Effective potential

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

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 = Emin: circular orbit (A4 as radius). Points A1, ..., A4 are called turning points.

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

U eff ( r ) = L 2 2 μ r 2 + U ( r ) , {\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:

F eff = − ∇ U eff ( r ) = L 2 μ r 3 r ^ − ∇ U ( r ) {\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 r ^ {\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

U eff ≤ E . {\displaystyle U_{\text{eff}}\leq E.}

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

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

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

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

ω = U eff ″ m , {\displaystyle \omega ={\sqrt {\frac {U_{\text{eff}}''}{m}}},}
where the double prime indicates the second derivative of the effective potential with respect to r {\displaystyle 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)[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 = 1 2 m ( r ˙ 2 + r 2 ϕ ˙ 2 ) − G m M r , {\displaystyle E={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)-{\frac {GmM}{r}},}
L = m r 2 ϕ ˙ {\displaystyle L=mr^{2}{\dot {\phi }}}
when the motion of the larger mass is negligible. In these expressions,

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

m r ˙ 2 = 2 E − L 2 m r 2 + 2 G m M r = 2 E − 1 r 2 ( L 2 m − 2 G m M r ) , {\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),}
1 2 m r ˙ 2 = E − U eff ( r ) , {\displaystyle {\frac {1}{2}}m{\dot {r}}^{2}=E-U_{\text{eff}}(r),}
where
U eff ( r ) = L 2 2 m r 2 − G m M r {\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).

See also

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.

Further reading

  • José, JV; Saletan, EJ (1998). Classical Dynamics: A Contemporary Approach (1st ed.). Cambridge University Press. ISBN 978-0-521-63636-0..
  • Likos, C.N.; Rosenfeldt, S.; Dingenouts, N.; Ballauff, M.; Lindner, P.; Werner, N.; Vögtle, F.; et al. (2002). "Gaussian effective interaction between flexible dendrimers of fourth generation: a theoretical and experimental study". J. Chem. Phys. 117 (4): 1869–1877. Bibcode:2002JChPh.117.1869L. doi:10.1063/1.1486209. Archived from the original on 2011-07-19.
  • Baeurle, S.A.; Kroener J. (2004). "Modeling Effective Interactions of Micellar Aggregates of Ionic Surfactants with the Gauss-Core Potential". J. Math. Chem. 36 (4): 409–421. doi:10.1023/B:JOMC.0000044526.22457.bb.
  • Likos, C.N. (2001). "Effective interactions in soft condensed matter physics". Physics Reports. 348 (4–5): 267–439. Bibcode:2001PhR...348..267L. CiteSeerX 10.1.1.473.7668. doi:10.1016/S0370-1573(00)00141-1.