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10.3 Physical Basis

10.3.1 Power Law Relationships

There is no complete physical description of the structure and kinematics of elliptical galaxies. However, starting from some simple physical assumptions a rudimentary justification of the method can be provided. Sargent et al. (1977) pointed out that the assumptions that all elliptical galaxies have the same surface brightness, L propto R2 (Fish 1964) and mass-to-light ratio (M/L), together with the virial theorem relation M propto Rsigma2, lead to the Faber-Jackson relation L propto sigma4. These ideas were extended by Djorgovski & Davis (1987) and Dressler et al. (1987b) who developed the concept of a fundamental plane in the 3-space of log Ie log Re log sigma where Ie is the galaxy surface brightness within the half-light radius, Re (or diameter Ae). Combining the results of these three studies, the equation of the fundamental plane is:

equation 25 (25)

or expressed in terms of a modified Faber-Jackson relation:

equation 26 (26)

As Dressler et al. (1987b) show, there are only two independent parameters in the three observables Ae, Dn, and Ie, so that Dn / Ae is a function of the normalizing surface brightness only. They find Dn / Ae propto Iex, where x = 0.8. This, together with the empirical (Dn-sigma) relation, leads to a formulation of the modified Faber-Jackson relation which is almost identical to equation (26). Thus the definition of Dn combines the parameters of the fundamental plane in such a way that it is viewed edge-on and Dn is therefore a close to optimal distance indicator. This is illustrated in Figure 22 which shows the fundamental plane and its projections, taken from Faber et al. 1987. They point out that the existence of the fundamental plane implies there are no constraints on the global parameter relations for elliptical galaxies other than the Virial Theorem. This is perhaps the most important difference between the Dn-sigma relation and the Tully-Fisher methods; the latter, being a single parameter relationship, requires the action of an extra physical constraint which is presumably set during galaxy formation (Gunn 1988). Thus the Dn-sigma relation depends on the physics of galaxy equilibrium rather than the physics of galaxy formation.

Figure 22c Figure 22d
Figure 22a Figure 22b
Figure 22. Structural parameters for group galaxies in the 7S survey, taken from Faber et al. (1987). (a) log sigma vs log Re; the diagonal line of slope -0.5 is the constant mass locus. (b) Surface brightness vs log Re, showing the fundamental plane almost edge-on. (c) log sigma vs surface brightness, showing the fundamental plane nearly face-on; the dots show the effect viewing an E3.6 oblate or prolate galaxy seen at maximum elongation and as round in projection. (d) log sigma vs (log Re + 0.84 log Ie) showing the plane exactly edge-on.

The Dn-sigma relation cannot be justified in terms of a detailed physical model such as can be formulated for Type Ia supernovae or Cepheids. A general physical description has been formulated using power law relationships between the structural and kinematic variables that illustrates the sensitivity of the method to the assumptions.

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