© CAMBRIDGE UNIVERSITY PRESS 2000 |

**22.1. Four approaches**

It may be convenient to distinguish four different ways of working on the construction of stellar dynamical models. It should be noted that in some cases the differences are more in the perception than in the actual methods and a given model may be reached from different directions.

**22.1.1. Search for interesting exact models**

One simple way to proceed is to collect all possible self-consistent
models. Usually here the emphasis is on the exactness or on the analytical
simplicity of the models
^{(1)}.
By producing a larger and larger set of
potential galaxy models, one may hope to find some that are able to give a
reasonable match to the observed profiles (see Chapter 4), others that
are best
suited for the numerical simulation of a specific dynamical mechanism
(e.g., see
Chapter 23), or models that are convenient to test special features,
such as the structure of the line-of-sight velocity profiles (see
section 22.1.3).

This "blind" search
operates between two extremes. In one limit, spherical symmetry, with
all the
desired classical integrals of the motion readily available, appears to be a
degenerate case. There is an infinite number of ways of specifying a
distribution
function and of constructing the related self-consistent model. Even
here we may
note that full self-consistency puts strong limitations to the set of
solutions
allowed by the Jeans theorem alone. In particular, much like for the
classification
of quantum states, the Jeans theorem would in principle allow any
*f* = *f* (*E*, *J*^{2},
*J*_{z}). In practice, if we require that the spherical
potential in
*E* be generated by the distribution function itself, then we have
to drop the use of
*J*_{z}, i.e. we should work with
*f* = *f* (*E*, *J*^{2}) (in principle, an odd
dependence on *J*_{z} would not break the symmetry of the
mass distribution; on the other hand,
the resulting internal streaming would correspond to the existence of a
preferred
axis, with no natural counterpart in the assumed symmetry of the
problem). Still,
within this framework we may construct models with a wide variety of
density and
pressure anisotropy profiles, and in many different ways we may match
the main
observed characteristics. Some notable models will be mentioned later in the
context of distribution function priority models.

At the other extreme, the general triaxial case looks like an almost
impossible
task, at least from the analytical point of view. If we anticipate the
possibility
of three good integrals of the motion (in order to allow for a fully
anisotropic pressure tensor), almost any combination
*f* = *f* (*E*, *I*_{2},
*I*_{3}) is going to lead to a
density distribution that is incompatible with the symmetry required for the
existence of *I*_{2}, *I*_{3}. Stäckel
potentials have three isolating integrals of
the motion (see previous chapter) and offer a very wide choice of
options (being
specified in term of three free functions). They might thus be
considered the
natural environment for the construction of self-consistent triaxial
models ^{(2)}.
Unfortunately, a discussion of Stäckel
type density-potential pairs has shown the undesired feature that the
associated density profile does not have sufficiently concentrated
central regions (with the
empirically suggested *r*^{-2} density behavior). The
application of a theorem
^{(3)}, which
guarantees the construction of a Stäckel potential with a desired
set of axial ratios starting from a specified density profile along the
*z*-axis, has shown that
realistic density profiles can be obtained only at the cost of introducing a
relatively massive disk
^{(4)}. The issue of
construction of triaxial models with central cusps is the focus of much
work currently in progress
^{(5)}.
We should also recall that Stäckel potentials have been proved to be
free of isophotal twisting
^{(6)}, and so they are
unable to reproduce the strongest empirical
argument in favor of triaxiality.

Some more general triaxial models can be constructed by numerical superposition of stellar orbits in a density priority approach, in line with studies mentioned in section 21.4.

**22.1.2. Density priority models**

In the construction of models giving priority to the density distribution one specifies in advance the potential-density pair and then tries to invert the integral relation

(22.1) |

One motivation at the basis of this point of view is very simple. The
quantity that is best constrained by the observations, under the
assumption of
a constant mass-to-light ratio, is the projected density profile. If we
add an
assumption on the three-dimensional geometry of the system, we thus have an
empirical determination of the volume density profile which appears on the
l.h.s. of the above equation (further discussion of this point in connection
with the empirical *R*^{1/4} law is postponed to
section 22.3.4 below). It is thus
natural to ask, for various geometrical configurations whether a (positive
definite) function of the integrals of the motion can be found
compatible with
the *imposed* density profile suggested by the
observations. Furthermore we
may ask whether, or under what conditions, such a solution is unique.

In the case of spherical symmetry, various options can be taken.
Since the density profile depends only on the radial coordinate, the
possibility of a unique inversion prescription is naturally expected
when the
distribution function (or the part of the function that is left free to
vary) depends on one integral only. For example, if we take *f* =
*f* (*E*), the solution is provided by an Abel inversion
^{(7)}
(cf. Eq. (14.15) in Chapter 14)

(22.2) |

In this inversion formula the density profile is assumed to be
monotonic, so that the radial coordinate can be eliminated to give
=
(). The function *f*
(*E*) resulting from the inversion should be checked
to be positive definite. The model thus produced is characterized by
isotropic pressure. Another possibility is to consider a dependence of
*f* on a combination *Q* = *Q*(*E*,
*J*^{2}) specified in advance, thus producing anisotropic
models according to some desired criteria. In this respect, the case
where *Q* is a linear combination of *E* and
*J*^{2} has often been used
especially to produce models with radially biased anisotropy
profiles ^{(8)},
with some difficulties appearing when tried for the construction of
tangentially biased anisotropic models
^{(9)}.
Clearly, many other options can be
considered and lead to several interesting models
^{(10)}. Some
separations with a rather intuitive basis can be adopted, focusing on a
*circularity function*
^{(11)} (which measures
the distance from the relation *J*_{circ}(*E*), given
for a few potentials in the previous chapter) or even on the
epicyclic energy for the case where one aims at tangentially biased
models ^{(12)}.

For the case of axial symmetry a very elegant generalization of the Abel
inversion has been devised
^{(13)} able to yield
*f* (*E*, *J*^{2}_{z}),
for a given analytical axisymmetric potential; the method is based on a
suitable contour integral in the complex plane. The most general
triaxial case appears to require the use of direct numerical
superposition of stellar orbits,
as briefly indicated at the end of the previous chapter
^{(14)}.

We mentioned above that one motivation of the density priority approach
is that of
imposing the quantity that is best constrained by the observations. This
viewpoint
may be generalized in procedures where all the available observational
constraints are imposed. In other words, models can be constructed
basically by deprojection
of the available photometric and spectroscopic profiles. In this
respect, one aspect that has recently attracted much interest is the
information contained in the line-of-sight velocity profiles
^{(15)}; this is reminiscent of
analogous interest in the 21 cm line profiles in radio observations of
spiral galaxies ^{(16)}.

In order to define the deprojection process, one may thus work with a
*mathematically* convenient basis of distribution functions and try to
determine empirically the combination leading to a best fit for all the
available data. A simple class of these techniques involves the use of
distributions that are polynomials
^{(17)} in *E* and
*J*.

One may even bypass the explicit use of distribution functions and refer to
numerical combinations of stellar orbits, with the additionl guidance of
"cost functions" or "entropy functions"
^{(18)}.
An even more "abstract", non-parametric method has been proposed, with the
important feature of ensuring that the derived distributions in phase
space be smooth ^{(19)}.

An alternative method resorts to the use of the moment "fluid"
equations (see Chapter 8). The modeling can be carried through
relatively easily in the axisymmetric case
^{(20)}. The moment
equations can be simplified considerably if
*f* = *f* (*E*, *J*_{z})
(without assuming any specific form for the underlying *f*), and become
(using cylindrical coordinates *R*, *z*)

(22.3) |

(22.4) |

The equations can be closed by making some
assumption on the pressure anisotropy profile (often taken to be
constant) and on the part of
<*v*^{2}_{}> that is associated with systematic
streaming motions. The photometric profiles taken at various position
angles are fitted to provide information on the three dimensional
structure of the galaxy and
the density-potential pair thus inserted in the fluid equations gives
the resulting kinematical profiles that are then compared with the
spectroscopic data. In practical cases, the method may show whether
reasonable fits can be provided with
constant mass-to-light ratios and whether significant pressure
anisotropies are
involved. It is basically by means of these tools that it has been
established ^{(21)}
that most ellipticals are probably not too far from
oblate systems with *f* = *f* (*E*,
*J*_{z}). It is to be stressed that there is no
guarantee that the solutions thus found admit a positive definite
supporting distribution function. In addition, even when the solution
selected by the data exists, it may require a rather artificial
distribution of stellar orbits.
In fact, we have already noted that descriptive methods of this kind tend to
postpone the discussion of their physical justification.

As noted earlier in this chapter, one should be aware that the data points sample only limited spatial regions with good accuracy and that the inversion process is generally unstable, so that one may often be led to unrealistic or physically implausible distributions of stellar orbits.

**22.1.4. Distribution function priority models**

From the physical point of view, the methods mentioned so far remain unsatisfactory. For a collisionless stellar system all the physics is contained in the distribution function in phase space and most methods mentioned earlier leave the resulting distribution function "out of control" and without critical examination. To some extent it is as if, in making models of stellar structure, we first assigned the density profile inside the star and then we asked whether there is an equation of state able to justify the imposed density distribution, limiting the physical discussion to a study of uniqueness or to checking that the pressure profile is positive definite.

The fourth approach, i.e. the line of work better developed in this
chapter, tries
to overcome this criticism by attacking the problem at its roots. The
idea is thus
to proceed through a *physically based choice* of distribution
function. This
approach is thus *predictive*, in the sense that it tries to look for
implications, both in terms of density profiles and of kinematical
profiles, that
should follow as a result of some physical scenarios. Current or future
data may *disprove* the arguments put forward in this modeling
process.

The main difficulty of this line of thinking is that the statistical mechanics of self-gravitating systems is poorly known and that there are very few empirical clues that we may take advantage of. Thus so far the most successful solutions are found for cases where the system is supposed to have relaxed significantly (see section 22.2) or for partially relaxed models of ellipticals where one exploits important clues offered by numerical simulations of collisionless collapse (see section 22.3). Note that in general the physical assumptions that offer a guidance in the choice of the relevant distribution function are unable to determine its form uniquely. This is also true for the choice of models described by Eq. (5) below, which represents only one available option within the physical scenario of isotropic quasi-Maxwellian truncated models.

The truncated "Maxwellian" distribution functions described in
section 22.2 have
found excellent applications to describe the structure and evolution of
globular clusters, where indeed relaxation processes occur on a
relatively fast
timescale. The models define a very interesting one-parameter equilibrium
sequence. In order to give a stronger physical justification to this and
other models invoked to describe weakly collisional systems, the
distribution
functions have been compared directly to self-similar solutions of the
Fokker-Planck
equation ^{(22)}.
The application of these
models to elliptical galaxies is less justified but it is often considered.

For the description of collisionless stellar systems, such as elliptical
galaxies, it would be important to produce, at least when the system is
not too far from spherical symmetry, a model or classes of models under
physical justification
with the capability of realistic luminosity and kinematical
profiles. Since one operates between the two extremes (spherical
degeneracy and
triaxial "impossibility") mentioned in
section 22.1.1, the case of
*quasi-spherical symmetry* is found to be simple enough to lead to some
workable solutions, and, on the other hand, being away from strict spherical
symmetry, to be
able to teach us something beyond the spherical degenerate case. The models
described in section 22.3 go in the
desired direction, but so far they
cannot be easily extended to cover cases where significant departures
from spherical symmetry are involved.

Interesting attempts at constructing non-spherical
models have been carried out, some focusing on the algorithm for the
numerical solution of the Poisson equation
^{(23)}
and others on the more interesting aspect of including
approximate integrals of the motion
^{(24)}.
These attempts, giving priority to the
distribution function, follow the most natural procedure of a physical
approach, as is generally done in plasma physics
^{(25)}.

^{1} See Dejonghe, H. (1986), *Phys. Rep.*,
**133**, 217
Back.

^{2} See discussion at the end of Chapter
21; Bishop, J.L. (1986), op. cit.; Statler, T.S. (1987), op. cit.
Back.

^{3} Kuzmin, G.G.
(1956), *Astr. Zh.*, **33**, 27
Back.

^{4} de Zeeuw, P.T., Peletier, R., Franx, M.
(1986), *Mon. Not. Roy. Astron. Soc.*, **221**, 1001
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^{5} See Merritt, D.
(1997), *Astrophys. J.*, **486**, 102;
Sridhar, S., Touma, J.
(1997), *Mon. Not. Roy. Astron. Soc.*, **292**, 657
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^{6} Franx, M.
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^{7} Eddington, A.S.
(1916), *Mon. Not. Roy. Astron. Soc.*, **76**, 572;
see also Tricomi, F.G.
(1985), **Integral inversions**, Dover, New York
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^{8} Osipkov, L.P.
(1979), *Pis'ma Astr. Zh.*, **5**, 77;
Merritt, D.
(1985), *Astron. J.*, **90**, 1027;
Merritt, D.
(1985), *Mon. Not. Roy. Astron. Soc.*, **214**, 25p
Back.

^{9} See Saha, P.
(1991), *Mon. Not. Roy. Astron. Soc.*, **248**, 464
Back.

^{10} Hénon, M.
(1973), *Astron. Astrophys.*, **24**, 229;
Cuddeford, P.
(1991), *Mon. Not. Roy. Astron. Soc.*, **253**, 414;
Louis, P.D.
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^{11} Gerhard, O.
(1991), *Mon. Not. Roy. Astron. Soc.*, **250**, 812
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^{12} In the context of tangentially biased
models, Bertin, G., Leeuwin, F., Pegoraro, F., Rubini, F.
(1997), *Astron. Astrophys.*, **321**, 703,
compare a density priority model construction with a distribution
function priority method
Back.

^{13} Hunter, C., Qian, E.
(1993), *Mon. Not. Roy. Astron. Soc.*, **262**, 401
Back.

^{14} See also Dejonghe, H., Laurent, D.
(1990), *Mon. Not. Roy. Astron. Soc.*, **252**, 606
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^{15} Franx, M., Illingworth, G.
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Bender, R.
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Rix, H.-W., White, S.D.M.
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Gerhard, O.
(1993), *Mon. Not. Roy. Astron. Soc.*, **265**, 213;
van der Marel, R., Franx, M.
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Kuijken, K., Merrifield, M.R.
(1993), *Mon. Not. Roy. Astron. Soc.*, **264**, 712.
The results of Winsall, M.L., Freeman, K.
(1993), *Astron. Astrophys.*, **268**, 443;
Bender, R., Saglia, R.P., Gerhard, O.
(1994), *Mon. Not. Roy. Astron. Soc.*, **269**, 785 on
galaxies such as NGC 4472 argue for small departures from gaussian
profiles and limited amounts of pressure anisotropy
Back.

^{16} For example, see the discussion of
radio emission from M 31 by van de Hulst, H.C., Raimond, E., van Woerden, H.
(1957), *Bull. Astron. Inst. Netherlands*, **14**, no. 480, 1
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^{17} Dejonghe, H.
(1989), *Astrophys. J.*, **343**, 113;
Dejonghe , H., de Zeeuw, P.T.
(1988), *Astrophys. J.*, **333**, 90;
see also Merritt, D., Saha, P.
(1993), *Astrophys. J.*, **409**, 75;
Gerhard, O.
(1991), *Mon. Not. Roy. Astron. Soc.*, **250**, 512;
Saha, P.
(1992), *Mon. Not. Roy. Astron. Soc.*, **254**, 132
Back.

^{18} Richstone, D.O., Tremaine, S.
(1984), *Astrophys. J.*, **286**, 27;
(1985), *Astrophys. J.*, **296**, 370
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^{19} Merritt, D., Tremblay, B.
(1994), *Astron. J.*, **108**, 514;
Merritt, D.
(1996), *Astron. J.*, **112**, 1085
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^{20} Binney, J.J., Davies, R.L., Illingworth,
G.D. (1990), *Astrophys. J.*, **361**, 78;
van der Marel, R.P., Binney, J.J., Davies, R.L.
(1990), *Mon. Not. Roy. Astron. Soc.*, **245**, 582;
van der Marel, R.P.
(1991), *Mon. Not. Roy. Astron. Soc.*, **253**,
710 and many following papers
Back.

^{21} See Franx, M., Illingworth, G.D., de
Zeeuw, P.T.
(1991), *Astrophys. J.*, **383**, 112
Back.

^{22} Spitzer, L., Härm, R.
(1958), *Astrophys. J.*, **127**, 544;
Hénon, M.
(1961), *Ann. d'Astrophys.*, **24**, 369;
King, I.R.,
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^{23} Prendergast, K.H., Tomer, E.
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Wilson, C.P.
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^{24} Petrou, M.
(1983), *Mon. Not. Roy. Astron. Soc.*, **202**, 1195
and 1209;
Lupton, R.H., Gunn, J. F.
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^{25} Harris, E.G.
(1962), *Nuovo Cim.*, **XXIII**, 115;
see section 9.1 in Part two
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