effective potential

{{short description|Net potential energy encountered in orbital mechanics.}}

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

File:Effective potential.png

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),

: U(r) is the general form of the potential.

The effective force, then, is the negative gradient of the effective potential:

$\begin{align}
\mathbf{F}_\text{eff} &= -\nabla U_\text{eff}(\mathbf{r}) \\
&= \frac{L^2}{\mu r^3} \hat{\mathbf{r}} - \nabla U(\mathbf{r}),
\end{align}$

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 $r_0$ :

$\frac{d U_\text{eff}}{dr} = 0.$

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,

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

the orbit is stable.

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 is evaluated at a minimum.

Gravitational potential

File:Restricted Three-Body Problem - Energy Potential Analysis.png

File:Lagrangian points equipotential.pngs (red) and a planet (blue) orbiting a star (yellow){{cite journal |title=The Roche Problem: Some Analytics |journal=The Astrophysical Journal |volume=603 |pages=283–284 |doi=10.1086/381315 |year=2004 |last1=Seidov |first1=Zakir F. |arxiv=astro-ph/0311272 |bibcode=2004ApJ...603..283S}}]]

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,

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

effective potential

Definition

Important properties

Gravitational potential

See also

Notes

References

Further reading