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 ${R_{\rm obs}}$, a velocity ${V_{\rm obs}}$, and a mean surface brightness ${\langle I \rangle_{\rm obs}}$. One particular choice of observables is ${R_{\rm obs}}= {r_{\rm e}}$, ${V_{\rm obs}}= \sigma$, and ${\langle I \rangle_{\rm obs}}= {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$. Other choices could be used, for example ${R_{\rm obs}}$ 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

$$- {E_{\rm pot}}= {k_{\rm E}}{E_{\rm kin}}, \quad {k_{\rm E}}> 1 \enspace .$$ (2.10)

For a virialized system ${k_{\rm E}}$ has the value 2. We define ${\langle R \rangle}$ and ${\langle V^2 \rangle}$ as the mean radius and mean square velocity that enters the expressions for the potential and kinetic energy, respectively, i.e.

$$\frac{GM }{{\langle R \rangle}} M = - {E_{\rm pot}}\quad {\rm... ...d \frac{{\langle V^2 \rangle}}{2 } M = {E_{\rm kin}}\enspace .$$ (2.11)

We can now write the energy equation () as

$$\frac{GM}{{\langle R \rangle}} = {k_{\rm E}}\frac{{\langle V^2 \rangle}}{2} \enspace .$$ (2.12)

We relate the observable quantities ${R_{\rm obs}}$, ${V_{\rm obs}}$, and ${\langle I \rangle_{\rm obs}}$ to the physical quantities ${\langle R \rangle}$, ${\langle V \rangle}$, and luminosity L through

   

$${R_{\rm obs}} {k_{\rm R}}{\langle R \rangle}$$ = (2.13)

$${V^2_{\rm obs}} {k_{\rm V}}{\langle V^2 \rangle}$$ = (2.14)

$${k_{\rm L}}{\langle I \rangle_{\rm obs}}{R^2_{\rm obs}}$$ L = (2.15)

The parameters ${k_{\rm R}}$, ${k_{\rm V}}$, and ${k_{\rm L}}$ 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 r^1/4 half-light radius ${r_{\rm e}}$ or the King core radius $r_{\rm c}$ is used for ${R_{\rm obs}}$). From the energy equation () we can find the mass M as

$$M = \frac{{k_{\rm E}}}{2 G {k_{\rm V}}{k_{\rm R}}} {V^2_{\rm obs}}{R_{\rm obs}}\enspace .$$ (2.16)

We can now find a relation for ${R_{\rm obs}}$, and compare it with the the FP

$$\arraycolsep=2pt % \begin{array}{lllccc} \mbox{Theory} & : & ... ..._{\rm obs}}& {\langle I \rangle^\beta_{\rm obs}}\\ \end{array}$$ (2.17)

where we have collected the three structural parameters in ${k_{\rm S}}$ as

$${k_{\rm S}}= \frac{1}{2G{k_{\rm R}}{k_{\rm L}}{k_{\rm V}}} \enspace .$$ (2.18)

Since the observations give $\alpha \approx 1.3$ and $\beta \approx -0.8$, it follows from Eq. () that ${k_{\rm S}}{k_{\rm E}}{\left( M/L \right)^{-1}}$ can not be constant, but has to be the following power law function of ${V_{\rm obs}}$ and ${\langle I \rangle_{\rm obs}}$

$${k_{\rm S}}{k_{\rm E}}{\left( M/L \right)^{-1}} \propto V^{\... ...-2}_{\rm obs} \langle I \rangle^{\beta+1}_{\rm obs} \enspace .$$ (2.19)

In other words, either the structure ${k_{\rm S}}$ (and ${k_{\rm E}}$) or the mass-to-light ratio ${\left( M/L \right)}$ (or both) need to vary in a systematic way to produce the observed FP slope. To explore the possibility of an ${\left( M/L \right)}$ variation further, we first find an expression for ${V_{\rm obs}}$ as function of L from the first line of Eq. () and Eq. ()

$${V_{\rm obs}}= k^{-1/2}_{\rm SR} k^{-1/4}_{\rm L} L^{1/4} \l... ...M/L \right)^{1/2} \langle I \rangle^{1/4}_{\rm obs} \enspace ,$$ (2.20)

where we have defined ${k_{\rm SR}}= {k_{\rm S}}{k_{\rm E}}$. We can now eliminate ${V_{\rm obs}}$ from Eq. () and instead get an expression involving L

$$\left( M/L \right) \propto k_{\rm SRL} L^{1/\alpha-1/2} \lan... ...{\rm SRL} \equiv {k_{\rm SR}}k^{1/2-\alpha}_{\rm L} \enspace .$$ (2.21)

For ${\langle I \rangle_{\rm obs}}= {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$, the exponent for ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ in the above equation turns out to be non-significantly different from zero. This is the case for the JFK96 values of $\alpha$ and $\beta$, where the ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ exponent is $0.02\pm0.04$, and this is also the case for the FPs studied by e.g. Faber et al. (1987). Also Prugniel & Simien (1996) found the ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ exponent to be non-significant. Therefore it can be stated, that the scalar virial theorem ( ${k_{\rm E}}= 2$) and structural homology ( ${k_{\rm S}}$ = constant) implies that the mass-to-light ratio varies with luminosity, or equivalently with mass, as

$${\left( M/L \right)}\propto L^\xi \, , \quad \xi = 1/\alpha-1... ... b = \frac{\xi}{\xi+1} = \frac{2-\alpha}{2+\alpha} \enspace .$$ (2.22)

For the value of $\alpha$ that we find in this study (in Gunn r), $\alpha = 1.35 \pm 0.07$, the result is

$${(M/L_{\rm r})}\propto L^{0.24\pm0.04} \, ; \quad \quad {(M/L_{\rm r})}\propto M^{0.19\pm0.04} \enspace .$$ (2.23)

To actually calculate ${\left( M/L \right)}$ from the data (under the above assumptions), we need to know the value of ${k_{\rm R}}$, ${k_{\rm V}}$, ${k_{\rm L}}$. For our choice of observables, ${k_{\rm L}}$ is simply $2 \pi$, i.e.

$$L = 2 \pi {< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}{r_{\rm e}}^2 \cdot 10^6 \, {\rm pc}^2/{\rm kpc}^2 \enspace ,$$ (2.24)

where the identity $1 \equiv 10^6 \, {\rm pc}^2/{\rm kpc}^2$ is inserted to denote that we in this study have ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ in units of $L_\odot/{\rm pc}^2$ and ${r_{\rm e}}$ in units of kpc.

Modeling is needed to calculate the constant in the equation for M, Eq. (). We will here write this equation as $M = c_2 \sigma^2 {r_{\rm e}}$. Note, that this dynamical determination of M gives the total mass $M_{\rm total}$, which includes luminous matter (stars) and dark matter. Bender, Burstein, & Faber (1992) calculated c₂ using models with King profiles and isotropic velocity dispersions. Assuming $M_{\rm total} = 10 M_{\rm luminous}$ they found

$$M_{\rm luminous} = 5.0 \, \sigma^2 {r_{\rm e}}/ G, \quad G = ... ...} \, ({\rm km/s})^{-2} \, {\rm kpc} \, M^{-1}_\odot \enspace .$$ (2.25)

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

$$\log(M_{\rm luminous}/L) = 2 \log \sigma - \log {< \hspace{-... ... I \hspace{-3pt}>_{\rm e}}- \log {r_{\rm e}}- 0.733 \enspace .$$ (2.26)

If we want to compare $\log(M_{\rm luminous}/L)$ 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 ${r_{\rm e}}$ 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, ${r_{\rm e}}$ in a common passband ``X'' should be used to calculate the mass, and ${r_{\rm e}}$ in the given passband ``Y'' to calculate the luminosity, i.e.

$$\log(M_{\rm luminous}/L) = 2 \log \sigma - \log {< \hspace{-... ...e}}+ \log r_{\rm e,X} - 2 \log r_{\rm e,Y} - 0.733 \enspace .$$ (2.27)

Both equations are independent of the Hubble constant H₀, but since both M and L depend on ${r_{\rm e}}$ in kpc, which depends on the distance (cf. Eq. ), which depends on H₀ if the distance is calculated from the redshift, $\log(M/L)$ becomes proportional to $\log H_0$.