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.
Description of the model
The density law of a Plummer model
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
,
{\displaystyle \rho _{P}(r)={\frac {3M_{0}}{4\pi a^{3}}}\left(1+{\frac {r^{2}}{a^{2}}}\right)^{-{\frac {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 distribution function is
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
)
{\displaystyle E({\vec {x}},{\vec {v}})={\frac {1}{2}}v^{2}+\Phi _{P}(r)}
is the specific energy .
Properties
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
{\displaystyle r_{c}=a{\sqrt {{\sqrt {2}}-1}}\approx 0.64a}
.
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
)
{\displaystyle 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}
.
Applications
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.[ 3]
References
^ Plummer, H. C. (1911), On the problem of distribution in globular star clusters , Mon. Not. R. Astron. Soc. 71 , 460.
^ Dejonghe, H. (1987), A completely analytical family of anisotropic Plummer models . Mon. Not. R. Astron. Soc. 224 , 13.
^ Aarseth, S. J., Henon, M. and Wielen, R. (1974), A comparison of numerical methods for the study of star cluster dynamics. Astronomy and Astrophysics 37 183.