1 ] 1012 MW. H I mass-to-light ratio was indeed exceptionally large with (M /L )º5.4. This is comparable to the H I rich but more

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1 THE ASTROPHYSICAL JOURNAL, 506:125È134, 1998 October 10 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. THE TOTAL ÏÏ MASS OF DDO 154 CLAUDE CARIGNAN De partement de Physique and Observatoire du Mont Me gantic, Universite de Montre al. C. P. 6128, Succ. centre-ville, Montre al, Que bec, Canada, H3C 3J7; claude=astro.umontreal.ca AND CHRIS PURTON Dominion Radio Astrophysical Observatory, P.O. Box 248, Penticton, B.C., Canada, V2A 6K3; crp=drao.nrc.ca Received 1995 November 1; accepted 1998 May 18 ABSTRACT Combining data from the Dominion Radio Astrophysical Observatory (DRAO) interferometer array with previous Very Large Array (VLA)1 observations, it was possible to recover all the single-dish Ñux of DDO 154, of which D1 was missing from the VLA data. The missing Ñux was found to be in an extended low surface brightness 3 component in the outer parts of this dwarf irregular galaxy. In the new combined data, the H I disk can be traced out to D6 optical radii (R ) or 21 optical scale lengths (a~1) at a level of D1.0 ] 1019 cm~2. Even at that level, there is no sign of Ho the abrupt truncation seen in most other galaxies, especially on the receding side. A total H I mass of 2.5 ] 108 M is derived, which gives _ an (M /L )^8, making DDO 154 one of the most gas-rich galaxies known. The recovered H I Ñux HI B makes it possible to extend the rotation curve by more than 33% in radii with the result that the rotation curve of DDO 154 is one of the most extensive rotation curves ever derived relative to the optical size of the galaxy. The new combined data conðrm that the rotation curve is declining in the outer parts. Detailed mass models show that we may have reached the edge of the mass distribution for this galaxy, which would allow, for the Ðrst time, an estimate of the total mass and the total extent of a galaxy. For DDO 154, this gives an estimated total mass of M ^ 3.0 ] 109 M and an upper limit to the radial tot _ extent of R ¹ 8 kpc. tot Subject headings: galaxies: fundamental parameters È galaxies: individual (DDO 154) È galaxies: ISM È galaxies: kinematics and dynamics È radio lines: galaxies 1. INTRODUCTION The total mass and total extent of galaxies (including their dark halos) are fundamental parameters that are completely unknown for all galaxies. The best estimates we have for spiral and dwarf irregular galaxies come from detailed mass models using extended H I rotation curves (Carignan & Freeman 1985; Begeman 1989; Broeils 1992). But, in every galaxy studied so far, such analysis has only allowed lower limits of the total mass and total extent of their dark halo to be derived out to the last measured velocity point of the rotation curves that are still Ñat or even rising, implying that more dark mass is present at larger radii. This situation was reviewed in detail by Ashman (1992). The only galaxy for which the questions of the total mass and total extent were tackled seriously is the Milky Way (see Fich & Tremaine 1991 for a review). In this case, dynamical techniques were used such as the determination of the tidal radii of globular clusters or satellite galaxies (Innanen, Harris, & Webbink 1983), a statistical form of the virial theorem (Little & Tremaine 1987) using known Galactocentric distances and radial velocities, or a timing argument for the Milky Way and M31 (Zaritsky et al. 1989). While those techniques are all resting on fairly solid theoretical grounds, they nevertheless yielded results ranging from M ^ 5 ] 1011 M and R ^ 46 kpc (Little & Tremaine MW 1987) to M ^ _ 1 ] 1012 MW M and R ^ 230 kpc (Kulessa & Lynden-Bell MW 1992). _ MW 1 The VLA is a telescope of the National Radio Astronomy Observatory, which is operated by Associated Universities, Inc., under a cooperative agreement with the National Science Foundation. 125 Especially, the virial theorem technique was shown to be very sensitive to the individual parameter determinations. For example, a new determination of the radial velocity of Leo I (Zaritsky et al. 1989) boosted the mass of the Milky Way by a factor of 4. While it is conceivable that the total mass of galaxies could be determined by dynamical methods (satellite galaxies, galaxies in pairs or in groups), the main hope to derive the total extent is that, at least in some galaxies, the H I (or some other probe) extends all the way to the edge of the halo (and is still detectable) so that the kinematics show the Keplerian decline expected when all the mass has been encountered. The large extent of the H I disk of DDO 154 was Ðrst discovered serendipitously by Krumm & Burstein (1984) using the Cornell single dish at Arecibo Observatory and conðrmed soon after by Hutchmeier & Seiradakis (1985) with the Bonn 100 m telescope. While those two studies were able to show unambiguously that the H I disk extends to several optical radii and that the (M /L ) is exceptionally large, those data were of too low resolution HI B to be able to derive a reliable rotation curve from which a detailed study of the mass distribution could have been done. This is why DDO 154 was observed with the Very Large Array VLA on 1985 November 23 for 12 hours in the D- conðguration. This Ðrst data set made it possible to trace the H I out to 4R (Holmberg radius: isophotal diameter at a level of k \ 26.6 Ho mag arcsec~2). From a detailed analysis of the data B (Carignan & Beaulieu 1989), it was found that its H I mass-to-light ratio was indeed exceptionally large with (M /L )º5.4. This is comparable to the H I rich but more HI B massive galaxy Malin I (Bothun et al. 1987), which has an (M /L )^5, or to the low-luminosity galaxies with HI B

2 126 CARIGNAN & PURTON extended H I envelopes observed by van Zee, Haynes, & Giovanelli (1995) at Arecibo. A study of the mass distribution (Carignan & Freeman 1988) showed that for r [ 1 kpc, the rotation curve is completely dominated by its dark halo component, which has a somewhat higher (^10 times) central density (o ^ 0.02 M pc~3) than what is normally seen in more 0 massive spirals. _ At the last measured velocity point of the rotation curve, more than 90% of the mass was provided by the dark component for a global (M/L ) at that point of 80. Some time after the study B dyn of DDO 154, other dwarf irregulars were studied. The H I disks of those dwarf irregulars, while generally more extended than the H I disks of spirals (compared with the optical disks), are not as extended as the H I disk of DDO 154. On the other hand, most of them showed a much higher dark-to-luminous mass ratio than what is seen in more massive spirals. For example, DDO 170 (Lake, Shommer, & van Gorkom 1990) has a dark-to-luminous mass ratio of 6 and a central density for its dark halo of o \ 0.01 M pc~3. Later, Magellanic- type spirals showed 0 similar trends. _ Studies of NGC 3109 (Jobin & Carignan 1990), NGC 5585 (Coü te, Carignan, & Sancisi 1991), and IC 2574 (Martimbeau, Carignan, & Roy 1994) all showed that those late-type spirals share the mass distribution properties of dwarf irregulars more than that of spirals with dark-to-luminous mass ratios of B10. One of the most interesting results of the earlier observations of DDO 154 is that the rotation curve appears to be declining in the outer parts, indicating that we may be close to the edge of the mass distribution. However, this decline was only based on the last three velocity points, and the gradient was still shallower than what would be expected of a Keplerian decline. Nevertheless, an alternate method of deriving the rotation curve using the orientation parameters derived from Ðtting the H I isophotes (Carignan & Beaulieu 1989) suggested that this decline might be real. More recently, Ho man et al. (1993), from Arecibo observations, also showed that the rotation curve is truly declining. Another result from the VLA observations that brought us to pursue the study of DDO 154 was that nearly 30% of the single-dish Ñux was missing from the synthesis observations. This is a well-known problem with nearby extended objects. Because the VLA, even in its most compact D-conÐguration, severely underrepresents baseline spacings shorter than 40 m, it is not sensitive to the largest scale structures (º9@). We thus assumed that the missing Ñux must be in such a large-scale low surface brightness component and decided to get the missing short spacings using the Dominion Radio Astrophysical Observatory (DRAO) interferometer array in Penticton, B.C. The DRAO Synthesis Telescope should be sensitive to structures up to D28@. Such a combination of the data from two arrays had been done successfully in the past for nearby galaxies (e.g., Westerbork & DRAO for NGC 6946; see Carignan et al. 1990). Hopefully, if that large-scale low surface brightness extended component was indeed present, it should be possible to extend the rotation curve and check if it is really declining. Finally, a grid was also obtained with the DRAO 26 m single-dish telescope. 2. OBSERVATIONS AND DATA REDUCTION The DRAO interferometer data were obtained during a period of 6 weeks from 1989 April 13 to May 25. The 26 m single-dish observations were done from 1990 May to July. The DRAO array consisted at the time of four dishes of 9 m diameter (the array now has seven dishes). Two dishes were moved every day to produce baselines ranging from 13 to 604 m during the whole observing session. This may be compared to baselines ranging from 40 to 840 m obtained with 25 VLA antennas in D-conÐguration. Both data sets were combined using the DRAO software. Care was taken to properly scale both data sets according to their respective beam areas. The two independent calibrations were checked by comparing the amplitudes of the UV data in the range 100 m ¹ r ¹ 0 m. UV 2.1. Recovering the Missing H I Flux Figure 1 shows the radial distribution of azimuthally averaged amplitudes of raw (uncleaned) UV data for one of the velocity channels. The dashed line shows the VLA data alone while the dotted line shows the VLA ] DRAO interferometer data. The solid line shows the interferometer data combined with the 26 m single-dish data. It can be seen clearly that, starting at D100 m, the amplitudes of the VLA data are smaller than the DRAO data and shortward of 40 m they drop precipitously. Baselines shorter than 40 m are severely underrepresented in the D-conÐguration. Cleaning provides data for the missing baselines but does so imperfectly in the central regions of the UV plane, as evidenced by the low value returned for the total Ñux in the cleaned data. Conversely, as expected, the DRAO amplitudes continue to increase all the way down to [ m. A lot of Ñux is involved in that region, and the e ect of that extra detected Ñux should be seen easily in the map plane. Finally, the 26 m data provide the zero point. However, the rest of the reduction showed that the 26 m data did not add anything FIG. 1.ÈAzimuthally averaged amplitudes of the UV data for channel 7(v\395.6 km s~1). The dashed curve is for the VLA data alone, the dotted curve for the VLA ] DRAO interferometric data, and the solid line for the 26 m ] the interferometric (VLA ] DRAO) data; in each case the data are raw and uncleaned. The plot demonstrates clearly that spacings less than 40 m are very poorly represented in the VLA data, even though the D-conÐguration was used over a wide range of hour angles. The 26 m data Ðll in the center of the UV plane but with poor S/N; the DRAO synthesis telescope samples baselines short enough to recover the total Ñux.

3 DECLINATION (1950) EP VEL E+05 IPOL D154_60.LMVCUB X RIGHT ASCENSION (B1950) RIGHT ASCENSION (B1950) RIGHT ASCENSION (B1950 Peak flux = E-01 JY/BEAM Levs = E-03 RIGHT * ( -2.50, ASCENSION 2.500, 5.000, (1950) FIG. 2.ÈMosaic of the H I channel maps for the combined VLA ] DRAO data. The 60A circular beam is shown in the bottom left-hand corner of the Ðrst channel, and the radial velocities (km s~1) of each channel are indicated in the upper left-hand corner. The contours are [0.75, 0.75 (2.5 p), 1.5, 3, 6, 12, and 24 K.

4 128 CARIGNAN & PURTON Vol RIGHT ASCENSION (B1950) Gre scale fl range Kilo FIG. 3.ÈTotal H I map superposed on the optical for the combined VLA ] DRAO data. The contours are 0.5, 1, 2, 4, 8, and 16 ] 10 cm~2. The circular beam size is 1@ to the interferometer data and thus will be ignored for the rest of this work H I Distribution The recovered H I Ñux is clearly seen in the radio maps. Figure 2 shows a mosaic of the channel maps. This can be compared with Figure 6 of Carignan & Beaulieu (1989), which shows the same for the VLA data alone. The di erence cannot come from the di erence in beam sizes since they are both very small compared with the galaxy scale. For example, the channel at km s~1 shows the larger extent on the southwest side, and the channel at km s~1 shows the larger extent on the northwest side. The total H I map given in Figure 3 also shows clearly the low surface brightness component that was recovered by adding the short spacings data. When compared with Figure 2 of Carignan & Freeman (1988), it can be seen that the galaxy extends nearly 3@ further to the southwest in the combined VLA ] DRAO data than in the VLA data alone. The total Ñux in the combined data is 105 Jy km s~1, which can be compared with the single-dish Ñuxes of 94 Jy km s~1 obtained by Allen & Shostak (1979) with the Dwingeloo dish and of 106 Jy km s~1 obtained by Krumm & Burstein (1984) at Arecibo. This means that, this time, the combined interferometer data have recovered all the Ñux. At our adopted distance of 3.2 Mpc (Carignan & Freeman 1988 used 4.0 Mpc), this gives a total H I mass of 2.5 ] 108 M for a total (M /L )^8. This makes of DDO 154 one of _ the most gas-rich HI galaxies B known. Contrary to what is written in the abstract of the Ho man et al. (1993) paper, but similar to what is said in their 4.3 (see also their Fig. 12), the H I disk of DDO 154

5 No. 1, 1998 TOTAL ÏÏ MASS OF DDO Kilo JY/B*M/S EP IPOL D154_60.MOM NE SW Kilo ARC ARC SEC SEC FIG. 4.ÈCut through the total H I map along the major axis (P.A. \ 218 ). It can be seen that the H I disk does not have a sharp cuto but decreases rather smoothly especially on the southwest side. still has a gentle exponential decline at a level of (1.0È 2.0) ] 1019 cm~2 and does not show the usual truncation of the H I disk seen in most other galaxies at these faint levels. This can be seen in Figure 4, which shows a cut through the total H I map along the major axis (P.A. \ 218 ). The low surface brightness component recovered by the short spacings data is again clearly seen. While the decline is steeper on the northeast side, it is very gentle on the southwest side. The large extent on the southwest side could suggest some kind of tidal distortion with nearby neighbors, which would render difficult the interpretation of the kinematics. However, looking in TullyÏs (1988) Nearby Galaxies Catalog, the nearest neighbors are NGC 4826 and UGC 7698, both at more than 350 kpc away, which allows us to consider that DDO 154 is a fairly well isolated system H I Kinematics The velocity Ðeld is shown in Figure 5. On the southwest side (receding side), where the extension is the greatest, it TABLE 1 OPTICAL AND H I PARAMETERS OF DDO 154 Parameter Value Morphological typea... IB(s)m IVÈV R.A. (1950)a... 12h51m39s.6 Decl. (1950)a... ]27 25@30A Adopted distanceb Mpc (1@\931 pc) Isophotal major diameterb... D \ 1@.8 Holmberg radiusb... R 25 \ 1@.5 (1.4 kpc) Exponential disk scale lengthb... a~1 Ho \ 0@.43 (0.4 kpc) Absolute blue magnitudeb... M0(B)\[13.33 Total blue luminosity... L0(B)\3.2 T ] 107 L Total H I Ñux T Jy km s~1 _ Total H I mass... (M /L )... HI B 2.5]108 M 8 _ a de Vaucouleurs et al b Carignan & Beaulieu can be seen that the isovelocity contours (p.e. 410 km s~1) are closing. While this could be entirely because of the warp of the H I plane, the tilted-ring model (for the 60A resolution data) of Figure 6 suggests that it is not and that the rotation curve is truly declining for 5@ ¹ r ¹ 9@. The warp is clearly seen with the inclination varying between 52 and 66, while the position angle h varies by D15 from 228 at r \ 1@.5 to 213 at r \ 9@. The only way that the rotation curve could stay Ñat in the outer parts would be if the inclination were smaller by D4 p for the last three velocity points. The main uncertainty comes from the fact that the kinematics in the outer parts is mainly deðned by the receding side (southwest), where the largest part of the recovered Ñux was found. This can be seen in Figure 6 where the P.A. for the approaching side is not very well deðned. This is understandable since, in the last ring, there are 10 times less velocity points on the approaching side than on the receding side. The main test that the tilted-ring model is a good representation of the data is to look at the residuals (model [ data) map given in Figure 7. This map shows convincingly that this is the case. Most of the residuals are ¹5kms~1 with only a few points along the minor axis with residuals between 5 and 10 km s~1 at the very edge of the disk where the signal-to-noise (S/N) ratio is low. However, those points have very little weight in the Ðnal solution since, as is the usual practice in deriving rotation curves, the points within a region of 45 about the minor axis (the size of the exclusion region increases with the inclination of the galaxy) are excluded because of the larger deprojection errors close to the minor axis (resulting from the Ðnite resolution). Moreover, as is also the usual practice, each velocity point is weighted by cos h, where h is the angle from the major axis (one would even use cos2h for very large inclinations). The main aim is to give the largest weights to the points with the smallest deprojection errors. It is thus normal to get larger residuals for the points along the minor axis. The main point is that the residuals are not larger on the approaching side (northeast) than on the receding side (southwest). This implies that the derived rotation curve is a good representation of the galaxy kinematics as a whole. The optical and H I properties, all calculated for our adopted distance of 3.2 Mpc, are summarized in Table MASS DISTRIBUTION The data analysis and the determination of the rotation curve were done as described by Carignan & Beaulieu (1989), and a description will not be repeated here. The new rotation curve derived from the combined data is given in Table 2. Three mass models will be presented. First, as a reminder, the rotation curve of the VLA data alone will be modeled using an isothermal halo as in Carignan & Freeman (1988). Based on the results of that Ðrst model, the VLA data alone and then the combined data will be modeled using a new density law for the halo that will reproduce the declining part of the rotation curve as well V L A Data, Isothermal Halo The top panel of Figure 8 presents the best-ðt mass model for the rotation curve derived from the VLA data and where the halo is represented by an isothermal sphere (Carignan 1985). The di erence in the radial scale between this plot and Figure 3 of Carignan & Freeman (1988) comes from adopting a distance of 3.2 Mpc (Carignan & Beaulieu

6 130 CARIGNAN & PURTON RIGHT ASCENSION (B1950) FIG. 5.ÈVelocity Ðeld for the combined VLA ] DRAO data. See especially the closing contours at 400 and 410 km s~1. TABLE 2 ROTATION CURVE OF DDO 154 (VLA ] DRAO) Radius Radius V rot Error (arcsec) (kpc) (km s~1) (km s~1) ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ) instead of 4.0 Mpc. The main results of the model are summarized in Table 3. As shown in Carignan & Freeman (1988), adding an isothermal halo to the contributions of the stellar and of the H I disks (the H I proðle has been multiplied by 4 to take into account primordial He) reproduces the observed 3 rotation curve fairly well, except for the last two points, which are declining. Since the isothermal sphere is inðnite in radius, there is no way that such a model can reproduce those two velocity points. The main characteristic of this model is that, contrary to more massive galaxies, the gravitational potential is completely dominated by the dark component at almost all radii (r [ 1 kpc). Even more interesting is the bottom panel of Figure 8, which shows the ratio of the integrated surface density of the dark matter halo to the integrated surface densities of the stellar and H I disks. It can be seen that, as expected, the

7 FIG. 6.ÈTilted-ring model representing the warp of the H I plane for the combined VLA ] DRAO data. The kinematical parameters are given for both sides (triangles) as well as separately for the approaching (circles) northeast side and for the receding (squares) southwest side. EP D154_60.ROCUR.1 FIG. 8.ÈBest-Ðt model for the VLA data using an isothermal halo (top panel). The contribution of each component is marked. Ratios of the integrated surface density of the dark halo to the integrated surface densities of the stellar and of the H I disks (bottom panel) TABLE 3 RESULTS FROM THE ISOTHERMAL HALO MODEL (VLA DATA) 30 Parameter Value RIGHT ASCENSION (B1950) Peak flux = E+04 M/S FIG. 7.ÈResidual map (model [ data) for the tilted-ring model. The contours are [10, [5, 5, and 10 km s~1. Luminous disk component: (M/L ) ^1.0 M /L B _ B_ M ]107 M _ M ]108 M HI`He _ M /M HI Dark halo component: r ^0.2 kpc c p ^ 0.5 km s~1 o M pc~3 0 _ At the Holmberg radius (R \ 1.4 kpc): o Ho M pc~3 M halo /M _ (M/L dark ) lum M /L M B dyn ] _ 108 B_ M At the dark`lum last velocity point (6.1 kpc): _ o M pc~3 M halo /M _ (M/L dark ) lum M /L M B dyn ]109 _ M B_ dark`lum _

8 132 CARIGNAN & PURTON Vol. 506 dark matter is not at all distributed as the stellar mass. On the other hand, the ratio of dark matter to H I is much more similar, not only in the outer parts, as already suggested in other galaxies (Bosma 1978; Sancisi 1983) but all the way to the center. The di erence of this ratio between the center and the outer parts is less than a factor of 4. This was already seen in galaxies ranging from the massive spiral NGC 6946 (Carignan et al. 1990) to the Magellanic-type spiral NGC 3109 (Jobin & Carignan 1990). This result suggests that dark matter may be distributed very similarly to the way the H I gas is and that instead of using a functional form such as the isothermal sphere (or any kind of r~2 type density law), one could use the form of the H I density proðle to represent the dark matter density law. This does not mean, however, that it is only a question of scaling the H I distribution, which would assume that the dark stu is distributed in a disk. As already shown in Figure 15 of Carignan & Beaulieu (1989), doing this failed to adequately represent the rotational velocities in the outer parts (r [ 4 kpc). The intrinsic axis ratio of the halo should be left as a free parameter V L A Data,HIDensity L aw Halo Figure 9 presents the next two models. The left-hand part is the best-ðt model for the VLA data alone. The curves coming from the top of the plot in Figure 9a and Figure 9d are Keplerian curves, assuming that all the mass present out to the last point is concentrated in the center. The results of the mass model are given in Table 4. The error on each parameter is deðned at the 99.0% conðdence limit in the s2 FIG. 9.ÈMass models for the H I density law halo. On the left is the model for the VLA data alone, and on the right for the VLA ] DRAO combined data. (a) and (d) are the mass models where the curve coming from the top is a Keplerian curve, assuming that the total mass present at the last point is concentrated in the center, (b) and (e) are the dark-to-luminous mass ratios, and (c) and ( f ) are the dynamical (M/L ) curves. B dyn

9 No. 1, 1998 TOTAL ÏÏ MASS OF DDO TABLE 4 RESULTS FROM THE H I DENSITY LAW HALO MODELS VLA DATA Luminous Component (M/L ) \ 1.9 ^ 0.6 M /L... B * _ B_ M \ 5.2 ] 107 M... * _ M \ 2.2 ] 108 M... HI`He _ Dark Halo Component VLA ] DRAO DATA (M/L ) \1.3 ^ 1.0 M /L B * _ B_ M \3.5 ] 107 M * _ M \ 3.3 ] 108 M HI`He _ X \11.7 ^ 0.3 (b/a \ 1.0)... X \8.7 ^ 0.4 (b/a \ 0.6) HI HI M /M \ M /M \ 7.8 dark lum dark lum M \ 2.9 ] 109 M... M \ 3.1 ] 109 M dark`lum _ dark`lum _ (M/L ) \ (M/L ) \ 115 B dyn B dyn minimization process. Each component of the mass distribution (stellar disk, H I disk, and halo) was computed using KalnajsÏs (1983) log-spiral potential solver code (see also Binney & Tremaine 1987). For the VLA data alone, the best solution is for a spherical halo (b/a \ 1.0). The other two free parameters are the mass-to-light ratio of the stellar disk (M/L ) \ 1.9 ^ 0.6 B * M /L and the scaling of the H I data proðle for the halo _ B_ X \11.7 ^ 0.3. Compared with the isothermal halo HI model, the best-ðt model of Figure 9a represents the declining part of the rotation curve much better. Finally, it can be seen that the slope of the model is shallower in the outer parts than the slope of the Keplerian curve, which means that more mass must still be present further out. Figures 9b and 9c show also that the dark-to-luminous mass ratio and the global (M/L ) are still increasing at the last measured B dyn velocity point VLA]DRAO Data,HIDensity L aw Halo The best-ðt model for the new rotation curve derived from the combined VLA ] DRAO data (Table 2) is given in Figure 9d. The Ðrst thing to notice is that there is no doubt with the combined data that the rotation curve is really declining in the outer parts (on D40% of its radial extent). It can be seen again that the scaled H I density law, used to represent the dark halo, reproduces the data very well. The value of (M/L ) \ 1.3 ^ 1.0 M /L found for the stellar disk is not very B * well constrained. _ B_ This is not surprising since, as already seen, the stellar disk is really a minor constituent of the mass. This time, the best Ðt is found for a slightly Ñattened halo with an axis ratio of b/a \ 0.6, instead of a spherical halo as in Figure 9a. The scaling factor is X \8.7 ^ 0.4. It can be HI seen that both the outer parts of the model and of the Keplerian curve have the same slope for r º 7 kpc. This means that by 7 kpc, most of the mass of DDO 154 has been encountered. This sets an upper limit to the total radial extent of DDO 154 at ¹8 kpc. Similarly, it can be seen from Figure 9e that the dark-to-luminous mass ratio has converged to M /M ^ 8.0 for an M ^ 3.0 ] 109 M, indicating dark that luminous we have derived not only tot the total extent _ of this galaxy but also its total mass, of which 90% is dark. 4. SUMMARY AND CONCLUSIONS Combining DRAO interferometer data with VLA data, it was possible to recover the D30% H I Ñux that was missing from the VLA data due to the lack of short spacings. The main results from the study of the H I distribution are the following: 1. The recovered missing Ñux was found to be in an extended low surface brightness component in the outer parts of DDO This brings the size of the H I disk to D6 D (10 D, 21 a~1), which makes the rotation curve of DDO Ho 154 one 25 of the most extended rotation curves ever derived relative to the optical size of the galaxy. 3. The total H I Ñux was found to be 105 Jy km s~1, very similar to the measured single-dish Ñux. For an adopted distance of 3.2 Mpc, this corresponds to a total H I mass of 2.5 ] 108 M. 4. From _ the measured absolute magnitude in B of [13.33, this gives an (M /L )^8, which makes DDO 154 one of the most gas-rich galaxies HI B known. 5. DDO 154 does not exhibit the abrupt truncation of the H I disk usually seen at the levels reached by the present observations. From the combined data, it was possible to extend the derived rotation curve by more than D33%. The main results of the rotation curve analysis are the following: 1. The new combined data conðrm that the rotation curve of DDO 154 is declining. 2. Using the form of the H I density proðle as the halo density law instead of an isothermal halo, it was possible to Ðt the declining part of the rotation curve if the dark matter is more spherically distributed than in a highly Ñattened disk. 3. The best-ðt model gives values of (M/L ) ^ 1.3 ^ 1.0 B * M /L for the stellar disk and X ^ 0.4 for the _ B_ HI^8.7 scaling parameter of the H I density proðle used to represent the halo. While the H I disk is assumed to be inðnitely thin, the model suggests that the dark halo has an axis ratio b/a ^ 0.6. However, that parameter along with the (M/L ) of the stellar disk is not well constrained. B * 4. By comparing the rotation curve to a Keplerian curve computed using the total mass derived at the last measured velocity point, it is found that they both run parallel to each other in the outer parts. From this, it is concluded that the edge of the mass distribution has been reached, which sets an upper limit to the total radial extent of DDO 154 at ¹8 kpc. 5. This makes it possible to derive for the Ðrst time the total mass of a galaxy. For DDO 154, we Ðnd M ^ 3.0 ] 109 M, of which 90% is dark. tot _ The main reason why it was possible to derive both the total extent and the total mass of DDO 154 is that, for the Ðrst time, it was possible to trace the H I gas all the way to the edge of the mass distribution. This does not mean that if we were to get very sensitive H I observations of other galaxies, this would also be the case. In fact, most galaxies show a very sharp cuto of their H I distribution (van Gorkom 1991) well before the edge of the mass distribution, as shown by the fact that the rotation curves are still Ñat or even rising at the last measured velocity point. It just so happens that in the case of DDO 154, the H I was detectable all the way to the edge. However, the sharp cuto seen in other galaxies does not mean that there is no more hydrogen further out. It just means that it is no longer

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