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 ρ P ( r ) = 3 M 0 4 π a 3 ( 1 + r 2 a 2 ) − 5 / 2 = 3 M 0 a 2 4 π ( 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 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 Φ P ( r ) = − G M 0 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 σ P 2 ( r ) = G M 0 6 r 2 + a 2 . {\displaystyle \sigma _{P}^{2}(r)={\frac {GM_{0}}{6{\sqrt {r^{2}+a^{2}}}}}.}

The isotropic distribution function reads f ( x → , v → ) = 24 2 7 π 3 a 2 G 5 M 0 4 ( − E ( x → , 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 E < 0 {\displaystyle E<0} , and f ( x → , v → ) = 0 {\displaystyle f({\vec {x}},{\vec {v}})=0} otherwise, where E ( x → , v → ) = 1 2 v 2 + Φ 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 r {\displaystyle r} is given by M ( < r ) = 4 π ∫ 0 r r ′ 2 ρ P ( r ′ ) d r ′ = M 0 r 3 ( r 2 + a 2 ) 3 / 2 . {\displaystyle M(<r)=4\pi \int _{0}^{r}r'^{2}\rho _{P}(r')\,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 r c {\displaystyle r_{c}} , where the surface density drops to half its central value, is at r c = a 2 − 1 ≈ 0.64 a {\textstyle r_{c}=a{\sqrt {{\sqrt {2}}-1}}\approx 0.64a} .^[3]

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

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

The 2D surface density is: Σ ( R ) = ∫ − ∞ ∞ ρ ( r ( z ) ) d z = 2 ∫ 0 ∞ 3 a 2 M 0 d z 4 π ( a 2 + z 2 + R 2 ) 5 / 2 = M 0 a 2 π ( 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: M ( R ) = 2 π ∫ 0 R Σ ( R ′ ) R ′ d R ′ = M 0 R 2 a 2 + R 2 . {\displaystyle M(R)=2\pi \int _{0}^{R}\Sigma (R')\,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: M ( R 1 / 2 ) = M 0 / 2 {\displaystyle M(R_{1/2})=M_{0}/2} .

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

The escape velocity at any point is v e s c ( r ) = − 2 Φ ( r ) = 12 σ ( r ) , {\displaystyle v_{\rm {esc}}(r)={\sqrt {-2\Phi (r)}}={\sqrt {12}}\,\sigma (r),}

For bound orbits, the radial turning points of the orbit is characterized by specific energy E = 1 2 v 2 + Φ ( r ) {\textstyle E={\frac {1}{2}}v^{2}+\Phi (r)} and specific angular momentum L = | r → × v → | {\displaystyle L=|{\vec {r}}\times {\vec {v}}|} are given by the positive roots of the cubic equation R 3 + G M 0 E R 2 − ( L 2 2 E + a 2 ) R − G M 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 R = r 2 + a 2 {\displaystyle R={\sqrt {r^{2}+a^{2}}}} , so that r = R 2 − a 2 {\displaystyle r={\sqrt {R^{2}-a^{2}}}} . This equation has three real roots for R {\displaystyle R} : two positive and one negative, given that L < L c ( E ) {\displaystyle L<L_{c}(E)} , where L c ( E ) {\displaystyle L_{c}(E)} is the specific angular momentum for a circular orbit for the same energy. Here 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 E _ L _ c 3 + ( 6 E _ 2 a _ 2 + 1 2 ) L _ c 2 + ( 12 E _ 3 a _ 4 + 20 E _ a _ 2 ) L _ c + ( 8 E _ 4 a _ 6 − 16 E _ 2 a _ 4 + 8 a _ 2 ) = 0 , {\displaystyle {\underline {E}}\,{\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 E _ = E r V / ( G M 0 ) {\displaystyle {\underline {E}}=Er_{V}/(GM_{0})} , L _ c = L c / G M r V {\displaystyle {\underline {L}}_{c}=L_{c}/{\sqrt {GMr_{V}}}} , and a _ = a / r V = 3 π / 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 ( ρ → 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]