2.2.2 Is the FP Universal?

To discuss whether the FP is universal, we will first need to make a point about ${r_{\rm e}}$ clear. What can be observed is an angle, measured in e.g. arcsec. To calculate the corresponding length, in e.g. kpc, the distance $d_{\rm A}$ needs to be known, i.e.

$$r_{\rm e,kpc} = d_{\rm A} \cdot \frac{ r_{\rm e,arcsec} }{ 206265\,{\rm arcsec/rad} } \enspace .$$ (2.6)

The subscript ``A'' on the distance indicates that it is a so-called angular diameter distance; this will be explained later. We can write the FP using both versions of ${r_{\rm e}}$, i.e.

  

$$\log r_{\rm e,arcsec} \alpha \log\sigma + \beta \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}+ {\gamma_{\rm cl}}$$ = (2.7)

$$\log r_{\rm e,kpc} \alpha \log\sigma + \beta \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}+ \Gamma$$ = (2.8)

For ${r_{\rm e}}$ in arcsec, the zero point ${\gamma_{\rm cl}}$ for the given cluster depends on the distance (in kpc) as

$${\gamma_{\rm cl}}= \Gamma - \log d_{\rm A} + \log 206265 \enspace .$$ (2.9)

In the following we will not explicitly denote whether ${r_{\rm e}}$ is an angle or a length, unless there is reason for confusion.

The question about universality can now be phrased as: a) are the FP coefficients ($\alpha$ and $\beta$) universal? b) is the FP zero point ($\Gamma$) universal? JFK96 addressed both questions. Based on data for 226 E and S0 galaxies in 10 nearby clusters, they determined $\alpha$, $\beta$, and ${\gamma_{\rm cl}}$ for each cluster. They found that $\alpha$ and $\beta$ were not significantly different from cluster to cluster, although variations of $\alpha$ of the order 10% could not be ruled out. Furthermore, they did not find $\alpha$ and $\beta$ to correlate with the distance to the cluster (more precisely $cz_{\rm CMB}$), the velocity dispersion of the cluster, or the intracluster gas temperature. This is remarkable, especially since their clusters spanned a large range in these cluster properties. JFK96 also found E and S0 galaxies to follow the same FP. The JFK96 values $\alpha = 1.24\pm0.07$ and $\beta=-0.82\pm0.02$ agree reasonably well with those from other studies in the literature of cluster E (and S0) galaxies. Different fitting methods and sample selection criteria make it difficult to compare FP coefficients from different studies in detail.

The universality of $\Gamma$ is harder to assess when the distances to the clusters are not known. JFK96 found that under the assumption that $\Gamma$ is constant, the derived peculiar velocities were small, mostly $< 1000\,{\rm km}\,{\rm s}^{-1}$. This means that $\Gamma$ can not be very different from cluster to cluster.

It is interesting to note, that some studies find that elliptical galaxies in the field are systematically different from elliptical galaxies in rich clusters. For example, de Carvalho & Djorgovski (1992) found that field ellipticals compared with cluster ellipticals had a larger value of the FP zero point, a larger intrinsic scatter in the FP, and perhaps also a different value of the FP slope. The larger value of the FP zero point for the field can also be phrased as a larger surface brightness (i.e. ${{{< \hspace{-3pt} I \hspace{-3pt}>}_{\rm e,field}}} > {{{< \hspace{-3pt} I \hspace{-3pt}>}_{\rm e,cluster}}}$) and/or a lower velocity dispersion at a given radius (cf. Eq. ; remember that $\beta<0$). de Carvalho & Djorgovski further found, that at a fixed radius (or luminosity) the field ellipticals were more blue and had a lower ${ {\rm Mg}_2}$ value. A possible interpretation is that the field galaxies have experienced merger-induced star formation.

Along those lines, Schweizer et al. (1990) found for a sample of 36 mostly field ellipticals that various line indices (such as ${ {\rm Mg}_2}$) were correlated with morphological fine-structure (ripples, jets, boxyness, and so-called X-structure). They found that the most probable interpretation was a variation in mean age with morphological fine-structure. This could be explained by merger-induced star formation. Gregg (1992) found that the peculiar velocities derived from the ${D_{\rm n}}$-$\sigma$ relation for the galaxies in the Schweizer et al. sample were correlated with the morphological fine-structure. It was concluded, that these differences in stellar population induced spurious peculiar velocities. Note, that in these studies a distance is determined for each field galaxy, as opposed to determining the distance to a cluster using many galaxies in the given cluster.

The findings of de Carvalho & Djorgovski (1992) could lead to the suspicion that there might be differences between rich and poor clusters, and between the central and outer regions of clusters. JFK96 tested this by plotting the residuals from the FP versus the projected cluster surface density. They did not find any correlation. The found stability of the zero point of the FP corresponds to $3\pm3$% if the FP is used for distance determinations. Lucey et al. (1991b) tested the stability of the zero point of the ${D_{\rm n}}$-$\sigma$ relation and found the derived distances to vary by only $6\pm9$%. Their sample of galaxies in the Coma cluster spanned a range of over 150 in projected cluster surface density. JFK96 found that the FP zero point for the Lucey et al. sample had a comparable stability to that of the JFK96 sample, i.e. $3\pm3$%.