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

The basic form of potential ${\displaystyle U_{\text{eff}}}$ is defined as:

${\displaystyle U_{\text{eff}}(\mathbf {r} )={\frac {L^{2}}{2\mu r^{2}}}+U(\mathbf {r} ),}$
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:

{\displaystyle {\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 ${\displaystyle {\hat {\mathbf {r} }}}$ denotes a unit vector in the radial direction.

Important properties

There are many useful features of the effective potential, such as

${\displaystyle U_{\text{eff}}\leq E.}$

To find the radius of a circular orbit, simply minimize the effective potential with respect to ${\displaystyle r}$, or equivalently set the net force to zero and then solve for ${\displaystyle r_{0}}$:

${\displaystyle {\frac {dU_{\text{eff}}}{dr}}=0}$
After solving for ${\displaystyle r_{0}}$, plug this back into ${\displaystyle U_{\text{eff}}}$ to find the maximum value of the effective potential ${\displaystyle U_{\text{eff}}^{\text{max}}}$.

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:

${\displaystyle {\frac {d^{2}U_{\text{eff}}}{dr^{2}}}>0}$

The frequency of small oscillations, using basic Hamiltonian analysis, is

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

Gravitational potential

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

${\displaystyle E={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)-{\frac {GmM}{r}},}$
${\displaystyle L=mr^{2}{\dot {\phi }}}$
when the motion of the larger mass is negligible. In these expressions,

• ${\displaystyle {\dot {r}}}$ is the derivative of r with respect to time,
• ${\displaystyle {\dot {\phi }}}$ 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

${\displaystyle m{\dot {r}}^{2}=2E-{\frac {L^{2}}{mr^{2}}}+{\frac {2GmM}{r}}=2E-{\frac {1}{r^{2}}}\left({\frac {L^{2}}{m}}-2GmMr\right),}$
${\displaystyle {\frac {1}{2}}m{\dot {r}}^{2}=E-U_{\text{eff}}(r),}$
where
${\displaystyle U_{\text{eff}}(r)={\frac {L^{2}}{2mr^{2}}}-{\frac {GmM}{r}}}$
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

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

References

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