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## 2.2.3 The Physics Underlying the FP

To interpret the empirical FP relation, we need to relate the observable quantities to the physical quantities. The observable quantities are a radius , a velocity , and a mean surface brightness . One particular choice of observables is , , and . Other choices could be used, for example could be taken as the core radius in a King model fit. The following considerations are inspired by Djorgovski, de Carvalho, & Han (1988).

For a bound system, such as a galaxy, the sum of the kinetic and potential energy must be less than zero. This can be written as

 (2.10)

For a virialized system has the value 2. We define and as the mean radius and mean square velocity that enters the expressions for the potential and kinetic energy, respectively, i.e.

 (2.11)

We can now write the energy equation () as

 (2.12)

We relate the observable quantities , , and to the physical quantities , , and luminosity L through

 = (2.13) = (2.14) L = (2.15)

The parameters , , and reflect the density structure, kinematical structure, and luminosity structure of the given galaxy. Obviously, they depend on the choice of observables (e.g. whether the r1/4 half-light radius or the King core radius is used for ). From the energy equation () we can find the mass M as

 (2.16)

We can now find a relation for , and compare it with the the FP

 (2.17)

where we have collected the three structural parameters in as

 (2.18)

Since the observations give and , it follows from Eq. () that can not be constant, but has to be the following power law function of and

 (2.19)

In other words, either the structure (and ) or the mass-to-light ratio (or both) need to vary in a systematic way to produce the observed FP slope. To explore the possibility of an variation further, we first find an expression for as function of L from the first line of Eq. () and Eq. ()

 (2.20)

where we have defined . We can now eliminate from Eq. () and instead get an expression involving L

 (2.21)

For , the exponent for in the above equation turns out to be non-significantly different from zero. This is the case for the JFK96 values of and , where the exponent is , and this is also the case for the FPs studied by e.g. Faber et al. (1987). Also Prugniel & Simien (1996) found the exponent to be non-significant. Therefore it can be stated, that the scalar virial theorem ( ) and structural homology ( = constant) implies that the mass-to-light ratio varies with luminosity, or equivalently with mass, as

 (2.22)

For the value of that we find in this study (in Gunn r), , the result is

 (2.23)

To actually calculate from the data (under the above assumptions), we need to know the value of , , . For our choice of observables, is simply , i.e.

 (2.24)

where the identity is inserted to denote that we in this study have in units of and in units of kpc.

Modeling is needed to calculate the constant in the equation for M, Eq. (). We will here write this equation as . Note, that this dynamical determination of M gives the total mass , which includes luminous matter (stars) and dark matter. Bender, Burstein, & Faber (1992) calculated c2 using models with King profiles and isotropic velocity dispersions. Assuming they found

 (2.25)

This is for = 100, with and being the tidal and core radii in the King model, respectively. For = 300, c2 would be 4.0/G, since c2 turns out not to be quite constant in their models. The ratio is about 100-300 for giant ellipticals (Bender et al. 1992). Equation () and () combine into

 (2.26)

If we want to compare for several passbands with for example prediction from stellar population synthesis models, the above equation cannot be used for all the passbands. This is because varies with wavelength, which is also to say that E and S0 galaxies have radial color gradients. The Bender et al. model does not take this into account. Instead, in a common passband X'' should be used to calculate the mass, and in the given passband Y'' to calculate the luminosity, i.e.

 (2.27)

Both equations are independent of the Hubble constant H0, but since both M and L depend on in kpc, which depends on the distance (cf. Eq. ), which depends on H0 if the distance is calculated from the redshift, becomes proportional to .

Next: 2.2.4 Variation of with Up: 2.2 The Fundamental Plane Previous: 2.2.2 Is the FP

Properties of E and S0 Galaxies in the Clusters HydraI and Coma
Master's Thesis, University of Copenhagen, July 1997

Bo Milvang-Jensen (milvang@astro.ku.dk)