const m = 1, k = 1.327e20, L = 4.46e15; // h = L/m for Earth
console.log('r_c =', circularRadius(k, L, m).toExponential(3), 'm'); // ~1.50e11
console.log('E_circ =', VeffMin(k, L, m).toExponential(3), 'J/kg');

// A bound orbit slightly above the well bottom -> two turning points
const E = 0.92 * VeffMin(k, L, m);   // less negative than the minimum
console.log(classify(E, k, L, m));   // 'bound ellipse'
console.log(turningPoints(E, k, L, m));

Where the effective potential shows up

• Celestial mechanics. Planet, comet, and satellite orbits; perihelion/aphelion; escape vs. capture; the Kepler problem's bound/unbound split.
• Lagrange points. In the co-rotating frame of two massive bodies, the effective potential's saddles and maxima locate the L1–L5 points and decide their stability — why JWST parks at L2.
• Atomic physics. The radial Schrödinger equation carries the same ℓ(ℓ+1)ℏ²/(2mr²) centrifugal term; it sets which orbitals can reach the nucleus (only s-states, with ℓ = 0).
• Nuclear and particle scattering. The Coulomb barrier plus centrifugal barrier together control fusion rates and resonance positions.
• General relativity. The Schwarzschild effective potential adds a −GML²/(c²r³) term that bends the curve at small r, producing the innermost stable circular orbit (ISCO) at 3 Schwarzschild radii and explaining Mercury's perihelion precession.
• Plasma and accelerator physics. Guiding-center motion and beam dynamics use effective potentials to fold fast gyration into slow drift.

Common mistakes

• Calling the centrifugal barrier a real force in the lab frame. It's the angular kinetic energy in disguise. Only in the rotating frame does it become a genuine pseudo-force potential.
• Forgetting that V_eff depends on L. Change the angular momentum and the whole curve changes shape — the well deepens or the barrier rises. There's a different V_eff for every value of L.
• Confusing turning points with the orbit's shape. Turning points are radial extremes (r_min, r_max), not the full elliptical geometry; the angle keeps advancing at both.
• Assuming circular orbits are always stable. The minimum must be a true minimum (positive curvature). For forces steeper than inverse-cube, the "well" is a hilltop and the circular orbit is unstable.
• Mixing up E = 0 and the well minimum. E = 0 is the escape threshold (parabolic); the minimum of V_eff is the most-bound, circular orbit, generally at a different, negative energy.
• Using the 1/r² Newtonian curve near a black hole. Relativity adds a steeply attractive small-r term; the barrier no longer wins at small r, which is precisely why an ISCO exists.