Triaxial Schwarzschild Models

I have created a flexible and efficient method to construct triaxial dynamical models of galaxies with a central black hole using Schwarzschild's (1979) orbital superposition approach. These computer models are fully general and can deal with realistic triaxial luminosity distributions, and two-dimensional stellar kinematic measurements (vdB et al. 2008). This is a non-trivial undertaking, as the parameter range to be explored is significantly larger than in axisymmetric geometry, and the internal dynamical structure is much more complicated as it includes four major orbit families, a host of minor families and chaotic orbits.

In these models, we use broadband photometry to construct the luminous mass models of the stars in each galaxy. Then, the gravitational potential is inferred from the combined luminous mass, black hole and dark halo. In this potential all the representative orbits are integrated numerically, while keeping track of the paths and orbital velocities of each orbit. We can then create an actual model of the galaxy by assigning each orbit an amount of mass, so that it recreates the total mass distribution, while fitting to simultaneously reproduce the observed stellar kinematics. In this way the stellar mass-to-light ratios (Cappellari et al. 2006, vdB et al. 2006), geometry & viewing angles (vdB & van de Ven 2009), black hole masses (vdB & de Zeeuw 2011), dark matter halos (Weijmans et al. 2009) and internal dynamical structure (van de Ven et al. 2008) can be recovered. These models have the advantage that they do not require any assumption for the orbital anisotropy of the galaxy. The ability to construct these models makes it possible to measure the dynamical structure of stellar systems, and opens the way to systematically explore their properties. Please email me if you are interested in using these models. The papers that use these models are grouped on my publication page. A selection is shown below:

Detailed Description of the Method

The triaxial Schwarzschild models are described in detail in my PhD thesis (2008, Leiden University) and have been published in Monthly Notices of the Royal Astronomical Society. There are four papers that describe the them in great detail.
  • Triaxial orbit based galaxy models with an application to the (apparent) decoupled core galaxy NGC 4365

    R. C. E. van den Bosch, G. van de Ven, E. K. Verolme, M. Cappellari, P. T. de Zeeuw

      We present a flexible and efficient method to construct triaxial dynamical models of galaxies with a central black hole, using Schwarzschild's orbital superposition approach. Our method is general and can deal with realistic luminosity distributions, which project to surface brightness distributions that may show position angle twists and ellipticity variations. The models are fit to measurements of the full line-of-sight velocity distribution (wherever available). We verify that our method is able to reproduce theoretical predictions of a three-integral triaxial Abel model. In the companion paper by van den Ven, de Zeeuw & van den Bosch (2008), we demonstrate that the method recovers the phase-space distribution function. We apply our method to two-dimensional observations of the E3 galaxy NGC 4365, obtained with the integral-field spectrograph sauron, and study its internal structure, showing that the observed kinematically decoupled core is not physically distinct from the main body and the inner region is close to oblate axisymmetric.

      Published in mnras, 2008, 385, 647

  • Recovery of the internal orbital structure of galaxies

    G. van de Ven, P. T. de Zeeuw & R. C. E. van den Bosch,

      We construct axisymmetric and triaxial galaxy models with a phase-space distribution function that depends on linear combinations of the three exact integrals of motion for a separable potential. These Abel models, first introduced by Dejonghe & Laurent and subsequently extended by Mathieu & Dejonghe, are the axisymmetric and triaxial generalizations of the well-known spherical Osipkov-Merritt models. We show that the density and higher order velocity moments, as well as the line-of-sight velocity distribution of these models can be calculated efficiently and that they capture much of the rich internal dynamics of early-type galaxies. We build a triaxial and oblate axisymmetric galaxy model with projected kinematics that mimic the two-dimensional kinematic observations that are obtained with integral-field spectrographs such as SAURON. We fit the simulated observations with axisymmetric and triaxial dynamical models constructed with our numerical implementation of Schwarzschild orbit-superposition method. We find that Schwarzschild method is able to recover the internal dynamics and three-integral distribution function of realistic models of early-type galaxies.

      Published in mnras, 2008, 385, 614

  • Recovering the intrinsic shape of Early-type galaxies

    Remco C. E. van den Bosch & Glenn van de Ven

      We investigate how well the intrinsic shape of early-type galaxies can be recovered when both photometric and two-dimensional stellar kinematic observations are available. We simulate these observations with galaxy models that are representative of observed oblate fast-rotator to triaxial slow-rotator early-type galaxies. By fitting realistic triaxial dynamical models to these simulated observations, we recover the intrinsic shape (and mass-to-light ratio), without making additional (ad hoc) assumptions on the orientation. For (near) axisymmetric galaxies, the dynamical modelling can strongly exclude triaxiality, but the regular kinematics do not further tighten the constraint on the intrinsic flattening significantly, so that the inclination is nearly unconstrained above the photometric lower limit even with two-dimensional stellar kinematics. Triaxial galaxies can have additional complexity in both the observed photometry and kinematics, such as twists and (central) kinematically decoupled components, which allows the intrinsic shape to be accurately recovered. For galaxies that are very round or show no significant rotation, recovery of the shape is degenerate, unless additional constraints such as from a thin disc are available.

      Published in mnras, 2009, 398, 1117

  • Estimating super massive black hole masses using triaxial Schwarzschild models

    Remco C. E. van den Bosch & P. Tim de Zeeuw

      Most of the super massive black hole mass (M) estimates based on stellar kinematics use the assumption that galaxies are axisymmetric oblate spheroids or spherical. Using fully general triaxial orbit based models we explore the effect of relaxing the axisymmetric assumption on the previously studied galaxies M32 and NGC 3379. We find that M32 can only be accurately modeled using an axisymmetric shape viewed nearly edge on and our black hole mass estimate is identical to previous studies. When the observed 5 degree kinematical twist is included in our model of NGC 3379, then the best shape is mildly triaxial and we find that our best-fitting black hole mass estimate doubles, with respect to the axisymmetric model. While this individual result is still consistent with the original estimate and the M-σ relation, it could have significant impact on the black hole demography as around a third of the most massive galaxies are triaxial.

      Published in mnras, 2011, 401, 1770

The code uses the quadratic programming from the Galahad package and HSL (2011), which is a collection of Fortran codes for large scale scientific computation.