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The accurate measurement of extragalactic distances is a central challenge of modern astronomy, being required for any realistic description of the age, geometry and fate of the Universe. The measurement of relative extragalactic distances has become fairly routine, but estimates of absolute distances are rare 1 . In the vicinity of the Sun, direct geometric techniques for obtaining absolute distances, such as orbital parallax, are feasible, but such techniques have hitherto been dif®cult to apply to other galaxies. As a result, uncertainties in the expansion rate and age of the Universe are dominated by uncertainties in the absolute calibration of the extragalactic distance ladder 2 . Here we report a geometric distance to the galaxy NGC4258, which we infer from the direct measurement of orbital motions in a disk of gas surrounding the nucleus of this galaxy. The distance so deter-minedÐ7:2 6 0:3 MpcÐis the most precise absolute extragalactic distance yet measured, and is likely to play an important role in future distance-scale calibrations.NGC4258 is one of 22 nearby active galactic nuclei (AGN) known to possess nuclear water masers (the microwave equivalent of lasers). The enormous surface brightnesses ( ) 10 12 K), relatively small sizes ( ( 10 14 cm) and narrow linewidths (a few km s -1 ) of these masers make them ideal probes of the structure and dynamics of the molecular gas in which they residue. Very-long-baseline interferometry (VLBI) observations of the NGC4258 maser have provided the ®rst direct images of an AGN accretion disk, revealing a thin, subparsec-scale, differentially rotating warped disk in the nucleus of this relatively weak Seyfert 2 AGN 3±6 . Two distinct populations of masers exist in NGC4258. The ®rst are the highvelocity masers. These masers amplify their own spontaneous emission and are offset 61,000 km s -1 and 4.7±8.0 mas (0.16± 0.28 pc for a distance of 7.2 Mpc) on either side of the disk centre. The keplerian rotation curve traced by these masers requires a central binding mass (M), presumably in the form of a supermassive black hole, of 3:9 6 0:1 3 10 7 D=7:2 Mpcsin i s =sin 82 2 2 Figure 1 The NGC4258 water maser. The upper panel shows the best-®tting warped-disk model superposed on actual maser positions as measured by the VLBA of the NRAO, with top as North. The ®lled square marks the centre of the disk, as determined from a global disk-®tting analysis 8 . The ®lled triangles show the positions of the high-velocity masers, so called because they occur at frequencies corresponding to Doppler shifts of ,61,000 km s -1 with respect to the galaxy systemic velocity of ,470 km s -1 . This is apparent in the VLBA total power spectrum (lower panel). The inset shows line-of-sight (LOS) velocity versus impact parameter for the best-®tting keplerian disk, with the maser data superposed. The high-velocity masers trace a keplerian curve to better than 1%. Monitoring of these features indicates that they drift by less than ,1 km s -1 yr -1 (refs 14±16) and requires that they lie within 5±1...

We report phase-referencing VLBA observations of H 2 O masers near the starforming region W3(OH) to measure their parallax and absolute proper motions. The measured annual parallax is 0.489 ± 0.017 milli-arcseconds (2.04 ± 0.07 kpc), where the error is dominated by a systematic atmospheric contribution. This distance is consistent with photometric distances from previous observations and with the distance determined from CH 3 OH maser astrometry presented in a related paper. We also find that the source driving the H 2 O outflow, the "TW-object", moves with a 3-dimensional velocity of > 7 km s −1 relative to the ultracompact H II region W3(OH).

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Recent Very Long Baseline Interferometer (VLBI) observations determined the distances and proper motions of star-forming regions in spiral arms directly. They showed that star-forming regions and young stars have large peculiar motions, as large as 30 km s −1 with complex structures. Such a large peculiar motion is incompatible with the prediction of the standard theory of quasi-stationary spiral arms. We use a high-resolution, self-consistent N -body+hydrodynamical simulation to explore how the spiral arms are formed and maintained, and how star-forming regions move. We found that arms are not quasi-stationary but transient and recurrent, as suggested in alternative theories of spiral structures. Because of this transient nature of the spiral arms, star-forming regions exhibit a trend of large and complex non-circular motions, which is qualitatively consistent with the VLBI observations. Owing to this large non-circular motion, a kinematically estimated gas map of our Galaxy has a large systematic errors of ∼ 2 − 3 kpc in the distance from the Sun.