[Paper Review] Schwarzschild modelling of elliptical galaxies and their black holes

Jens Thomas|arXiv (Cornell University)|Jul 21, 2010

Galaxies: Formation, Evolution, Phenomena3 citations

TL;DR

This paper applies the Schwarzschild orbit-superposition method to model 19 Coma cluster early-type galaxies, revealing that their dark matter halos are 13 times denser than those of spirals with similar stellar mass. The results indicate that ellipticals formed at a redshift of $ z_{\text{form}} \approx 2-3 $, roughly twice that of spirals, and that galaxies with younger stars than their halos show dissipative orbital structures, suggesting secondary star formation after main halo assembly.

ABSTRACT

This article describes the Schwarzschild orbit superposition method. It is the state-of-the-art dynamical modelling tool for early-type galaxies. Tests with analytic models show that masses and orbital anisotropies of not too face-on galaxies can be recovered with about 15 percent accuracy from typical observational data. Applying Schwarzschild models to a sample of Coma galaxies their dark matter halos were found to be 13 times denser than those of spirals with the same stellar mass. Since denser halos assembled earlier, this result indicates that the formation redshift 1+z of ellipticals is about two times higher than of spirals. Roughly half of the sample galaxies have halo assembly redshifts in agreement with their stellar-population ages. Galaxies where stars appear younger than the halos show strong phase-space density gradients in their orbital structure, indicative for dissipational evolution and possibly connected with secondary star-formation after the main halo assembly epoch. The importance of considering dark-matter in dynamical models aimed to measure black-hole masses is briefly discussed.

Motivation & Objective

• To measure dark matter content and orbital structure in early-type galaxies using dynamical modelling.
• To determine the formation redshift of elliptical galaxies by comparing dark matter halo densities with stellar population ages.
• To assess the impact of dark matter on black hole mass measurements in the central regions of early-type galaxies.
• To investigate whether phase-space density gradients in orbital distributions can distinguish between dissipative and collisionless formation histories.
• To test the reliability of Schwarzschild modelling in recovering masses and anisotropies from typical observational data.

Proposed method

• The Schwarzschild orbit-superposition method constructs phase-space distribution functions by superposing individual orbits in a given gravitational potential.
• A library of several thousand orbits is computed in a trial axisymmetric potential, with each orbit's light and kinematic properties stored.
• The final model is a linear combination of orbits, weighted to match observed surface brightness and line-of-sight velocity distributions.
• Orbits are classified as spheroidal ($ |\vartheta|_{\text{max}} > 70^\circ $) or disk-like ($ |z|_{\text{max}} < r_{\text{eff}}/4 $) based on their geometry relative to the equatorial plane.
• Phase-space density is computed per orbit and compared across galaxies to infer orbital structure and dynamical history.
• The method accounts for dark matter by including it in the gravitational potential during model fitting, ensuring accurate mass-to-light ratio estimates.

Experimental results

Research questions

• RQ1 What is the formation redshift of early-type galaxies, and how does it compare to that of spiral galaxies?
• RQ2 Can the orbital structure of stars in elliptical galaxies reveal evidence of dissipative evolution, such as secondary star formation?
• RQ3 How accurately can the Schwarzschild method recover black hole masses and dynamical masses when dark matter is neglected?
• RQ4 To what extent do dark matter halos in elliptical galaxies exceed those in spirals of similar stellar mass?
• RQ5 What is the relationship between the age of stellar populations and the assembly redshift of dark matter halos in early-type galaxies?

Key findings

• The Schwarzschild method recovers masses and orbital anisotropies with approximately 15% accuracy for not too face-on galaxies using typical observational data.
• Dark matter halos in Coma ellipticals are 13 times denser than those in spirals of the same stellar mass, indicating earlier formation.
• The formation redshift of ellipticals is estimated at $ z_{\text{form}} \approx 2-3 $, roughly twice that of spirals ($ z_{\text{form}} \approx 1 $).
• In about half the sample, the halo assembly redshift matches the stellar population age, suggesting consistent formation epochs.
• Galaxies where stars appear younger than the halo show strong phase-space density gradients on near-circular orbits, indicating dissipative evolution and possible secondary star formation.
• Neglecting dark matter in black hole mass models can lead to overestimation of the stellar mass-to-light ratio and, consequently, underestimation of the black hole mass—this effect is most significant in massive galaxies.