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The pattern speed of the bar in NGC 936

We have used the Tremaine-Weinberg method to measure the angular speed of rotation for the bar in the SB0 galaxy NGC 936. With this technique, the bar's pattern speed, Omega_p, can be derived from the luminosity and stellar-kinematic information in long-slit spectral observations taken parallel to the major axis of the galaxy. The kinematic measurement required is the mean line-of-sight velocity of all stellar light entering the slit. This quantity can only be calculated reliably if any asymmetry in the shape of the broadening function of the spectral lines is also measured, and so we present a method which allows for such asymmetry. The technique also returns a true measure of the RMS uncertainty in the estimate. Application of the analysis to a set of long-slit spectra of NGC 936 returns four separate measures of Omega_p which are mutually consistent. Combining these data produces a best estimate for the bar pattern speed of Omega_p = 60 +/- 14 km/s/kpc (assuming a distance of 16.6 Mpc). This result refines the only previous attempt to make this measurement, which yielded an estimate for Omega_p in NGC 936 of 104 +/- 37 km/s/kpc (Kent 1987). The new measurement places the co-rotation radius just beyond the end of the bar, in agreement with theoretical calculations.


INTRODUCTION
Bar-like structures are found at the centres of approximately 30 per cent of disk galaxies (Sellwood & Wilkinson 1993). It is believed that these features rotate rapidly and maintain their shapes over many rotation periods. Hence, a de ning parameter of any galactic bar is the angular velocity at which it rotates, its \pattern speed" p. Orbit calculations show that a bar can only be built self-consistently if it lies entirely within its co-rotation radius { the point at which the pattern rotates at the same speed as a star on a circular orbit at that radius (e.g. Contopoulos 1980). Indeed, N-body simulations con rm that long-lived bar structures can form in disk galaxies, and that they extend only to near the corotation radii of their host galaxies (e.g. Combes & Sanders 1981, Sellwood 1981).
If we are to relate these theoretical ideas to observations of real galaxies and model their properties in any detail, then we must measure their pattern speeds. Unfortunately, this quantity has proved somewhat elusive to observation. In the case of simulations, the pattern speed can be measured directly by following the motions of the bar over time, but we do not have this luxury for a snapshot view of a real galaxy. Several methods have been suggested for estimat-ing the pattern speed from the available data. For example, morphological features around the bar such as rings have been associated with the various resonant locations (such as the Lindblad and 4:1 resonances), and this information has been inverted to infer the fundamental parameters of the bar including p (see, for example, Combes & Gerin 1985, or Buta 1986. Where there are extensive observations of gas kinematics, it is possible to try to match the observations to hydrodynamic models with varying pattern speeds; the model that ts best then provides an estimate of p (e.g. Sanders & Tubbs 1980). Although these approaches seem to work quite well see Sellwood & Wilkinson (1993) for an extensive review, or Sempere et al. (1994) for a recent application to NGC 4321], some more direct model-independent measure of the pattern speed is clearly highly desirable. Tremaine & Weinberg (1984, henceforth TW) derived just such a measure by invoking the continuity equation to obtain p from the kinematic properties of a barred galaxy which are accessible to observation. If the galaxy lies at an inclination i, and if cartesian x and y axes are aligned parallel to its apparent major and minor axes such that the nucleus is located at x = x0, then p can be calculated from the relation (1) where the integrals are performed along any cut parallel to the apparent major axis, vlos(x) is the mean line-of-sight velocity of the stars at position x on the slit, v0 is the systemic velocity of the galaxy, and angled brackets denote the average value along the slit weighted by l(x), the luminosity of the stars at each point. All of the quantities on the right-hand side of this equation can, at least in principle, be measured directly from observation: the integrated luminosity in a long-slit spectrum parallel to the major axis, weighted by position x along the slit, allows us to calculate hxi; and the Doppler shifts in the spectral lines in the same spectrum can be used to measure hvlosi. The inclination i can be measured from photometric observations if it is assumed that the galaxy becomes intrinsically circular at large radii.
In practice, it has proved di cult to apply this formula to observational data. TW showed that the method could not be applied reliably to Hi observations, at least partly because Hi gets converted into H2 and stars in galaxies, and so the equation of continuity is not applicable. They suggested that optical observations of SB0 galaxies might prove more fruitful, as there is little gas or star formation in such systems to confuse the issue. Kent (1987) made the rst attempt to follow up this suggestion by obtaining images and long-slit spectra of the SB0 galaxy NGC 936. This galaxy, shown in Fig. 1, is at an ideal inclination of 41 (Kormendy 1984), and its bar position angle lies 45 from the major axis, which optimizes the signal in the integrals in equation (1). The galaxy also appears to be nicely symmetric about its center, and its outer isophotes are elliptical with constant axis ratio, consistent with a bar embedded in an otherwise axisymmetric disk. Kent obtained an estimate for NGC 936's pattern speed of p = 104 37 km s 1 kpc 1 . This value places the co-rotation radius within the bar, contrary to the predictions of simple bar models, although the large uncertainty means that the measurement is still just consistent with a co-rotation radius beyond the end of the bar.
Kent's spectroscopic data were obtained using image tube observations recorded on photographic plates, and he expressed the desirability of obtaining CCD spectra of higher quality in order to reduce the errors. A further large gain in the signal-to-noise ratio of the spectral data can be obtained by noting that vlos(x) only occurs inside the integral in equation (1). The long slit data can therefore be coadded along the slit to produce a single high signal-to-noise ratio one-dimensional spectrum, and the mean Doppler shift in the spectral lines then gives exactly the weighted value of hvlosi required in the TW formula. The di culty in this approach is that the spectral line pro les in the co-added spectrum will be broadened in a signi cantly asymmetric manner, re ecting the distribution of vlos(x) along the slit. The mean velocity cannot, therefore, be extracted by simple techniques such as cross-correlation (Tonry & Davies 1979) which assume that any broadening of the galaxy's spec- Figure 1. Contour plot of a I-band CCD image of NGC 936 obtained with the FLWO 48-inch telescope. The contours are spaced with 0.5 magnitude intervals. The slit positions for the spectral observations are overlayed: the solid line indicates the part of the slit that was co-added to produce the integrated mean velocity, and the dotted line shows the region that was used to de ne the background spectrum.
tral lines is symmetric. Recently developed techniques (e.g. Franx & Illingworth 1988, Bender 1990, van der Marel & Franx 1993, Kuijken & Merri eld 1993 permit us to obtain the complete shape of the Doppler broadening in a galaxy spectrum, and so we can now allow for asymmetric broadening to obtain the requisite unbiased estimate for hvlosi. ?
In this paper, we present the analysis of new CCD spectral observations of NGC 936 which allow us to calculate an improved estimate for its pattern speed. Section 2 describes the data, and the application of a stabilized spectral deconvolution method to obtain an unbiassed estimate for the weighted mean value of hvlosi. We also show how this method can be used to obtain a direct measure of the uncertainty in hvlosi, and apply the technique to our data set. Combining this analysis with the derived value of hxi provides a wellconstrained measure of the pattern speed in NGC 936, and we discuss the implications of the result in Section 3.

OBSERVATIONS AND ANALYSIS
We obtained long-slit spectra of NGC 936 on the night of 1993 December 9 using the Red Channel Spectrograph on ? As Kent (1987) recognised, this e ect was also present when he analyzed the full two-dimensional spectrum, since it is unlikely that the stellar motions at any single point in a bar will result in a symmetric velocity distribution. The values for v los (x) derived by cross-correlation are therefore likely to be systematically incorrect. However, with the lower signal-to-noise ratio of the non-co-added spectra, it is impossible to correct reliably for any such asymmetries. the Multiple Mirror Telescope at Mt. Hopkins, Arizona. The 12x8mmt CCD was used as a detector, and the spectrograph was con gured with a 1200 lines per mm grating and a 1:25 arcsec 3 arcmin slit, which provided a spectral resolution of 2.6 A (FWHM). The CCD pixels each correspond to 0.7 A, and so the instrumental broadening function is adequately over-sampled.
The spectral region near the Mg b triplet of lines ( 5170 A) was observed. The slit was aligned to be parallel to the major axis of the outer disk isophotes at a position angle of 135 (Kent 1987), and exposed at various o sets from the nucleus (see Table 1). The telescope was o set perpendicular to the slit position angle, ensuring that the zero point of position along the slit did not change between exposures: because of the near-perfect NW-SE alignment of the outer disk isophotes, these o sets were simply achieved by moving the telescope equal amounts N and E or S and W. To guard against loss of positional accuracy in these blind o sets, we con rmed that a nal move back to the nucleus realigned the galaxy at its original position with respect to the slit.
Integration times were chosen so that the total signal obtained for each slit would be comparable: thus the outer, fainter positions were observed for somewhat longer (see Table 1). The ve integrations were taken in sequence, with exposures of an arc lamp for wavelength calibration bracketing each one. Dome ats and biases were taken in the afternoons before and after the observations, and exposures of the twilight sky were taken in the early morning.
Standard data reduction was performed using iraf. The spectra were corrected for bias and at eld variations. The bracketing arc lamp exposures were used to determine the wavelength scale of each galaxy spectrum, after which the data were rebinned on to a logarithmic wavelength scale running from 4760 A to 5660 A. The twilight sky exposures were used to determine the slit vignetting pattern on the detector, and this e ect was then corrected for. In order to remove the night sky spectrum from the data, the spectra in the outermost 10 arcsec of slit were subtracted from the remainder of each frame. While there is still some ux from the galaxy at these radii, its contribution is quite minor; varying the exact region from which the sky background spectrum was extracted made no signi cant di erence to any of the results.
The resulting reduced data frames were then analyzed as described below in order to yield the two averaged quantities required for each slit o set in the TW formula equation (1)].

Mean positions along the slit
In order to calculate hxi, the total ux in the slit between wavelengths of 4970 A and 5360 A was calculated as a function of position x along the slit by simply summing the twodimensional data in the spectral direction. The resulting slit luminosity pro les are shown in Figure 2. Mean positions were calculated from these pro les, and are marked on the gure and tabulated in Table 1. The statistical errors in hxi are totally negligible compared to those in hvlosi.

Mean line-of-sight velocities
The second quantity required for the TW analysis, the mean luminosity-weighted line-of-sight velocity of the stars that illuminate the slit hvlosi, is rather less straightforward to calculate. For each slit position, we need to extract the Doppler broadening function of the spectra after they have been coadded in the spatial direction; the value of hvlosi is then just the mean value of this distribution. As a rst step, we therefore summed the spectra in each slit which lay less than 70 arcsec from the galaxy's minor axis.
We measured the broadening function of each of these co-added spectra using the Unresolved Gaussian Decomposition (UGD) algorithm (Kuijken & Merri eld 1993). Brie y, this technique involves a simple least squares t to the continuum-subtracted galaxy spectrum G(ln ). The model that we t consists of a sum of Doppler-shifted copies of a single component spectrum S: The parameterization in terms of logarithmic wavelength turns the Doppler shift into a constant o set, and velocity broadening into a linear convolution. The spectrum S is formed by continuum-subtracting a stellar template spectrum and then broadening it by convolving it with a Gaussian of dispersion . The coe cients aj are optimized to give the closest match to the observed galaxy spectrum. The best-t model is then the convolution of the original stellar template spectrum with the broadening function, By choosing the separation of the gaussians' means, vj, to be slightly less than 2 , we can model galaxy spectra which have been doppler broadened by any function that is smooth on the scale of . As well as smoothness, we place two further constraints on the models. Firstly, the broadening function must be nonnegative at all velocities. This requirement arises from the physical requirement that the stars in the galaxy all make positive contributions to the total luminosity; it e ectively suppresses high frequency noise in the derived broadening function. This linear constraint on the solution converts the least-squares analysis into a quadratic programming problem, which can be solved e ciently even for large numbers of components using a variation of the simplex algorithm. The algorithm also returns a matrix with components Cij, the covariance between the tted coe cients ai and aj.
The second imposed constraint is that we use gaussians in only a limited range around the systemic velocity of the galaxy. We thus avoid spurious high-velocity features in the broadening function which might arise from crosstalk between di erent spectral lines. In order to choose the appropriate range for the broadening function to be non-zero, we have adopted a variant of the 3 clipping algorithm which has previously been applied to velocity distributions of discrete objects such as galaxies in clusters (e.g. Yahil & Vidal 1977). We rst t a single gaussian broadening function to the data to obtain a crude estimate for the dispersion of the distribution, est. We then carry out the full velocity distribution analysis using ve gaussian components. We place the central component at the systemic velocity of the galaxy, v0, and position the other four components uniformly on either side so as to cover the range between v0 3 est and v0 + 3 est. The widths of the gaussians are set to approximately 0.6 times the separation between components. The adopted parameters are given in Table 1. Note that the symmetric choice of values for vj about the systemic velocity of the galaxy avoids biasing the derived means systematically: the outer parts of the galaxy, which are axisymmetric, should have velocity distributions which are symmetric about the systemic velocity, and our choice of components allows these outer parts of the slit to contribute equally to the broadening function.
The resulting broadening functions are shown in Fig. 3. Clearly, the positivity constraint and 3 clipping have resulted in distributions from which robust estimates of the Figure 3. Line pro les for the coadded spectra for each o set position. The error bars, while correlatedbetween adjacent pixels, are the 1-error on the best-t velocity distribution with the smoothing listed in Table 1. The mean of each distribution is shown with a vertical dashed line. mean can be obtained. The functions are also obviously nongaussian, and slits located on opposite sides of the galaxy show oppositely skewed pro les as expected in an equilibrium system.
Given the best-t amplitudes, ai, for the gaussian components and their covariance matrix elements, Cij, we can calculate the mean velocity of the broadening function and its error. The mean velocity is simply hvlosi = X i aivi X i ai: (4) Its error can be calculated by noting that the covariance between two linear combinations of the ai's is given by The pattern speed of NGC 936 5 Using the result, valid for small variances, that Var(ln X) = X 2 Var(X), and setting T = P i aivi, B = P i ai, we nd that Var(hvlosi) = hvlosi 2 Var(lnhvlosi) = hvlosi 2 Var (ln T ln B) = hvlosi 2 Var(ln T ) 2Cov(ln T; ln B) + Var(ln B)] = B 2 Var(T) 2hvlosiCov(T; B) + hvlosi 2 Var(B) : (6) By adjusting the velocity zeropoint so that the mean velocity is zero, (an operation which does not a ect the variance of hvlosi), the last two terms can be made to vanish, leaving us with the equivalent but more compact expression, We can thus propagate the errors on the t to the galaxy spectrum through the entire analysis to derive the uncertainty in the mean line-of-sight velocity.
We have applied this analysis to the ve co-added spectra of NGC 936. The resulting mean velocities and their corresponding uncertainties are tabulated in Table 1.  (1) requires that these points lie on a straight line whose slope is p sin i. Figure 4 shows that the points do, indeed, approximately follow a straight line. A linear regression on these data gives a best-t slope of p sin i = 3:1 0:75 km s 1 arcsec 1 : (8) The quality of the straight-line t in Fig. 4, and the reasonable value of 2 (1.6 with 3 degrees of freedom) imply that the observations from di erent slit positions are all consistent with the single bar pattern speed of equation (8). Further, we can use the 2 statistic to conclude that the error analysis returns a realistic measure of the uncertainty in each of the estimates of hvlosi.

The pattern speed
It is also heartening that essentially identical slopes are derived by just considering the data from the pair of slit positions 14 arcseconds from the major axis or the pair 28 arcseconds from the major axis: TW pointed out that by using data in such pairs, systematic errors due to any uncertainty in the systemic velocity and centre of the galaxy vanish. Such pairing also eliminates any systematic error in the mean velocity which might arise from imperfect charge transfer in the CCD and other factors which might distort the galaxy's broadening function. Note, however, that the symmetry of the major axis broadening function (see Fig. 3) means that such skewing e ects must, in any case, be small. A further possible source of systematic error can arise from \template mismatch" { the spectrum of the template star used in the analysis of Section 2.2 might be signi cantly different from that of the galaxy (and the galaxy's spectrum might also vary with radius). However, we have repeated the kinematic analysis using stellar templates with a range of spectral types, and we nd that the derived values of hvlosi are insensitive to di erences between the spectra of the galaxy and the templates. Similarly, poor subtraction of the sky contribution from the spectra could produce systematic errors in the derived kinematics, but varying the region of the slit from which the sky spectrum was extracted makes no signi cant di erence to the derived values of hxi and hvlosi.
Hence, the value that we derive for p is not compromised by these possible systematic e ects.

DISCUSSION
The good t to a straight line in Fig. 4 provides direct evidence for the validity of the assumptions in the TW method. It con rms that the continuity equation is satis ed for the stars in NGC 936, which is to be expected in this SB0 system since it contains no evidence for dust obscuration or on-going star formation. The analysis also shows that the bar seems to be a long-lived feature with a single well-de ned pattern speed: if there were signi cant shearing e ects in the bar which might arise from dynamical friction forces that vary with radius, then the bar would evolve rapidly by winding up, and Fig. 4 would no longer show a straight-line relation between hxi and hvlosi (Engstr om 1991).
The value that we obtain for p sin i equation (8)] is somewhat lower than Kent's (1987) estimate of 5:4 1:9 km s 1 arcsec 1 . However, the combined error on these two measurements means that the di erence is not statistically signi cant. By using higher signal-to-noise ratio CCD spectra, and by coadding along the slit and analyzing the full galaxy broadening function, we have been able to reduce the uncertainty in the measurement considerably.
Using the disk inclination listed by Kormendy (1984) of i = 41 , the value for the bar pattern speed in the plane of NGC 936 is p = 4:8 1:1 km s 1 arcsec 1 . At an assumed distance of 16.6 Mpc (Kormendy 1984), this value corresponds to 60 14 km s 1 kpc 1 . Combining our pattern speed estimate with Kormendy's (1984) rotation curve for this galaxy, we can infer that co-rotation lies at a radius of 69 15 arcsec. Kent & Glaudell (1989) found that the bar extends to a radius of 50 arcsec (see also Fig. 1); we therefore nd that the bar ends at a point close to, but within, its own co-rotation radius, in good agreement with the predictions of N-body simulations (e.g. Combes & Sanders 1981, Sellwood 1981. The result also ts very well with Kent & Glaudell's (1989) orbit calculations for the bar of NGC 936, since they found that they could best model the structure of this system if its pattern speed were 64 km s 1 kpc 1 . Any signi cantly higher pattern speed is essentially ruled out by their modelling, since it would result in a bar which extends to beyond its co-rotation radius: such bars are very di cult to build self-consistently since the principal orbit family outside the co-rotation radius is aligned perpendicular to the bar.
By applying new analysis techniques to high quality spectral data, we have shown that it is possible to derive stellar kinematic properties of galaxies with reasonable precision and well-de ned uncertainties. It therefore now appears practical to use the Tremaine-Weinberg formula in order to obtain direct, accurate measurements of the pattern speeds of early-type barred galaxies. In the case of NGC 936, our ndings are in good agreement with the theoretically-driven belief that galactic bars end just inside their co-rotation radii.