NML Cyg Research Articles

The Arizona Radio Observatory 1 mm Spectral Survey of the Hypergiant Star NML Cygni (215–285 GHz)

A sensitive (1σ rms ≤ 3 mK; 2 MHz resolution) 1 mm spectral survey (214.5–285.5 GHz) of the envelope of the oxygen-rich supergiant star NML Cygni (NML Cyg) has been conducted using the 10 m Submillimeter Telescope of the Arizona Radio Observatory. These data represent the first spectral line survey of NML Cyg and are complementary to a previous 1 mm survey of the envelope of a similar hypergiant, VY Canis Majoris (VY CMa). The complete NML Cyg data set is presented here. In the survey, 104 emission lines were observed, arising from 17 different molecules and 4 unidentified features. Many of the observed features have complex line profiles, arising from asymmetric outflows characteristic of hypergiant stars. While most of the lines in the survey arise from SiO, SO, SO2, and SiS, CO had the strongest emission. Five other C-bearing species are identified in the survey (HCN, CN, HCO+, CS, and HNC), demonstrating an active carbon chemistry despite the O-rich environment. Moreover, NS was observed, but not NO, although favorable transitions of both molecules lie in the surveyed region. Sulfur chemistry appears to be prominent in NML Cyg and plays an important role in the collimated outflows. The refractory species observed, NaCl and AlO, have narrow emission lines, indicating that these molecules do not reach the terminal expansion velocity. NaCl and AlO likely condense into dust grains at r < 50 R *. From NaCl, the chlorine isotope ratio was determined to be 35Cl/37Cl = 3.85 ± 0.30.

Multiple components in the molecular outflow of the red supergiant NML Cyg

ABSTRACT Despite their large impact on stellar and galactic evolution, the properties of outflows from red supergiants are not well characterized. We used the Onsala 20m telescope to perform a spectral survey at 3 and 4 mm (68–116 GHz) of the red supergiant NML Cyg, alongside the yellow hypergiant IRC + 10420. Our observations of NML Cyg were combined with complementary archival data to enable a search for signatures of morphological complexity in the circumstellar environment, using emission lines from 15 molecular species. The recovered parameters imply the presence of three distinct, coherent, and persistent components, comprised of blue-shifted and red-shifted components, in addition to an underlying outflow centred at the stellar systemic velocity. Furthermore, to reproduce 12CO emission with 3D radiative transfer models required a spherical outflow with three superposed conical outflows, one towards and one away from the observer, and one in the plane of the sky. These components are higher in density than the spherical outflow by up to an order of magnitude. We hence propose that NML Cyg’s circumstellar environment consists of a small number of high-density large-scale coherent outflows embedded in a spherical wind. This would make the mass-loss history similar to that of VY CMa, and distinct from μ Cep, where the outflow contains many randomly distributed smaller clumps. A possible correlation between stellar properties, outflow structures, and content is critical in understanding the evolution of massive stars and their environmental impact.

Molecules and Outflows in NML Cygni: New Insights from a 1 mm Spectral Line Survey

Abstract A sensitive (1σ rms ≤ 3 mK) 1 mm spectral survey (214.5–285.5 GHz) of the oxygen-rich circumstellar envelope of the red hypergiant NML Cygni (NML Cyg) has been conducted using the Sub-millimeter Telescope (SMT) of the Arizona Radio Observatory (ARO). Over 100 spectral lines were detected, arising from 17 different molecules, including the carbon-bearing species CO, HCN, HCO+, CN, and HNC; sulfur- and silicon-containing compounds H2S, SO, SO2, SiO, and SiS; and more exotic NaCl and AlO. The 1 mm spectrum of NML Cyg closely resembles that of VY Canis Majoris (VY CMa) suggesting that the chemistries of hypergiant stars are similar. The line profiles in NML Cyg consist of multiple velocity features, particularly evident in SO2 and SO. In addition to a spherical wind at the star’s systemic velocity, the spectra suggest an asymmetric, blueshifted component near V LSR = −21 ± 3 km s−1 and a collimated, redshifted component near 15 ± 3 km s−1, positioned ∼34° and ∼12°, respectively, from the line of sight. The red- and blueshifted flows appear to be randomly oriented, and likely trace sporadic mass loss events. Their LSR velocities align closely with those of 22 GHz water masers, suggesting an NE–SE orientation. The winds may also be associated with the asymmetric nebula in F555W HST images but extending to 5″ (∼600R *). NML Cyg appears to be another example of rare, massive stars with collimated, episodic ejections, analogous to Betelgeuse and VY CMa, lending support for an important new mass loss mechanism—surface activity.

Searching for Cool Dust. II. Infrared Imaging of The OH/IR Supergiants, NML Cyg, VX Sgr, S Per, and the Normal Red Supergiants RS Per and T Per∗

Abstract New MMT/MIRAC (9–11 μm), SOFIA/FORCAST (11–37 μm), and Herschel/PACS (70 and 160 μm) infrared (IR) imaging and photometry is presented for three famous OH/IR red supergiants (NML Cyg, VX Sgr, and S Per) and two normal red supergiants (RS Per and T Per). We model the observed spectral energy distributions (SEDs) using radiative-transfer code DUSTY. Azimuthal average profiles from the SOFIA/FORCAST imaging, in addition to dust mass distribution profiles from DUSTY, constrain the mass-loss histories of these supergiants. For all of our observed supergiants, the DUSTY models suggest that constant mass-loss rates do not produce enough dust to explain the observed infrared emission in the stars’ SEDs. Combining our results with Shenoy et al. (Paper I), we find mixed results with some red supergiants showing evidence for variable and high mass-loss events while others have constant mass loss over the past few thousand years.

New Circumstellar Sources of PO and PN: The Increasing Role of Phosphorus Chemistry in Oxygen-rich Stars

Abstract PO and PN have been newly identified in several oxygen-rich circumstellar envelopes, using the Submillimeter Telescope of the Arizona Radio Observatory. The J = 5 → 4 and J = 6 → 5 transitions of PN near 235 and 282 GHz, and the lambda doublets originating in the J = 5.5 → 4.5 and J = 6.5 → 5.5 lines of PO at 240 and 284 GHz, have been detected toward the shells of asymptotic giant branch (AGB) stars TX Cam and R Cas. A similar set of lines has been observed toward the supergiant NML Cyg, and new transitions of these two molecules were also measured toward the AGB star IK Tau. Along with the previous data from VY Canis Majoris (VY CMa), these spectral lines were analyzed using the non-local thermodynamic equilibrium (non-LTE) circumstellar modeling code, ESCAPADE. For the AGB stars, peak abundances found for PN and PO were f ∼ (1–2) × 10−8 and (0.5–1) × 10−7, respectively, while those for the supergiants were f(PN) ∼ (0.3–0.7) × 10−8 and f(PO) ∼ (5–7) × 10−8. PN was well modeled with a spherical radial distribution, suggesting formation near the stellar photosphere, perhaps enhanced by shocks. PO was best reproduced by a shell model, indicating a photochemical origin, except for VY CMa. Overall, the abundance of PO is a factor of 5–20 greater than that of PN. This study suggests that phosphorus-bearing molecules are common in O-rich envelopes, and that a significant amount of phosphorus (&gt;20%) remains in the gas phase.

Herschel/HIFI observations of red supergiants and yellow hypergiants

Red supergiant stars (RSGs) and yellow hypergiant stars (YHGs) are believed to be the high-mass counterparts of stars in the AGB and early post-AGB phases. We study the mass-loss in the post main-sequence evolution of massive stars, through the properties of their envelopes in the intermediate and warm gas layers. These are the regions where the acceleration of the gas takes place and the most recent mass-loss episodes can be seen. We used the HIFI instrument on-board the Herschel Space Observatory to observe sub-mm and FIR transitions of CO, water, and their isotopologues in a sample of two RSGs (NML Cyg and Betelgeuse) and two YHGs (IRC+10420 and AFGL 2343) stars. We present an inventory of the detected lines and analyse the information revealed by their spectral profiles. On the basis of the results presented in an earlier study, we model the CO and 13CO emission in IRC+10420 and compare it to a set of lines ranging from the mm, to the FIR. Red supergiants have stronger high-excitation lines than the YHGs, indicating that they harbour dense and hot inner shells contributing to these transitions. Consequently, these high-J lines in RSGs originate from acceleration layers that have not yet reached the circumstellar terminal velocity and have narrower profiles than their flat-topped lower-J counterparts. The YHGs tend to lack this inner component, in line with the picture of detached, hollow envelopes derived from studies at longer wavelengths. NH3 is only detected in two sources (NML Cyg, IRC+10420), which are also observed to be the strongest water-line emitters of the studied sample. In contrast, OH is detected in all sources and does not seem to correlate with the water line intensities. We show that the IRC+10420 model derived solely from mm low-J CO transitions is capable of reproducing the high-J transitions when the temperature in the inner shell is simply lowered by about 30%.

The distance and size of the red hypergiant NML Cygni from VLBA and VLA astrometry

We have measured the annual parallax and proper motion of NML Cyg from multiple epoch VLBA observations of the circulstellar H2O and SiO masers. The measured parallax of NML Cyg is 0.620+/-0.047 mas, corresponding to a distance of 1.61+/-0.12 kpc. The measured proper motion of NML Cyg is mu_x = -1.55+/-0.42 mas/yr eastward and mu_y= -4.59+/-0.41 mas/yr northward. Both Both the distance and proper motion are consistent with that of Cyg OB2, within their joint uncertainty, confirming their association. Taking into consideration molecular absorption signatures seen toward NML Cyg, we suggest that NML Cyg lies on the far side of the Cyg OB2 association. The stellar luminosity revised with our distance brings NML Cyg significantly below the empirical luminosity limit for a red supergiant. Using the VLA observation the radio photosphere and the SiO maser as a phase reference, we partially resolve the radio photosphere of NML Cyg at 43 GHz and find its diameter is about 44 mas, suggesting an optical stellar diameter of 22 mas, if the size of radio photosphere is 2 times the optical photosphere. Based on the position of circumstellar SiO masers relative to the radio photosphere, we estimate the absolute position of NML Cyg at epoch 2008.868 to be R.A. = 20h46m25.5382s +/- 0.0010s, Decl. = 40d06'59.379" +/- 0.015". The peculiar motions of NML Cyg, the average of stars in Cyg OB2, and four other star-forming regions rules out that an expanding "Stroemgren sphere" centered on Cyg OB2 is responsible for the kinematics of the Cygnus X region.

Distance and Size of the Red Hypergiant NML Cyg

AbstractWe present astrometric results of phase-referencing VLBI observations of 22 GHz H2O maser and 43 GHz SiO maser emission towards the red hypergiant NML Cyg using VLBA. We obtained an annual parallax of 0.62 ± 0.04 mas, corresponding to a distance of 1.61+0.13−0.11 kpc. With a VLA observation in its largest (A) configuration at 43 GHz, we barely resolve the radio photosphere of NML Cyg, and find a uniform-disk diameter of 44 ± 16 mas.

CIRCUMSTELLAR ION-MOLECULE CHEMISTRY: OBSERVATIONS OF HCO+IN THE ENVELOPES OF O-RICH STARS AND IRC+10216

Millimeter-wave observations of HCO+ have been conducted toward oxygen-rich circumstellar envelopes, as well as IRC+10216, using the facilities of the Arizona Radio Observatory (ARO). The J = 1 → 0 and 2 → 1 transitions of this molecule were measured with the ARO 12 m antenna, while the J = 3 → 2 and 4 → 3 lines were observed using the ARO Sub-Millimeter Telescope. HCO+ was detected toward the supergiant NML Cyg and the asymptotic giant branch (AGB) stars IK Tau, TX Cam, and W Hya in at least two transitions. The J = 2 → 1 and 3 → 2 lines of this ion were also detected toward IRC+10216, confirming the identification of HCO+ in this object. The line profiles measured for HCO+ toward NML Cyg consist of red- and blueshifted components, suggesting a non-spherical shell. Based on a radiative transfer analysis, the abundances in the O-rich envelopes were f(HCO+/H2) ∼ 0.15–1.3 × 10−7, with the AGB stars typically exhibiting the higher values. In IRC+10216, f(HCO+/H2) ∼ 4.1 × 10−9, lower than the O-rich counterparts. The abundances of HCO+ were also found to peak at considerable distances from the star, indicative of an outer envelope molecule. Comparison with H2O and CO, the main precursor species, suggests that HCO+ is more prevalent in envelopes that have substantial water, but CO also plays a role in its formation. The abundance of HCO+ appears to increase inversely with mass-loss rate, provided the rate is >10−6 M☉ yr−1. The common appearance of HCO+ in circumstellar gas indicates that, at some level, ion–molecule reactions influence the chemistry of evolved stellar envelopes.

The mass-loss rates of red supergiants and the de Jager prescription

Mass loss of red supergiants (RSG) is important for the evolution of massive stars, but is not fully explained. Several empirical prescriptions have been proposed, trying to express the mass-loss rate (Mdot) as a function of fundamental stellar parameters (mass, luminosity, effective temperature). Our goal is to test whether the de Jager et al. (1988) prescription, used in some stellar evolution models, is still valid in view of more recent mass-loss determinations. By considering 40 Galactic RSGs presenting an infrared excess and an IRAS 60-mu flux larger than 2 Jy, and assuming a gas-to-dust mass ratio of 200, it is found that the de Jager rate agrees within a factor 4 with most Mdot estimates based on the 60-mu signal. It is also in agreement with 6 of the only 8 Galactic RSGs for which Mdot can be measured more directly through observations of the circumstellar gas. The two objects that do not follow the de Jager prescription (by an order of magnitude) are mu Cep and NML Cyg. We have also considered the RSGs of the Magellanic Clouds. Thanks to the works of Groenewegen et al. (2009) and Bonanos et al. (2010), we find that the RSGs of the SMC have Mdots consistent with the de Jager rate scaled by (Z/Zsun)**(alpha), where Z is the metallicity and alpha is 0.7. The situation is less clear for the LMC RSGs. In particular, for luminosties larger than 1.6E+05 Lsun, one finds numerous RSGs (except WOH-G64) having Mdot significantly smaller than the de Jager rate, and indicating that Mdot would no longer increase with L. Before this odd situation is confirmed through further analysis of LMC RSGs, we suggest to keep the de Jager prescription unchanged at solar metallicity in the stellar evolutionary models and to apply a (Z/Zsun)**0.7 dependence.

IMAGING THE COOL HYPERGIANT NML CYGNI'S DUSTY CIRCUMSTELLAR ENVELOPE WITH ADAPTIVE OPTICS

We present subarcsecond angular resolution, high-Strehl ratio mid-IR adaptive optics images of the powerful OH/IR source and cool hypergiant NML Cyg at 8.8, 9.8, and 11.7 ?m. These images reveal once more the complexity in the dusty envelope surrounding this star. We spatially resolve the physical structures (radius ~ 014, ~240 AU adopting a distance of 1.74 kpc) responsible for NML Cyg's deep 10 ?m silicate dust absorption feature. We also detect an asymmetric excess, at separations of ~03-05 (~520-870 AU), northwest from the star. The colors of this excess are consistent with thermal emission of hot, optically thin dust. This excess is oriented in the direction of the Cyg OB2 stellar association, and is likely due to the disruption of NML Cyg's dusty wind with the near-UV radiation flux from the massive hot stars within Cyg OB2. This interaction was predicted in our previous paper to explain the geometry of an inverted photodissociation region observed at optical wavelengths.

CARBON CHEMISTRY IN THE ENVELOPE OF VY CANIS MAJORIS: IMPLICATIONS FOR OXYGEN-RICH EVOLVED STARS

Observations of the carbon-bearing molecules CO, HCN, CS, HNC, CN, and HCO+ have been conducted toward the circumstellar envelope of the oxygen-rich red supergiant star, VY Canis Majoris (VY CMa), using the Arizona Radio Observatory (ARO). CO and HCN were also observed toward the O-rich shells of NML Cyg, TX Cam, IK Tau, and W Hya. Rotational transitions of these species at 1 mm, 0.8 mm, and 0.4 mm were measured with the ARO Submillimeter Telescope, including the J = 6 → 5 line of CO at 691 GHz toward TX Cam and W Hya. The ARO 12 m was used for 2 mm and 3 mm observations. Four transitions were observed for HCO+ in VY CMa, the first definitive identification of this ion in a circumstellar envelope. Molecular line profiles from VY CMa are complex, indicating three separate outflows: a roughly spherical flow and separate red- and blueshifted winds, as suggested by earlier observations. Spectra from the other sources appear to trace a single outflow component. The line data were modeled with a radiative transfer code to establish molecular abundances relative to H2 and source distributions. Abundances for CO derived for these objects vary over an order of magnitude, f ∼ 0.4–5 × 10−4, with the lower values corresponding to the supergiants. For HCN, a similar range in abundance is found (f ∼ 0.9–9 × 10−6), with no obvious dependence on the mass-loss rate. In VY CMa, HCO+ is present in all three outflows with f ∼ 0.4–1.6 × 10−8 and a spatial extent similar to that of CO. HNC is found only in the red- and blueshifted components with [HCN]/[HNC] ∼ 150–190, while [CN]/[HCN] ∼ 0.01 in the spherical flow. All three velocity components are traced in CS, which has a confined spatial distribution and f ∼ 2–6 × 10−7. These observations suggest that carbon-bearing molecules in O-rich shells are produced by a combination of photospheric shocks and photochemistry. Shocks may play a more prominent role in the supergiants because of their macroturbulent velocities.

H2O Maser Outflow from the Red Supergiant Star NML Cygni Observed with Japanese VLBI Network

Abstract We present the proper motions of H$_2$O masers in NML Cygni, observed with the Japanese VLBI Network at three epochs spanning 455d. We detected about 15 maser features at each epoch. Overall, 13 features that were detected at least twice were tracked by their radial velocities and proper motions. The three-dimensional kinematics of the maser features indicate the presence of an expanding outflow. The major axis of the outflow is estimated to be at a position angle of $\sim\;$108°, and an inclination angle of $\sim\;$8° with respect to the line of sight. The H$_2$O masers are located between an apparent minimum radius of $\sim\;9.6 \times 10^{12}$m (64AU) and a maximum radius of $\sim\;3.0 \times 10^{13}$m (202AU), where the expansion velocity increases from 12 to 27kms$^{-1}$. A comparison with the distributions of SiO, H$_2$O, and OH masers suggests that the outflow of NML Cygni is expanding outside a radius of $\sim\;$1.5$\times$10$^{13}$m (100AU). This radius corresponds to 6 stellar radii, and is consistent with the radius of the inner boundary for the dust shell.

The Evolutionary State of the Cool Hypergiants – Episodic Mass Loss, Convective Activity and Magnetic Fields

AbstractThe evolved cool stars near the empirical upper luminosity boundary in the HR Diagram all show evidence for considerable instability perhaps due to their proximity to this stability limit and/or their evolutionary state. Recent high resolution imaging and spectroscopy of several of these stars have revealed a subset characterized by complex ejecta and evidence for episodic mass loss driven by convective activity and magnetic fields. This group includes famous stars such as the red supergiants VY CMa, NML Cyg and the post RSG IRC +10420. I will review the observational evidence and discuss the implications for the final stages of these evolved stars, their mass loss mechanism, and evolutionary state.

Excited-State OH Main-Line Masers in AU Geminorum and NML Cygni

Excited-state OH maser emission has previously been reported in the circumstellar envelopes of only two evolved stars: the Mira star AU Geminorum and the hypergiant NML Cygni. We present Very Large Array (VLA) observations of the 1665, 1667, and excited-state 4750 MHz main-line OH transitions in AU Gem and Expanded Very Large Array (EVLA) observations of the excited-state 6030 and 6035 MHz OH main-line transitions in NML Cyg. We detect masers in both main-line transitions in AU Gem but no excited-state emission in either star. We conclude that the excited-state OH emission in AU Gem is either a transient phenomenon (such as for NML Cyg, outlined below) or possibly an artifact in the data and that the excited-state OH emission in NML Cyg was generated by an episode of enhanced shock between the stellar mass loss and an outflow of the Cyg OB2 association. With these single exceptions, it therefore appears that excited-state OH emission indeed should not be predicted nor observable in evolved stars as part of their normal structure or evolution.

The hypergiant masers: episodic mass loss, convection and magnetic fields

AbstractThe cool hypergiants are the most luminous known stars in the upper HR Diagram in the apparent temperature range represented by spectral types A to M. Most of the stars in this regime are unstable as evidenced by their high mass loss rates, variability, and in some cases large IR excesses and circumstellar ejecta. We have obtained high resolution multi-wavelength images with HST/WFPC2 of several of the most known evolved cool stars including several well known stellar masers. VX Sgr and S Per were marginally resolved, while NML Cyg has a peculiar asymmetric envelope that has been shaped by its environment. The powerful maser sources IRC+10420 and VY CMa have extensive and complex circumstellar ejecta due to high mass loss episodes apparently driven by large-scale convective activity.

Bringing VY Canis Majoris Down to Size: An Improved Determination of Its Effective Temperature

The star VY CMa is a late-type M supergiant with many peculiarities, mostly related to the intense circumstellar environment due to the star's high mass-loss rate. Claims have been made that would imply that this star is considerably more luminous (L ~ 5 × 105 L☉) and larger (R ~ 2800 R☉) than other Galactic red supergiants (RSGs). Indeed, such a location in the H-R diagram would be well within the Hayashi forbidden zone, where stars cannot be in hydrostatic equilibrium. These extraordinary properties, however, rest on an assumed effective temperature of 2800-3000 K, far cooler than recent work has shown RSGs to be. To obtain a better estimate, we fit newly obtained spectrophotometry in the optical and NIR with the same MARCS models used for our recent determination of the physical properties of other RSGs; we also use V - K and V - J from the literature to derive an effective temperature. We find that the star likely has a temperature of 3650 K, a luminosity L ~ 6 × 104 L☉, and a radius of ~600 R☉. These values are consistent with VY CMa being an ordinary, evolved 15 M☉ RSG and agree well with the Geneva evolutionary tracks. We find that the circumstellar dust region has a temperature of 760 K, and an effective radius of ~130 AU, if spherical geometry is assumed for the latter. What causes this star to have such a high mass loss and large variations in brightness (but with little change in color) remains a mystery at present, although we speculate that perhaps this star and NML Cyg are simply normal RSGs caught during an unusually unstable time.

The Circumstellar Environments of NML Cygni and the Cool Hypergiants

We present high-resolution HST WFPC2 images of compact nebulosity surrounding the cool M-type hypergiants NML Cyg, VX Sgr, and S Per. The powerful OH/IR source NML Cyg exhibits a peculiar bean-shaped asymmetric nebula that is coincident with the distribution of its H2O vapor masers. We show that NML Cyg's circumstellar envelope is likely shaped by photodissociation from the powerful, nearby association Cyg OB2 inside the Cygnus X superbubble. The OH/IR sources VX Sgr and S Per have marginally resolved envelopes. S Per's circumstellar nebula appears elongated in a northeast/southwest orientation similar to that for its OH and H2O masers, while VX Sgr is embedded in a spheroidal envelope. We find no evidence for circumstellar nebulosity around the intermediate-type hypergiants ρ Cas, HR 8752, and HR 5171a, nor around the normal M-type supergiant μ Cep. We conclude that there is no evidence for high mass loss events prior to 500–1000 years ago for these four stars.

Erratum: “Observations of Water Vapor Outflow from NML Cygnus” (ApJ, 610, 427 [2004

For the values of the outflow velocities vinner and radius Rinner that we adopted at the inner edge of the outflows, and vf , the final velocities we cited for large radial distances from the stars, this requires revision of the calculated stellar masses. The corrected masses for NML Cyg and VY CMa become MNML 53 M and MVY 27 M , respectively. If we assume parabolic outflow, instead of FWHM velocities, the terminal velocities vf can be taken to be 26 and 20 km s , respectively, instead of 19 and 17 km s , and the corresponding stellar masses become MNML 41 M and MVY 22 M . Neither the calculated mass-loss rates that we cite in our paper, ṀNMLk 2:4 ; 10 4 M yr 1 and ṀVYk 3:3 ; 10 4 M yr , nor any other aspects or conclusions of the paper are affected by this correction.

Observations of Water Vapor Outflow from NML Cygnus

We report new observations of the far-infrared and submillimeter water vapor emission of NML Cygnus based on data gathered with the Infrared Space Observatory and the Submillimeter Wave Astronomy Satellite. We compare the emission from NML Cyg to that previously published for VY CMa and W Hya in an attempt to establish the validity of recently proposed models for the outflow from evolved stars. The data obtained support the contention by Ivezic & Elitzur that the atmospheres of evolved stars obey a set of scaling laws in which the optical depth of the outflow is the single most significant scaling parameter, affecting both the radiative transfer and the dynamics of the outflow. Specifically, we provide observations comparing the water vapor emission from NML Cyg, VY CMa, and W Hya and find, to the extent permitted by the quality of our data, that the results are in reasonable agreement with a model developed by Zubko & Elitzur. Using this model we derive a mass loss based on the dust opacities, spectral line fluxes, and outflow velocities of water vapor observed in the atmospheres of these oxygen-rich giants. For VY CMa and NML Cyg, we also obtain an estimate of the stellar mass.