Effective potential

Effective potential in the gravitational field

The effective potential is a mechanical term that is useful in dealing with central forces such as the gravitational force in planetary motion . In the effective potential, the potential energy and the azimuthal kinetic energy of the rotating object are combined. Despite its name, the effective potential is not strictly speaking a potential , but has the dimension of an energy . ${\displaystyle V_{\text{eff}}}$

Non-relativistic mechanics

A body of mass , which moves in a central force field at a distance from the center of force, has a total energy that is composed of the potential energy and the kinetic energy . In polar coordinates we get: ${\displaystyle m}$ ${\displaystyle r}$ ${\displaystyle V}$ ${\displaystyle E_{\mathrm {kin} }}$ ${\displaystyle (r,\varphi )}$

${\displaystyle {\begin{aligned}E&=V(r)+E_{\mathrm {kin} }\\&=V(r)+{\frac {1}{2}}m(r{\dot {\varphi }})^{2}+{\frac {1}{2}}m\cdot {\dot {r}}^{2}\end{aligned}}}$

The azimuthal portion of the kinetic energy can be expressed by the amount of angular momentum , which is a conservation quantity for a central force , and summarized with the potential energy to form the effective potential : ${\displaystyle {\frac {1}{2}}m(r{\dot {\varphi }})^{2}}$ ${\displaystyle L=m\cdot r^{2}\cdot {\dot {\varphi }}}$ ${\displaystyle V_{\text{eff}}}$

${\displaystyle {\begin{alignedat}{2}\Rightarrow E&=V(r)+{\frac {1}{2}}{\frac {L^{2}}{m\cdot r^{2}}}&&+{\frac {1}{2}}m\cdot {\dot {r}}^{2}\\&=V_{\text{eff}}(r)&&+{\frac {1}{2}}m\cdot {\dot {r}}^{2}\,\,(*)\end{alignedat}}}$

whereby the effective potential is defined as:

${\displaystyle V_{\text{eff}}(r)=V(r)+{\frac {1}{2}}{\frac {L^{2}}{m\cdot r^{2}}}}$

The second term on the right-hand side of this equation is also known as the centrifugal potential or the angular momentum barrier .

In the equation one now only has to do with an ordinary differential equation in the radial coordinate . The solution to this is done by using the method of separating the variables (d t and d r ) with the constants of motion and as parameters. Your solution is through the elliptic integral ${\displaystyle (*)}$ ${\displaystyle r}$ ${\displaystyle E}$ ${\displaystyle L}$

${\displaystyle t-t_{0}=\int _{r_{0}}^{r}\mathrm {d} r'{\frac {1}{\sqrt {{\frac {2}{m}}\left(E-V_{\text{eff}}(r')\right)}}}}$

given. For another, clearer solution, in which the radius is represented as a function of the angle, see under two-body problem .

The curve of the effective potential clearly shows without further mathematical considerations for initially two points of intersection and with the effective potential curve , between which the body moves on its path. For the minimum of the effective potential, both distances coincide and a circular path is obtained . For the body describes an unbound movement with only a minimal distance. ${\displaystyle E<0}$ ${\displaystyle r_{\mathrm {min} }}$ ${\displaystyle r_{\mathrm {max} }}$ ${\displaystyle E\geq 0}$

general theory of relativity

In general relativity , the effective potential is given higher order correction terms. The constants of movement in the Schwarzschild metric are no longer and , but and , where is the so-called Schwarzschild radius . ${\displaystyle E}$ ${\displaystyle L}$ ${\displaystyle p_{0}c=E\left(1-{\frac {r_{\text{s}}}{r}}\right)}$ ${\displaystyle L}$ ${\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}}}$

The following applies:

${\displaystyle (p_{0}c)^{2}=m^{2}{\dot {r}}^{2}c^{2}+\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(1+{\frac {L^{2}}{m^{2}r^{2}c^{2}}}\right)m^{2}c^{4}}$,

so that the effective potential in general relativity is called

${\displaystyle V_{\text{eff}}(r)=mc^{2}{\sqrt {\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(1+{\frac {L^{2}}{m^{2}r^{2}c^{2}}}\right)}}}$

can be represented. This potential contains the constant term of the rest energy , towards which the potential for also tends, and is imaginary for. Objects with a radius smaller than their Schwarzschild radius are called black holes . ${\displaystyle r\to \infty }$ ${\textstyle r<r_{\text{s}}}$

While in classical physics any narrow orbits around the central body are possible, as there is a minimum for each , this is not the case in the Schwarzschild solution. The effective potential possesses for a maximum and a minimum in the places ${\displaystyle L^{2}>0}$ ${\displaystyle L^{2}>3m^{2}c^{2}r_{\text{s}}^{2}}$

${\displaystyle r_{0}={\frac {L^{2}}{m^{2}c^{2}r_{\text{s}}}}\mp {\sqrt {{\frac {L^{4}}{m^{4}c^{4}r_{\text{s}}^{2}}}-3{\frac {L^{2}}{m^{2}c^{2}}}}}\,}$;

below this value for the angular momentum it increases monotonically. A minimal stable orbit thus results at

${\displaystyle r_{\text{min}}=3r_{\text{s}}}$.

literature

• Herbert Goldstein: Classical Mechanics . Addison-Wesley, 1980, ISBN 0-201-02918-9 , pp. 76 f . (English).
• Volker Meden (RWTH Aachen): Script for the lecture Theoretical Physics I (Mechanics) . 2012, p. 11 ( rwth-aachen.de [PDF; accessed December 19, 2016]).