Plummer potential

The Plummer potential is an abstract mathematical potential . It was named after HC Plummer , who introduced it in 1911 to calculate globular clusters . It is useful in numerical treatment of problems that involve terms such as the limit value . ${\displaystyle \textstyle \lim _{x\to 0}{\frac {1}{x}}}$

Due to its close relationship with the Coulomb and gravitational potential - both are special cases of the Plummer potential - most of the applications of this potential can be found in electrodynamics and gravitation theory .

The potential function has the form

${\displaystyle \Phi (r,\epsilon )=\Phi _{\epsilon }(r)=-{\frac {\mathrm {const.} }{\sqrt {(r^{2}+\epsilon ^{2})}}}}$

If you now set , you get the classical (Coulomb) potential, which plays an important role in Newton's theory of gravity and in electrodynamics: ${\displaystyle \epsilon =0}$ ${\displaystyle {\tfrac {1}{r}}}$

${\displaystyle \varphi _{\mathrm {G} }=G{\frac {M}{r}}}$ (Gravitational potential)

or.

${\displaystyle \varphi _{\mathrm {C} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r}}}$ (Coulomb potential)

It is therefore advisable to compare the Plummer potential and the Coulomb potential: In contrast to the Coulomb potential, the Plummer potential has no singularity at this point , but has a finite value ; the normal potential, on the other hand, results for the undefined expression . This means that the Plummer potential at the zero point is continuous and differentiable , which is of interest for analytical calculations. ${\displaystyle r=0}$ ${\displaystyle \Phi (r=0,\epsilon )=-{\tfrac {\mathrm {const.} }{\epsilon }}}$ ${\displaystyle {\tfrac {1}{r}}}$ ${\displaystyle r=0}$ ${\displaystyle {\tfrac {1}{0}}}$

application

An important application for the Plummer potential can be found in astronomy , in the simulation of the dynamics of star clusters and galaxies. In simulations with a large number of bodies (so-called multi - body simulations), one is often not primarily interested in the collisions or near-collisions of individual bodies, but in the structures that develop over a large area. Since it is practically impossible to rule out such collisions in the initial conditions of a simulation, the Plummer potential is often used, as it is a good approximation of the gravitational potential for large distances and does not grow beyond all limits for small distances. If two bodies come too close, they practically fly through each other without excessive force.