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Astron. Astrophys. 339, 409-422 (1998)

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2. Data analysis

2.1. Surface photometry and structural decomposition

The techniques used to reduce and analyze the photometric data are detailed in Paper I. Briefly, we decomposed the surface brightness distributions of each sample galaxy into the two components of bulge and disk. Two independent methods were used: a) a parametric fit assuming a generalized exponential bulge plus an exponential thin disk; and b) an iterative non-parametric (np) decomposition algorithm. We emphasize that both procedures decompose the entire two-dimensional brightness distribution, rather than the brightness profiles, and take into account the effects of seeing. The bulge is assumed to be an oblate rotation ellipsoid coaxial with the disk. In addition to the NIR images, r band brightness profiles and M/L ratios (Kent 1988) are available for ten galaxies of our sample.

2.2. Observed rotation curves

Rotation curves (RC's) exist for all the galaxies in our sample; the references for them are shown in Table 1, together with general information regarding the sample. More information about the sample objects (coordinates, etc.) can be found in Paper I. HI data are available for six objects; for the remainder, RC's were obtained from optical emission lines. We discarded absorption data since they are likely to yield a lower limit rather than a true estimate of the rotational velocity, due to the effect of velocity dispersion and projection along the line of sight. All distances were scaled to match our values, and circular speeds were corrected when estimated from inclinations differing from ours, with the exception of extended HI RC's derived from interferometric maps.


Table 1. Galaxy sample and RC references.
Column 2 : see Paper I.
Columns 3-4 : values from the Third Reference Catalog of Bright Galaxies (de Vaucouleurs et al. 1991)
Column 5 : References for the RC's (O = optical, R = radio).
1: Rubin et al. 1985 (O);
2: Begeman 1987 (R);
3: Bertola et al. 1996 (O);
4: Demoulin 1969 (O);
5: Krumm & Salpeter 1979 (R);
6: Fillmore, Boroson & Dressler 1986 (O). For NGC 4450 we considered just the rising part of their observed curve, since the declining outer part is inconsistent with the 21 cm data from reference 8;
7: van Driel & van Woerden 1991 (R);
8: Guhathakurta et al. 1988 (R);
9: Warmels 1986 (R);
10: Broeils & van Woerden 1994 (R).

2.3. Model rotation curves

To determine the shape of the RC for our galaxies we assumed: a) optical transparency (which was also assumed for the surface brightness decompositions); and b) the M/L ratio of bulge and disk to be constant within each component. The circular velocity in the equatorial plane of an axisymmetric mass distribution is given by


where [FORMULA] is the gravitational energy at radius r. In this case [FORMULA] is the sum of the contributions of bulge and disk and, when necessary, also of a dark halo.

2.3.1. The bulge

The circular speed of an axisimmetric ellipsoidal distribution of matter is given by (Binney & Tremaine 1987):


where [FORMULA] is the bulge M/L ratio, [FORMULA] the luminosity density at distance a from the center in the equatorial plane, and [FORMULA] the constant intrinsic eccentricity. To determine [FORMULA] from the brightness distribution, when [FORMULA] we cannot invert the Abel equation (e.g., Binney & Tremaine 1987) appropriate for spherical distributions. An alternative approach (see Appendix) considers the "strip brightness", [FORMULA], defined as the integral of the bulge brightness distribution along a path orthogonal to the line of nodes, at distance a from the center. Under the hypothesis of transparency, this quantity is independent of the inclination of the galaxy. Since


it is straightforward to determine [FORMULA] (and hence the rotation speed from Eq. 2) once [FORMULA] has been evaluated from the brightness distribution. Eq. 3, in turn, can be expressed either in terms of a parametric or np profile respectively, depending on the kind of decomposition considered.

2.3.2. The disk

In the case of an infinitely thin exponential disk, the rotation speed is given by (Freeman 1970):


where [FORMULA], [FORMULA] and [FORMULA] are the modified Bessel functions of first and second kind, respectively; [FORMULA] is the central disk surface density, and [FORMULA] the disk scale length.

For the np disk we have adopted the following expression for the circular speed (Toomre 1963; Kent 1986):


where [FORMULA] is again the disk surface density, and


Here [FORMULA] and [FORMULA] are the complete elliptic integrals of first and second kind, respectively. Equation 5 can be rewritten in the more condensed form


which we have used in our computations.

2.3.3. The dark halo

We have considered two different spherical distributions of matter to model the dark component. The first is a pseudo-isothermal sphere (Kent 1986) with density profile:


and circular speed:


where [FORMULA], the asymptotic circular speed, and [FORMULA], the scale length, are parameters of the distribution. The second is a sphere of constant density [FORMULA], with


where [FORMULA] is the only parameter of the distribution. This distribution, with [FORMULA] linearly increasing with r, is suited in some cases to represent the inner part of the halo RC; it is in fact the limit of Eq. 10 for [FORMULA]. We have adopted it whenever measurements of circular speed did not extend beyond the optical radius. In most cases the two models are mutually exclusive, since the latter cannot fit a flat RC, whereas [FORMULA] is not constrained if the data are restricted to the rising part of the RC.

2.4. Modified Newtonian dynamics

To test the predictions of modified Newtonian dynamics (MOND: Milgrom 1983; Sanders 1990; Begeman et al. 1988 - hereafter BBS), we have altered our model RC's according to the prescriptions given by BBS. In particular, the relation between Newtonian ([FORMULA]) and modified ([FORMULA]) acceleration




can be used to derive the modified expression for the circular velocity.

Although in most cases the available RC's are not very extended, in six cases (namely NGC 1024, NGC 2841, NGC 3593, NGC 4698, NGC 5879, and IC 724), the critical acceleration ([FORMULA] cm s-2) is achieved within the radius sampled by the RC. These objects therefore can provide a test for MOND.

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© European Southern Observatory (ESO) 1998

Online publication: October 21, 1998