Mon. Not. R. Astron. Soc. 296, 569–578 (1998)
Degenerate equilibrium states of collisionless stellar systems
P.-H. Chavanis and J. Sommeria
Ecole Normale Supérieure de Lyon, 69364 Lyon Cedex 07, France
Accepted 1997 December 2. Received 1997 June 13; in original form 1996 October 7
A BSTR ACT
We study spherically symmetrical equilibrium states of collisionless stellar systems
confined to a spherical box. These equilibrium states correspond to the statistics
introduced by Lynden-Bell in his theory of ‘violent relaxation’, and are described by
a Fermi–Dirac distribution function. We compute the corresponding equilibrium
diagram and show that a global entropy maximum exists for any accessible control
parameter. This equilibrium state shows a pronounced separation between a
degenerate core and a halo. We therefore check that degeneracy is able to stop the
gravitational collapse (of a collisionless system), and we propose a simple model for
the ‘core–halo’ structure. We also discuss the relevance of our study for real galaxies
or other astrophysical systems such as massive neutrinos.
Key words: celestial mechanics, stellar dynamics – galaxies: general.
1
INTRODUCTION
It is remarkable that elliptical galaxies look very much alike.
This is revealed in the empirical R 1/4 law of de Vaucouleurs
(1948) characterizing their surface brightness. It has long
been recognized that a Maxwell–Boltzmann distribution
function of the form f (e)1e Ðbe, where e is the stellar
energy, provides good fits to observations (except for
energies close to the escape energy). At first sight, we could
believe that this Maxwell–Boltzmann distribution function
is the natural outcome of a collisional relaxation. However,
the collisional time tcoll for ellipticals is larger than the age of
the Universe by many orders of magnitude, so that collisions
(i.e. close encounters) are quite irrelevant for such systems.
Moreover, a collisional relaxation would lead to a segregation by mass, which is not observed. As a result, the quasiuniversal regularity of elliptical galaxies does not
correspond to a collisional equilibrium.
However, the rapid fluctuations of the gravitational field
in the early stage of a galaxy can lead to a ‘quasi-equilistcoll . In this
brium’ after only a few dynamical times tD s
process, known as violent relaxation, the stars exchange
energy via the time-varying gravitational field, which provides a mechanism analogous to that of relaxation in a gas.
The importance of this form of relaxation was first stressed
by Hénon (1964) and King (1966) and is quite likely to
justify the apparent regularity of collisionless systems. Lynden-Bell (1967) introduced the concept of phase mixing and
showed the relevance of a statistical description. He predicted a Fermi–Dirac distribution function when the initial
© 1998 RAS
condition consists of a patch with uniform phase density,
and a complicated sum of Fermi–Dirac distributions for
non-uniform initial states (here, the degeneracy results not
from quantum mechanics but from the conservation of
phase density). In the non-degenerate limit, these reduce to
Maxwellians.
The case of a self-gravitating gas described by a Maxwell–
Boltzmann distribution has been studied extensively and
may exhibit a ‘gravothermal catastrophe’ (Lynden-Bell &
Wood 1968): for some values of the control parameters,
there is no equilibrium; the system takes a ‘core–halo’ structure and the core becomes denser and denser with no limit
(leading presumably to a black hole singularity). In the case
of collisionless systems, there is also a gravitational collapse,
but when degeneracy is taken into account Lynden-Bell &
Wood (1968) argue that the ‘exclusion principle’ saturates
this gravitational collapse and that a true equilibrium is
finally reached. However, they did not actually compute this
equilibrium state explicitly. Such degenerate equilibrium
states have been observed in many numerical simulations of
violent relaxation (Hohl & Campbell 1968; Janin 1971; Fujiwara 1983; Melott 1983). For real stellar systems, degeneracy (if there is any) is harder to prove because it may
affect only the very central core, which is poorly resolved. It
may be of greater importance in the case of dark matter
made of massive neutrinos (Kull, Treumann & Böhringer
1996).
The object of the present paper is to compute degenerate
equilibrium states and investigate their stability. As isothermal spheres have infinite mass and energy, it is neces-
570
P.-H. Chavanis and J. Sommeria
sary to introduce a cut-off to remedy this difficulty. This
cut-off is justified physically by the realization that the relaxation is necessarily incomplete (i.e. restricted to some region
of space). In this context, a truncated model was introduced
by Michie (1963) and King (1966). More recently, Hjorth &
Madsen (1993) have improved this model with a star escape
scenario and obtained an excellent fit with NGC 3379. In
this paper, we consider a system confined to a spherical box
as a simple and classical way of dealing with incomplete
relaxation. It can be shown that the physical results are not
very sensitive to the model (see, e.g., Katz 1980). We find
that a global maximum of entropy exists for any accessible
value of the control parameters. For high energies this
global maximum is similar to the ordinary (Maxwellian)
isothermal sphere, for moderate energies the centre
becomes partially degenerate and for low energies we
obtain a ‘core–halo’ structure consisting of a completely
degenerate nucleus surrounded by a maxwellian atmosphere. Other solutions exist and are either local entropy
maxima or unstable critical points. We compute the general
bifurcation diagram and propose a simple model for the
‘core–halo’ structure that is convenient for investigating
asymptotic limits.
plete mixing (which would result in a uniform distribution
function) is forbidden by the conservation of energy. As a
result, the equilibrium state will be the most probable outcome compatible with the energy constraint. If only this
limitation were imposed, the statistical mechanics would
give a Maxwellian distribution function. The collisionless
character of the system under consideration imposes an
additional constraint, however, which is the conservation of
phase density. The macroscopic distribution function f̄ can
then only decrease by internal mixing (because vacuum has
been incorporated into the system) and must satisfy f̄Rn0
everywhere. This constraint, specific to collisionless systems,
acts like an ‘exclusion principle’ similar to Pauli’s exclusion
principle in quantum mechanics but arising here for different reasons. It is therefore expected that collisionless
stellar systems will achieve a Fermi–Dirac equilibrium distribution (when examined at a coarse resolution).
To be more precise, we introduce the probability density
r (r, v) of finding the level of phase density n0 in a small
neighbourhood of the position r, v in phase-space (the probability of the level 0 is the complementary 1Ðr). The locally
averaged distribution function is expressed in terms of this
probability density by
f̄ (r, v)\r (r, v) n0 ,
(2.1)
2 E Q U I L I B R I U M S TAT E S O F
COLLISIONLESS STELL A R SYSTEMS
and the associated (macroscopic) gravitational potential
satisfies the Newton–Poisson equation
2.1
DF\4pGn̄,
Lynden-Bell’s statistical theory
For a wide variety of galaxies, the collisions (i.e. close
encounters) between stars are negligible (the corresponding
relaxation time tcoll exceeds the age of the Universe) and the
galaxy dynamics is well modelled by the Vlasov equation.
This equation simply states that, in the absence of collision,
the distribution function f is conserved by the flow in phase
space [the distribution function is defined such that f (r, v,
t) d3 r d3 v represents the total mass of stars at location r with
velocity v at time t].
Let us consider an initial condition consisting of a patch
of uniform distribution function f0 (r, v)\n0 outside of
which f0 (r, v)\0. As time goes on, this patch is stirred by the
violent fluctuations of the gravitational field, but its total
hypervolume g (n0) is conserved, as a property of the Vlasov
equation (this results from the conservation of the phase
density and the incompressibility of the ‘flow’ in phasespace). We could try to follow the dynamics of its contour
(this was attempted in 1+1 dimensions with water-bag
models) but the problem becomes progressively more and
more intricate and the use of statistical methods is preferable. Indeed, while it is clear that the exact (fine-grained)
distribution function f never achieves equilibrium but builds
up a very finely striated structure in phase-space, we expect
that the coarse-grained distribution function f̄ (averaged
over these striations) will reach a certain equilibrium. This
equilibrium state was investigated by Lynden-Bell (1967) in
the context of ‘violent relaxation’ and justified mathematically by Michel & Robert (1994) with the concept of Young
measures (providing some hypothesis of ergodicity).
At a macroscopic level, the violent relaxation can be
interpreted as a ‘mixing process’: the initial patch f\n0
tends to mix with the surrounding fluid f\0, but the com-
(2.2)
where
h
n̄ (r)= f̄ (r, v) d3 v
(2.3)
is the local mass density. It is then possible to express the
conserved quantities of the Vlasov equation as integrals of
the macroscopic fields. For a uniform initial state, these
conserved quantities are simply the total energy,
h
E\
1
2
f̄ v 2 d3 r d3 v+
1
2
h
f̄ F d3 r d3 v,
(2.4)
and the total mass of the system,
h
M\ f̄ d3 r d3 v.
(2.5)
The linear impulse and the angular momentum are also
conserved by the Vlasov equation. The constraint on the
linear momentum can always be suppressed by taking the
origin at the centre of mass of the system. Moreover, in this
paper we restrict our analysis to non-rotating spherical
galaxies for which the angular momentum is zero (as is the
corresponding Lagrange multiplier). As a result, within our
previous assumptions, the equilibrium state depends only
on three control parameters, which are the mass M, the
energy E and the maximum accessible value n0 of the distribution function.
Following Lynden-Bell (1967), the most likely distribution to be reached at equilibrium is obtained by maximizing
the mixing entropy,
© 1998 RAS, MNRAS 296, 569–578
Degenerate equilibrium states 571
h
S\Ð [r ln r+(1Ðr) ln (1Ðr)] d3 r d3 v,
(2.6)
under the constraints (2.4) and (2.5) of energy and mass. To
that purpose, we introduce Lagrange multipliers and write
the variational problem in the form
dSÐ
b
n0
dEÐadM\0,
(2.7)
where b is an inverse temperature and a is a ‘chemical
potential’. The resulting equilibrium distribution function is
a Gibbs state of the form
n0
f̄ (r, v)\
,
1+e b (eÐeF)
(2.8)
which belongs to Fermi–Dirac statistics. In this expression,
e\v 2/2+F is the individual energy of the stars (per unit of
mass) and eF=Ðan0 /b is traditionally called the Fermi
energy.
The distribution function (2.8) depends on the gravitational potential F, which in turn depends on the distribution
function through the Newton–Poisson equation (2.2). The
procedure to break this circularity is to solve the differential
equation
DF\4pG
h
n0
1+e
b (v 2/2+FÐeF)
d3 v
(2.9)
with F1ÐGM/r at infinity, and substitute the solution back
into formula (2.8) in order to obtain the explicit value of
f̄.
2.2
Limits of strong and weak degeneracy
There are two special limits in which the distribution function (2.8) can be simplified. As the gravitational potential F
increases with distance (F/m\ÐdF/drs0), the distribution function will rapidly satisfy f̄s
sn0 and we can then
make the approximation
f̄3n0 e Ðb (eÐeF)
(2.10)
1
corresponding to Maxwell–Boltzmann statistics. In that
case, the system is said to be ‘non-degenerate’ or ‘diluted’.
This approximation is expected to be valid in most part of
the galaxy, except possibly at the very centre. Indeed, as F
decreases towards the centre, the distribution function
increases and it may become necessary to take into account
the ‘degeneracy’ limiting its rise. One case of great interest
is that in which the central potential F0 is very negative. In
that case, the centre is ‘completely degenerate’ and the
distribution function is almost uniform, with f̄3n0 (the maximum value it can reach). Of course, this is true only for
stars with velocity vs
sZ2 (eF ÐF); stars with velocity
va
aZ2 (eF ÐF) are, in comparison, very scarce and we can
take f̄30. Thus, in this ‘completely degenerate’ limit, the
1
Although this is only an approximate distribution function in the
case of collisionless systems, the Maxwell–Boltzmann distribution
(without the bar on f ) is the true equilibrium state of a collisional
system (for stars of equal mass).
© 1998 RAS, MNRAS 296, 569–578
distribution function can be approximated by a step function
f̄3n0 H (eÐeF),
(2.11)
where H (x)\1 if xs0 and H (x)\0 if xa0 (H is the Heaviside function). This distribution function has been widely
studied in the context of white dwarf stars (see, e.g., Shapiro
& Teukolsky 1983).
2.3
Existence of equilibrium states
The previous analysis gives a well-defined procedure with
which to compute and classify the equilibrium states of
collisionless stellar systems (if they exist!). The gravitational
field is obtained by solving the Newton–Poisson equation
(2.2) with the Fermi–Dirac distribution function (2.8) (or its
local approximations 2.10 and 2.11). The solution depends
on the Lagrange multipliers b and eF , which must be related
to the constraints of energy E and mass M using equations
(2.4) and (2.5). Finally, we have to make sure that this
solution is a true entropy maximum by investigating the
second-order variations of the entropy.
In fact, it is well known that no equilibrium can exist in a
spatially infinite domain; the density can spread indefinitely
while conserving its energy and increasing its entropy. More
precisely, there is no critical point of entropy dS\0 in the
whole space: the solution of equation (2.9) has an infinite
mass, in contradiction to the initial assumptions. However,
this is not a serious problem because in practice, as a result
of kinetic effects, relaxation occurs only in a subdomain of
space and statistical theory must be applied only in this
subdomain. One way of taking into account this incomplete
relaxation is to use truncated models like the King model
(1966) or more refined ones (Hjorth & Madsen 1991).
Another possibility is to work in a spherical box of radius R
with stars bouncing elastically against the wall. Although
not very realistic, this model is very convenient in order to
go further into theoretical work. Moreover, it can give good
insight into the central structure of galaxies, which may not
be influenced too much by what happens at the periphery
(since n̄h0 at large distances).
This problem has been tackled by several authors (Antonov 1962; Lynden-Bell & Wood 1968; Padmanabhan 1990)
in the context of collisional systems described by Maxwell–
Boltzmann statistics. It is possible to prove the following
results (see Figs 1 and 2, solid line).
(i) There is no global maximum for entropy.
(ii) There is not even a critical point of entropy if ÐER/
GM 2 a0.335.
(iii) A local entropy maximum exists if ÐER/
GM 2 s0.335 and corresponds to a density contrast
R\n (0)/n (R)s709.2
Conclusions (i) and (ii) have been called ‘gravothermal
catastrophe’ or ‘Antonov instability’. In that case, the system takes a ‘core–halo’ structure and can always increase its
entropy (without changing energy) by making its centre
2
In the case where ER/GM 2 s0.335, other solutions satisfying
dS\0 exist. These solutions spiral towards the singular sphere
n11/r 2 of coordinate (1/4, 2) and have a density contrast Ra709.
It can be shown that they are unstable.
572
P.-H. Chavanis and J. Sommeria
Figure 1. Diagram of equilibrium states for collisionless stellar systems described by the Fermi–Dirac statistics (equation 2.8). The radius of
the box is fixed and the (normalized) energy is varied on the abscissa. The dot–dashed curve corresponds to a (normalized) phase level m\103
(we have also represented the non-degenerate case with a solid line for comparison). The degeneracy makes it possible to exceed the
Antonov critical value L c\0.335. There are solutions for LsL max36.418 (see equation 5.1), and these solutions are global entropy maxima.
When LhL max , the system has the same structure as a white dwarf star.
Figure 2. Same as Fig. 1, but for higher values of density levels m\105 (dashed line) and m\107 (dotted line). The curves continue to the right
until L max\138 and 2979 respectively (see formula 5.1), at which point GMb/R diverges (as in Fig. 1). In the range L (m)RLRL c , we find
*
several solutions for the same value of the energy. Their structure and stability are discussed in the text.
denser and denser. Lynden-Bell & Wood (1968) and Lynden-Bell & Lynden-Bell (1977) have shown this phenomenon is related to the negative heat capacity of
self-gravitating systems: by losing heat, the system automatically grows hotter and evolves away from equilibrium.
This famous instability can possibly initiate the formation of
black holes at the centre of galaxies (see, e.g., Lynden-Bell
& Eggleton 1980, Heggie & Stevenson 1988).
These stability results would also apply to collisionless
systems if the ‘non-degenerate’ approximation (2.10) were
valid everywhere in the galaxy. However, because gravity
tends to create systems with very high density, this approximation will fail in the central region of the galaxy and the
degeneracy will saturate the gravitational collapse. In fact,
we can prove (Robert, in preparation) that when degeneracy is taken into account, there exists a global entropy
© 1998 RAS, MNRAS 296, 569–578
Degenerate equilibrium states 573
maximum for any accessible value of the control parameters
E, M and box radius R.
3 D E G E N E R AT E I S O T H E R M A L S T E L L A R
S Y S T E M S : F O R M U L AT I O N O F T H E
PROBLEM
In this section, we solve the mean field equation (2.9)
numerically and relate the Lagrange multipliers b, eM to the
conserved quantities E, M (and the box radius R). Using the
method of Katz (1978), we then deduce from the bifurcation
diagram which solutions are stable and unstable.
If we assume spherical symmetry [in fact Antonov (1962)
argues that only spherically symmetrical states can correspond to entropy maxima of non-rotating systems], the differential equation to be solved is
1 d
2
r dr
2 3
r2
dF
dr
\4pG
h
n0
+l
0
1+e
g [v 2/2+F (r)ÐeF]
4pv 2 dv,
(3.1)
with F (R)\ÐGM/R at the wall. To specify the boundary
conditions completely, we must also discuss the behaviour
of F as rh0. For non-degenerate systems, a singular solution F; Ð2 ln r exists and has a finite mass (when confined
to a box): this is the so-called ‘singular isothermal sphere’
(see, e.g., Binney & Tremaine 1987) with density profile
n;1/r 2. In the case of degenerate systems, a logarithmic
singularity does not satisfy the differential equation (3.1)
and a power-law singularity F1Ð1/r a yields a\4.3 This
solution therefore has an infinite mass and must be rejected.
We cannot rigorously exclude the possibility of another kind
of singularity, but this is unlikely, and we shall assume here
that the gravitational potential remains finite at the origin
and set F (0)=F0 . We also introduce the notations
c (r)=b [F (r)ÐF0] and k=e b (F0 ÐeF). With these notations,
*
the local mass density (corresponding to the integral on the
right-hand side of equation 3.1) takes the form
4p Z2
b 3/2
n0 I1/2 [k e c* (r)],
(3.2)
where
In (t)\
h
+l
x
n
1+t e x
0
dx
(3.3)
is the Fermi integral of order n. When th+l we have
I1/2 (t)1Zp/2t (corresponding to the non-degenerate case
2
2.10) and when th0 we have I1/2 (t)13 (Ðln t)3/2 (corresponding to the limit 2.11 of complete degeneracy).
Let us also introduce the scaled radius
2
r\
3
16p 2 Z2Gn0
b
1/2
1/2
r.
(3.4)
r dr
2
Because FhÐl at the centre, we can use the approximation
(2.11) for the distribution function. The right-hand side of equation
(3.1) reduces to 1(ÐF)3/2 and we obtain a classical Lane–Emden
equation, like that for white dwarf stars.
© 1998 RAS, MNRAS 296, 569–578
2 3
r2
dc
rsa,
\I1/2 [k e c (r)],
dr
(3.5)
c (0)\cp (0)\0,
and its solution ck (r) depends on a unique parameter k.
This parameter k as well as the normalized box radius a can
be related to the mass M and the energy E of the system
(and the box radius R) by applying the constraints (2.4) and
(2.5).
In the case of spherically symmetrical systems, the mass
constraint (2.4) can be expressed equivalently as
dr
(R)\
GM
R2
,
(3.6)
which is a well-known consequence of the Gauss theorem.
With our notation, this becomes
bGM
acpk (a)\
R
.
(3.7)
Now, the inverse temperature b can be related to a (and the
control parameters) with the aid of equation (3.4) applied at
r\R. Introducing the ‘degeneracy parameter’ 4
m=n0 Z512p 4 G 3 MR 3,
(3.8)
the mass constraint takes the form
a 5 cpk (a)\m 2.
(3.9)
The second constraint is given by the conservation of
energy. After some rearrangements, the kinetic energy can
be written as
REc
a7
GM
m4
\
2
h
a
I3/2 [k e ck (r)] q 2 dr,
(3.10)
0
where I3/2 (t) is a Fermi integral defined by equation (3.3).
The potential energy can be calculated using the Virial
theorem. As the system is confined to a box, with stars
bouncing elastically at the wall, we must take the pressure
into account and use the complete expression
Ep\3pVÐ2Ec ,
(3.11)
where V\ pR is the total volume and p\ q f̄!r\R v d3 v is
the pressure of the stars against the wall. After some manipulations, the potential energy can be written as
4
3
REp
2 a 10
GM
3 m
4
3
d
1
dF
rsR,
n̄ (r)\
The value of r at the edge r\R deserves a name; we call it
a. Then r\ar/R and we set c (r)=c (r)\c (Rq/a). For
*
*
r\0, we have c (0)\0 by construction. We also have
wp (0)\0, because the gravitational field must vanish at the
centre of a spherically symmetrical system. With these new
variables, the differential equation (3.1) becomes
\
2
4
1
3
3
I3/2 [k e ck (a)]Ð2
REc
GM 2
.
2
(3.12)
The typical value of the distribution function at equilibrium scales
as , f .1M/L3 V 3, where L1R is a typical length and V1(GM/R)1/2
is a typical velocity. Therefore m1n0 /, f . can be considered as the
ratio of the initial value of the distribution function to its typical
equilibrium value.
P.-H. Chavanis and J. Sommeria
574
Finally, the total energy is obtained by simply summing
equations (3.10) and (3.12). This yields
L=Ð
a7
\
GM 2 m 4
ER
h
a
I3/2 [k e ck (r)] q 2 drÐ
0
2 a 10
3 m4
I3/2 [k e ck (a)].
(3.13)
As a result, for given control parameters L and m we can
solve the system of equations (3.9) and (3.13) to obtain the
values of k and a (possibly not unique) and determine the
corresponding equilibrium state. As the Fermi–Dirac distribution (2.8) only cancels the first-order variations of the
entropy, it is important to investigate the stability of the
solutions in relation to the second-order variations. If
several stable solutions subsist, we must compare their
entropy
7 GMb
GMb
\Ð L
+c (a)+
M
3
R
R
S
+ln kÐ
2a 6
9m 2
I3/2 [k e c (a)],
(3.14)
S3Ð
h
f̄
ln
n0
f̄
n0
d3 r d3 v,
(4.4)
n̄
2
3/2
(2pT )
e Ðv /(2T ).
(4.5)
After slight manipulations, we can express the entropy as a
function of the mass M of the nucleus and the temperature
*
T of the envelope by the formula
&
3
n0 S\(MÐM ) ln (2pT )Ðln(MÐM )
* 2
*
'
3
+ln (VÐV )+ln n0 + .
*
2
It will appear in the following section that, for low energies,
the system tends spontaneously to form a very dense
nucleus (completely degenerate) surrounded by a Maxwellian atmosphere. By shrinking, the nucleus releases an enormous amount of energy and heats the envelope. The
envelope therefore behaves like an ordinary gas (of positive
energy) maintained by the walls of the box, and its density is
approximately uniform. Our model thus consists of a completely degenerate nucleus (core) of mass M surrounded
*
by an atmosphere (halo) of constant density n̄\(MÐM )/
*
Vatm and temperature T. The mass of the nucleus is not
arbitrary but will be obtained by maximizing the entropy
(considered as a function of M ) at fixed total energy and
*
total mass.
It is well known from the theory of white dwarf stars that
the mass M of a completely degenerate system, i.e. a poly*
trope with index 3/2 (modelling the nucleus), is a decreasing
function of its radius R (see, e.g., Shapiro & Teukolsky
*
1983). More precisely, we have the relation
(4.1)
0
where x35.97Å10Ð3 is a constant. The total energy of the
nucleus can be calculated using the Virial theorem (neglecting the pressure of the atmosphere) and is simply
3 GM 2
*.
E \Ð
*
7 R
*
(4.3)
which we will prove to be excellent. On the other hand,
because the atmosphere is very diluted, its entropy (2.6)
reduces to
f̄\
4 A S I M P L E M O D E L F O R T H E ‘ C O R E–
H A LO’ STRUCTUR E
M R3 \ 2 3,
* * n G
3
Eatm3 (MÐM ) T,
*
2
where f̄ is the Maxwellian:
obtained from equations (2.6) and (2.8), in order to determine the global maximum. Before solving this problem in
the general case, we shall first investigate an approximation
that will help us to interpret the general diagrams.
x
Finally, the nucleus has no contribution to the entropy
because a completely degenerate system is unmixed
(S \0).
*
Concerning the envelope, we shall assume that it is very
hot (because of the central collapse) so that its energy is
dominated by the kinetic term. We shall therefore make the
approximation
(4.2)
(4.6)
As the temperature is determined by the energy constraint
2
3 n 2/3
3
0 G
E=E +Eatm\Ð
M 7/3 + (MÐM ) T,
1/3
*
*
*
7 x
2
(4.7)
the entropy is in fact a function of M alone. The mass of the
*
nucleus at equilibrium is then obtained by maximizing S
versus M . To have a simple solution, we shall make the
*
(licit) assumption that the radius of the nucleus is very small
compared with the box radius, so that VÐV 3V in formula
*
(4.6). In that case, the mass of the nucleus that maximizes
S (M ) is determined implicitly by the equation
*
G 2 n 2/3
M 4/3
3
0
*
ln (MÐM )Ð ln T+ 1/3
*
2
T
x
3
\ln n0 +ln VÐ1+ ln (2p),
2
(4.8)
where T depends on M through equation (4.7).
*
For given control parameters, the left-hand side of equation (4.8) is a function of M that we call F (M ). As the
*
*
temperature T is positive, the energy constraint (4.7)
implies that
2 3
7E
M EM0= Ð
*
3
3/7
x 1/7
G 6/7 n 2/7
0
,
(4.9)
© 1998 RAS, MNRAS 296, 569–578
Degenerate equilibrium states 575
where M0 is the mass of a completely degenerate system
with energy E. Therefore, the mass of the nucleus must lie in
the range M0RM RM. When M hM, F (M )hÐl and
*
*
*
when M hM0 , Th0 and F (M )h+l. More precisely,
*
*
F (M ) is a strictly decreasing function of M . Its intersec*
*
tion
with
the
horizontal
line
of
ordinate
Fs=ln n0 +ln VÐ1+3/2Åln(2p) determines the (unique)
solution of equation (4.8), i.e. the mass of the nucleus.
There are two cases worth considering.
(i) The box radius is fixed but the energy EhÐl. In that
case M0 h+l, which shows that the system tends to concentrate all its mass in the nucleus. This is possible as long
as
EEEmin=Ð
2
3 n 2/3
0 G
7
x 1/3
M 7/3,
(4.10)
where Emin (minimum accessible energy) is the energy of a
completely degenerate system with mass M. When E\Emin ,
all the mass and energy are in the nucleus and the atmosphere has been swallowed. In that case, the system has the
same structure as a white dwarf star.
(ii) The energy is fixed but the box radius Rhl. In that
case Fsh+l so that M hM0 and Th0: all the energy is in
*
the nucleus and the remaining mass MÐM0 is diluted in the
envelope. If we now take the limit of ‘weak degeneracy’,
4/7
n0h+l, we find that M 11/n 2/7
0 h0, R 11/n 0 h0 and
*
*
E hE. The mass and the radius of the nucleus tend to zero,
*
is infinite: in the limit of
but its typical density M /R 3 1n 10/7
0
* *
a very large box and a very low degeneracy, the equilibrium
state is a gas of uniform density M/V with a central singularity.
5 DESCR IPTION OF THE GENER A L
B I F U R C AT I O N D I A G R A M
5.1
Fixed box radius
We now come back to the general problem. We first assume
that the system is confined to a box of given radius R, and we
progressively decrease its energy from positive to negative
values. We shall represent the equilibrium temperature
bGM/R as a function of the energy L\ÐRE/(GM 2) for
different values of the phase level m\n0 (512p 4 G 3 MR 3)1/2
(each quantity being normalized by parameters involving M
and R, which do not change in the process). For specified m
and for each value of L ranging from Ðl to
3
m 2/3
L max (m)=
7 (512p 4 x)1/3
(5.1)
(corresponding to formula 4.10), we can solve in principle
the system of equations (3.9) and (3.13) to obtain the corresponding values of a and k and then compute bGM/R, using
equation (3.7). However, to avoid solving this rather complicated system we have varied the parameter k instead of L.
For each value of k, we have solved numerically the differential equation (3.5) until the mass constraint (3.9) was
satisfied, thus determining the value of a. The corresponding values of L and GMb/R followed from equations (3.13)
and (3.7). The curves of Figs 1 and 2 are therefore parametrized by k ranging from +l (corresponding to
© 1998 RAS, MNRAS 296, 569–578
LhÐl) to 0 + (corresponding to L max). The quantity m
determines the importance of the degeneracy (the strength
of the constraint f̄Rn0). We have considered three
cases corresponding to m\103, 105 and 107.
For m\103, there is only one solution for each value of
the energy (see Fig. 1). For positive energies (negative L),
the density tends to be uniform in the whole domain
because, in that case, the system is confined by the walls
rather than by self-gravity. For slightly negative energies,
the density is a smooth decreasing function of radius. The
density contrast R\n̄ (0)/n̄ (R) is still very weak (R19 for
L\0.05) and the system does not feel the exclusion principle: in this region, the curve bÐL coincides with that of a
non-degenerate system (solid line). For more negative
energies, the centre of the system becomes denser and partially degenerate, which makes possible to exceed Antonov’s
critical values L c (this is impossible for non-degenerate systems). The density profile is similar to the previous one, but
with a stronger density contrast (R11800 for L\0.45). If
we keep decreasing the energy, the centre becomes completely degenerate and concentrates more and more mass
(R13Å108 for L\5.94). When LhL max (m), all the mass
and energy are in the nucleus (and Th0; see Fig. 1) in
agreement with the simple model of Section 4. In that limit,
the system has the same structure as a white dwarf star.
For higher values of m (corresponding to a weaker degeneracy), the curves in Fig. 2 follow the non-degenerate spiral
for longer (i.e. to higher density contrasts), then proceed
backwards along the upper branch until they finally turn
over at L (m) and come back to the right towards L max (m).
*
In the limit mh+l, L (m) and L max (m) are rejected to
*
infinity (L minhÐl and L maxh+l). We now have several
solutions for each single value of the energy in the range
L (m)sLsL c .
*
(i) The solution on the upper branch (point A) is everywhere non-degenerate with a smooth density profile (see
curve A in Fig. 3).
(ii) The solution on the lower branch (point C) shows a
very strong separation between a completely degenerate
nucleus and a non-degenerate atmosphere. The nucleus is
very dense [n̄ (0)/,n.33Å1010, where ,n.\M/V is a typical density] and very small (R /R33Å10Ð4). It contains
*
about 15 per cent of the mass and concentrates a very
negative energy E /E3800. As a result of this gravitational
*
collapse, the atmosphere acquires a very positive energy and
its density becomes almost uniform. All these results are in
very good agreement with the simple model of Section 4
(yielding M /M\0.154, R /R\2.6Å10Ð4, E /E\805).
*
*
*
(iii) The solution on the intermediate branch (point B) is
a little more complex (see curve B in Fig. 3). It also contains
a completely degenerate nucleus, but this central nucleus
concentrates less mass (M /M15Å10Ð3) and less energy
*
(E /E10.2). This nucleus is surrounded by a non-degener*
ate gas consisting of several parts: for R RrR(M /M)1/3 R,
*
*
the atmosphere is sustained by the gravity of the nucleus
and its density decreases as
n̄\n̄ exp
*
& 2 3'
GM
1
T
r
*
Ð
1
R
.
(5.2)
*
There is first a large cusp of length 1GM /T, then the
*
576
P.-H. Chavanis and J. Sommeria
Figure 3. Density profiles of points A, B and C corresponding to a degeneracy parameter m\107 and an energy L\0.05. Point A (local
entropy maximum) is non-degenerate with a smooth density profile. Point C (global entropy maximum) has a ‘core–halo’ structure with a
degenerate nucleus surrounded by a non-degenerate atmosphere. The nucleus is very dense and concentrates much mass. Point B (unstable)
is similar to point A, with an embryonic nucleus concentrating little mass. This ‘germ’ can initiate a gravitational collapse leading from point
A to point C.
density becomes almost uniform with value n̄\n̄ e ÐGM* /TR*,
*
which is responsible for the ‘plateau’ in Fig. 3. This plateau
finishes at the point where self-gravity cannot be ignored
any more: this happens when the mass contained in the
region of uniform density becomes comparable to the mass
M of the central object. Assuming n̄1M/V on the plateau,
*
we obtain a rough estimate of its length by L1(M /M)1/3 R.
*
As the mass of the central object is small, the plateau connecting the nucleus to the self-gravitating atmosphere does
not extend very far from the centre (this is not very clear in
Fig. 3 because the use of logarithmic coordinates gives the
central region a disproportionate importance). In fact, solutions A and B are very similar (they have almost the same
temperature) except at their very centre, where solution B
develops an embryonic nucleus of high density (but weak
mass).
The stability of the solutions can be deduced from Fig. 2
using the method of Katz (1978) for series of equilibria. The
parameter conjugate to the entropy with respect to energy is
the inverse temperature b\(qS/qE)M, R . Then, if we plot b
as a function of ÐE (at fixed mass and radius) as in Fig. 2,
we have the following results.
(i) A change of stability can occur only at a limit point,
where E is an extremum (db/dE infinite).
(ii) A mode of stability is gained when the curve rotates
anticlockwise, but lost otherwise.
Now, we know that for E sufficiently large the solutions are
stable because, in this limit, self-gravity is negligible and the
system behaves like an ordinary gas. From point (i) we
conclude that the whole upper branch (point A) is stable. As
the curve spirals inward, more and more modes of stability
are lost. However, a mode of stability is recovered each time
we turn anticlockwise. From this very powerful result, we
deduce that the lower branch (point C) is stable whereas the
intermediate branch (point B) is unstable. Now, by comparing the entropy (equation 3.14) of points A (S/M\16.2) and
C (S/M\19.8), we find that the lower branch has more
entropy than the upper branch.
As a result, point A is only a local entropy maximum. By
contrast, point C is a global entropy maximum, but it is not
necessarily reached by the system. To go from point A to
point C, we have to overcome an entropic barrier represented by the unstable solution B. As mentioned previously,
this solution is similar to A but with a small nucleus at its
centre. This embryonic nucleus can be interpreted as a
‘germ’ in the language of phase transitions: starting from the
local maximum A, if we could create a nucleus of type B we
would initiate a gravitational collapse and the nucleus
would capture more and more stars until the global maximum C is reached. We may suspect, however, that the
entropic barrier is too hard to cross so that in general the
system prefers the local maximum on the upper branch.
Nevertheless, when we approach the critical Antonov value
L c , the upper branch disappears: in this case, the system will
inexorably undergo gravitational collapse and fall on to the
global maximum (point D). This global maximum clearly
shows a ‘core–halo’ structure, so that we have a physical
picture (in terms of equilibrium states) of this self-organization. If we keep decreasing energy (point E), more and
more mass is concentrated into the nucleus until the atmosphere has been completely swallowed at L\L max .
5.2
Fixed energy
We now assume that the energy of the system is fixed, and
we slowly increase the radius of the box. We want to plot the
inverse temperature GMb/R as a function of the box radius
L\ÐER/(GM 2) for different values of n0 in order to compare it with Figs 1 and 2. As m depends on R, we must find
© 1998 RAS, MNRAS 296, 569–578
Degenerate equilibrium states 577
Figure 4. Diagram of equilibrium states for collisionless stellar systems described by the Fermi–Dirac statistics (equation 2.8). The energy
is fixed and the (normalized) radius of the box is varied on the abscissa. The dashed curve corresponds to a (normalized) phase level v\10 Ð12
(we have also represented the non-degenerate case with a solid line for comparison).
another normalization for n0 , involving only parameters like
E or M which are held fixed in the process. Let us take
v=
L3
(ÐE)3
.
\
m 2 512p 4 G 6 M 7 n 20
(5.3)
For any value of L and v, we can solve the system of equations (3.9) and (3.13) and determine the inverse temperature bGM/R from equation (3.7). An easier way to proceed
is to parametrize the curve by k as indicated previously. The
results are reported in Fig. 4 (assuming Es0). This curve is
rather similar to that of Fig. 2 (and the same discussion
applies) except for Lh0 and Lh+l.
The case Lh0 is obtained by shrinking the box so much
that very high densities are obtained and degeneracy comes
into play. This is responsible for the discrepancy with Fig. 2.
However, this limit Rh0 is very artificial.
The case Lh+l is obtained by expanding the box to
infinity. As L increases, more and more (negative) energy is
concentrated in the nucleus, in agreement with the simple
model of Section 4. In the limit Lh+l, all the energy is in
the nucleus of mass
M
23
7
*\
M
3
3/7
(512p 4 xv)1/7,
(5.4)
corresponding to formula (4.9), and the rest of the mass is
diluted in an atmosphere of uniform density.
6
DISCUS SION A ND CONCLUSION
We have calculated statistical equilibrium states of collisionless stellar systems corresponding to the statistics introduced by Lynden-Bell (1967). These equilibrium states are
described by a Fermi–Dirac distribution function where
© 1998 RAS, MNRAS 296, 569–578
degeneracy results from the conservation of phase density.
A similar equilibrium exists for two-dimensional Euler flows
(this analogy is studied in Chavanis, Sommeria & Robert
1996) and has been proposed to explain the persistence of
vortices such as the Great Red Spot of Jupiter (Sommeria et
al. 1991a). In both cases, the degeneracy is not an exotic
phenomenon but has been vindicated by many numerical
simulations (Hohl & Campbell 1968, Janin 1971, Fujiwara
1983 and Melott 1983 for the Vlasov equation; Sommeria,
Staquet & Robert 1991b and Robert & Rosier 1997 for the
Euler equation) and fluid laboratory experiments (Denoix,
Sommeria & Thess 1993). In the case of stellar systems, the
degeneracy is able to stop the collisionless gravitational collapse and the final product is a structure consisting of a
‘core’ and a ‘halo’. For low energies, the core is completely
degenerate and very massive whereas the halo is more diffuse and non-degenerate. Our model shares similar properties with the model of isothermal spheres with short
distance cut-off (described by a Van der Waals equation of
state) studied by Padmanabhan (1990). Both models show
very clearly the tendency of gravitational systems to form a
central nucleus surrounded by a halo of stars.
There is a wide belief that degeneracy is not relevant in
the case of galaxies and that the Maxwell–Boltzmann distribution (2.10) should be used instead of equation (2.8).
Clearly, the answer depends on the specification of the
initial condition which, in practice, is not well known (the
details of galaxy formation are still in dispute). Lynden-Bell
(1967) assumes that the phase-space density, nstar , is equal to
the phase-space density of the gas immediately prior to star
formation, ngas (a quantity which can be derived from the
Jeans instability criterion), and concludes that galaxies are
non-degenerate. However, this estimate is criticized by
sngas and that degenMadsen (1987), who argues that nstar s
eracy must be taken into account. Shu (1978) also considers
578
P.-H. Chavanis and J. Sommeria
the question of degeneracy, but from another point of view.
His argument is that when degeneracy occurs, two-body
encounters cannot be neglected any more, so that the Vlasov equation (and consequently the degeneracy) breaks
down. His argument is not very convincing because it does
not rely explicitly on the initial condition. It is, however,
clear that collisions will little by little erase the degeneracy,
but this may occur on (very) different time-scales. In fact, in
the course of its history, a galaxy may first achieve a collisionless equilibrium with central degeneracy, until collisional effects become important and drive the system to the
ordinary Maxwell–Boltzmann equilibrium state (or cause
the gravothermal catastrophe).
We may wonder whether observations can reveal the
presence of partially degenerate nuclei at the centres of
galaxies. It seems difficult to detect them directly, because
the centre is poorly resolved; however, we may have
evidence of their presence indirectly from stability arguments. Indeed, it is well known (see Section 2.3) that a
stellar system imprisoned in a box of radius R and described
by a Maxwell–Boltzmann distribution function can be
stable only if the density contrast R\n (0)/n (R) between
the centre and the periphery is less than 709. Of course, the
box is not realistic and we would like another condition of
stability that could be directly compared with observations.
For instance, we may compute the corresponding surface
brightness contrast I (0)/I (Re) between the centre and the
effective radius Re (the radius of the isophote containing
half of the total luminosity). With an assumption of constant
mass-to-light ratio, the surface brightness contrast is equal
to the surface density contrast (projected along a line of
sight). Then, using the box model, we can easily show that a
Maxwell–Boltzmann equilibrium state can be stable only
for I (0)/I (Re)s11.7. Now, in the case of NGC 3379, the
brightness contrast is I (0)/I (Re)3130a11.7 (see Hjorth &
Madsen 1993), well above the limit of stability. As a result,
this elliptical galaxy cannot be in a pure Maxwell–Boltzmann equilibrium state. In contrast, such a high brightness
is compatible with degenerate equilibrium states. Therefore, degeneracy may play a role in real galaxies. Our study
could be also relevant for massive neutrinos in dark matter
models (Madsen 1987; Shu 1987; Kull et al. 1996).
ACKNOWLEDGMENTS
We thank R. Robert and P. Y. Longaretti for very interesting discussions.
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