Plummer model

The Plummer model or Plummer sphere is a density law that was first used by H. C. Plummer to fit observations of globular clusters.^[1] It is now often used as toy model in N-body simulations of stellar systems.

The Plummer 3-dimensional density profile is given by $${\displaystyle \rho _{P}(r)={\frac {3M_{0}}{4\pi a^{3}}}\left(1+{\frac {r^{2}}{a^{2}}}\right)^{-{5}/{2}}={\frac {3M_{0}a^{2}}{4\pi (a^{2}+r^{2})^{{5}/{2}}}},}$${\displaystyle \rho _{P}(r)={\frac {3M_{0}}{4\pi a^{3}}}\left(1+{\frac {r^{2}}{a^{2}}}\right)^{-{5}/{2}}={\frac {3M_{0}a^{2}}{4\pi (a^{2}+r^{2})^{{5}/{2}}}},} where ${\displaystyle M_{0}}${\displaystyle M_{0}} is the total mass of the cluster, and a is the Plummer radius, a scale parameter that sets the size of the cluster core. The corresponding potential is $${\displaystyle \Phi _{P}(r)=-{\frac {GM_{0}}{\sqrt {r^{2}+a^{2}}}},}$${\displaystyle \Phi _{P}(r)=-{\frac {GM_{0}}{\sqrt {r^{2}+a^{2}}}},} where G is Newton's gravitational constant. The velocity dispersion is $${\displaystyle \sigma _{P}^{2}(r)={\frac {GM_{0}}{6{\sqrt {r^{2}+a^{2}}}}}.}$${\displaystyle \sigma _{P}^{2}(r)={\frac {GM_{0}}{6{\sqrt {r^{2}+a^{2}}}}}.}

The isotropic distribution function reads $${\displaystyle f({\vec {x}},{\vec {v}})={\frac {24{\sqrt {2}}}{7\pi ^{3}}}{\frac {a^{2}}{G^{5}M_{0}^{4}}}(-E({\vec {x}},{\vec {v}}))^{7/2},}$${\displaystyle f({\vec {x}},{\vec {v}})={\frac {24{\sqrt {2}}}{7\pi ^{3}}}{\frac {a^{2}}{G^{5}M_{0}^{4}}}(-E({\vec {x}},{\vec {v}}))^{7/2},} if ${\displaystyle E<0}${\displaystyle E<0}, and ${\displaystyle f({\vec {x}},{\vec {v}})=0}${\displaystyle f({\vec {x}},{\vec {v}})=0} otherwise, where ${\textstyle E({\vec {x}},{\vec {v}})={\frac {1}{2}}v^{2}+\Phi _{P}(r)}${\textstyle E({\vec {x}},{\vec {v}})={\frac {1}{2}}v^{2}+\Phi _{P}(r)} is the specific energy.

The mass enclosed within radius ${\displaystyle r}${\displaystyle r} is given by $${\displaystyle M(<r)=4\pi \int _{0}^{r}r'^{2}\rho _{P}(r'),円dr'=M_{0}{\frac {r^{3}}{(r^{2}+a^{2})^{3/2}}}.}$${\displaystyle M(<r)=4\pi \int _{0}^{r}r'^{2}\rho _{P}(r'),円dr'=M_{0}{\frac {r^{3}}{(r^{2}+a^{2})^{3/2}}}.}

Many other properties of the Plummer model are described in Herwig Dejonghe's comprehensive article.^[2]

Core radius ${\displaystyle r_{c}}${\displaystyle r_{c}}, where the surface density drops to half its central value, is at ${\textstyle r_{c}=a{\sqrt {{\sqrt {2}}-1}}\approx 0.64a}${\textstyle r_{c}=a{\sqrt {{\sqrt {2}}-1}}\approx 0.64a}.^[3]

Half-mass radius is ${\displaystyle r_{h}=\left({\frac {1}{0.5^{2/3}}}-1\right)^{-0.5}a\approx 1.3a.}${\displaystyle r_{h}=\left({\frac {1}{0.5^{2/3}}}-1\right)^{-0.5}a\approx 1.3a.}

Virial radius is ${\displaystyle r_{V}={\frac {16}{3\pi }}a\approx 1.7a}${\displaystyle r_{V}={\frac {16}{3\pi }}a\approx 1.7a}.

The 2D surface density is: $${\displaystyle \Sigma (R)=\int _{-\infty }^{\infty }\rho (r(z))dz=2\int _{0}^{\infty }{\frac {3a^{2}M_{0}dz}{4\pi (a^{2}+z^{2}+R^{2})^{5/2}}}={\frac {M_{0}a^{2}}{\pi (a^{2}+R^{2})^{2}}},}$${\displaystyle \Sigma (R)=\int _{-\infty }^{\infty }\rho (r(z))dz=2\int _{0}^{\infty }{\frac {3a^{2}M_{0}dz}{4\pi (a^{2}+z^{2}+R^{2})^{5/2}}}={\frac {M_{0}a^{2}}{\pi (a^{2}+R^{2})^{2}}},} and hence the 2D projected mass profile is: $${\displaystyle M(R)=2\pi \int _{0}^{R}\Sigma (R'),円R'dR'=M_{0}{\frac {R^{2}}{a^{2}+R^{2}}}.}$${\displaystyle M(R)=2\pi \int _{0}^{R}\Sigma (R'),円R'dR'=M_{0}{\frac {R^{2}}{a^{2}+R^{2}}}.}

In astronomy, it is convenient to define 2D half-mass radius which is the radius where the 2D projected mass profile is half of the total mass: ${\displaystyle M(R_{1/2})=M_{0}/2}${\displaystyle M(R_{1/2})=M_{0}/2}.

For the Plummer profile: ${\displaystyle R_{1/2}=a}${\displaystyle R_{1/2}=a}.

The escape velocity at any point is $${\displaystyle v_{\rm {esc}}(r)={\sqrt {-2\Phi (r)}}={\sqrt {12}},円\sigma (r),}$${\displaystyle v_{\rm {esc}}(r)={\sqrt {-2\Phi (r)}}={\sqrt {12}},円\sigma (r),}

For bound orbits, the radial turning points of the orbit is characterized by specific energy ${\textstyle E={\frac {1}{2}}v^{2}+\Phi (r)}${\textstyle E={\frac {1}{2}}v^{2}+\Phi (r)} and specific angular momentum ${\displaystyle L=|{\vec {r}}\times {\vec {v}}|}${\displaystyle L=|{\vec {r}}\times {\vec {v}}|} are given by the positive roots of the cubic equation $${\displaystyle R^{3}+{\frac {GM_{0}}{E}}R^{2}-\left({\frac {L^{2}}{2E}}+a^{2}\right)R-{\frac {GM_{0}a^{2}}{E}}=0,}$${\displaystyle R^{3}+{\frac {GM_{0}}{E}}R^{2}-\left({\frac {L^{2}}{2E}}+a^{2}\right)R-{\frac {GM_{0}a^{2}}{E}}=0,} where ${\displaystyle R={\sqrt {r^{2}+a^{2}}}}${\displaystyle R={\sqrt {r^{2}+a^{2}}}}, so that ${\displaystyle r={\sqrt {R^{2}-a^{2}}}}${\displaystyle r={\sqrt {R^{2}-a^{2}}}}. This equation has three real roots for ${\displaystyle R}${\displaystyle R}: two positive and one negative, given that ${\displaystyle L<L_{c}(E)}${\displaystyle L<L_{c}(E)}, where ${\displaystyle L_{c}(E)}${\displaystyle L_{c}(E)} is the specific angular momentum for a circular orbit for the same energy. Here ${\displaystyle L_{c}}${\displaystyle L_{c}} can be calculated from single real root of the discriminant of the cubic equation, which is itself another cubic equation $${\displaystyle {\underline {E}},円{\underline {L}}_{c}^{3}+\left(6{\underline {E}}^{2}{\underline {a}}^{2}+{\frac {1}{2}}\right){\underline {L}}_{c}^{2}+\left(12{\underline {E}}^{3}{\underline {a}}^{4}+20{\underline {E}}{\underline {a}}^{2}\right){\underline {L}}_{c}+\left(8{\underline {E}}^{4}{\underline {a}}^{6}-16{\underline {E}}^{2}{\underline {a}}^{4}+8{\underline {a}}^{2}\right)=0,}$${\displaystyle {\underline {E}},円{\underline {L}}_{c}^{3}+\left(6{\underline {E}}^{2}{\underline {a}}^{2}+{\frac {1}{2}}\right){\underline {L}}_{c}^{2}+\left(12{\underline {E}}^{3}{\underline {a}}^{4}+20{\underline {E}}{\underline {a}}^{2}\right){\underline {L}}_{c}+\left(8{\underline {E}}^{4}{\underline {a}}^{6}-16{\underline {E}}^{2}{\underline {a}}^{4}+8{\underline {a}}^{2}\right)=0,} where underlined parameters are dimensionless in Henon units defined as ${\displaystyle {\underline {E}}=Er_{V}/(GM_{0})}${\displaystyle {\underline {E}}=Er_{V}/(GM_{0})}, ${\displaystyle {\underline {L}}_{c}=L_{c}/{\sqrt {GMr_{V}}}}${\displaystyle {\underline {L}}_{c}=L_{c}/{\sqrt {GMr_{V}}}}, and ${\displaystyle {\underline {a}}=a/r_{V}=3\pi /16}${\displaystyle {\underline {a}}=a/r_{V}=3\pi /16}.

The Plummer model comes closest to representing the observed density profiles of star clusters ^{[citation needed ]}, although the rapid falloff of the density at large radii (${\displaystyle \rho \rightarrow r^{-5}}${\displaystyle \rho \rightarrow r^{-5}}) is not a good description of these systems.

The behavior of the density near the center does not match observations of elliptical galaxies, which typically exhibit a diverging central density.

The ease with which the Plummer sphere can be realized as a Monte-Carlo model has made it a favorite choice of N-body experimenters, in spite of the model's lack of realism.^[4]

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