5.3 Parameters Quantifying the Global Deviations From Ellipses

We determined parameters that quantify the global deviations from elliptical isophotes in Gunn r. From the Fourier coefficient profiles s_n(r) and c_n(r) we determined intensity-weighted mean Fourier coefficients ${< \hspace{-4pt} s_n \hspace{-4pt}>}$ and ${< \hspace{-4pt} c_n \hspace{-4pt}>}$ as

$${< \hspace{-4pt} s_n \hspace{-4pt}>}= \frac{ \int_{r_{\rm min... ...m max}}I(r) \, {\rm d}r } \, , \quad n = 1,\ldots,6 \enspace .$$ (5.3)

Uncertainties are calculated on basis of min-max variations in s_n(r) and c_n(r), respectively. Following JF94, we used ${r_{\rm min}}= 2 \cdot {\rm FWHM}$ and ${r_{\rm max}}$ as the radius where $\mu = 23.35{^{\rm m} /{\rm arcsec}^{2}}$. This ensures that seeing effects are small and that the signal-to-noise is sufficient.

We determined a characteristic value of c₄(r), denoted c₄, as the mean value of the three points around the extremum of c₄(r). In case of no well-defined extremum, the mean value of the three points around the effective radius was used. This is the definition used by JF94 and JFK95a. We only looked for an extremum in the radius interval from $2 \cdot {\rm FWHM}$ to the radius where the uncertainty on $\mu$ exceeded $0.5{^{\rm m} /{\rm arcsec}^{2}}$. We calculated the uncertainty on c₄ as half the min-max variation of the 3 points. A few of the galaxies had a minimum and a maximum in the c₄(r)-profile of comparable amplitude. In these cases we still calculated c₄ at the extremum with the largest amplitude, and if a tie, at the most regular one. Examples are R194 (p. ; c₄ = -0.043) and R238 (p. ; c₄ = -0.029). Both galaxies are boxy at low radii, and disky further out. For 4 of the 64 galaxies we did not determine c₄, since it would be too uncertain ( $\sigma_{c_4} > 0.05$). These galaxies had no extrema and ${r_{\rm e}}$ at or beyond the maximum radius used for the c₄ determination.

In Figure  we compare c₄ with ${< \hspace{-4pt} c_4 \hspace{-4pt}>}$. The relation between them is ${< \hspace{-4pt} c_4 \hspace{-4pt}>}= 0.33 \cdot c_4$ (dashed line). JF94 found ${< \hspace{-4pt} c_4 \hspace{-4pt}>}= 0.6 \cdot c_4$ (dotted line). The discrepancy is likely due to the poorer spatial resolution of JF94, which affects c₄ but not ${< \hspace{-4pt} c_4 \hspace{-4pt}>}$; see also Sect. . That $\vert{< \hspace{-4pt} c_4 \hspace{-4pt}>}\vert$ per se is smaller than |c₄| is because ${< \hspace{-4pt} c_4 \hspace{-4pt}>}$ is integrated over a larger range in r than just the 3 points around the extremum as c₄ is. Some of the galaxies have quite different values of c₄ and ${< \hspace{-4pt} c_4 \hspace{-4pt}>}$. For example, R188 (p. ) and R250 (p. ) have $c_4 \approx 0.07$ and ${< \hspace{-4pt} c_4 \hspace{-4pt}>}\approx 0.01$, cf. Fig. (b). These two galaxies have a small c₄(r) minimum at low radii and a large c₄(r) maximum at large radii. Since ${< \hspace{-4pt} c_4 \hspace{-4pt}>}$ is an intensity-weighted mean, ${< \hspace{-4pt} c_4 \hspace{-4pt}>}$ is small for these galaxies, while c₄ is large.