# Plummer model

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Plummer_model
> Markdown URL: https://mediated.wiki/source/Plummer_model.md
> Source: https://en.wikipedia.org/wiki/Plummer_model
> Source revision: 1321609988
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Mathematical model in astronomical systems}}
The '''Plummer model''' or '''Plummer sphere''' is a density law that was first used by [H. C. Plummer](/source/Henry_Crozier_Keating_Plummer) to fit observations of [globular cluster](/source/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], ''[Mon. Not. R. Astron. Soc.](/source/Monthly_Notices_of_the_Royal_Astronomical_Society)'' '''71''', 460.</ref> It is now often used as [toy model](/source/toy_model) in [N-body simulation](/source/N-body_simulation)s of stellar systems.

== Description of the model ==
thumb|right|220px|The density law of a Plummer model

The Plummer 3-dimensional density profile is given by
<math display="block">\rho_P(r) = \frac{3M_0}{4\pi a^3} \left(1 + \frac{r^2}{a^2}\right)^{-{5}/{2}} = \frac{3M_0a^2}{4\pi (a^2 + r^2)^{{5}/{2}}},</math>
where <math>M_0</math> is the total mass of the cluster, and ''a'' is the '''Plummer radius''', a [scale parameter](/source/scale_parameter) that sets the size of the cluster core. The corresponding potential is
<math display="block">\Phi_P(r) = -\frac{G M_0}{\sqrt{r^2 + a^2}},</math>
where ''G'' is [Newton](/source/Isaac_Newton)'s [gravitational constant](/source/gravitational_constant). The [velocity dispersion](/source/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 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](/source/specific_energy).

== Properties ==

The mass enclosed within radius <math>r</math> is given by 
<math display="block">M(<r) = 4\pi\int_0^r r'^2 \rho_P(r') \,dr' = M_0 \frac{r^3}{(r^2 + a^2)^{3/2}}.</math>

Many other properties of the Plummer model are described in [Herwig Dejonghe](/source/Herwig_Dejonghe)'s comprehensive article.<ref>Dejonghe, H. (1987), [http://adsabs.harvard.edu/abs/1987MNRAS.224...13D A completely analytical family of anisotropic Plummer models]. ''[Mon. Not. R. Astron. Soc.](/source/Monthly_Notices_of_the_Royal_Astronomical_Society)'' '''224''', 13.</ref>

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} \approx 0.64 a</math>.<ref>{{cite OEIS|A154747|2=Decimal expansion of sqrt(sqrt(2) - 1), the radius vector of the point of bisection of the arc of the unit lemniscate (x^2 + y^2)^2 = x^2 - y^2 in the first quadrant.}}</ref>

[Half-mass radius](/source/Half-mass_radius) is <math>r_h = \left(\frac{1}{0.5^{2/3}} - 1\right)^{-0.5} a \approx 1.3 a.</math>

[Virial radius](/source/Virial_Theorem) 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>.

For the Plummer profile: <math>R_{1/2} = a</math>.

The escape velocity at any point is
<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](/source/specific_energy) <math display="inline">E = \frac{1}{2} v^2 + \Phi(r)</math> and [specific angular momentum](/source/specific_relative_angular_momentum) <math>L = |\vec{r} \times \vec{v}|</math> are given by the positive roots of the [cubic equation](/source/cubic_function)
<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](/source/circular_orbit) for the same energy. Here <math>L_c</math> can be calculated from single real root of the [discriminant of the cubic equation](/source/Cubic_function), which is itself another [cubic equation](/source/cubic_function)
<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 [Henon units](/source/N-body_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](/source/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.

The ease with which the Plummer sphere can be realized as a [Monte-Carlo model](/source/Monte_Carlo_method) has made it a favorite choice of [N-body experimenters](/source/N-body_simulation), in spite of the model's lack of realism.<ref>Aarseth, S. J., Henon, M. and Wielen, R. (1974), [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1974A%26A....37..183A A comparison of numerical methods for the study of star cluster dynamics.] ''[Astronomy and Astrophysics](/source/Astronomy_and_Astrophysics)'' '''37''' 183.</ref>

==References==
{{reflist}}

Category:Astrophysics
Category:Equations of astronomy

---
Adapted from the Wikipedia article [Plummer model](https://en.wikipedia.org/wiki/Plummer_model) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Plummer_model?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.