Made-to-measure dynamical particle models of elliptical galaxies

NMAGIC modelling:
Syer and Tremaine (1996) proposed a made-to-measure (M2M) algorithmto construct N-particle
systems from observational data and used it to generate a triaxial model from density observables.
The first practical application was made by Bissantz et al. (2004), who constructed a
dynamical model of the projected face-on density distribution of the Milky Way. So far, only
density constraints have been considered and all models have been evolved in predetermined
potentials.
In chapter 2, a χ2-made-to-measure (χ2M2M) algorithm was developed, extending earlier
ideas by Syer and Tremaine. An important component of the new method is the use of the
standard χ2 merit function at the heart of the algorithm, which allows to assess the quality of a
model fit directly. In addition, kinematic observables including higher order moments have been
incorporated. Hence, kinematic and density (or surface density) constraints can be used to tailor
particle models. The new χ2M2M method was implemented in a fast, parallel code, NMAGIC.
This code also incorporates an optional but fast potential solver, allowing it to recompute the
potential during a model fit and, in addition, to test the stability of the final particle model.
The NMAGIC implementation of the χ2M2M algorithm is highly efficient, with a sequential
fraction of only ∼ 1%.
The geometric flexibility and performance of NMAGIC was illustrated with a number of
tests using spherical, oblate and triaxial target models. In the spherical experiments, the correct
isotropic target model was recovered, independently of the adopted initial conditions. The initial
model with density closer to the density of the final model had smaller final deviations from the
target observables, and a narrower distribution of weights.
The oblate tests showed that a large phase-space gradient can be recovered if present, and
illustrated the advantage of integral field data over slit data for constraining the model.
The triaxial experiments demonstrated that it is possible to start from a spherical model and
converge to a triaxial target, and illustrated NMAGIC’s ability in constructing models for triaxial
elliptical galaxies with which nature confronts us. A second triaxial experiment, in which a
slowly rotating model was used as a target for a non-rotating model, revealed that the residuals
in the first order kinematic moment are correlated. This gives a signature of tumbling which,
at least for this triaxial system, allows to distinguish between internal stellar streaming and
pattern rotation within Re, provided a full velocity field is available. However, a more complete
study of this problem is needed to firm up this result. This experiment also demonstrates the
usefulness of the χ2M2M algorithm for modelling mock (rather than real) galaxies in order to
learn about their dynamics. Such an experiment would not have been practical with standard
N-body simulations.
In chapter 3, I extended NMAGIC to account for seeing effects and proposed and implemented
an efficient method to estimate the mass-to-light ratio �. Tests of this scheme using
isotropic rotator input models have shown that the method recovers � within a few percent both
for self-consistent and dark matter dominated target galaxies. In addition, a likelihood scheme
was implemented, by which discrete velocity measurements can be taken into account without
binning them beforehand.
The modelling of NGC 4697 and NGC 3379, presented in chapters 3 and 4, respectively,
showed that the χ2M2M/NMAGIC particle method is a very promising modelling technique. In
fact, it has already gone further than the Schwarzschild method in that the gravitational potential
of the stars has been allowed to vary in the modelling, the mass-to-light ratio has been adapted
on the fly and the stability of the models has been checked.
Compared to the Schwarzschild method, the main advantages of the χ2M2M algorithm
as implemented in NMAGIC are that no symmetry restrictions have to be made and that no
complicated procedure for orbit sampling is needed. Another advantage is that the gravitational
potential can be evolved self-consistently, which further allows to test the stability of a model
after the correction phase. Every χ2M2M model corresponds to a computation of an orbit
library in the Schwarzschild method. In problems where the same orbit library can be reused,
Schwarzschild’s method is more efficient.
The present implementation of NMAGIC is optimized for modeling nearly spherical systems.
This is mainly due to the potential solver and the density observables (Alm), both based
on a spherical harmonics decomposition of the density distribution.
In the next two sections below, I will discuss the astronomical results from the dynamical
modelling of NGC 4697 and NGC 3379.
Astronomical results:
NGC 4697:
Chapter 3 presented a dynamical study of the E4 galaxy NGC 4697 using surface brightness
measurements and a combined kinematic data set, which runs from the center of the galaxy to
about 4.5 effective radii. The kinematic data set consists of long slit spectroscopic data and
discrete PNe velocity measurements.
Even though NMAGIC does not require any symmetry assumptions for the modelling, I have
forced the method to generate axisymmetric particle models for NGC 4697. Both self-consistent
models without dark matter, and models following a sequence of circular speed curves with
increasing dark halo contributions have been investigated. The PN data have been used both
binned on two different spatial grids, as well as with the new likelihood scheme, to make sure
that the results are not biased by the way the PNe data are incorporated.
The main result is that models both with and without dark matter are consistent with all
the data. These models fit all kinematic data with χ2/N < 1, both in potentials with only
luminous matter and in potentials including sufficiently massive halos to generate nearly flat
circular rotation curves. The massive dark halo models tend to fit the data slightly better in
the sense of lower χ2/N, for both the slit kinematics and the PN data, but these variations are
small and not yet statistically significant. To exclude models without dark matter would require
PN velocities at even larger radii than currently available, out to an estimated ≃ 6Re from the
center.
These models differ from earlier studies performed byM´endez et al. (2001) in the sense that
we generate axisymmetric models instead of spherical ones and that our models are flexible with
regard to anisotropy. The best-fitting models are slightly radially anisotropic, with β ≃ 0.3 at the
center, increasing to β ≃ 0.5 at∼
> 2Re. This is consistent with the value given by Dekel et al.
(2005) obtained from merger simulations carried out within the �CDM cosmology framework.
NGC 3379:
In chapter 4, the dynamical modelling of the intermediate luminosity E1 galaxy NGC 3379 was
presented. The models were constructed using photometric and kinematic observations for this
galaxy. Again, a combined kinematic data set was used, consisting of long slit spectroscopic
data with SAURON integral field absorption line kinematics and PN velocity measurements
with the PN.S instrument from Douglas et al. (2007). The combined data set runs from the
center of NGC 3379 to about 7 effective radii. This is the first time that integral field SAURON
kinematic data of a real galaxy has been incorporated in NMAGIC.
Both self-consistent models without dark matter, and models following a sequence of circular
speed curves with increasing dark halo contributions have been investigated.
Several dynamical models, with and without DM, produce a viable fit to all the data. For
assumed spherical symmetry the data is consistent with near-isotropic models dominated by
stellar mass and with radially anisotropic models in moderately massive halos with DMfractions
≃ 60 percentage at 7 Re. In addition, a series of of oblate models have been constructed which
essentially confirm the spherical results.
The main conclusion is that the steeply declining PNe velocity dispersion profile is consistent
with a variety of DM halos. It is difficult to constrain the potential in this galaxy with the
present data. This is mainly due to the well known mass anisotropy degeneracy, which masks
the DM distribution by preferentially populating radial orbits. Hence the possibility remains
that NGC 3379 has the kind of dark matter halo that is consistent with the current �CDM
paradigm.