7.2 The Fundamental Plane in Gunn r

We fitted a plane to the distribution of galaxies in $({\log{r_{\rm e}}}, {\log\sigma}, {\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}})$. This was done as an ``orthogonal fit''; we seek the vector normal to the plane, $\vec n =(-1,\alpha ,\beta )$, that minimizes the sum of the absolute residuals perpendicular to the plane. The equivalent equation for the FP is

$${\log{r_{\rm e}}}= \alpha \log\sigma + \beta \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}+ \gamma _{\rm cl} \enspace .$$ (7.4)

${r_{\rm e}}$ is in arcsec. ${\gamma_{\rm cl}}$ was taken as the median value. For photometry in Gunn r we find

$$\arraycolsep=2pt % \begin{array}{lllllllll} {\rm HydraI} & : ... ... & N=114 \\ & & & \pm & 0.06 & \pm & 0.03 & & \\ \end{array}$$ (7.5)

${\sigma_{\rm fit}}$ is the rms scatter in the ${\log{r_{\rm e}}}$ direction and N is the number of galaxies involved in the fit. The numerical values of ${\gamma_{\rm cl}}$ have been omitted since they are only of interest when the two clusters are fitted with the same coefficients $\alpha$ and $\beta$. The uncertainties on $\alpha$ and $\beta$ were derived using the bootstrap method (Efron 1979; see also Efron & Tibshirani 1986, 1993).

The above-mentioned fitting method treats the variables symmetrically, which is preferred when we want to establish the physical relation between them, as opposed to when we want to predict one variable from the other variable(s). The fact that we minimize the sum of the absolute residuals and not the sum of the square of the residuals makes the determination more robust against a few galaxies with large deviations from the relation, as does the use of median zero points instead of mean zero points. This fitting method has been used in the literature by e.g. JFK96, Baggley (1996), and Mohr & Wegner (1997). Unless otherwise stated, all fits presented in the following are of this type.

Figure  shows the FP edge-on. The coefficients of the two FPs are not significantly different, from Eq. () we find $\Delta \alpha = 0.24 \pm 0.17$ and $\Delta \beta = 0.04 \pm 0.06$. If we fix $\beta$ at the value -0.82 and only fit $\alpha$, we find $\alpha_{\rm HydraI} = 1.66 \pm 0.16$ and $\alpha_{\rm Coma} = 1.32 \pm 0.05$. The difference, $\Delta \alpha = 0.34 \pm 0.17$, is significant at the 2 sigma level.

 

The difference becomes non-significant if we impose the same limiting magnitude of ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$ (the Coma completeness limit) on the two samples, reducing the number of galaxies as $N_{\rm HydraI}\!\!: 45 \rightarrow 28$ and $N_{\rm Coma}\!\!: 114 \rightarrow 105$. We find $\alpha_{\rm HydraI} = 1.51 \pm 0.24$ and $\alpha_{\rm Coma} = 1.38 \pm 0.08$ (still for $\beta \equiv -0.82$). The difference is non-significant, $\Delta \alpha = 0.13 \pm 0.25$.

However, we do not find any significant evidence that the FP coefficients depend on the limiting magnitude. If we fit HydraI alone and only galaxies brighter than ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$, we find $\alpha = 1.42 \pm 0.24$ and $\beta = -0.79 \pm 0.08$ (with ${\sigma_{\rm fit}}= 0.097$), which is not significantly different from the fit to the full HydraI sample, see Eq. (). If we fit Coma alone and only galaxies brighter than ${M_{\rm r_T}}=-21\hbox{$.\!\!^{\rm m}$ }58$ (the point that split the sample in half, N=57), we find $\alpha = 1.32 \pm 0.10$ and $\beta = -0.83 \pm 0.05$ (with ${\sigma_{\rm fit}}= 0.093$), which is not significantly different from the fit to the full Coma sample, see Eq. ().

We conclude, that no significant differences in the coefficients for the FP for the HydraI and Coma samples can be detected. Differences in $\alpha$ on the 10% level cannot be ruled out. A common fit to the full HydraI and Coma samples gives

$$\arraycolsep=2pt % \begin{array}{lllllllll} {\rm HydraI} & : ... ... & N=114 \\ & & & \pm & 0.07 & \pm & 0.03 & & \\ \end{array}$$ (7.6)

This is not significantly different from the coefficients found by JFK96, $\alpha = 1.24\pm0.07$, $\beta=-0.82\pm0.02$, based on 226 E and S0 galaxies in 10 clusters.

In the following we adopt the values of $\alpha$ and $\beta$ from Eq. (). At this point, we determine the peculiar velocity implied by the FP. The peculiar velocity for a given cluster is given by $v_{\rm pec} \approx {cz_{\rm CMB}}- cz$ to first order in z, where z is the expected redshift in the CMB frame in the absence of peculiar velocities. Since we do not have an accurate calibration of the intrinsic FP zero point, we use the observed FP zero point difference to calculate the relative distance between HydraI and Coma. We assume Coma to have no peculiar velocity. From the FP zero points, $\gamma_{\rm HydraI} = 0.189 \pm 0.0155$ and $\gamma_{\rm Coma} = - 0.044 \pm 0.0092$, we then find the peculiar velocities $v_{\rm pec,HydraI} = -93 \pm 152 \,{\rm km}\,{\rm s}^{-1}$ and $v_{\rm pec,Coma} = 0 \pm 160 \,{\rm km}\,{\rm s}^{-1}$. The peculiar velocity for HydraI is non-significant.

For the JFK96 FP, we get the same peculiar velocity for HydraI within the uncertainties. We find FP zero points $\gamma_{\rm HydraI} = 0.410 \pm 0.0153$ and $\gamma_{\rm Coma} = 0.184 \pm 0.0086$, and peculiar velocities $v_{\rm pec,HydraI} = -157 \pm 152 \,{\rm km}\,{\rm s}^{-1}$ and $v_{\rm pec,Coma} = 0 \pm 149 \,{\rm km}\,{\rm s}^{-1}$.

We have calculated the uncertainty on $\gamma$, which is a median value, as ${\sigma_{\rm fit}}/\sqrt{N}$ (where ${\sigma_{\rm fit}}$ is the rms scatter, cf. above). It is known from the statistical literature (e.g. Stuart & Ord 1987) that the uncertainty on the mean is ${\rm rms}/\sqrt{N}$ regardless of the probability distribution that the data points are drawn from, whereas the uncertainty on the median depends on the distribution. For the special case of a normal distribution, the uncertainty on the median is $\sqrt{\pi/2} \cdot {\rm rms}/\sqrt{N} \approx 1.25 \, {\rm rms}/\sqrt{N}$. For distributions with increasingly larger tails than the normal distribution, the uncertainty on the median becomes increasingly less than the uncertainty on the mean. Mohr & Wegner (1997) used the bootstrap method to calculate the uncertainty on the FP zero point difference, and found the value ${\rm rms}/\sqrt{N}$ to be a conservative estimate on the uncertainties on the individual FP zero points. Throughout this work we calculate the uncertainty on the median as ${\rm rms}/\sqrt{N}$.

We now calculate ${r_{\rm e}}$ in kpc using Eq. (). The FP, Eq. (), then becomes

$$\arraycolsep=2pt % \begin{array}{lllll} \quad {\log{r_{\rm e}... ...$\space in kpc)} \\ & \pm & 0.07 & \pm & 0.03 \\ \end{array}$$ (7.7)

In order to plot the FP face-on, we define a new 3-space (x,y,z) by

$$\arraycolsep=2pt % \begin{array}{ccccc} x & = & \left[ (\alph... ...+ {{\alpha }^2} + {{\beta }^2} \right) }^{0.5}} \\ \end{array}$$ (7.8)

This is an orthonormal transformation of the original 3-space $({\log{r_{\rm e}}}, {\log\sigma}, {\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}})$. The (x,y) projection shows the FP face-on, since z is constant for the FP. Furthermore, x is proportional to ${\log{r_{\rm e}}}$ for the FP. For our values of $\alpha$ and $\beta$ we get

$$\arraycolsep=2pt % \begin{array}{ccccc} x & = & \left[ 2.51 \... ...pt} I \hspace{-3pt}>_{\rm e}}\right] & / & 1.87 \\ \end{array}$$ (7.9)

Figure  shows the FP face-on for the two samples. The dashed line marks ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$, the completeness limit of the Coma data. The equation for the plane in 3-space of constant absolute magnitude ${M_{\rm r_T}}$ is given by $0.4 (M_\odot-{M_{\rm r_T}}) = \log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}+ 2 \log{r_{\rm e}}+ \log(2\pi) + 6$ (with ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ in $L_\odot/{\rm pc}^2$ and ${r_{\rm e}}$ in kpc, as noted earlier). The intersection with the (x,y) plane can be found using the equation for the FP and the equations for x and y by eliminating ${\log{r_{\rm e}}}$, ${\log\sigma}$, and $\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$. The galaxies brighter and fainter than ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$ are shown in Fig.  as filled and open symbols, respectively. The fact that a few of the points are on the wrong side of the line marking ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$ is because the plane of constant ${M_{\rm r_T}}$ intersects the FP at an angle of 132$^\circ$, not 90$^\circ$. This is most easily seen in $({\log{r_{\rm e}}}, {\log\sigma}, {\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}})$ space where the FP and the plane of constant ${M_{\rm r_T}}$ have normal vectors $(-1,\alpha,\beta)$ and (2,0,1), respectively. Since the galaxies scatter somewhat perpendicular to the plane, the projection onto the plane might render them on the ``wrong'' side.

 

Bender et al. (1992) noted, that the region occupied within the FP by luminous ellipticals was delimited by the line $y \approx -0.56 x + 4.13$^7.1, which is shown as the dotted line in Fig. . The region beyond this line was coined the exclusion zone by Burstein, Bender, & Faber (1992), and recently the zone of exclusion (ZOE) by Burstein et al. (1997).

The existence of the exclusion zone is not caused by selection effects. Rather, it is a physical constraint in addition to the FP, corresponding to (note, that the sign of the ${< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$ exponent is wrong in Bender et al. 1992 and Burstein et al. 1992).

As can be seen from Fig. , the distribution within the FP is similar for the HydraI and Coma samples when imposing the same absolute magnitude limit. This can be quantified by means of the 2-dimensional 2-sample Kolmogorov-Smirnov test (Fasano & Franceschini 1987; as implemented by Press el al. 1992). This test gives the probability ${P_{\rm same\;distr.}}$ that the two samples are drawn from the same distribution. We find ${P_{\rm same\;distr.}}$ = 25% for HydraI vs. Coma for galaxies brighter than ${M_{\rm r_T}}= -20\hbox{$.\!\!^{\rm m}$ }75$. This test is not invariant under a rotation of the coordinate system. In our case, we might as well have used another coordinate system (x',y') that was rotated by an angle $\theta$ with respect to (x,y) to describe the distribution of galaxies within the FP. If we perform the above-mentioned test using coordinates rotated by $\theta = 0.0^\circ, 0.1^\circ, 0.2^\circ, \ldots, 89.9^\circ$, we obtain values of ${P_{\rm same\;distr.}}$ in the range 14%-67%, with a mean value of 31%. Fortunately, all the values agree in the sense that they all indicate a non-significant difference. For comparison, for a normal distribution, a two sigma deviation has a probability of 4.6%, so in that sense the differences we find are not significant at the two sigma level.

Figure  shows the FP edge-on along ${\log{r_{\rm e}}}$. Since the galaxies span a larger range in ${\log{r_{\rm e}}}$ than in ${\log\sigma}$, 1.8 and 1.0, respectively, the scatter looks smaller along ${\log{r_{\rm e}}}$ (Fig. ) than along ${\log\sigma}$ (Fig. ). Nevertheless, the scatter (in the ${\log{r_{\rm e}}}$ direction) of the two FPs is somewhat higher than found earlier, though the difference in the scatter is not statistically significant. For the Coma sample we find $\sigma_{\rm fit}=0.095 \pm 0.009$. JFK96 found $\sigma_{\rm fit} = 0.079 \pm 0.009$ for a sample of 79 galaxies in the Coma cluster (their sample is a subset of ours). The two values of ${\sigma_{\rm fit}}$ are not significantly different. If galaxies with ${\log\sigma}< 2.0$ are omitted from our two samples, we get $\sigma_{\rm fit} = 0.090$ for HydraI and 0.088 for Coma. We note, that of the 8 Coma galaxies with residuals > 0.19, 4 have been observed by Caldwell et al. (1993), and two of these, NGC4853 and D15, were classified as peculiar (starburst or post-starburst). NGC4853 has an FP residual more than twice as large as any of the other galaxies, see Fig. (b) and (b). NGC4853 and D15 also have large residuals from the ${ {\rm Mg}_2}$-$\sigma$ relation, Fig. (b), and the ${ <{\rm Fe}>}$-$\sigma$ relation, Fig. (b). The high residuals are most likely caused by the presence of young stars in these two galaxies.

 

The FP has significant intrinsic scatter ( ${\sigma_{\rm int}}$).  We estimate ${\sigma_{\rm int}}$ by subtracting the typical measurement errors in quadrature from ${\sigma_{\rm fit}}$, taking into account the correlation between the errors in ${\log{r_{\rm e}}}$ and $\log{< \hspace{-3pt} I \hspace{-3pt}>_{\rm e}}$. We do this as ${\sigma_{\rm int}}= ({\sigma_{\rm fit}}^2 - [\sigma_{({\log{r_{\rm e}}}-\beta\... ...ce{-1pt} I \hspace{-1pt}>_{\rm e}})}^2 + (\alpha\sigma_{\log\sigma})^2])^{1/2}$. When we insert $\sigma_{({\log{r_{\rm e}}}-\beta\log{< \hspace{-1pt} I \hspace{-1pt}>_{\rm e}})} = 0.017$ and $\sigma_{\log\sigma}= 0.032$ (Table , p. ; weighted mean values for HydraI and Coma), and ${\sigma_{\rm fit}}= 0.099$ and $\alpha = 1.35$ (Eq. ), we get ${\sigma_{\rm int}}= 0.087$. JFK96 found an intrinsic scatter of 0.070, which is lower than our value at the 2 sigma level.

Unless our estimates of the measurement errors are a factor of two too low, which seems unlikely, there is significant intrinsic scatter in the FP. We will search for the source of this scatter in Sect. , where we investigate correlations between the FP residuals and a number of available parameters.

Do galaxies classified as E and S0 follow the same FP? JFK96 found, that E and S0 galaxies had similar FP zero points. This is also the case for the samples studied here, we find an FP zero point difference of $0.000 \pm 0.015$. In Fig.  we plot the FP face on with E and S0 galaxies in separate panels. The dot-dashed line on the figure marks ${M_{\rm r_T}}= -23\hbox{$.\!\!^{\rm m}$ }1$. JF94 found no S0 galaxies brighter than this magnitude in their Coma sample. Their Coma sample is the one we use here. The visual impression from Fig.  might be that E and S0 galaxies are not distributed in the same way within the FP. However, a statistical test gives the opposite result. When we select galaxies with ${M_{\rm r_T}}$ between $-20\hbox{$.\!\!^{\rm m}$ }75$ and $-23\hbox{$.\!\!^{\rm m}$ }1$ to get a complete sample and to take into account the effect found by JF94, a 2D K-S test gives ${P_{\rm same\;distr.}}$ in the range 13%-44%, with a mean value of 26%, indicating a non-significant difference.