2.2.6 The FP as a Distance Indicator

If the FP is universal (or the deviations from universality are known to be within certain limits), it can be used to determine distances.

If the FP is established for a given cluster of galaxies and in addition the distance to the cluster is known from some other method, the intrinsic FP zero point $\Gamma$ can be calculated, cf. Eq. (). A recent example of this is Hjorth & Tanvir (1997), who calibrated the intrinsic FP zero point using the observed FP zero point for 5 E and S0 galaxies in the Leo-I group and the HST cepheid distance to the Leo-I galaxy M96.

Without knowing the intrinsic FP zero point $\Gamma$, the FP can be used to determine relative distances. For example, if we have two clusters HydraI and Coma, it follows from Eq. () that their relative distance is related to their observed FP zero point difference as

$$\frac{ d_{\rm A,Coma} }{ d_{\rm A,HydraI} } = 10^{\Delta\gamm... ...amma \equiv \gamma_{\rm HydraI} - \gamma_{\rm Coma} \enspace .$$ (2.28)

In order to compare different zero points ${\gamma_{\rm cl}}$, i.e. to calculate a meaningful $\Delta\gamma$, the same values of $\alpha$ and $\beta$ should be used, since ${\gamma_{\rm cl}}$ is very sensitive to the choice of $\alpha$ and $\beta$. This follows from the fact that ${\gamma_{\rm cl}}$ is the intersection of the fundamental plane with the ${\log{r_{\rm e}}}$ axis (cf. Eq. ), and that the galaxies are not symmetrically distributed around this axis, but rather displaced somewhat to the side. The zero point differences $\Delta {\gamma_{\rm cl}}$, on the other hand, are quite stable for somewhat different values of $\alpha$ and $\beta$, so the only requirement is to use common values of $\alpha$ and $\beta$ when comparing ${\gamma_{\rm cl}}$ for different clusters. However, for some values of $\alpha$ and $\beta$, the residuals from the FP are correlated with absolute magnitude, which will cause systematic errors on the derived distances if the different clusters have different limiting absolute magnitudes (cf. JFK96 and Sect. ).

The subscript ``A'' on the distances d in Eq. () indicates that they are so-called angular diameter distances, cf. Weinberg (1972). $d_{\rm A}$ is defined as ${d_{\rm A}}\equiv D/\delta$, where D is the linear diameter and $\delta$ is the angular diameter of the object. Another distance is the luminosity distance ${d_{\rm L}}$, which is defined as ${d_{\rm L}}\equiv [L/(4 \pi l)]^{1/2}$, where L is the (intrinsic) luminosity and l is the apparent luminosity of the object. In Euclidian geometry the two distances agree with each other and with the true distance. In an expanding universe (here given by the Robertson-Walker metric), this is not the case. Rather, $d_{\rm A}$ and ${d_{\rm L}}$ are related through the redshift z as

$$d_{\rm A} = d_{\rm L} (1+z)^{-2}$$ (2.29)

(Weinberg 1972). The luminosity distance is needed to calculate the distance modulus,

$$(m-M) = 5 \log(d_{\rm L}/10\,{\rm pc}) \enspace .$$ (2.30)

In the absence of peculiar motions (i.e. deviations from the pure Hubble expansion), ${d_{\rm L}}$ can be calculated from the redshift z as

$$d_{\rm L} \approx \frac{c}{H_0} \left[z+0.5(1-q_0)z^2\right] \enspace$$ (2.31)

(Weinberg 1972), neglecting terms of higher than second order in z. The approximation is very good at the redshift of Coma, with a relative error of less than 0.01%.

Later (in Sect. ) we will determine the distance to Coma and HydraI in the following way. The distance to Coma will be derived from the redshift, assuming Coma to be at rest in the CMB frame. The distance modulus can then be found from Eq. () and (). The distance to HydraI relative to Coma will be calculated from the observed FP zero point difference. By combining Eq. (), (), and (), we get the following equation for $z_{\rm HydraI}$

$$\frac { \left[z_{\rm Coma}+0.5(1-q_0)z_{\rm Coma}^2\right] (1... ...ght] (1+z_{\rm HydraI})^{-2} } = 10^{\Delta\gamma} \enspace ,$$ (2.32)

where $z_{\rm HydraI}$ is the CMB redshift for HydraI if HydraI has zero peculiar velocity. Equation () can be solved numerically. The distance modulus can then be found from Eq. () and (). In the same framework, the conversion of ${r_{\rm e}}$ from arcsec to kpc becomes

 

$$\log r_{\rm e,kpc} \log r_{\rm e,arcsec} - \log (206265\,{\rm arcsec/rad}) + \log (10^3\,{\rm kpc}/{\rm Mpc})$$ =

$$\log \left( \frac{c}{H_0} \left[z_{\rm Coma}+0.5(1-q_0)z_{\rm Com... ...or Coma} \\ \Delta\gamma & \mbox{for HydraI} \\ \end{array}\right. \enspace .$$ + (2.33)

using Eq. (), (), and (), and for HydraI in addition Eq. (). The identity $1 \equiv 10^3\,{\rm kpc}/{\rm Mpc}$ enables us to insert H₀ in units of ${\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ and get ${r_{\rm e}}$ in units of kpc. c needs to be in units of ${\rm km}\,{\rm s}^{-1}$.