Plummer model: Difference between revisions

Revision as of 21:23, 26 January 2020

Latest revision as of 16:16, 30 June 2025

(11 intermediate revisions by 9 users not shown)

Line 1:

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 ==

Line 5:

The Plummer 3-dimensional density profile is given by

: <math display="block">\rho_P(r) = \frac{3M3M_0}{4\pi a^3} \left(1 + \frac{r^2}{a^2}\right)^{-\frac{5}/{2}},</math>

where ''M''<math>M_0</math> 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

: <math display="block">\Phi_P(r) = -\frac{G MM_0}{\sqrt{r^2 + a^2}},</math>

where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]]. The [[velocity dispersion]] is

: <math display="block">\sigma_P^2(r) = \frac{G MM_0}{6\sqrt{r^2 + a^2}}.</math>

The isotropic distribution function is reads

: <math>\rho_P(r) = \frac{3M}{4\pi a^3} \left(1 + \frac{r^2}{a^2}\right)^{-\frac{5}{2}},</math>

: <math display="block">f(\vec{x}, \vec{v}) = \frac{24\sqrt{2}}{7\pi^3} \frac{N a^2}{G^5 MM_0^54} (-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}) = \frac12frac{1}{2} v^2 + \Phi_P(r)</math> is the [[specific energy]].

where ''M'' 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

: <math>\Phi_P(r) = -\frac{G M}{\sqrt{r^2 + a^2}},</math>

where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]]. The velocity dispersion is

: <math>\sigma_P^2(r) = \frac{G M}{6\sqrt{r^2 + a^2}}.</math>

The distribution function is

: <math>f(\vec{x}, \vec{v}) = \frac{24\sqrt{2}}{7\pi^3} \frac{N a^2}{G^5 M^5} (-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>E(\vec{x}, \vec{v}) = \frac12 v^2 + \Phi_P(r)</math> is the [[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' = MM_0 \frac{r^3\over\left}{(r^2 + a^2\right)^{3/2}}.</math>.

Many other properties of the Plummer model are described in [[Herwig Dejonghe]]'s comprehensive paperarticle.<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>

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} \approx0approx 0.64a64 a</math>.

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

The radial turning points of an orbit characterized by [[specific energy]] <math>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]]

<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>.

:<math>R^3 + \frac{GM}{E}R^2 - \left(\frac{L^2}{2E} + a^2 \right) R - \frac{GMa^2}{E} = 0 </math>.

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

The escape velocity at any point is

:<math>\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>

<math display="block">v_{\rm esc}(r)=\sqrt{-2\Phi(r)}=\sqrt{12}\,\sigma(r) ,</math>

TheFor bound orbits, the radial turning points of anthe 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]]

where underlined parameters are dimensionless in Henon units defined as <math>\underline{E}=Er_V/(GM)</math>, <math>\underline{L}_c = L_c / \sqrt{G\, M\, r_V}</math>, and <math>\underline{a} = a / r_V = 3\pi/16</math>.

:<math display="block">R^3 + \frac{GMGM_0}{E} R^2 - \left(\frac{L^2}{2E} + a^2 \right) R - \frac{GMaGM_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} =Er_V E r_V / (GMG 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 ==

Line 56 ⟶ 57:

[[Category:Astrophysics]]

[[Category:Equations of astronomy]]