6 The rotation curve

The rotation curve is determined by fitting a tilted ring model to the velocity field (Begeman 1989). It is assumed that the H  I gas can be described by a set of concentric rings. Each ring has a certain position, systemic velocity, rotation velocity, and two orientation angles: inclination and position angle. The widths and separations of the rings were taken to be 10$\arcsec$ for the full resolution data and 20$\arcsec$ for the data with resolution of 30$\arcsec$. When fitting the model velocity field to the observed field a weighting factor has been taken proportional the the cosine of the angle measured from the major axis.

An iterative strategy was followed. First a guess has been made of the rotation, inclinations, and position angles for all rings. With these parameters held fixed the central positions and the systemic velocities of the rings were determined. These parameters appeared to be nicely constant as a function of radius and were fixed at (RA, dec, $v_{\rm sys}$) of (11$^{\rm h}$55$^{\rm m}$0$\fs$59, 53$\degr$39$\arcmin$10$\farcs$9, $1049 \pm 2$ km s^-1). The optical position of the bulge was measured and lies within 0$\farcs$2 from the kinematic position while both positions can be determined with an accuracy of approximately 2$\arcsec$. Consequently the bulge is exactly at the kinematic centre even though that was determined by an extrapolation inwards because of the H  I hole. This proves the overall regularity and symmetry of the velocity field.

Further steps in the iteration are illustrated in Fig. 9. As a first step the remaining parameters, PAs, inclinations, and rotational velocities were left free and the resulting position angles were considered. As can be seen in Fig. 9, both for the full resolution and the smoothed data the position angle slightly changes as a function of radius. In addition, for the full resolution data there is a wiggle superposed which is caused by the spiral arm streaming motions. In first instance a constant position angle of 248$\degr$ was assumed and rotation curve determined. However, for that case the residual velocity field showed a large scale systematic pattern which could be attributed to a wrong position angle. Therefore it was decided to adopt a position angle which is slightly changing as a function of radius, as indicated by the dashed line in Figs. 9b and 9d. Considering the errors on the data points this change seems indeed real. Moreover, in this case the systematics in the residual velocity field disappear. As a second step the position angles were fixed and the inclinations and rotational velocities were left as free parameters. The fitting procedure was rerun and resulting inclinations were considered. For the full resolution data there appears to be a slight increase in the inclination from 57 to 60$\degr$ between radii of 280 to 320$\arcsec$. However, for the smoothed data this increase is not present and this effect in the full resolution data probably has to be ascribed to patchiness and small irregularities in the velocity field at high resolution. Over all, the data are consistent with a constant inclination of 57 $\degr \pm 1\degr$ indicated by the dashed line in Figs. 9a and 9c. This constant value has been adopted. As a third and final step in the iteration, having the PAs and inclinations fixed, the rotation velocity was fitted of which the result is displayed in Fig. 9, lower panel.

Figure 9: Determination of the rotation curve by a tilted ring fit; filled circles for full resolution data ( a), b), and e)), open circles for smoothed data ( c), d), and e)). In first instance the orientation angles and rotation velocity were all left as free parameters producing position angles as a function of radius ( b) and d)). These PAs were fixed at the values indicated by the dashed line and the fitting procedure was rerun producing the inclinations ( a) and c)). Fixing these at 57$\degr$ finally gives the rotation curves ( e)).

At radii between 270 and 320$\arcsec$ which is at the outermost radii where the full resolution observations still give rotational values, the full resolution rotation velocities are lower by some 6 km s^-1 compared to the smoothed data. A possible explanation is that at those positions the full resolution only pics up the brightest emission regions at the spiral arms. Velocities over there might deviate somewhat from the average velocities because of streaming motions associated with the spiral arms. Also at these radii the stellar disc ends, which might have some influence on the radial velocities when changing from a situation of stars plus gas to pure gas arms. Anyway, from 240$\arcsec$ outwards the smoothed data have been adopted as best representation of the rotation. Inwards from 240$\arcsec$ until 70$\arcsec$, where the hole begins, the full resolution rotation is determined with sufficient certainty. As a summary the rotation curve data are given in Table 3.

 

 

Table 3: The rotation curve of NGC 3992.

R	$V_{\rm rot}$	${\varepsilon}_{\rm vrot}$	R	$V_{\rm rot}$	${\varepsilon}_{\rm vrot}$
($\arcsec$)	(km s-1)	(km s-1)	($\arcsec$)	(km s-1)	(km s-1)
70	234	6.8	210	272	3.1
80	245	7.8	220	270	4.3
90	253	7.3	230	264	3.4
100	259	6.0	240	260	5.0
110	261	4.3	260	252	9.0
120	264	3.2	280	249	10.4
130	267	3.3	300	247	5.5
140	267	3.5	320	244	3.2
150	267	3.4	340	251	5.5
160	269	3.4	360	252	8.6
170	269	3.2	380	247	5.9
180	270	3.1	400	249	8.7
190	273	3.1	420	248	3.5
200	273	3.1	440	259	4.3
Pos. of dynamical centre
RA (1950)			11$^{\rm h}$55$^{\rm m}$0$\fs$59
Declination (1950)			53$\degr$ 39$\arcmin$ 10$\farcs$9
$V_{\rm sys}$ (Hel.)			1049 $\pm$ 2 km s-1
Inclination			57$\degr$ $\pm$ 1$\degr$
PA			245$\degr$ < PA < 255$\degr$

The least squares fitting method gives errors, but these are only formal errors, which are not always a good representation of the true deviation from the data. To come up with a more realistic error of the rotational velocity, the fitting procedure has been repeated for the receding and approaching side of the galaxy separately. Positions and orientation angles were kept fixed at the same values as for the whole galaxy and rotation velocities were determined. The difference in rotation of the two sides gives a better representation of the true error. The final error is then given by the quadratic sum of the formal fit error plus half the difference between the two sides, plus the error generated by an error of 1$\degr$ in the inclination. These values are also given in Table 3 and in Fig. 9. To illustrate the reliability and consistency of the method, in Fig. 10, the rotation curve, converted to radial velocities is overplotted on a position-velocity cut through the smoothed data along the major axis. Note that the globally determined rotational values may deviate slightly from the local kinematics given in x,v diagram.

The present rotational parameters have been compared with the ones of Verheijen & Sancisi (2001). They have produced a rotation curve from the velocity field at a resolution of only 60$\arcsec$ and therefore they have a rather course sampling of the rotation curve. Only five independent and reliable PA and inclination values could be determined between 160 and 330$\arcsec$ from which a constant value of the orientation angles was concluded. The rotation curve deduced goes out to 400$\arcsec$ and is qualitatively the same as determined presently but shows a lot less details and features.

A comparison was made with the observations and derived kinematics in the paper of G84. They have observed NGC 3992 with the VLA at a FWHM resolution of 23$\arcsec$ spatially and of 41.4 km s^-1 in velocity. Their rotation curve extends out to a radius of 300$\arcsec$ and is similar in shape as the present curve. There is a difference, however; G84 determined an inclination of 53$\fdg$4 which is smaller than our value of 57$\degr$. Consequently G84's rotational velocities are a bit larger. It is claimed, though uncertain, that there is a small amount of gas with associated radial velocities detected in the central hole. In our opinion this detection is not real and is probably an artifact of the employed moment method to construct the velocity field.

Figure 10: The rotation curve converted to radial velocities overplotted on a position - velocity map along the major axis. This illustrates the consistency of the tilted ring fitting method. Contour levels are at -1.65 and -0.82 K (dashed), and at 0.82, 1.65, 2.93, 6.59, and 9.89 K, and the resolution (30 $\arcsec \times 33.3$ km s-1 FWHM) is indicated in the lower left corner.

A model velocity field has been created assuming the fitted rotation curve parameters. This field was subtracted from the observed velocity field producing the residual velocity field which is depicted in Fig. 8. As can be seen, in the full resolution case there are some residuals associated with the streaming motions along the spiral arms. Other systematic residuals are not present demonstrating that a proper rotation curve fit has been made. The random deviations are generally smaller than 10 km s^-1 except for a few patches bordering on the central hole. Could these deviating patches have been generated by the central bar?