Effective potential

The effective potential (also known as effective potential energy) combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the 'opposing' centrifugal potential energy with the potential energy of a dynamical system. It may be used to determine the orbits of planets (both Newtonian and relativistic) and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions.

Definition

Effective potential. E > 0: hyperbolic orbit (A₁ as pericenter), E = 0: parabolic orbit (A₂ as pericenter), E < 0: elliptic orbit (A₃ as pericenter, A₃' as apocenter), E = E_min: circular orbit (A₄ as radius). Points A₁, ..., A₄ are called turning points.

The basic form of potential $U_{\text{eff}}$ is defined as: $U_{\text{eff}}(\mathbf {r} )={\frac {L^{2}}{2\mu r^{2}}}+U(\mathbf {r} ),$ where

L is the angular momentum
r is the distance between the two masses
μ is the reduced mass of the two bodies (approximately equal to the mass of the orbiting body if one mass is much larger than the other); and
U(r) is the general form of the potential.

The effective force, then, is the negative gradient of the effective potential: ${\begin{aligned}\mathbf {F} _{\text{eff}}&=-\nabla U_{\text{eff}}(\mathbf {r} )\\&={\frac {L^{2}}{\mu r^{3}}}{\hat {\mathbf {r} }}-\nabla U(\mathbf {r} )\end{aligned}}$ where ${\hat {\mathbf {r} }}$ denotes a unit vector in the radial direction.

Important properties

There are many useful features of the effective potential, such as $U_{\text{eff}}\leq E.$

To find the radius of a circular orbit, simply minimize the effective potential with respect to $r$ , or equivalently set the net force to zero and then solve for $r_{0}$ : ${\frac {dU_{\text{eff}}}{dr}}=0$ After solving for $r_{0}$ , plug this back into $U_{\text{eff}}$ to find the maximum value of the effective potential $U_{\text{eff}}^{\text{max}}$ .

A circular orbit may be either stable or unstable. If it is unstable, a small perturbation could destabilize the orbit, but a stable orbit would return to equilibrium. To determine the stability of a circular orbit, determine the concavity of the effective potential. If the concavity is positive, the orbit is stable: ${\frac {d^{2}U_{\text{eff}}}{dr^{2}}}>0$

The frequency of small oscillations, using basic Hamiltonian analysis, is $\omega ={\sqrt {\frac {U_{\text{eff}}''}{m}}},$ where the double prime indicates the second derivative of the effective potential with respect to $r$ and it is evaluated at a minimum.

Gravitational potential

Visualisation of the effective potential in a plane containing the orbit (grey rubber-sheet model with purple contours of equal potential), the Lagrangian points (red) and a planet (blue) orbiting a star (yellow)^[1]

Consider a particle of mass m orbiting a much heavier object of mass M. Assume Newtonian mechanics, which is both classical and non-relativistic. The conservation of energy and angular momentum give two constants E and L, which have values $E={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)-{\frac {GmM}{r}},$ $L=mr^{2}{\dot {\phi }}$ when the motion of the larger mass is negligible. In these expressions,

${\dot {r}}$ is the derivative of r with respect to time,
${\dot {\phi }}$ is the angular velocity of mass m,
G is the gravitational constant,
E is the total energy, and
L is the angular momentum.

Only two variables are needed, since the motion occurs in a plane. Substituting the second expression into the first and rearranging gives $m{\dot {r}}^{2}=2E-{\frac {L^{2}}{mr^{2}}}+{\frac {2GmM}{r}}=2E-{\frac {1}{r^{2}}}\left({\frac {L^{2}}{m}}-2GmMr\right),$ ${\frac {1}{2}}m{\dot {r}}^{2}=E-U_{\text{eff}}(r),$ where $U_{\text{eff}}(r)={\frac {L^{2}}{2mr^{2}}}-{\frac {GmM}{r}}$ is the effective potential.^{[Note 1]} The original two-variable problem has been reduced to a one-variable problem. For many applications the effective potential can be treated exactly like the potential energy of a one-dimensional system: for instance, an energy diagram using the effective potential determines turning points and locations of stable and unstable equilibria. A similar method may be used in other applications, for instance determining orbits in a general relativistic Schwarzschild metric.

Effective potentials are widely used in various condensed matter subfields, e.g. the Gauss-core potential (Likos 2002, Baeurle 2004) and the screened Coulomb potential (Likos 2001).

Notes

^ A similar derivation may be found in José & Saletan, Classical Dynamics: A Contemporary Approach, pgs. 31–33

References

^ Seidov, Zakir F. (2004). "The Roche Problem: Some Analytics". The Astrophysical Journal. 603: 283–284. arXiv:astro-ph/0311272. Bibcode:2004ApJ...603..283S. doi:10.1086/381315.