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Both stars and planets can lose mass through an expansive wind outflow, often constrained or channeled by magnetic fields that form a surrounding magnetosphere. The very strong winds of massive stars are understood to be driven by line-scattering of the star's radiative momentum, while in the Sun and even lower-mass stars a much weaker mass loss arises from the thermal expansion of a mechanically heated corona. In exoplanets around such low-mass stars, the radiative heating and wind interaction can lead to thermal expansion or mechanical ablation of their atmospheres. Stellar magnetospheres result from the internal trapping of the wind outflow, while planetary magnetospheres are typically shaped by the external impact from the star's wind. But in both cases the stressing can drive magnetic reconnection that results in observable signatures such as X-ray flares and radio outbursts. This review will aim to give an overview of the underlying physics of these processes with emphasis on their similarities and distinctions for stars vs. planets., Comment: 20 pages, 15 figures, plenary review for IAUS 370: Winds from Stars and Exoplanets

Many stars across all classes possess strong enough magnetic fields to influence dynamical flow of material off the stellar surface. For the case of massive stars (O and B types), about 10\% of them harbour strong, globally ordered (mostly dipolar) magnetic fields. The trapping and channeling of their stellar winds in closed magnetic loops leads to {\it magnetically confined wind shocks} (MCWS), with pre-shock flow speeds that are some fraction of the wind terminal speed that can be a few thousand km s$^{-1}$. These shocks generate hot plasma, a source of X-rays. In the last decade, several developments took place, notably the determination of the hot plasma properties for a large sample of objects using \xmm\ and \ch, as well as fully self-consistent MHD modelling and the identification of shock retreat effects in weak winds. In addition, these objects are often sources of H$\alpha$ emission which is controlled by either sufficiently high mass loss rate or centrifugal breakout. Here we review the theoretical aspects of such magnetic massive star wind dynamics., Comment: Accepted for publication invited chapter of the Handbook of X-ray and Gamma-ray Astrophysics published by Nature Springer. arXiv admin note: text overlap with arXiv:1509.06482, arXiv:1605.04979

Most of a star's mass is bound in a hydrostatic equilibrium in which pressure balances gravity. But if at some near-surface layer additional outward forces overcome gravity, this can transition to a supersonic, outflowing wind, with the sonic point, where the outward force cancels gravity, marking the division between hydrostatic atmosphere and wind outflow. This talk will review general issues with such transonic initiation of a stellar wind outflow, and how this helps set the wind mass loss rate. The main discussion contrasts the flow initiation in four prominent classes of steady-state winds: (1) the pressure-driven coronal wind of the sun and other cool stars; (2) line-driven winds from OB stars; (3) a two-stage initiation model for the much denser winds from Wolf-Rayet (WR) stars; and (4) the slow "overflow" mass loss from highly evolved giant stars. A follow on discussion briefly reviews eruptive mass loss, with particular focus on the giant eruption of eta Carinae., Comment: 14 pages, 11 figures; to appear in proceedings of IAUS 366, "The Origin of Outflows from Evolved Stars", Leen Decin and Clio Gielen, editors

The most massive stars are thought to lose a significant fraction of their mass in a steady wind during the main-sequence and blue supergiant phases. This in turn sets the stage for their further evolution and eventual supernova, with consequences for ISM energization and chemical enrichment. Understanding these processes requires accurate observational constraints on the mass-loss rates of the most luminous stars, which can also be used to test theories of stellar wind generation. In the past, mass-loss rates have been characterized via collisional emission processes such as H$\alpha$ and free-free radio emission, but these so-called "density squared" diagnostics require correction in the presence of widespread clumping. Recent observational and theoretical evidence points to the likelihood of a ubiquitously high level of such clumping in hot-star winds, but quantifying its effects requires a deeper understanding of the complex dynamics of radiatively driven winds. Furthermore, large-scale structures arising from surface anisotropies and propagating throughout the wind can further complicate the picture by introducing further density enhancements, affecting mass-loss diagnostics. Time series spectroscopy of UV resonance lines with high resolution and high signal-to-noise are required to better understand this complex dynamics, and help correct "density squared" diagnostics of mass-loss rates. The proposed Polstar mission easily provides the necessary resolution at the sound-speed scale of 20 km s$^{-1}$, on three dozen bright targets with signal-to noise an order of magnitude above that of the celebrated IUE MEGA campaign, via continuous observations that track structures advecting through the wind in real time., Comment: white paper

PolStar is an Explorer-class far ultraviolet (FUV) spectropolarimetry mission designed to target massive stars and their environments. PolStar will take advantage of resonance lines only available in the FUV to measure for the first time the magnetic and wind environment around massive stars to constrain models of rotation and mass loss.

In studying the formation of massive stars, it is essential to consider the strong radiative feedback on the stars' natal environments from their high luminosities ($10^4 \sim 10^6 L_\odot$). Given that massive stars contract to main-sequence-like radii before accretion finishes, one form this feedback takes is UV line-acceleration, resulting in outflows much like those expected from main-sequence massive stars. As shown by the prior papers in this series, in addition to driving stellar winds, such line forces also ablate the surface layers off of circumstellar discs within a few stellar radii of the stellar photosphere. This removal of material from an accretion disc in turn results in a decreased accretion rate onto the forming star. Quantifying this, however, requires accounting for the continuum optical depth of the disc along the non-radial rays required for the three-dimensional line-acceleration prescription used in this paper series. We introduce the "thin disc approximation", allowing these continuum optical depths arising from an optically thick but geometrically thin disc to be dynamically treated in the context of radiation-hydrodynamics simulations. Using this approximation in full dynamical simulations, we show that such continuum optical depth effects only reduce the disc ablation by 30 percent or less relative to previous simulations that ignored continuum absorption., Comment: 10 pages, 9 figures

In this contribution, we present BRITE observations of the early-B supergiants $\epsilon$ Ori and $\kappa$ Ori. We perform a preliminary analysis of the data acquired over the first two Orion observing runs. We evaluate whether they are compatible with co-rotating bright spots and discuss the challenges of such an approach., Comment: 6 pages, 3 figures, to be published in the proceedings of the 3rd BRITE Science Conference held in Saint-Michel-des-Saints (QC, Canada), 2017 August 7-10 -- Proceedings of the Polish Astronomical Society

We extend our previous study of radiative ablation of circumstellar disks by line-scattering of the star's radiation, accounting now for the effect of multiple line resonances off the stellar limb. For an analytic, three-dimensional model of the velocity structure of an equatorial Keplerian disk bounded at higher latitudes by a radially accelerating stellar wind outflow, we use root-finding methods to identify multiple resonances from a near-disk circumstellar location along starward rays both on and off the stellar core. Compared to our previous study that accounted only for the effect of on-core resonances in reducing the radiative driving through the scattering of radiation away from a near-disk circumstellar location, including off-limb resonances leads to additional radiative driving from scattering toward this location. Instead of the up-to-50% reduction in line-acceleration previously inferred from multiple resonance effects, we now find a more modest 15-20% net reduction., Comment: 8 pages, 6 figures

In this paper we report 23 magnetic field measurements of the B3IV star HD 23478: 12 obtained from high resolution Stokes $V$ spectra using the ESPaDOnS (CFHT) and Narval (TBL) spectropolarimeters, and 11 from medium resolution Stokes $V$ spectra obtained with the DimaPol spectropolarimeter (DAO). HD 23478 was one of two rapidly rotating stars identified as potential "centrifugal magnetosphere" hosts based on IR observations from the Apache Point Observatory Galactic Evolution Experiment survey. We derive basic physical properties of this star including its mass ($M=6.1^{+0.8}_{-0.7}\,M_\odot$), effective temperature ($T_{\rm eff}=20\pm2\,$kK), radius ($R=2.7^{+1.6}_{-0.9}\,R_\odot$), and age ($\tau_{\rm age}=3^{+37}_{-1}\,$Myr). We repeatedly detect weakly-variable Zeeman signatures in metal, He and H lines in all our observations corresponding to a longitudinal magnetic field of $\langle B_z\rangle\approx-2.0\,$kG. The rotational period is inferred from Hipparcos photometry ($P_{\rm rot}=1.0498(4)\,$d). Under the assumption of the Oblique Rotator Model, our obsevations yield a surface dipole magnetic field of strength $B_d\geq9.5\,$kG that is approximately aligned with the stellar rotation axis. We confirm the presence of strong and broad H$\alpha$ emission and gauge the volume of this star's centrifugal magnetosphere to be consistent with those of other H$\alpha$ emitting centrifugal magnetosphere stars based on the large inferred Alfv\'en to Kepler radius ratio., Comment: 12 pages, 12 figures

We report on both high-precision photometry from the MOST space telescope and ground-based spectroscopy of the triple system delta Ori A consisting of a binary O9.5II+early-B (Aa1 and Aa2) with P = 5.7d, and a more distant tertiary (O9 IV P > 400 yrs). This data was collected in concert with X-ray spectroscopy from the Chandra X-ray Observatory. Thanks to continuous coverage for 3 weeks, the MOST light curve reveals clear eclipses between Aa1 and Aa2 for the first time in non-phased data. From the spectroscopy we have a well constrained radial velocity curve of Aa1. While we are unable to recover radial velocity variations of the secondary star, we are able to constrain several fundamental parameters of this system and determine an approximate mass of the primary using apsidal motion. We also detected second order modulations at 12 separate frequencies with spacings indicative of tidally influenced oscillations. These spacings have never been seen in a massive binary, making this system one of only a handful of such binaries which show evidence for tidally induced pulsations., Comment: Accepted to Apj. Part of a 4 paper series. 11 figures, 4 tables

This work explores a Pulsationally Driven Orbital Mass Ejection (PDOME) model for the launching of Classical Be star disks. Under this model, a combination of rapid rotation and non-radial pulsation modes contribute to placing material into the circumstellar environment. Several varieties of non-radial pulsation modes, characterized by their propagation direction and the relative phase of their velocity and density perturbations, are considered. As well, the orbital stability of material launched by such a mechanism is investigated., Comment: Bright Emissaries Conference 2014, London, ON, Canada

We present results from a study of the eclipsing, colliding-wind binary V444 Cyg that uses a combination of X-ray and optical spectropolarimetric methods to describe the 3-D nature of the shock and wind structure within the system. We have created the most complete X-ray light curve of V444 Cyg to date using 40 ksec of new data from Swift, and 200 ksec of new and archived XMM-Newton observations. In addition, we have characterized the intrinsic, polarimetric phase-dependent behavior of the strongest optical emission lines using data obtained with the University of Wisconsin's Half-Wave Spectropolarimeter. We have detected evidence of the Coriolis distortion of the wind-wind collision in the X-ray regime, which manifests itself through asymmetric behavior around the eclipses in the system's X-ray light curves. The large opening angle of the X-ray emitting region, as well as its location (i.e. the WN wind does not collide with the O star, but rather its wind) are evidence of radiative braking/inhibition occurring within the system. Additionally, the polarimetric results show evidence of the cavity the wind-wind collision region carves out of the Wolf-Rayet star's wind., Comment: 19 pages, 14 figures, accepted A&A

Magnetically confined winds of early-type stars are expected to be sources of bright and hard X-rays. To clarify the systematics of the observed X-ray properties, we have analyzed a large series of Chandra and XMM observations, corresponding to all available exposures of known massive magnetic stars (over 100 exposures covering ~60% of stars compiled in the catalog of Petit et al. 2013). We show that the X-ray luminosity is strongly correlated with the stellar wind mass-loss-rate, with a power-law form that is slightly steeper than linear for the majority of the less luminous, lower-Mdot B stars and flattens for the more luminous, higher-Mdot O stars. As the winds are radiatively driven, these scalings can be equivalently written as relations with the bolometric luminosity. The observed X-ray luminosities, and their trend with mass-loss rates, are well reproduced by new MHD models, although a few overluminous stars (mostly rapidly rotating objects) exist. No relation is found between other X-ray properties (plasma temperature, absorption) and stellar or magnetic parameters, contrary to expectations (e.g. higher temperature for stronger mass-loss rate). This suggests that the main driver for the plasma properties is different from the main determinant of the X-ray luminosity. Finally, variations of the X-ray hardnesses and luminosities, in phase with the stellar rotation period, are detected for some objects and they suggest some temperature stratification to exist in massive stars' magnetospheres., Comment: 31p, 11 figures, 8 tables - including online material and erratum, accepted for publication by ApJ

For 30 years, cyclical wind variability in OB stars has puzzled the astronomical community. Phenomenological models involving co-rotating bright spots provide a potential explanation for the observed variations, but the underlying physics remains unknown. We present recent results from hydrodynamical simulations constraining bright spot properties and compare them to what can be inferred from space-based photometry. We also explore the possibility that these spots are caused by magnetic fields and discuss the detectability of such fields., Comment: 2 pages, 1 figure, IAU Symposium 307

X-ray satellites since Einstein have empirically established that the X-ray luminosity from single O-stars scales linearly with bolometric luminosity, Lx ~ 10^-7 Lbol. But straightforward forms of the most favored model, in which X-rays arise from instability-generated shocks embedded in the stellar wind, predict a steeper scaling, either with mass loss rate Lx ~ Mdot ~ Lbol^1.7 if the shocks are radiative, or with Lx ~ Lx ~ Mdot^2 ~ Lbol^3.4 if they are adiabatic. We present here a generalized formalism that bridges these radiative vs. adiabatic limits in terms of the ratio of the shock cooling length to the local radius. Noting that the thin-shell instability of radiative shocks should lead to extensive mixing of hot and cool material, we then propose that the associated softening and weakening of the X-ray emission can be parametrized by the cooling length ratio raised to a power m, the "mixing exponent." For physically reasonable values m ~= 0.4, this leads to an X-ray luminosity Lx ~ Mdot^0.6 ~ Lbol that matches the empirical scaling. We conclude by noting that such thin-shell mixing may also be important for X-rays from colliding wind binaries, and that future numerical simulation studies will be needed to test this thin-shell mixing ansatz for X-ray emission., Comment: 5 pages, to appear in "Four Decades of Research on Massive Stars", eds. L. Drissen, C. Robert, and N. St-Louis, ASP Conference Series

Perhaps the most extreme examples of "Active OB stars" are the subset of high-mass X-ray binaries -- consisting of an OB star plus compact companion -- that have recently been observed by Fermi and ground-based Cerenkov telescopes like HESS to be sources of very high energy (VHE; up to 30 TeV) gamma-rays. This paper focuses on the prominent gamma-ray source, LS5039, which consists of a massive O6.5V star in a 3.9-day-period, mildly elliptical (e = 0.24) orbit with its companion, assumed here to be a black-hole or unmagnetized neutron star. Using 3-D SPH simulations of the Bondi-Hoyle accretion of the O-star wind onto the companion, we find that the orbital phase variation of the accretion follows very closely the simple Bondi-Hoyle-Lyttleton (BHL) rate for the local radius and wind speed. Moreover, a simple model, wherein intrinsic emission of gamma-rays is assumed to track this accretion rate, reproduces quite well Fermi observations of the phase variation of gamma-rays in the energy range 0.1-10 GeV. However for the VHE (0.1-30 TeV) radiation observed by the HESS Cerenkov telescope, it is important to account also for photon-photon interactions between the gamma-rays and the stellar optical/UV radiation, which effectively attenuates much of the strong emission near periastron. When this is included, we find that this simple BHL accretion model also quite naturally fits the HESS light curve, thus making it a strong alternative to the pulsar-wind-shock models commonly invoked to explain such VHE gamma-ray emission in massive-star binaries., Comment: To appear in "Active OB Stars: Structure, Evolution, Mass Loss & Critical Limits", Proceedings of IAUS 272, held July 2010 in Paris, France. 7 pages; 3 figures. This version 2 corrects an alignment error in figure 2

We report near-IR spectroscopic observations of the Eta Carinae massive binary system during 2008-2009 using VLT/CRIRES. We detect a strong, broad absorption wing in He I 10833 extending up to -1900 km/s across the 2009.0 spectroscopic event. Archival HST/STIS ultraviolet and optical data shows a similar high-velocity absorption (up to -2100 km/s) in the UV resonance lines of Si IV 1394, 1403 across the 2003.5 event. UV lines from low-ionization species, such as Si II 1527, 1533 and C II 1334, 1335, show absorption up to -1200 km/s, indicating that the absorption with v from -1200 to -2100 km/s originates in a region markedly faster and more ionized than the nominal wind of the primary star. Observations obtained at the OPD/LNA during the last 4 spectroscopic cycles (1989-2009) also display high-velocity absorption in He I 10833 during periastron. Based on the OPD/LNA dataset, we determine that material with v < -900 km/s is present in the phase range 0.976 < phi < 1.023 of the spectroscopic cycle, but absent in spectra taken at phi < 0.947 and phi > 1.049. Therefore, we constrain the duration of the high-velocity absorption to be 95 to 206 days (or 0.047 to 0.102 in phase). We suggest that the high-velocity absorption originates from shocked gas in the wind-wind collision zone, at distances of 15 to 45 AU in the line-of-sight to the primary star. Using 3-D hydrodynamical simulations of the wind-wind collision zone, we find that the dense high-velocity gas is in the line-of-sight to the primary star only if the binary system is oriented in the sky so that the companion is behind the primary star during periastron, corresponding to a longitude of periastron of omega ~ 240 to 270 degrees. We study a possible tilt of the orbital plane relative to the Homunculus equatorial plane and conclude that our data are broadly consistent with orbital inclinations in the range i=40 to 60 degrees., Comment: 18 pages, 15 figures, accepted for publication in A&A; high-resolution PDF version available also at http://www.mpifr.de/staff/jgroh/etacar.html

Dimtri Mihalas' textbooks in the 70's and 80's on "Stellar Atmospheres" and "Foundations of Radiation Hydrodynamics" helped lay the early groundwork for understanding the moving atmospheres and winds of massive, luminous stars. Indeed, the central role of the momentum of stellar radiation in driving the mass outflow makes such massive-star winds key prototypes for radiation hydrodynamical processes. This paper reviews the dynamics of such radiative driving, building first upon the standard CAK model, and then discussing subtleties associated with the development and saturation of instabilities, and wind initiation near the sonic point base. An overall goal is to illuminate the rich physics of radiative driving and the challenges that lie ahead in developing dynamical models that can explain the broad scaling of mass loss rate and flow speed with stellar properties, as well as the often complex structure and variability observed in massive-star outflows., Comment: 14 pages. to appear in "Recent Directions in Astrophysical Quantitative Spectroscopy and Radiation Hydrodynamics"

Results from the most extensive study of the time-evolving dust structure around the prototype "Pinwheel" nebula WR 104 are presented. Encompassing 11 epochs in three near-infrared filter bandpasses, a homogeneous imaging data set spanning more than 6 years (or 10 orbits) is presented. Data were obtained from the highly successful Keck Aperture Masking Experiment, which can recover high fidelity images at extremely high angular resolutions, revealing the geometry of the plume with unprecedented precision. Inferred properties for the (unresolved) underlying binary and wind system are orbital period 241.5 +/- 0.5 days and angular outflow velocity of 0.28 +/- 0.02 mas/day. An optically thin cavity of angular size 13.3 +/- 1.4 mas was found to lie between the central binary and the onset of the spiral dust plume. Rotational motion of the wind system induced by the binary orbit is found to have important ramifications: entanglement of the winds results in strong shock activity far downstream from the nose of the bowshock. The far greater fraction of the winds participating in the collision may play a key role in gas compression and the nucleation of dust at large radii from the central binary and shock stagnation point. Investigation of the effects of radiative braking pointed towards significant modifications of the simple hydrostatic colliding wind geometry, extending the relevance of this phenomena to wider binary systems than previously considered. Limits placed on the maximum allowed orbital eccentricity of e < 0.06 argue strongly for a prehistory of tidal circularization in this system. Finally we discuss the implications of Earth's polar (i < 16 deg) vantage point onto a system likely to host supernova explosions at future epochs., Comment: 35 pages, 8 figures, Accepted for publication in Astrophysical Journal

The migration of profile sub-peaks identified in time-monitored optical emission lines of Wolf-Rayet star spectra provides a direct diagnostic of the dynamics of their stellar winds via a measured line-of-sight velocity change per unit time. Inferring the associated wind acceleration scale from such an apparent acceleration then relies on the adopted intrinsic velocity of the wind material at the origin of this variable pattern. We develop radiative transfer tools to characterise the Line Emission Region (LER) of the program lines, including turbulence and line-optical depth effects. We find that monitored-lines can be fitted well with a pure optically thin formation mechanism, line-broadening resulting from the finite velocity extent of the LER rather than turbulence.Our new estimates of LER velocity centroids are systematically shifted outwards closer to terminal velocity compared to previous determinations, now suggesting WR-wind acceleration length scales of ca. 10-20Rsun, a factor of a few smaller than previously inferred. Based on radiation-hydrodynamics simulations of the line-driven-instability mechanism, we compute synthetic line profile variability for CIII5696A, for WR111 The results match well the measured observed migration of 20-30 m/s2, However, our model stellar radius of 19 Rsun, typical of an O-type supergiant, is a factor 2--10 larger than generally expected for WR core radii.Such small radii leave inferred acceleration scales to be more extended than expected from dynamical models of line driving, but the severity of the discrepancy is substantially reduced compared to previous analyses. We conclude with a discussion of how using lines formed deeper in the wind would provide a stronger constraint on the key wind dynamics as well as information on the onset of wind clumping., Comment: 16 pages, 13 figures

This talk summarizes results from recent MHD simulations of the role of a dipole magnetic field in inducing large-scale structure in the line-driven stellar winds of hot, luminous stars. Unlike previous fixed-field analyses, the MHD simulations here take full account of the dynamical competition between the field and the flow. A key result is that the overall degree to which the wind is influenced by the field depends largely on a single, dimensionless `wind magnetic confinement parameter', $\eta_\ast (= B_{eq}^2 R_{\ast}^2/\dot{M} v_\infty$), which characterizes the ratio between magnetic field energy density and kinetic energy density of the wind. For weak confinement, $\eta_\ast \le 1$, the field is fully opened by wind outflow, but nonetheless, for confinement as small as $\eta_\ast=1/10$ it can have significant back-influence in enhancing the density and reducing the flow speed near the magnetic equator. For stronger confinement, $\eta_\ast > 1$, the magnetic field remains closed over limited range of latitude and height above the equatorial surface, but eventually is opened into nearly radial configuration at large radii. Within the closed loops, the flow is channeled toward loop tops into shock collisions that are strong enough to produce hard X-rays. Within the open field region, the equatorial channeling leads to oblique shocks that are again strong enough to produce X-rays and also lead to a thin, dense, slowly outflowing ``disk'' at the magnetic equator., Comment: To be published in the proceedings of International Conference on magnetic fields in O, B and A stars, ASP Conference Series, Vol. 216, 1003, L.A. Balona, H. Henrichs & T. Medupe, eds.; for alternate upload go to: http://wonka.physics.ncsu.edu/~auddoul/publications/udDoula.ps

We summarize recent 2D MHD simulations of line-driven stellar winds from rotating hot-stars with a dipole magnetic field aligned to the star's rotation axis. For moderate to strong fields, much wind outflow is initially along closed magnetic loops that nearly corotate as a solid body with the underlying star, thus providing a torque that results in an effective angular momentum spin-up of the outflowing material. But instead of forming the ``magnetically torqued disk'' (MTD) postulated in previous phenemenological analyses, the dynamical simulations here show that material trapped near the tops of such closed loops tends either to fall back or break out, depending on whether it is below or above the Keplerian corotation radius. Overall the results raise serious questions about whether magnetic torquing of a wind outflow could naturally result in a Keplerian circumstellar disk. However, for very strong fields, it does still seem possible to form a %``magnetically confined, centrifugally supported, rigid-body disk'', centrifugally supported, ``magnetically rigid disk'' (MRD), in which the field not only forces material to maintain a rigid-body rotation, but for some extended period also holds it down against the outward centrifugal force at the loop tops. We argue that such rigid-body disks seem ill-suited to explain the disk emission from Be stars, but could provide a quite attractive paradigm for circumstellar emission from the magnetically strong Bp and Ap stars., Comment: To be published in the proceedings of International Conference on magnetic fields in O, B and A stars, ASP Conference Series, Vol. 216, 1003, L.A. Balona, H. Henrichs & T. Medupe, eds.; for alternate upload go to: http://www.bartol.udel.edu/~owocki/preprints/Owocki-SA.pdf

We carry out an extended analytic study of how the tilt and faster-than-radial expansion from a magnetic field affect the mass flux and flow speed of a line-driven stellar wind. A key motivation is to reconcile results of numerical MHD simulations with previous analyses that had predicted non-spherical expansion would lead to a strong speed enhancement. By including finite-disk correction effects, a dynamically more consistent form for the non-spherical expansion, and a moderate value of the line-driving power index $\alpha$, we infer more modest speed enhancements that are in good quantitative agreement with MHD simulations, and also are more consistent with observational results. Our analysis also explains simulation results that show the latitudinal variation of the surface mass flux scales with the square of the cosine of the local tilt angle between the magnetic field and the radial direction. Finally, we present a perturbation analysis of the effects of a finite gas pressure on the wind mass loss rate and flow speed in both spherical and magnetic wind models, showing that these scale with the ratio of the sound speed to surface escape speed, $a/v_{esc}$, and are typically 10-20% compared to an idealized, zero-gas-pressure model., Comment: Accepted for publication in ApJ, for the full version of the paper go to: http://www.bartol.udel.edu/~owocki/preprints/btiltdiv-mdotvinf.pdf