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The '''Plummer model''' or '''Plummer sphere''' is a density law that was first used by [[Henry Crozier Keating Plummer|H. C. Plummer]] to fit observations of [[globular cluster]]s.<ref>Plummer, H. C. (1911), [http://adsabs.harvard.edu/abs/1911MNRAS..71..460P On the problem of distribution in globular star clusters], ''[[Monthly Notices of the Royal Astronomical Society|Mon. Not. R. Astron. Soc.]]'' '''71''', 460.</ref> It is now often used as [[toy model]] in [[N-body simulation]]s of stellar systems.
== Description of the model ==
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The Plummer 3-dimensional density profile is given by
where
where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]]. The [[velocity dispersion]] is▼
<math display="block">\sigma_P^2(r) = \frac{G M_0}{6\sqrt{r^2 + a^2}}.</math>
The isotropic distribution function reads
▲:<math>\rho_P(r) = \bigg(\frac{3M}{4\pi a^3}\bigg)\bigg(1+\frac{r^2}{a^2}\bigg)^{-\frac{5}{2}}\,,</math>
<math display="block">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},</math>
if <math>E < 0</math>, and <math>f(\vec{x}, \vec{v}) = 0</math> otherwise, where <math display="inline">E(\vec{x}, \vec{v}) = \frac{1}{2} v^2 + \Phi_P(r)</math> is the [[specific energy]].
== Properties ==▼
▲where ''M'' is the total mass of the cluster, and ''a'' is the '''Plummer radius''', a scale parameter which sets the size of the cluster core. The corresponding potential is
The mass enclosed within radius <math>r</math> is given by ▼
▲:<math> \Phi_P(r) = -\frac{G M}{\sqrt{r^2+a^2}}\,,</math>
Many other properties of the Plummer model are described in [[Herwig Dejonghe]]'s comprehensive
▲where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]].
Core radius <math>r_c</math>, where the surface density drops to half its central value, is at <math display="inline">r_c = a \sqrt{\sqrt{2} - 1} \
▲== Properties ==
[[Half-mass radius]] is <math>r_h = \left(\frac{1}{0.5^{2/3}} - 1\right)^{-0.5} a \approx 1.3 a.</math>▼
▲The mass enclosed within radius <math>r</math> is given by
▲:<math>M(<r) = 4\pi\int_0^r r^2 \rho_P(r) dr = M{r^3\over\left(r^2+a^2\right)^{3/2}}</math>.
[[Virial Theorem#Galaxies and cosmology (virial mass and radius)|Virial radius]] is <math>r_V = \frac{16}{3 \pi} a \approx 1.7 a</math>.▼
The 2D surface density is:
<math display="block"> \Sigma(R) = \int_{-\infty}^{\infty}\rho(r(z))dz=2\int_{0}^{\infty}\frac{3a^2M_0dz}{4\pi(a^2+z^2+R^2)^{5/2}} = \frac{M_0a^2}{\pi(a^2+R^2)^2},</math>
and hence the 2D projected mass profile is:
<math display="block">M(R)=2\pi\int_{0}^{R}\Sigma(R')\, R'dR'=M_0\frac{R^2}{a^2+R^2}.</math>
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: <math>M(R_{1/2}) = M_0/2</math>.
▲Many other properties of the Plummer model are described in [[Herwig Dejonghe]]'s comprehensive paper.<ref>Dejonghe, H. (1987), [http://adsabs.harvard.edu/abs/1987MNRAS.224...13D A completely analytical family of anisotropic Plummer models]. ''[[Monthly Notices of the Royal Astronomical Society|Mon. Not. R. Astron. Soc.]]'' '''224''', 13</ref>
For the Plummer profile: <math>R_{1/2} = a</math>.
▲Core radius <math>r_c</math>, where the surface density drops to half its central value, is at <math>r_c =a\sqrt{\sqrt{2}-1}\approx0.64a</math>.
The escape velocity at any point is
▲[[Half-mass radius]] is <math>r_h \approx 1.3 a</math>
<math display="block">v_{\rm esc}(r)=\sqrt{-2\Phi(r)}=\sqrt{12}\,\sigma(r) ,</math>
For bound orbits, the radial turning points of the orbit is characterized by [[specific energy]] <math display="inline">E = \frac{1}{2} v^2 + \Phi(r)</math> and [[specific relative angular momentum|specific angular momentum]] <math>L = |\vec{r} \times \vec{v}|</math> are given by the positive roots of the [[cubic function|cubic equation]]
▲[[Virial Theorem#Galaxies and cosmology (virial mass and radius)|Virial radius]] is <math>r_V = \frac{16}{3 \pi} a \approx 1.7 a</math>
<math display="block">R^3 + \frac{GM_0}{E} R^2 - \left(\frac{L^2}{2E} + a^2\right) R - \frac{GM_0a^2}{E} = 0,</math>
where <math>R = \sqrt{r^2 + a^2}</math>, so that <math>r = \sqrt{R^2 - a^2}</math>. This equation has three real roots for <math>R</math>: two positive and one negative, given that <math>L < L_c(E)</math>, where <math>L_c(E)</math> is the specific angular momentum for a [[circular orbit]] for the same energy. Here <math>L_c</math> can be calculated from single real root of the [[Cubic function #The discriminant|discriminant of the cubic equation]], which is itself another [[cubic function|cubic equation]]
<math display="block">\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,</math>
where underlined parameters are dimensionless in [[N-body units|Henon units]] defined as <math>\underline{E} = E r_V / (G M_0)</math>, <math>\underline{L}_c = L_c / \sqrt{G M r_V}</math>, and <math>\underline{a} = a / r_V = 3 \pi/16</math>.
== Applications ==
The Plummer model comes closest to representing the observed density profiles of [[star clusters]]{{citation needed|date=April 2018}}, although the rapid falloff of the density at large radii (<math>\rho\rightarrow r^{-5}</math>) 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.
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[[Category:Astrophysics]]
[[Category:Equations of astronomy]]
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