14. Equilibria of elliptical galaxies and dark matter halos¶
Just like for spherical systems and disk galaxies, steady-state configurations of elliptical galaxies and (non-spherical) dark-matter halos are of high interest. Indeed, in many ways, understanding equilibrium states and performing equilibrium dynamical modeling is more important for elliptical galaxies than it is for disks, because we can learn much about the dynamical structure and mass distribution of disk galaxies (including the existence of dark matter as discussed in Chapter 8 !) from the kinematics of closed, circular gas orbits. Gas or any sort of disk component is rare or subdominant in elliptical galaxies (e.g., Rix & White 1990), so we cannot simply observe tracers on circular orbits and directly constrain the mass profile using the relation between circular velocity and enclosed mass. Modeling spheroidally or ellipsoidally distributed tracers with equilibrium modeling is therefore one of the only ways to constrain the orbital structure, mass profiles, and dark matter content of elliptical galaxies (the other main method being gravitational lensing)
But equilibrium modeling of elliptical galaxies is significantly more complicated than in the case of spherical mass distributions or disks, where the existence of multiple integrals of motion (especially in the spherical case where there are four) and the ordered, close-to-circular nature of orbits (for disks) makes it possible to analyze realistic systems with analytical tools. While similar analytical results can be obtained for spheroidal or even triaxial systems—for example, one can work out the Jeans equations in spheroidal or ellipsoidal coordinates, or write down distribution functions for oblate and prolate models, or triaxial models of Staeckel form—these results are far less useful, because they require so many assumptions as to not be applicable to realistic galaxies. (However, they can be very useful for testing the more general, numerical tools used to model elliptical galaxies).
In this chapter, we discuss two main topics useful in the study of spheroidal or ellipsoidal systems: a generalization of the virial theorem first derived in Chapter 5.2 to tensor form, the tensor-virial theorem, that provides a relation between the integrated radial profile, shape, and 3D kinematics that allows us to understand how the shape of elliptical galaxies is sustained by their internal kinematics. Then we will discuss the main numerical technique used to investigate possible equilibrium configurations of complex elliptical galaxies and dark-matter halos and to perform dynamical modeling of observations of the surface density and kinematics of elliptical galaxies, Schwarzschild modeling. Examples of the dynamical modeling techniques discussed in this chapter to observations of elliptical galaxies are discussed further in Chapter 16.
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