1 Introduction

Rotation curves derived from neutral hydrogen observations at the outer regions of spiral galaxies unambiguously show that substantial amounts of dark matter are required (Bosma 1978; Begeman 1987, 1989). Any physically reasonable distribution of this dark matter necessitates the presence of at least some of that in the inner optical disc region, contributing in some degree to the total rotation in that region. Unfortunately, from the observed rotation curve and light distribution one cannot a priori determine the ratio of dark to luminous matter (Van Albada et al. 1985). There are arguments, mainly theoretical, that the contribution of the disc has to be maximized, leading to the so called maximum disc hypothesis (Van Albada & Sancisi 1986; Salucci et al. 1991; Sellwood & Moore 1999). On the other hand, observations of disc stellar velocity dispersions (Bottema 1993, 1997) lead to the conclusion that the disc contributes, on average, 63% to the total rotation at the position where the disc has its maximum rotation. This finding is supported by a statistical analysis of rotation curve shapes in relation to the compactness of discs (Courteau & Rix 1999). Every detailed rotation curve of any galaxy may give clues as to the ratio of dark to luminous matter in a galaxy. It has been argued that the observed correlation of features in the rotation curve with features in the photometry excludes a sub maximum disc case (van Albada & Sancisi 1986). Since it was already known that NGC 3992 exhibits such specific rotation curve features, this warranted more detailed observations and a proper decomposition of the curve into the contributions of the galactic constituents.

At least a substantial fraction of spiral galaxies has no bar. Yet an isolated cold stellar disc can never be stable (Ostriker & Peebles 1973). Various criteria for galaxies have been put forward that should be obeyed in order to avoid a bar instability. Toomre's (1964) Q criterion for local stability can be applied for a global situation assuming a minimum Q value for a stellar disc (Sellwood & Carlberg 1984). Already in 1973 Ostriker & Peebles showed by numerical experiments that a substantial spherical dark halo can stabilize a disc. Their criterion states that ${\langle} E_{\rm kin} \rangle / E_{\rm pot} < 0.14$, or the ratio of the average kinetic energy of a disc to its potential energy should be less than 0.14 in order to be stable. As an alternative to a dark halo, a disc can be stabilized by a substantial bulge (Sellwood & Evans 2001). In addition to adding potential energy the bulge also makes the rotation curve flat in the inner regions, creating an ILR which may inhibit the bar formation mechanism (Toomre 1981). Calculations for a specific set of galaxies by Efstathiou et al. (1982) showed that for

$$\varepsilon = \frac{v_{\rm max}}{\sqrt{G M_{\rm disc} /h}} \la 1.1,$$ (1)

the disc was unstable to bar formation. This criterion has recently been reviewed and reanalysed by Syer et al. (1999) for a 3D situation. For discs with a ratio of scalelength (h) to sech-squared thickness z₀ of five and Q = 1.2 at R = 2.4 h they found that the numerical value of 1.1 in Eq. (1) has to be decreased to 0.7. Though this criterion may have its merits, it is difficult to imagine that disc stability does not depend on its stellar velocity dispersion. Hence it may be dangerous to apply it to a randomly observed galaxy.

As one can see, a robust, well established stability criterion does not (yet) exist. Nevertheless, from the criteria just mentioned some general relations can be extracted. A galaxy becomes more unstable when its disc mass to dark matter ratio, or its disc to bulge mass ratio is larger. It is also more unstable when its stellar velocity dispersion is lower. In the extreme one then has a cold, thin, maximum disc situation, which is certainly unstable. Even a disc with h/z₀ of five, a representative value for spiral galaxies (Van der Kruit & Searle 1982), at the maximum disc limit forms a large bar (Bottema & Gerritsen 1997). A fair fraction of galaxies has a bar. At least for the long lived bars it might then be logical to assume that barred galaxies are close to maximum disc while non-barred galaxies are sub maximum, though there is no observational evidence for this assumption. Yet the situation must be more complicated; the stellar velocity dispersion is a factor of importance as is the gas content and formation history of a galaxy.

From basic principles it can be shown that every bar has to end within the corotation radius (Teuben & Sanders 1985). This implies that the angular pattern speed of the bar ( ${\Omega}_{\rm p}$) always has to be smaller than the angular speed at corotation ( ${\Omega}_{\rm cr}$). A bar is defined as fast when its pattern speed approaches the speed at corotation; it is slow when ${\Omega}_{\rm p} \ll {\Omega}_{\rm cr}$. Theoretical arguments related to the observed morphology of bars indirectly lead to the conclusion that bars must rotate fast (Sanders & Tubbs 1980; Athanassoula 1992). Recently, by applying the Tremaine & Weinberg (1984) method it has been demonstrated directly by observations of two galaxies (Merrifield & Kuijken 1995; Gerssen et al. 1999) that the bar is indeed a fast rotator.

Observations of neutral hydrogen gas show that in general there is a depression in the H  I surface density at the position of the bar. This can be explained because gas in a barred potential will experience strong shocks at the leading side of the bar. As a result gas will be transported inwards (Athanassoula 1992) and the bar region gets depleted of gas. The existence of a severe H  I hole then means that for a reasonable amount of time the central region is not disturbed by gas accretion. In addition the bar must be rather long lived in order to transport all the gas to the centre.

A barred disc does not exist on its own but is embedded in a dark halo which should respond to the barred potential in some way. Starting with a fast rotating bar in a nonrotating isotropic dark matter halo Debattista & Sellwood (1998) showed that the bar is quickly slowed down by dynamical friction. At least when the dark halo contributes significantly to the mass in the inner region; for a maximum disc situation this slowing down mechanism has almost disappeared. The initial setup of Debattista & Sellwood is, of course, rather specific. In reality a galaxy will gradually built up and dark matter will acquire rotation by a number of mechanisms (Tremaine & Ostriker 1999). If the dark halo is co-rotating with the bar the dynamical friction process has disappeared. Still it has to be kept in mind that a fast rotating bar cannot exist in a substantial non-rotating dark halo. At this stage let us summarize a number of properties and processes that are important for the existence of a barred structure. Preventing a bar or destroying it can be done by:

• A large non-corotating spherical dark matter (DM) contribution.
• A substantial bulge.
• Large amounts of gas will destroy any triaxiality (Shlosman & Noguchi 1993).
• A high stellar velocity dispersion.

Making or allowing a bar can be done by:

• A limited amount of spherical dark or luminous matter in the inner regions.
• Rotating dark matter.
• Low stellar velocity dispersions or alternatively a thin disc.
• Small amounts of gas (see also Noguchi 1996).

There are a number of neutral hydrogen observations of barred galaxies preferentially of late types. Without trying to be complete we mention NGC 5383 (Sancisi et al. 1979), NGC 4731 (Gottesman et al. 1984), NGC 3359 (Ball 1986), NGC 1365 (Ondrechen & Van der Hulst 1989), NGC 1097 (Ondrechen et al. 1989), NGC 1300 (England 1989), and NGC 1073 (England et al. 1990). Observations of NGC 3992 were carried out by Gottesman et al. (1984, hereafter G84) with the VLA. Their FWHM resolution amounted to 23$\arcsec$ in the spatial direction and 41 km s^-1 in the velocity direction. They noted a drop in the rotational velocities near the end of the disc light and this drop was ascribed to the effect of a truncation of the disc mass. It is claimed that at least some gas was detected in the bar region with a certain radial velocity, although this might be an artifact of the way the data have been interpreted. The velocity field and fit to that resulted in a systemic velocity of 1046 km s^-1, a position angle of 248$\degr$, and an inclination of 53$\degr$. A rotation curve was determined out to a radius of 5$\arcmin$.

In a later study Hunter et al. (1988) made an additional analysis of these observations. Their aim was to explain the observed spiral arm pattern as a result of gas moving in the barlike potential. A model for NGC 3992 was used employing a rigid potential consisting of a dark halo, a disc, a bar, and an oval distortion beyond the perimeter of the bar. It appeared that this model was not able to explain the observed tightly wound, star-formation arms. Instead a strong two armed spiral structure was generated with large pitch angle. Hunter et al. concluded that: "either the model is incomplete or, other, nondynamical processes cause the spiral arm pattern''.

NGC 3992 was also observed in H  I by Verheijen (1997) and by Verheijen & Sancisi (2001) as part of a study of the Ursa Major cluster. In fact, NGC 3992 is one of the most massive members of this small and non-concentrated cluster. Because of the limited integration time Verheijen could only study this galaxy at a resolution of one arcminute. Consequently the derived rotation curve was sparsely sampled. Yet, these observations showed that the rotation curve exhibits some specific features and that neutral hydrogen gas was present rather far beyond the optical edge. When comparing the derived M/L ratios of the galaxies in the Ursa Major cluster, the M/L ratio of NGC 3992 stands out, in the sense that it is a factor of two larger than the average value of the cluster galaxies. These facts motivated a much longer observation aiming to shed more light on several matters. The main aim of this project was to derive a high quality and extended rotation curve for a barred galaxy. Of this rotation curve a decomposition can be made and the result can be compared with that of other galaxies. For example, is there evidence for a specific luminous to dark mass ratio and does that differ from the ratio for non-barred galaxies? Can the large M/L ratio be confirmed and if so can it be explained? More generally, the number of well determined rotation curves is still limited and extension of the sample allows a better (statistical) analysis of galaxy parameters.

For convenience a high quality photograph of NGC 3992 is presented in Fig. 1 and a listing of the main parameters of the four galaxies is given in Table 1.

 

 

Table 1: Galaxy parameters.

NGC 3992
Hubble type	SBb(rs)I	a
Brightness (in B)	10.86 mag.	b
Brightness (in I)	8.94 mag.	b
Opt. incl. ( q0 = 0.11)	57$\degr$	b
Opt. PA major axis	68$\degr$ (=248$\degr$)	b
PA major axis bar	37$\degr$	c
Deprojected bar length	145$\arcsec$	d
Scalelength	undef
Total H  I mass	$5.9 \times 10^9$  ${M}_{\odot}$	d
21 cm cont. flux	43.2 mJy	d
UGC 6923
Brightness (in B)	13.91 mag	b
Brightness (in I)	12.36 mag	b
Opt. incl. ( q0 = 0.11)	66$\degr$	b
Opt. PA major axis	354$\degr$	b
Scalelength (in I)	20$\farcs$9	b
Total H  I mass	$0.64 \times 10^9$  ${M}_{\odot}$	e
21 cm cont. flux	<2.6 mJy	b
UGC 6940
Brightness (in B)	16.45 mag	b
Brightness (in I)	15.44 mag	b
Opt. incl. ( q0 = 0.11)	75$\degr$	b
Opt. PA major axis	135$\degr$	b
Scalelength (in I)	8$\farcs$52	b
Total H  I mass	$0.16 \times 10^9$  ${M}_{\odot}$	e
21 cm cont. flux	<1.3 mJy	b
UGC 6969
Brightness (in B)	15.12 mag	b
Brightness (in I)	14.04 mag	b
Opt. incl. ( q0 = 0.11)	73$\degr$	b
Opt. PA major axis	330$\degr$	b
Scalelength (in I)	11$\farcs$65	b
Total H  I mass	$0.44 \times 10^9$  ${M}_{\odot}$	e
21 cm cont. flux	<3.8 mJy	b

a Sandage & Tammann (1981).
b Verheijen (1997).
c Measured from photograph.
d This paper.
e Paper II.

The distance to the UMa cluster as a whole and to NGC 3992 in particular has not yet been determined precisely. Sakai et al. (2000) give a distance to the UMa cluster of $20.7 \pm 3.2$ Mpc following from a Tully-Fisher analysis using the cepheid distances to local galaxies. On the other hand, for a similar analysis, Tully & Pierce (2000) derive a distance of 18.6 Mpc, probably with the same error as that of Sakai et al. In a recent re-evaluation of the HST distance scale project (Freedman et al. 2001) the distances to the local calibrator galaxies have decreased by $\sim$5% and consequently the distances to UMa of 20.7 and 18.6 Mpc should also be decreased by that amount. As for now, a distance of 18.6 Mpc seems reasonable and has been adopted in the present paper. This distance differs, however, from the 15.5 Mpc used in earlier studies of the UMa cluster by Tully et al. (1996) and by Verheijen (1997).

Figure 1: Optical image of NGC 3992 reproduced from the Carnegie atlas of galaxies (Sandage & Bedke 1994). The photograph was taken with a blue sensitive emulsion. Total size on the sky is 10 $\farcm{7} \times {7}\farcm$7, North at top, East on left.