3 Kinematics

In the H$\alpha$ intensity map in Fig. 2 we can pick out the brightest H II regions, which are also visible in the channel maps (Fig. 1), which do not show special features, apart from those associated with the normal rotation pattern of the galaxy. In the rest of this section we will study in detail the velocity and velocity dispersion maps, and derive the rotation curve.

3.1 The velocity map

The velocity map at the highest angular resolution (1.5 $^{\prime \prime }$) is shown in Fig. 3, calculated as described above using the MOMENTS program in GIPSY. In this map we show the overall velocity field in H II emission: the first moment of the H$\alpha$ emission for each pixel with measurable emission.

Figure 4: Low resolution velocity map for NGC 3359 (16 $^{\prime \prime }$) presenting more clearly the global kinematics of the galaxy. The dynamical centre is marked with an asterisk and isovelocity contours are overlaid on the velocity map. See the text for details of how the map was produced

In practice, the global kinematics of the galaxy can be seen more clearly in the map smoothed to 16 $^{\prime \prime }$ resolution shown in Fig. 4, in which isolevels are overlaid on a gray scale representation. The velocity field is generally quite regular although there are some deviations which we will deal with below. In a number of positions on the map, we can detect kinks in the otherwise regular contours, which are recognizable as due to streaming motions related to the behaviour of a density wave in the region of a spiral arm. From the deviations in the isolevels we can estimate amplitudes along the line of sight for these motions, which correspond to $\sim$ 50 km s^-1 in the plane of the galaxy disc, deprojected using an inclination angle of 53$^\circ$.

Although these values are quite high, they are not outside the range of values found in other galaxies, (Visser 1980 for M 81; Rots 1990 for M 51; Knapen et al. 1999 for M 100 and Knapen 1997 for NGC 3631). As well as these deviations in the external part of the disc there are others of greater amplitude which dominate the internal part of the disc and which can reach values as high as 35-40 km s^-1 projected along the line of sight. Below we show that these are due to non-circular motions around the bar. Ball (1986) found such motions in H I in the same zone.

The velocity field shows also an interesting feature in the immediate NW and SE of the centre of the bar, which are twin peaks in the H$\alpha$ emission of the galaxy. However the residual velocities in these peaks do not have identical projected values (in the plane of the sky); in the NW peak the residual velocities reach 45 km s^-1, but 25 km s^-1 is the maximum residual value in the peak located in the SE. This feature does not fit the global disc kinematics and it may be due to a velocity component not in alignment with the bar, even perpendicular to it. A study of the dynamical properties of NGC 3359 will include a more detailed study of this feature (Sempere & Rozas 2000).

3.2 The rotation curve

To obtain the rotation curve we use the velocity maps at different angular resolutions, following the procedure described by Begeman (1989), in which the galaxy is divided into a series of concentric elliptical annuli, each described by a set of parameters: i (inclination angle), PA (position angle of the major axis), and $V_{{\rm c}}$ (rotational velocity). Additional parameters include the systemic velocity of the galaxy, $V_{{\rm sys}}$, and the coordinates of the centre of each annulus: $x_{\rm c}$and $y_{\rm c}$. These parameters are then fitted using a least squares algorithm, which uses the function:

$$V = V_{{\rm sys}} + V_{{\rm c}} {\rm cos}(\theta) {\rm cos}(i)$$ (1)

where $\theta$ is the azimuthal angle in the plane of the galaxy measured from the major axis. The points within each annulus are weighted by cos$(\theta)$, and we eliminated all the points within 15$^\circ$ of the minor axis, since the errors in deprojecting these points onto the major axis are unacceptable.

In the first instance, to obtain the curve we used the maps smoothed to low resolution (16'' and 10'') to fit the position of the kinematic centre (as this depends only on the symmetry of the velocity field), using in this procedure fixed values of i = 53$^\circ$, PA = -10$^\circ$, and $V_{{\rm sys}}=1008$ km s^-1 (all the values taken from RC3). The results were: R.A. (2000) = 10h46m 35.55s ($\pm$ 0.02 s) and dec. (2000) =  63d13m26.1s ($\pm$ 0.3''). This position was defined using the annuli of radius bigger than 40 $^{\prime \prime }$, in which the number of independent points for a fit was large, giving rise to small errors in the fit. The kinematic centre coincides with the optical centre in R.A. but there is a shift in dec. of 4 $^{\prime \prime }$.

Figure 5: Rotation curve (obtained from the 6'' resolution data cube) for NGC 3359 from H$\alpha$ velocities (upper graph) and computed major axis position angle (lower graph)

The second step was a fine tuning of the system velocity of the galaxy. To do this, we fixed the position of the kinematic centre, found above, and the inclination angle, deriving a satisfactory value for the systemic velocity of 1006.8 ($\pm$ 0.3) km s^-1, very similar to the value found by Ball (1986) from H I, of 1009 km s^-1, and to the previous optical determination of 1008 km s^-1 given in RC3.

As an initial choice, we use the inclination angle of 53$^\circ$ found in the RC3 catalogue. Since the inclination of the galaxy is not especially low, we made a series of test fits using sets of values of i and $V_{{\rm c}}$, and finding inclination angles between 48$^\circ$ and 58$^\circ$. The best fits, in fact, coincided with the value of 53$^\circ$ given in RC3, in fair agreement with those used by Ball (1986) and Gottesman (1982) of 51$^\circ$.

As there are no signs of a warp in this galaxy, we can use a constant value of i for the whole disc; in any case the effect of using a different value is not to change the shape of the rotation curve but to rescale the values of $V_{{\rm c}}$. Using the value of i = 53$^\circ$, we find values for $V_{{\rm c}}$ at large galactocentric radii of 145 km s^-1, which fits within the range of values of the synthetic curves produced by Rubin et al. (1985) for a galaxy of this morphological type and luminosity.

We then used the same low resolution (16 $^{\prime \prime }$) map to derive an initial rotation curve: the values of $V_{{\rm c}}$ and PA fits for each radius using a radial interval of 8 $^{\prime \prime }$ between annuli.

Figure 6: Rotation curve for NGC 3359 derived in the present study superposed on that obtained by Ball (1986) from H Iobservations

Figure 7: Position-velocity diagram along the major axis of NGC 3359. Superposed is the inner portion of our computed mean rotation curve

Afterwards rotation curves were derived for the whole disc, and separately for the approaching and receding halves, using the maps at different resolutions, and employing widths for the annuli of half the resolution limit for the corresponding map. Following the same procedure as for the low resolution map, we checked the values for the kinematic centre position and the systemic velocity, finding the same values as before, and then adjusted the values of $V_{{\rm c}}$ and PA for each of the higher resolution cubes. The final rotation curve (obtained with the 6 $^{\prime \prime }$ resolution map) is shown in Fig. 5, and was obtained after a full analysis to check the effects of fixing or freeing the position angles of the annuli and their widths, and to see what would be the effect of including expansion motion of any of the rings in Eq. (1). The definitive curve was then taken by setting the expansion velocity in the disc to zero, and with annuli of 3 $^{\prime \prime }$ width. The position angle was fixed for radii of less than 20 $^{\prime \prime }$, (to avoid the effects of the relatively strong non-circular motions in this zone) and free at larger radii.

The upper panel of Fig. 5 shows the rotation curve for the whole disc, and separately for the approaching and receding halves, while the lower panel shows the position angle of the major axis v. radius, measured from North through East. Out to a radius of 50 $^{\prime \prime }$ we see the clear coincidence between the curves for the blue-shifted and red-shifted sides of the galaxy, showing that the position of the kinematic-dynamic centre is valid. Out to this radius the curve can be described as that of rigid body motion, with $V_{\rm c} = a + br$, where a = -1 $\pm$ 4, and b = 142 $\pm$ 6 km s^-1 arcmin^-1. The curve is reliable out to 80-90 $^{\prime \prime }$ radius; beyond this range, the paucity of points available to make a fit means that detailed features of the curve may not be valid, but the general trend agrees well with that derived from H I by Ball (1986), and the result is shown in Fig. 6. Although the two curves are similar, we note that our curve shows, on average, values 10 km s^-1higher than the H I curve in the radial range beyond 50 $^{\prime \prime }$ from the centre. In Fig. 7 we show a position-velocity diagram along the major axis. In this figure we have superposed the inner part of the mean rotation curve and the good coincidence between this curve and the gas distribution shown indicates that the curve derived is reasonable.

3.3 Model velocity map, and residual velocity field

The derived values of $V_{{\rm c}}$ and PA have been used to construct an axisymmetric model of the velocity field, with the remaining parameters as specified above: (i, $V_{{\rm sys}}$, $x_{\rm c}$, $y_{\rm c}$). The most external parts of the model velocity field (beyong 80''-90'') have been calculated using the velocity value to which the rotation curve tends at large galactocentric radii. The model is shown in Fig. 8. The strongest deviations from regularity, in a clockwise direction could be artificial, due to a point with a value of PA lower than the true value. The model is quite smooth, though the deviations due to streaming motions across the arms can be see at $\sim$1.5$^\prime$ in from the kinematical centre.

By subtracting the two-dimensional model of the regular velocity field from the observed field we obtain a map of the residual velocity field which is shown in Fig. 9. This map allows us to study the global field of the non-circular components of the motion of the IS gas, which can be either radial or vertical (only by using external information can we try to distinguish between these cases, though this information does exist for key features of the galaxy). The largest residuals coincide with the bar, along which there is a clear velocity gradient. The residuals attain values of up to 35 km s^-1 along the line of sight. This velocity gradient represents a real dynamically induced field of motion in the ionized gas, and is not an artifact of the model subtraction. We have derived residual maps using a range of possible fits to the rotation curve on which the central zone of the model is based, and all of these give rise to the same form of residual field around the bar.

In the disc, the residual velocity field shows a zone of negative values close to the minor axis. This could be due to the exclusion of those points within a 15$^\circ$ angle of the minor axis from the set used to compute the rotation curve. In the residual velocity map we can pick out the spiral arms by the positive residuals due to the streaming motions which we could also identify in the model field. Finally some of the H II regions show a notable residual velocity signature due to their high internal velocity dispersions, (see the next section). These peaks of residual velocity are evidence for expansive motions associated with each of the regions concerned. In certain dynamical models of the most active star forming regions, containing O and B stars, motions of this type, giving rise to "chimneys'' were proposed by Norman & Ikeuchi (1989).

3.4 The velocity dispersion map

The velocity dispersion is the last parameter whose behaviour is described from one of the moment maps of the disc of NGC 3359: in this case the second moment. The original map was obtained using the MOMENTS program in GIPSY, with the values of the Gaussian widths of the emission lines fitted in each pixel. The velocity dispersion map in Fig. 10 is not directly taken from the observations. It has been corrected for the instrumental width, the natural line width, and the thermal line width at each point. These widths must be subtracted in quadrature from the observed width, $\sigma_{\rm obs}$, to yield the width due to the mean internal motions of each region, which we assume to be dominated by turbulence. The expression used in this subtraction is:

$$\sigma_{\rm obs} = \left [ \sigma_{\rm N}^{2} + \sigma_{\rm Ins}^{2}+ \sigma_{\rm t}^{2}+\sigma_{\rm nt}^{2} \right ]^{1/2}$$ (2)

where the natural, instrumental, and thermal widths are termed $\sigma_{\rm N}$, $\sigma_{\rm Ins}$ and $\sigma_{\rm t}$, respectively.

Figure 8: Model rotational velocity map of NGC 3359, computed using fits to a set of inclined annuli (see text for more details). The isolevels have a separation of 10 km s-1

Figure 9: Map of residual velocities for NGC 3359 obtained by subtracting off the two-dimensional model computed from the rotation curve from the observed velocity map. The resolution is 1.5''

The natural width is virtually constant for hydrogen, taking a value of 0.16 Å, which is the equivalent of 3 km s^-1 (O'Dell & Townsley 1988). The instrumental width was calculated knowing the intrinsic width of the laboratory source which was used to produce the calibration cube. After close analysis of this cube we showed that there were no systematic structures due to calibration anomalies, and that the instrumental dispersion map showed a constant width of 16.2 $\pm$ 0.3 km s^-1, which we used for $\sigma_{\rm Ins}$. The thermal width, $\sigma_{\rm t}$ was taken by assuming a temperature of 10⁴ K for the ionized gas (Spitzer 1978; Osterbrock 1989), and is 9.1 km s^-1.

In the velocity dispersion map we can see that the highest dispersions correspond to the most luminous H II regions, and this gives rise to a correlation between the dispersion map and the map of residuals. A more detailed analysis will be needed to quantify these relations, (Zurita et al. 2000, in preparation) but if they are similar to those found in M 100 and M 101 (Rozas et al. 1998a) as we predict from a quick look, we will have further evidence for density bounding in these regions. The mode value for the dispersions in the galaxy is close to 15 km s^-1, as we can see if Fig. 11. This value remains rather constant outside a radius of 30 $^{\prime \prime }$. There are several possible heating mechanisms which might underly this behaviour, including the conversion of galactic rotational energy to random cloud-cloud dispersion through viscosity (Combes & Becquaert 1997). At smaller galactocentric distances the greater concentration of star formation would imply that this source is the dominant heating mechanism.