Effective potential

Template:Short description 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: $$U_{\text{eff}}(𝐫)=\frac{L^2}{2\mu r^2}+U(𝐫),$$ 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: $$\begin{array}{cc}{𝐅}_{\text{eff}}& =-\mathrm{\nabla }{U}_{\text{eff}}(𝐫)\\ & =\frac{L^2}{\mu r^3}\widehat{𝐫}-\mathrm{\nabla }U(𝐫)\end{array}$$ where $\widehat{𝐫}$ denotes a unit vector in the radial direction.

Important properties

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

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, the orbit is stable: $$\frac{d^2{U}_{\text{eff}}}{d{r}^2}>0$$

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

Components of the effective potential of two rotating bodies: (top) the combined gravitational potentials; (btm) the combined gravitational and rotational potentials

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)

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

• $\dot{r}$ is the derivative of r with respect to time,
• $\dot{\varphi }$ 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 $$m{\dot{r}}^2=2E-\frac{L^2}{m{r}^2}+\frac{2GmM}{r}=2E-\frac{1}{r^2}(\frac{L^2}{m}-2GmMr),$$ $$\frac{1}{2}m{\dot{r}}^2=E-U_{\text{eff}}(r),$$ where $$U_{\text{eff}}(r)=\frac{L^2}{2m{r}^2}-\frac{GmM}{r}$$ is the effective potential. 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).

See also

• Geopotential

Further reading

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