Theory and Diagnostics of Hot Star Mass Loss. Jorick S. Vink 2022. A Puke(TM) Audiopaper.

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Hot Star Mass Loss, or Radiatively driven Stellar winds from large, massive stars.

Annual Review of Astronomy and Astrophysics Theory and Diagnostics of Hot Star Mass Loss.

https://arxiv.org/abs/2109.08164v4

Jorick S. Vink Armagh Observatory and Planetarium, Armagh, Northern Ireland.
email: jorick.vink@armagh.ac.uk
Annual Rev. Astron. Astrophysics 2022. 60, 203 to 46

Keywords: stellar winds, stellar evolution, massive stars, black holes, clumps, supernovae

Abstract.

Massive stars have strong stellar winds that direct their evolution through the upper Hertzsprung Russell diagram and determine the black hole mass function. Furthermore, wind strength dictates the atmospheric structure that sets the ionizing flux. Finally, the wind directly intervenes with the stellar envelope structure, which is decisive for both single-star and binary evolution, affecting predictions for gravitational wave events. Key findings of current hot star research include:
The traditional line-driven wind theory, supplemented with Monte Carlo and co moving frame computations, is revealing a rich multivariate behavior of the mass-loss rate M dot in terms of M, L, Eddington Gamma, effective temperature, and chemical composition Z. Concerning the latter, M dot is shown to depend on the iron opacity, making Wolf Rayet populations, and gravitational wave events dependent on host galaxy Z.
On top of smooth mass-loss behavior, there are several transitions in the Hertzsprung Russell diagram, involving bi-stability jumps around Iron recombination temperatures, leading to quasi-stationary episodic, and not necessarily eruptive, luminous blue variable and pre-SN mass loss.
Furthermore, there are kinks. At 100 Solar Masses a high mass-loss transition implies that hydrogen-rich, very massive stars have higher mass-loss rates than commonly considered. At the other end of the mass spectrum, low-mass stripped helium stars no longer appear as Wolf Rayet stars but as optically thin stars. These stripped stars, in addition to very massive stars, are two newly identified sources of ionizing radiation that could play a key role in local star formation as well as at high redshift.

Section One. Introduction.
Accurate mass-loss rates, M dot, for hot massive stars are critical for our understanding of the evolution and fate of the most massive stars, M greater than 25 solar masses, in the Universe. Whereas the stellar initial mass function (IMF) dictates that there are more massive stars below 25 Solar masses than above it, due to their higher luminosities L star and mass-loss rates, it is these objects more massive than 25 Solar masses that dominate the radiative and mechanical feedback. Their light is also expected to be detected with new facilities such as the James Webb Space Telescope, JWST, and ground-based extremely large telescopes in the coming decade.
Furthermore, the interest in the most massive stars above 25 Solar masses has grown tremendously thanks to the discovery of heavy black holes BH’s of over 25 Solar masses via gravitational waves with LIGO, Laser Interferometer Gravitation-Wave Observatory, and Virgo, Abbott et al, 20 16, and 20 20.
Stars above 25 Solar masses may not produce SNe, but directly collapse into a BH instead. It is these stars above approximately 25 Solar masses where winds are sufficiently strong to affect stellar evolution, which this review is focused on.
More generally, though there is an established paradigm for the evolution of massive OB stars in the 8 to 25 solar mass range toward red super giants RSG’s leading to hydrogen, Hydrogen rich type Two P supernovae, SNe, Smartt 2009. These likely leave behind a neutron star. The evolutionary route for stars above 25 Solar masses is, despite significant theoretical and observational progress, still in its Infancy.

Answers regarding the evolution of the most massive stars must rely on theoretical and observational progress in our detailed understanding of stellar winds, as a function of metallicity, Z. M dot equals a function of Z. Also, the landscape of various SN subtypes, see the sidebar titled Taxonomy for Core Collapse Supernovae, is thought to involve a mass-loss sequence, with the common Two P SNe still having the largest Hydrogen envelope intact, whereas progressively higher-mass stars lose more of their envelopes. This mass loss could involve Z-dependent winds, eruptive mass loss that might be Z-independent, or alternatively be caused by Roche-Lobe overflow (RLOF) in a binary system. Mapping SN subtypes as a function of host galaxy Z may reveal the relative importance of Z-dependent versus Z-independent mass-loss mechanisms.
Stars up to around 25 Solar masses become RSG’s that keep their Hydrogen envelopes intact.
The evolutionary paths for more massive stars must be notably different as there is a complete absence of RSGs in the Hertzsprung Russell diagram (HRD) above a critical luminosity limit: the Humphreys Davidson (HD) limit, see Figure 1, at a luminosity of log (L over Solar luminosity equals 5.8, but see Levesque et al 2005. This bifurcation between the H-rich and H-poor Universe forms a distinct feature in the HRD. Although it has become clear that somewhere around 50 percent (Sana et al 20 13, of massive OB stars are part of a close binary system, there is no a priori reason that this would lead to such a sharp feature in the HRD. Instead, this tipping point is most likely related to the Eddington limit against radiation pressure.
Recent imaging studies suggest that the total binary fraction is close to 100 percent, for example Sana et al 20 14, but some companions are too distant to alter massive-star evolution. It is the close binaries of around 50 percent discovered through radial velocity (RV) variations that are relevant for stellar evolution, involving the stripping of the Hydrogen envelope, and even merging events on and off the main sequence.
In any case contrary to the lower-mass range, stars above 25 Solar Masses do not become RSG’s. This implies that it is fast hot star winds with Effective temperatures greater than 10,000 K that dominate the mass-loss history and fate of the most massive stars that need to be quantitatively mapped to understand Hydrogen rich and Hydrogen poor, superluminous, SNe and to accurately predict the BH mass function as a function of metallicity Z. Although during the last 50 years a theoretical paradigm for these hot star winds in terms of radiation pressure has been established by Castor, Abbott, and Klein, CAK, Castor et al 1975, there are still many uncertainties in even the absolute mass-loss rates predicted by the radiation-driven wind theory, RDWT.

Figure One. Hertzsprung Russell diagram of the VLT-Flames Tarantula Survey in the Large Magellanic Cloud (LMC). The solid and dashed black lines are the nonrotating Bonn stellar evolutionary tracks and isochrones from Brott et al 20 11 and Kohler et al 20 15.
The original HD limit is shown by the dot-dashed line in the top-right corner. Symbols are indicated in the legend. Slash stars are Of WN stars. Objects cooler than 9,000 Kelvin are shown as triangles. The luminosities of the classical WR stars are indicated by the arrows on the left. They actually have surface temperatures of over 90 thousand K. Figure adapted from Schneider et al.
Abbreviations: BSG, blue supergiant; HD, Humphreys-Davidson; LBV, luminous blue variable; RSG, red supergiant; VLT, Very Large Telescope; WN, nitrogen-type Wolf Rayet. WNh, Hydrogen rich WN stars. WR, Wolf Rayett.

In the 8 to 25 Solar Mass regime, observational results have been published in which the discrepancies between the RDWT of Vink et al, and observations have been up to two orders of magnitude in the weak-wind regime Martins et al two thousand and five.
As a definition, the Weak-wind regime is: an empirical regime where mass-loss rates below log L over Solar Luminosity equals 5.2 are around 2 orders of magnitude lower than predicted.
Very massive stars, VMS’s, are stars more massive than 100 Solar Masses, including WNh and, usually, Of WN slash stars.
By contrast, on the other extreme, for very massive stars VMS’s on the order of 100 Solar Masses and above, we know the absolute mass-loss rate relatively accurately, within about 30 percent, but because the cumulative effect of mass loss on stellar evolution models is more dominant in this regime, the uncertainties in mass-loss history are nonetheless profound.
In the local Universe, these are WNh stars: N-type Wolf-Rayett stars that still contain H. The less dense-winded Of WN “slash” stars are usually also included.
This is especially true given that VMS’s are close to the Eddington limit, and there is not yet any settled solution for stellar structure models as to how the envelopes and stellar winds interact with one another. This implies that VMS radii and effective temperatures are highly uncertain, and thus we need to unravel how the mass loss rates depend on T eff. This is absolutely critical for establishing whether VMS’s are important sources of ionizing radiation in the Universe. Despite the fact that mass-loss rates for around 100 Solar mass stars have been accurately calibrated at the transition mass-loss rate, see Section 2 point 2 point 4, for stars in the canonical O-star regime above 25 Solar masses there are uncertainties in mass-loss rates of a factor of 2 to 3. These uncertainties are mostly related to wind clumping, the radiative transfer treatment in subsonic regions, and wind hydrodynamics beyond CAK. Finally, we note that an uncertainty of just a factor of 2 in the mass-loss rate in stellar evolution of a canonical O star has long been known to be crucial.

Such a relatively small difference in Mass loss rate on the main sequence can make a 60 Solar Mass star either remain intact and die as an around 25 solar mass BH or almost completely evaporate, dying as a neutron star or light BH of just a few solar masses.
In other words, it is pertinent that we develop an accurate framework for radiation-driven winds over the entire upper HRD and that we predict how the mass-loss rate depends on the stellar mass M, luminosity L (or Eddington parameter Gamma), the effective temperature T eff, and chemical composition. Furthermore, stars may not remain in the same HRD position, but change their T eff locations on relatively short timescales, on the order of years and decades for the case of luminous blue variables LBV’s.
This could lead to episodic mass loss that could be described by stationary wind models or, in other situations, the mass-loss rates are so humongous about 10 to the minus 2 Solar masses per year or higher, that nonstationary processes such as eruptive mass loss or dynamical phenomena such as explosions need to be considered. This is especially the case for giant eruptions of LBV’s, for example Shaviv, Owocki, and certain SN progenitors, such as the superluminous Two n SN 2006 gy.
Note that the existence of these extreme cases, in which super-Eddington winds and explosions are required, does not imply that each and every case of episodic or pre-SN mass loss demands such physical extremes. It is quite possible that for the vast majority of pre-SN mass-loss cases quasi stationary winds can explain most episodic mass loss. We return to this in Section 5. For convenience, I provide an overview of mass-loss recipes for hot star winds in the Appendix. For a recent overview on cooler RSG winds, I simply refer to Decin. This review focuses on hot star winds from massive stars only, but general principles should apply to low-mass stars as well.
Section Two. Radiation-driven wind theory.
Although it had already been established in the early parts of the twentieth century that radiation pressure was capable of ejecting atoms from stellar atmospheres, the theory matured in the nineteen seventies thanks to observational impetus from ultraviolet (UV) spectroscopy of O-type stars with satellites such as the International Ultraviolet Explorer (IUE). These data showed the widespread presence of P Cygni profiles in UV resonance lines, such as carbon four, silicon four, and nitrogen five, which were interpreted as stationary outflows that might be sufficiently strong to affect massive star evolution. A P Cygni profile is a characteristic spectral resonance line involving almost symmetric line emission and blue shifted absorption, named after the luminous blue variable P Cygni.
The first theoretical models predicted modest rates, but when Castor et al 1975 introduced line-distribution functions the predicted O-star rates grew by 2 orders of magnitude to 10 to the minus 6 Solar masses per year, which is sufficient to affect stellar evolution. This led to the so-called Conti (1975) scenario, in which the loss of significant amounts of envelope mass would result in the formation of classical Wolf Rayet stars (WR).
Spectroscopic designation based on emission lines. WR’s come in many flavors, but the classical WR stars are of types WN and WC: the nitrogen-rich WN stars have enhanced nitrogen (N) due to the CNO cycle, whereas the more evolved WC stars show the product of Helium burning, in the form of carbon (C), at the surface.
The mass-loss rates predicted by CAK theory are now thought to be somewhat on the high side but still correct within a factor of a few. In the 1980s, the Munich group, Pauldrach et al 19 86, updated the atomic data with the inclusion of millions of iron lines and established a mass-loss metallicity (Z) relation, Abbott 19 82, Kudritzki et al 1987. These models up to the year 2,000 are referred to as modified CAK (or M-CAK) models, as the CAK theory was modified to include the effects of the finite disk of the star with non radially streaming photons.

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In the new millennium, most stellar evolution models switched to using the Monte Carlo (MC) predictions of Vink et al 2000, that were based on a technique developed by Abbott and Lucy in 19 85. These models included the effect of multiple photon scattering in a realistic way, but the hydrodynamics was simplified in that the models relied on a global energy approach. Muller and Vink in two thousand and eight improved this assumption, providing locally consistent models through the use of the Lambert W function but still relying on the Sobolev approximation, which is a questionable assumption in the subsonic portions of the wind.
The Lambert W function is a function that allows one to solve mathematical problems in which the variable is both a base and an exponent.
The Sobolev approximation is used to simplify radiative transfer when dv dr is large and conditions over the Sobolev length v, dv dr hardly change.
This is why during the past five years progress has been made using non-local thermodynamic equilibrium (non-LTE) comoving frame (CMF) approaches using codes such as CMFGEN, PoWR, FASTWIND, and METUJE that no longer rely on the Sobolev approximation.
2 point 1. Castor Abbott Klein Theory.
The basic idea of CAK theory, Castor et al, 1975, is to describe the radiative force, “A” rad, in terms of a force multiplier M. This is a way to describe the radiative force on the opacity resulting from Doppler-shifted lines through multiples of the electron scattering opacity. For hot O and WR stars, Hydrogen and Helium are fully ionized, and the electron scattering opacity is, consequently, easily established. The challenge is to compute M in terms of the Sobolev velocity gradient (dv dr Sob, to the power alpha, where the famous CAK-parameter alpha gives the line force from optically thin lines to the total line force, related to the line-strength distribution. By entering the radiative acceleration “A” rad into the equation of motion, and neglecting the gas pressure, an excellent assumption in the supersonic part of the wind, one only needs to consider the following equation one:
V times d v d r equals minus gm star over r squared plus c times d v, d r to the power of alpha.
Where C is a constant. By equating the velocity gradient in the inertia term on the left-hand side of Equation one to the Sobolev velocity gradient on the right-hand side, CAK, could solve the equation of motion analytically and predict the wind velocity to behave as a beta law, equation two:
v (Of r) equals v infinity times one minus R star over r to the power of beta.
With Beta equals 0.5 and a mass-loss rate M˙ that scales with both the stellar mass M star and Eddington factor Gamma, or L star. Through the nineteen eighties and nineties, the CAK models were modified using more complex spectral line lists. Furthermore, the point source approximation was lifted, resulting in somewhat larger beta values, on the order of 0.8 to 1, and with terminal wind velocities now 2 to 3 times the stellar escape velocity, numerically corresponding to values in the correct range of 2 to 3 thousand kilometers per second for O-type stars. In the Point source approximation the star’s finite-sized disk is assumed to be infinitely small, with photons streaming out radially.
I refer to Puls et al and Owocki for more detailed derivations and discussions regarding M CAK theory, but I end this subsection by mentioning a clever trick of Kudritzki et al. They constructed the so-called modified wind momentum rate, D mom equals Mass loss times v infinity times the square root of R over R sol. Given that v infinity scales with the escape velocity, and as long as the CAK parameter alpha is constant and of the order of two thirds, the stellar mass conveniently cancels out, which is known as Kudritzki et al’s trick, resulting in a momentum product that only scales with stellar luminosity:

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Equation three, Log D mom equals x log L over L sol plus D zero.

With slope x and offset D zero, the wind momentum luminosity relationship WLR, Kudritzki et al 19 95 was born. Although initially proposed to be a cosmological distance indicator, the WLR played an instrumental role in determining the empirical mass-loss metallicity dependence for O stars in the Small Magellanic Cloud (SMC) and LMC predicted WLR’s can be compared to test the validity of the theory in different physical regimes, helping to highlight potential shortcomings, for example, regarding wind clumping. However, one should be aware that in reality Mass loss rate is a function of not only Luminosity of the star but also parameters such as mass M star and T eff, and one should properly account for this multivariate behavior of Mass loss rate when comparing observations with theory. Finally, CAK-type relations are only valid for spatially constant CAK force multiplier parameters, which is an assumption that does not hold in more realistic models.
2 point 2. Beyond Castor Abbott Klein.
Despite the plethora of achievements by the MCAK theory discussed above, it was clear that in order to achieve a more realistic framework of the RDWT, a number of assumptions needed to be improved upon. The main drawbacks of the MCAK theory are the usage of the Sobolev approximation, see Section 2 point 2 point 2, the simplified hydrodynamics in terms of a two-parameter force multiplier approach, the reliance on the CAK critical point, and the neglect of multiple scattering.
2 point 2 point 1. Multiple scattering and the Monte Carlo method.
The first alternative approach to CAK to be discussed is the MC method developed by Abbott and Lucy 1985. In this approach, photon packets are tracked on their journey from the photosphere to the outer wind. At each interaction, momentum and energy are transferred from the photons to the gas particles. One of the major advantages of the MC method is that it includes multiline scatterings in a relatively straight forward manner, see Schmutz et al 1990. Prior to the year 2000, M CAK mass-loss rates fell short of the observed rates for the denser O-star and WR winds. The crucial point is that multiple scattered photons add radially outward momentum to the wind, and the momentum may exceed the single-scattering limit.
Phrased differently, the wind efficiency eta equals Mass loss rate times v infinity over Luminosity of the star over c, can exceed unity, see Section 2 point 2 point 4. Vink et al used the MC method to derive a mass-loss recipe, see the Appendix), where for objects hotter than the so-called bistability jump (BS-jump; see Figure 2, at around 20 to 25 thousand Kelvin, the rates roughly scale as equation four in the text.
The main success of these models was that they were equally successful for relatively weak winds (with Mass loss rate down to ten to the minus seven solar masses per year as they were for dense O-star winds, with Mass loss rates as high as ten to the minus four solar masses per year.
Note that the positive Mass loss rates dependence with T eff translates into a dropping mass loss rate with T eff during redward stellar evolution.

Figure 2. Monte Carlo mass-loss rate predictions as a function of effective temperature across the bistability regime, where Mass loss rate is predicted to jump upward by a factor of approximately 5, even for this moderately massive star. In these older global models, the ratio of v infinity over v escape was assumed to drop by a factor of 2 from 2.6 to 1.3. Figure adapted from Vink et al in 19 99.

The reason is that while the flux distribution shifts to longer wavelengths, most line-transition opacities remain at shorter wavelengths, resulting in a growing mismatch between the flux and the opacity. This changes abruptly when Iron four recombines to Iron three in the stellar atmosphere. With the increased amount of line transitions in Iron three, the flux-weighted opacity increases, which defines the BS-jump. Note that the dual dependence of Mass loss rate on both M star and stellar luminosity in Equation 4 can in this theoretical setting be simplified to the somewhat shallower relation Mass loss rate is proportional to stellar luminosity to the power of 1.6 using the mass luminosity relation. One has to take care when stellar masses are not known, and one cannot simply replace an L dependence that includes a stellar mass dependence with another one that does not.
The predictions of the mass-loss recipe are only valid for hot star winds at a sufficient distance from the Eddington limit, with Gamma less than 0.5. There are two regimes where this is no longer the case:
(a) Stars that have formed with high initial masses and luminosities, meaning, VMS’s, and
(b) Less extremely luminous stars that approach the Eddington limit when they have evolved significantly.
Examples of this latter category are LBV’s and classical WR stars. Vink and de Koter in 2002 showed that the mass-loss rate for LBV’s increases more rapidly with L over M than Equation 4 would indicate. For VMS’s, Vink et al 2011 discovered a kink, see Section 2 point 3 point 2 in the mass-loss versus relation, with the slope jumping from 2 to 5, right at the transition from optically thin O-type to optically thick WNh winds, see Section 2 point 2 point 4.
The main drawback of the MC method but simultaneously also its strength, was the initial assumption of a predetermined velocity structure. As the terminal wind velocity is an accurate empirically derived number, around 10 percent from the blue edge of UV P Cygni lines, the velocity structure in the supersonic wind part, where the Sobolev approximation is excellent, was basically enforced to be correct, and predicted mass-loss rates from the MC method could thus be considered rather accurate. This also implied that they were useful as input for stellar evolution modeling. However, from a purely theoretical hydrodynamical perspective, the global MC approach was rather unsatisfactory, which is why the modeling was improved involving a locally consistent Lambert W framework by Muller and Vink for one dimension and Muller and Vink 20 14 for two dimensions.

Solving the equation of motion self-consistently, and no longer relying on free parameters, Muller and Vink in 2008 determined the velocity field through the use of the Lambert W function with a line acceleration that only depended on radius (rather than explicitly on the velocity gradient dv dr as in CAK theory. As a result of this simplified “A”rad (Of r) dependency, the CAK critical point was no longer the critical point of the wind, but this had now become the sonic point, as in the Parker (1958) solar wind theory. For the isothermal case, Muller and Vink derived an analytic solution of the velocity law that was compared with the beta law and utilized to derive v infinity values. These were around 25 to 40 percent higher than observed, and due to the invariance of the momentum product Mass loss times v infinity, mass-loss rates were somewhat lower. Similar small over predictions in terminal wind velocity occur in M-CAK and recent CMF computations by Bjorklund et al, possibly highlighting that some physics is still missing in 1D models. Muijres et al tested the Muller and Vink wind solutions through explicit numerical integrations of the fluid equation, also accounting for a temperature stratification, obtaining results that were in good agreement with the Müller and Vink solutions.
2 point 2 point 2. Co moving frame method.
Despite the many successes of the CAK and MC methods, one of the remaining drawbacks is the use of the Sobolev approximation. The Sobolev approximation is considered excellent in the supersonic parts of stellar outflows, but it falls short around and below the sonic point, where the mass-loss rate is initiated. It is thus pertinent that a full CMF approach to the radiative transfer is contemplated. Over the past about 5 years, a number of CMF calculations of the radiative transfer have been published.
These are based on complex non-LTE atmospheric codes, such as CMFGEN, Hillier and Miller 1998, PoWR, and FASTWIND, that had traditionally only been used to simulate the spectra of hot massive stars for comparison with observations, with the aim of empirically determining the stellar parameters such as log g and T eff as well as wind parameters such as Mass loss rate.
However, these empirical studies have until recently always been performed through the use of an adopted constant beta velocity law v(r). Although the task of constructing non-LTE model atmospheres would deserve a review in its own right, here I discuss the new hydrodynamical aspects of these sophisticated codes. I refer to the main publications of the codes for their specific physical inputs, such as the treatment of atomic data, the temperature stratification. For the art of spectral analysis employing these model atmosphere codes, see the recent review by Simon-Diaz. Instead of parameterizing the radiative acceleration, the current state of the art in terms of hydrodynamics relies on a direct integration of the radiative acceleration in the CMF, equation five:

“A” rad equals one over c times the integral of kappa nu time F nu d nu.

where kappa nu is the frequency dependent opacity and F nu is the flux. This full integration has been performed using PoWR, FASTWIND and METUJE, and Gormaz-Matamala et al, using the Lambert W concept in CMFGEN.
The advantage over previous approaches is that the Sobolev approximation is no longer used, and as long as the detailed atmospheric physics allows for it, these CMF calculations are equally applicable for optically thin as for optically thick winds, such as the winds of WR stars, Sander et al, although a monotonic velocity field is enforced.

2 point 2 point 3. Optically thick winds.

Classical WR stars have strong winds with large mass-loss rates, that are typically a factor of 10 larger than those O-star winds with the same luminosity, and they are not at all explicable by optically thin CAK theory involving a dv dr gradient.

Figure 3. Comparison of the flux-weighted mean opacity (red solid line) to the Rosseland mean opacity (blue dashed line) in a model atmosphere model for a hydrogen-free N-type Wolf Rayet star (see Sander et al for parameters) at solar metallicity. Note that the Rosseland and flux-mean opacities are almost equal to each other at the sonic (and critical) point, but they diverge by an order of magnitude in the outer wind. The two iron bumps are notably present.

By contrast, in these optically thick wind models dv dr is approximately zero. The observed wind efficiency eta values are typically in the range of 1 to 5, well above the single scattering limit.
As the state of ionization decreases outward, photons may interact with opacity from a variety of different ions, such as iron, on their way out, whereas gaps between lines become filled in due to the layered ionization structure. The initiation of classical WR outflows relies on the condition that the winds are optically thick at the sonic point and that the line acceleration due to the high opacity iron peak is able to overcome gravity, driving an optically thick WR wind.
The crucial point of such an optically thick wind analysis is that due to their huge mass-loss rates, the atmospheres become so extended that the sonic point of the wind is already reached at large flux-mean optical depth, which implies that the radiation can be treated in the diffusion approximation. The equation for the radiative acceleration can then be approximated to equation six:

“A” rad is equal to one over c times the integral of kappa nu times F nu d nu, which is approximately:
Kappa Ross times Stellar Luminosity over four pi r squared c.
Where Kappa Ross is the Rosseland mean opacity, which can be taken from the OPAL opacity tables of Iglesias and Rogers 1996. The Eddington limit with respect to the Rosseland mean opacity is being crossed at the sonic point. Nugis and Lamers showed that the condition that kappa Ross needs to increase outward with decreasing density could be fulfilled at the hot edges of two Iron opacity peaks, the cool bump at around 70 thousand Kelvinand the hot bump above 160 thousand Kelvin, which can be identified as the two bumps in Figures 3 and 4.

Figure 4. Contributions of the main driving ions to the radiative acceleration “A”, as a function of Newtonian gravity, g of a hydrodynamically consistent N-type Wolf Rayet model with T equals 141,000 Kelvin. The main contributions are shown with different ions indicated by a combination of different colors and symbols. For comparison, the total radiative acceleration “A” rad, the Thomson acceleration from free electrons, and the contribution from gas (and turbulent) pressure are also shown.
Instead of “A” rad, it is probably more insightful to express the radiative acceleration “A” rad (Of r) through the Eddington ratio, equation seven.
Gamma rad (Of r) equals “A” rad (Of r) over g newton (Of r), which equals:
Kappa F (OF r) times the stellar luminosity, over four pi c G Mass star.

where the radial dependency of Gamma rad is due to the flux-weighted mean opacity kappa F (Of r). The quantity Kappa F should not be confused with the Rosseland opacity that is used in stellar structure calculations or wind initiation studies.
In the deeper layers of the atmosphere where the diffusion approximation is valid, kappa F is approximately Kappa Ross as shown in Figure 3, but the difference between the two quantities becomes significant further out in the wind where they become discrepant by more than an order of magnitude. Therefore, the construction of a consistent wind stratification boils down to an accurate calculation of Kappa F (Of r) through the entire atmosphere including both the deep optically thick regions and the outer optically thin ones. Grafener and Hamann, and others included the Opacity Project Iron peak opacities from the ions Irons nine to eight in sophisticated PoWR models for classical WR stars.
An example from Sander et al for a hydrodynamically consistent model is shown in Figure 4, where the radiative acceleration has been dissected in individual ions, and compared to other accelerations relevant in the equation of motion. It is clear that Iron is most important in the inner wind where the mass-loss rate is set, whereas the opacities of CNO and intermediate-mass elements, such as Neon, Sulphur, Argon, and Chlorine, dominate the outer wind, Vink et al 1999.

It should be noted that wind clumping in these sophisticated non-LTE models is treated in the optically thin microclumping approach, see Section 3 point 2 point 1. For MC models both the optically thin and thick approaches have been studied. For less ad hoc approaches, multi dimensional hydrodynamical approaches such as those modeling the line-de-shadowing instability (LDI) are required. The point of the LDI physics is that a small perturbation in velocity leads, via “A” rad proportional to delta v, to a further velocity enhancement, meaning a runaway effect, usually referred to as the self-excited LDI, Owocki and Rybicki 19 84.

2 point 2 point 4. The transition mass-loss rate.
There are many uncertainties in the quantitative mass loss rates of both VMS’s as well as canonical 20 to 60 solar mass massive stars. One reason is related to the role of wind clumping, which is discussed later, but the most pressing uncertainty is actually still qualitative: Do VMS winds become optically thick? And if they do, could this lead to an accelerated enhancement of Mass loss rate? And at what location does this transition occur?
To find this transition point and its associated mass-loss rate, Vink and Grafener performed a full analytical derivation. Here, I only discuss some key aspects. The basic point is to solve the equation of motion (Equation 1) in integral form, and assuming that hydrostatic equilibrium is a good approximation for the subsonic part of the wind, while in the supersonic regime the gas pressure gradient becomes small.
Employing the mass continuity equation, and the wind optical depth tau equals the integral from zero to infinity of kappa F times density dr, one obtains equation eight:
Mass loss rate over L over c times d v equals Gamma minus one over Gamma, times d tau.
Assuming for O stars that Gamma is significantly larger than 1 in the supersonic region, such that the factor Gamma minus one over Gama becomes close to unity, one can show that equation nine:
Eta equals Mass loss rate time v infinity over L over c, equals tau.
One can now employ the unique condition eta equals tau equals one exactly at the transition from optically thin winds to optically thick WR winds. In other words, if one were to have a full dataset listing all luminosities for O and WNh stars, the corresponding transition mass-loss rate Mass loss rate trans is obtained by simply considering the transition luminosity L trans and the terminal velocity v infinity, equation ten:
Transitional Mass loss rate equals L trans over v infinity times c.

This transition mass-loss rate can be obtained by purely spectroscopic means, independent of any assumptions regarding wind clumping. Vink and Grafener 20 12 followed a model-independent approach, adopting beta type velocity laws as well as full hydrodynamic models, computing the integral tau equals the integral fromr s to infinity of kappa rho, dr, numerically using the flux-mean opacity kappa F (Of r).
The mean opacity kappa F follows from the resulting radiative acceleration “A” rad (Of r). The sole assumption entering the transition mass-loss analysis is that the winds are radiatively driven. The resulting mean opacity kappa F captures all physical effects that could affect the radiative driving, including clumping and porosity. The obtained values for the correction factor is 0.6 plus or minus 0.2. The transition between O and WNh spectral types should in reality occur at Mass loss equals f L trans over v infinity c is approximately 0.6 times the transitional Mass loss rate, Vink and Grafener 2012.
For the Arches cluster, the spectroscopic transition occurs at log L equals 6.05 and log Mass loss rate for eta equals one over Solar mass per year equals minus 4.95. This is related to the transition mass-loss rate for Galactic metallicity.

Table 1. Typical wind parameters for different stellar types.
For example, BSG, Blue super giant has an effective temperature of 15 to 25 thousand Kelvin, and a mass of 15 to 30 Solar masses. The V infinity is 500 to 1,500 kilometers per second, and the mass loss rate is around ten to the minus five to seven solar masses per year.
The Supernova type for a BSG is a two B to two P.

The only uncertainties result from relatively small uncertainties in the terminal wind velocity (of about 10 percent and the stellar luminosity L 30 percent at worst. With potential errors of around 40 percent at most, these are several orders of magnitude lower than the humongous uncertainties in the accuracy of empirical mass-loss rates resulting from the dual effects of clumping and porosity, which could easily amount to an order of magnitude or more.
2 to 3. Theoretical Mass-Loss Results.
In CAK-type O-star models, the terminal wind velocity scales with a constant value of about 2 to 3 times the escape velocity. For this reason, terminal wind velocities are often discussed in terms of the stellar effective escape velocity. Furthermore, as the escape velocity drops with larger radii during redward evolution, the wind velocity is anticipated to drop accordingly. Table 1 shows that to first order this is indeed the case: Lower Effective temperature types have generally lower terminal wind speeds, but this is only part of the story. Important modifications to this general principle occur when stars approach the Eddington limit or encounter temperature transitions in which the opacity changes drastically, such as at BS-jumps.
2 point 3 point 1. O-type stars.
The currently most used recipe in stellar evolution modeling is the MC predictions of Vink et al 2000, with an associated Z dependence from Vink et al in 2,001. There are several aspects to this recipe that form active areas of research. One of them involves the absolute levels of predicted mass-loss rates. Although these MC mass-loss rates are already a factor of 2 to 3 lower than unclumped empirical mass-loss rates, we see later that some current analyses, derive clumping factors larger than 4 to 10. For these reasons, new CMF models have been published in recent years that no longer employ the Sobolev approximation. These new hydrodynamical CMF models generally predict mass-loss rates that are another factor of 2 to 3 lower and would now fall below the Vink et al rates.
One example involves the radiation-driven wind predictions with the FASTWIND code by Bjorklund et al 20 21, where mass-loss rates are a factor of 2 to 3 lower than Vink, although the mass-loss rate predictions depend significantly on the assumed amount of microturbulence. Although the Z dependence is similar to Vink et al 2001, an effective temperature dependence for O-type stars is not present in this prescription. Also, the rapid increase of Mass loss rate by a factor of about 5 in Vink et al 1999, at the bistability location of 20 to 25 thousand Kelvin is not predicted by the Bjorklund et al approach.

The reason for the absence of a mass-loss jump has been relayed to be a lack of radiative opacity at the critical point. This is reminiscent of a potentially related issue identified by Muijres et al for low luminosity weak O-type winds in the so-called weak-wind regime, see also Lucy. One possible solution is that the lack of acceleration at the critical point results in genuinely weak or even nonexistent winds in these critical regimes. Another possibility is that nature somehow manages to find a way to push the gas through the critical point with a canonical wind solution establishing itself regardless. Vink and Sander suggested an empirical test with ULLYSES, Ultraviolet Legacy Library for Young Stars as Essential Standards, spectra to investigate the weaker, faster wind solutions in more detail.
The Ultraviolet Legacy Library for Young Stars as Essential Standards, ULLYSES, was a HST program involving 1,000 orbits with 500 orbits dedicated to massive stars in low-Z galaxies.
2 point 3 point 2. Very massive stars.
In addition to canonical O stars in the 20 to 60 solar mass range, we switch our attention to the more massive stars. Vink et al in 2011 discovered a kink in the slope of the mass-loss versus Gamma relation at the transition from optically thin O-type to optically thick WNh type winds, whereas Bestenlehner et al performed a homogeneous spectral analysis study of over 60 O Of, WN WNh stars in 30 Doradus as part of the Very Large Telescope Flames Tarantula Survey.
They confirmed the predicted kink empirically, although one could instead modify CAK parameters as an alternative. Figure 5 depicts empirical mass-loss predictions for VMSs as a function of the Eddington parameter Gamma from Bestenlehner et al. For ordinary O stars with relatively low Gamma, the Mass loss rate is proportional to Gamma to the power of x relationship is shallow, with x about 2.
There is a steepening at higher Gamma, where x becomes closer to 5. Alternative models for VMS’s have been presented by Grafener and Hamann for stars up to 150 solar masses and Pauldrach et al and Vink for stars up to 1,000 solar masses.
Whether and at what stage the winds of VMS’s become optically thick, is still debated and an active field of research. The mass-loss history is crucial for understanding the stellar upper mass limit, as currently observed VMS’s may have originated from a range of initial stellar masses. The reason is the rapid drop in mass already in their first few million years as depicted in Figure 6. Basically, VMS’s evaporate extremely rapidly if the mass-loss rates are higher than ten to the minus four solar masses per year.
2 point 3 point 3. Blue super giants and the bistability jumps.
With respect to B supergiants, BSG’s, it should be noted that the evolutionary state of these stars is still under debate. They might already be burning Helium in their cores, or they might be objects on an extended main sequence due to a large amount of core overshooting with alpha ov equals 0.3 to 0.5. Their winds are slower than those of their O-star counterparts, dropping by a factor of about 2 at spectral type B1 with an effective temperature of about 21 thousand Kelvin. This drop is usually referred to as the BS-jump. It has been attributed to a change of the main wind driver from Iron four to Iron three. In the original MC models the wind efficiency number eta was found to increase by a factor of 2 to 3, and with an assumed drop in v infinity by a factor of about 2, this implied an increase in the mass-loss rate of about 5.

Figure 5. Empirical Mass loss rate, Gamma relations, with similar slopes to those of the predicted mass-loss kink. The mass-loss rates are unclumped and could be corrected using the volume filling factor f. Gamma only accounts for electron scattering. Symbols highlight different stellar subclasses. The solid line indicates the O-star slope, approximately 2, whereas the steeper, approximately 5 dashed line covers the very massive Of WN and WNh stars. The kink occurs at log Gamma e equals minus 0.58. The gray asterisks indicate the position of the stars under the assumption of core Helium burning. The gray upside down triangles are from Mokiem et al, with only upper limits, they are excluded from the fits. Figure adapted from Bestenlehner et al.
Abbreviations: WN, Wolf Rayet, nitrogen-sequence. WNh, H-rich WN stars.

Contrary to the original global approach, the drop in v infinity has more recently been predicted directly with locally consistent models. Despite the important consequences for massive star evolution models, for instance, in terms of angular momentum loss, the predicted Mass loss rate increase has so far hardly been tested. This has become all the more pivotal now that new CMF modeling with FASTWIND by Bjorklund et al, 20 21, exhibits the complete lack of an increased Mass loss rate at the BS-jump. So far, empirical wind diagnostics tests have only been performed in H alpha and the radio regime. Although the first evidence for a local maximum in mass-loss rate at spectral type B1 was identified, empirical radio and H alpha rates below the BS-jump T eff are up to an order of magnitude lower than predicted. It still needs to be established exactly what happens at spectral type B1 when Iron four recombines to Iron three. One possibility is that when the mass-loss rate increases as predicted, this leads to optically thicker winds that have different diagnostics, see Section 3 point 2 point 3), as suggested by Petrov et al. Another possibility is that the MC calculations are inaccurate and that the recently predicted absence of a BS-jump by Bjorklund et al turns out to be more accurate. One should, however, note that very low mass-loss rate predictions would normally go hand in hand with very fast winds, which are not observed. In fact, the empirical drop in v infinity is well established. Whether it is abrupt or more gradual is a different matter.

Simplified mass evolution for the most massive stars. The effective upper-mass limit drops with time, as four different initial masses, 250, 500, 750, 1,000 solar masses represented by the four different lines, converge to very similar values after just a few million years. The plot highlights that it is not possible to determine the value of the upper-mass limit from present-day stars up to about 200 solar masses unless we have exact knowledge of the mass-loss history.

To make progress on the issue of the reality of the BS-jump, it would be helpful if we were able to devise a testing strategy that is model independent, as so far the issue is that when theory changes abruptly at the BS-jump, so do the diagnostics. A promising alternative empirical method might involve the bow-shock method, discussed in Section 3 point 1 point 5.
2 point 3 point 4. Cool super giants.
When considering cooler objects, another model prediction is that of a second BS-jump at effective temperatures lower than around 10,000 Kelvin. This is caused by a recombination of Iron three and Iron two, and its existence and size are even more uncertain than the first BS-jump, noting that also in this T eff range, corresponding to spectral type A0. Lamers et al in 1995 found a rapid drop in terminal wind speeds by a factor of 2 empirically in IUE spectra. The cool supergiants and hypergiants could be divided between red, 3 to 5 thousand Kelvin and yellow, 5 to 10 thousand Kelvin supergiants. The driving mechanism of RSG’s is usually assumed to be radiation pressure on dust, whereas hot star winds above 10,000 Kelvin are driven by radiation pressure on gas. The driving mechanism for yellow supergiants YSG’s is still unknown, but these objects find themselves at a critical HRD location: right at the boundary between the convection and or dust-driven RSGs and the line-driven blue supergiants, BSG’s. From an evolutionary perspective, YSG’s also play an important role, possibly as post-RSGs, which might imply their death could be imminent. Although convection, pulsation, and radiation pressure might all play a role in the driving of YSG winds, the theory is in dire need of further development, as their mass-loss rates are critical for understanding the HD limit.

Figure 7. Z-dependent mass-loss predictions for WR stars. Despite the overwhelming presence of C over the full Z range, the Iron dependent WR mass-loss rate does not show Z-independent behavior, as was previously assumed (dotted line). The newer computations show that WR mass loss depends on Iron, a key result for predicting a high occurrence of long-duration gamma-ray bursts at low Z. At very low Z, the slope becomes smaller, flattening off completely at Z over Z sol equals ten to the minus three. The Monte Carlo computations are from Vink and de Koter from 2,005.
Abbreviation: WR, Wolf Rayet.
2 point 3 point 5. Wolf Rayet and stripped Helium stars.
The first sophisticated theoretical model for a WR did not appear until 2005, which is why the stellar evolution community still had to rely on empirical mass-loss rates such as those of Nugis and Lamers.
More recently, the first grids of hydrodynamically consistent PoWR models have been published. During the WR phase, the atmospheres become chemically enriched with elements such as carbon (C), which may potentially modify WR mass loss. In a first attempt to investigate the effects of self-enrichment on the total wind strength, Vink and de Koter in 2,005 performed a pilot study of late-type WR mass loss versus Z. The key point here is that WR stars, especially Carbon types (WC stars), show the products of core-Helium burning in their outer layers, and it could have been the Carbon abundance that is most relevant for driving WC winds rather than Iron. However, Figure 7 shows that despite the fact that the Carbon ions overwhelm the amount of Iron, WR stars show a strong Mass loss rate, Z dependence, because Iron has a more complex atomic structure involving millions of spectral lines.
The implications of Figure 7 are two-fold. First, WR mass-loss rates decrease steeply with Z. This may be of key relevance for BH formation and the progenitor evolution of long-duration gamma-ray bursts, GRB’s. The 19 93 collapsar model of Woosley requires a rapidly rotating stellar core prior to collapse, but at solar metallicity stellar winds are expected to remove the bulk of the angular momentum. The WR Mass loss rate Z dependence from Figure 7 provides a route to maintain rapid rotation, as the winds are weaker at lower Z, even during the final phases toward collapse.
The second point is that mass loss does not decrease when Z falls below ten to the minus three of Solar metallicity, resulting from the dominance of radiative driving by carbon lines. This suggests that even for Population three stars with only trace amounts of metals, once the heavy objects enrich their outer atmospheres, radiation-driven winds might still exist.

Figure 8. Mass-loss rate M dot, as a function of L over M for sets of hydrodynamically consistent atmospheres for a wide range of Z. The onset of multiple scattering eta equals 1 in each sequence is marked with a gray cross and a dotted line. Gray diamonds denote the locations of eta equals 2.5.

Whether the mass-loss rates are sufficiently high to alter the evolutionary tracks of these first stars remains to be seen, but it is important to keep in mind that the mass-loss physics depends on not only Z but also the fact that other factors, such as the proximity to the Gamma limit, could also be relevant.

From more recent computational results it has also become clear that the M˙ dependence on WR parameters such as M is not a power law either. Figure 8 showcases the combined M and Z dependence of N-type WR stars from hydrodynamical self-consistent models. Interestingly, the figure also shows a Z-dependent break-down in which the winds become optically thin. This region appears to correspond to that of the Stripped Helium stars as these Helium stars, of just a few solar masses, have significantly weaker winds than the optically thicker WR winds. The transition luminosity for different Z galaxies between these optically thinner and thicker stars was discussed by Shenar et al. Note that the predictions in Figure 8 have so far only been computed for a single temperature, and the steep drop should therefore not be taken too literally. For now it might be recommendable to use the MC predictions of Vink as a lower bound, see the Appendix. Future hydrodynamic mass-loss calculations and observations are warranted for this Helium star regime, as the SN type ultimately produced, meaning, a type one b, two b, and so on. depends very critically on the included mass-loss rate, given that there is only a very small amount of Hydrogen on top, and extrapolations from optically thick empirical WR recipes may not be justified.

2 point 4. Summary.

One. The traditional CAK line-driven wind theory is being updated with MC and non-Sobolev comoving frame models that provide Mass loss rates as a function of M and L, as well as effective temperature, and appropriate scalings on the Iron abundance.

Two. These new models provide the velocity stratification v (Of r) showing that traditional beta law approaches need to be reevaluated.
Three. Different theoretical models, using different assumptions, do not agree on the absolute mass-loss rates, within approximate factors of 2 to 3, but for the most massive stars MC rates are of the right order of magnitude, as determined from calibrations using the transition mass-loss rate.
Section Three. Mass-loss diagnostics.
RDWT for smooth winds generally provides predictions for two global wind parameters: Mass loss rate and V infinity. Most diagnostics are based on the wind density, and the mass-loss rate then simply follows from the continuity equation. Up until recently, most diagnostics have been based on wind models that employ a prescribed beta-type velocity field, but with new hydrodynamical-consistent CMF approaches this is no longer necessary. Regarding the global wind parameters, v infinity is directly observable from P Cygni absorption profiles, with 10 percent accuracy, and for emission line WR stars the width of the line may also provide relatively direct wind velocity information. Note that this does not include the very broad wings that are caused by electron scattering and that provide interesting constraints on the amount of wind clumping in WR stars.
The mass-loss diagnostics are generally far more model dependent, although the transition mass-loss rate gives a model-independent value in the very high-mass range of around 80 to 100 solar masses. Ideally, we would be able to derive accurate model-independent mass-loss rates in the canonical 20 to 60 solar masses range as well. One recently proposed method is that of the bow-shock method, see Section 3 point 1 point 5.
An important issue to keep in mind when comparing wind theories to observations is that whenever the theory undergoes discontinuities, such as at BS-jumps, then the diagnostics might also get affected. For instance, in the case of the BS-jump temperature it is rather unlikely that the recombined winds on the cool side of the jump can be directly compared with those of the ionized diagnostics on the hot side of the jump. In the case of the H alpha line, Petrov in 20 14 found that for a reasonable range of parameters the stars on the cool side of the
BS-jump H alpha might become optically thick, and if the winds are clumped we see later that though optically thin clumping generally leads to an overestimate of unclumped empirical mass-loss rates, for optically thick cases the reverse might be true. These kinds of considerations need to be kept in mind when comparing theory with observations.
3 point 1. Multi wavelength Diagnostics.
Traditionally, the mass-loss rate can be obtained from either stellar continua at long wavelengths, millimeter and radio, or spectral line transitions in the optical or UV, and alternatively also in the X-ray and near-IR domains. Sticking to the optical and UV wave-bands, there is a profound difference between UV resonance lines and (optical) recombination lines such as H alpha. For resonance lines, the line opacity ae proportional to rho, whereas recombination involves a two-body process in which the line emissivities are proportional to rho squared. This becomes particularly relevant when wind clumping is accounted for in the analysis. Most empirical analyses rely on the use of non-LTE model atmospheres. In state-of-the-art modeling, the stellar and wind parameters, such as Mass loss rate, are determined by fitting the resonance and recombination lines simultaneously. A detailed discussion of the various methods to derive wind parameters is given by Puls et al. The most common line profiles in a stellar wind are UV P Cygni profiles with a blue absorption trough and a red emission peak, and optical emission lines such as H alpha.

In P Cygni scattering lines, the upper level is populated by the balancing act between absorption from, and spontaneous decay to, the lower level. An emission line is formed if the upper level is populated by recombinations from above. See, however, Puls et al 1998 and Petrov et al 2014 for the formation of P Cygni optical H alpha lines in the cooler BSG’s.
3 point 1 point 1. Ultraviolet P Cygni resonance lines.
P Cygni lines have been used to determine the terminal wind velocities v infinity for decades, see Prinja et al 1990 and Lamers et al 1995 for results with IUE. UV P-Cygni lines from hot stars, for example C four and P five are usually analyzed by means of the Sobolev optical depth. In principle, Mass loss rate can be derived from resonance line P Cygni profiles if the ionization fraction is known. However, for higher Z stars, most P Cygni lines are saturated, and mass-loss rate derivations become unreliable such that only lower limits on M˙ can be determined.
UV resonance lines instead have been considered relatively free from wind-clumping effects, but this would no longer hold if porosity effects become relevant, see Section 3 point 2 point 3. The Far Ultraviolet Spectroscopic Explorer (FUSE) was instrumental in identifying the phosphorus five problem, see Section 3 point 2 point 2, whereas the new Director’s Discretionary program ULLYSES is expected to enable tremendous progress on wind velocities of hundreds of stars in low Z environments with the Hubble Space Telescope.
3 point 1 point 2. The H alpha recombination emission line.
The most oft-used diagnostics to derive Mass loss rate for O-star winds involves H alpha, for which there is hardly any uncertainty due to Hydrogen ionization. As the Hα opacity scales with rho squared, any notable inhomogeneity results in an Mass loss rates overestimate if clumping were neglected in the analysis. An advantage of H alpha is the fact that it remains optically thin in the main part of the emitting wind, such that porosity effects can be neglected, which is not the case for UV resonance lines. A drawback of the H alpha line as a mass-loss diagnostic is that it is only sensitive for mass loss rates above ten to the minus seven solar masses per year. For weaker lines the near-IR regime might provide alternative rho squared diagnostics. Hence, a dual analysis of UV plus optical lines is critical for an accurate multi wavelength view of stellar winds, and for this reason the massive star community suggested a complementary optical, and near-IR counterpart to ULLYSES, called X-Shooting ULLYSES XSHOOTU, which in conjunction with the UV data is needed to disentangle the effects of clumping and porosity.
X-Shooting ULLYSES, or XSHOOTU, is a ESO-VLT Large Programme with X-Shooter, involving 250 low-Z massive stars; the optical counterpart to the UV Legacy Library.
3 point 1 point 3. Radio and submillimeter continuum emission.

A complementary approach to measure mass-loss rates is to utilize long wavelength thermal radio and submillimeter continua. In fact, this approach may lead to the most accurate results, as they are considered to be model independent. The basic concept is to measure the excess wind flux over that from the stellar photosphere. This excess flux is emitted by free-free and bound-free processes.
In the case of magnetic fields or binary interactions, nonthermal radio emission needs to be considered. See the 2,007 review by De Becker.
The reason the excess flux becomes more dominant at longer wavelengths is thanks to the lambda squared dependence of the opacities. As the continuum becomes optically thick in the wind in free-free opacity, the emitting wind volume increases as a function of lambda, leading to the formation of a radio photosphere in which the radio emission dominates the stellar photospheric emission. For a typical O supergiant this occurs at about 100 stellar radii. At such large distances, the outflow has already reached its terminal velocity, and an analytic solution of the radiative transfer is achieved involving the spectral index of thermal wind emission close to 0.6.

The method can be used for both submillimeter observations and updated radio facilities such as the EVLA, Expanded Very Large Array and e-MERLIN, extended Multi-Element Remote-Linked Interferometer Network, Morford et al, and should be utilized more frequently.
3 point 1 point 4. X-ray method.
The launch of the Chandra X-ray Observatory and XMM-Newton, X-ray Multi-Mirror Mission-Newton, X-ray telescopes in the past two decades enabled mass-loss rate constraints from high-resolution X-ray spectroscopy. For a canonical O-star wind one would expect to observe asymmetric line profiles, as the receding wind region attenuates more X-ray line emission than the region in front of the star. This would be expected to result in more line emission on the blue-shifted side than on the red-shifted side, resulting in an asymmetric line profile of X-ray lines. However, it turned out that the X-ray profiles were far less asymmetric than expected, suggesting a lower than canonical mass-loss rate. By contrast, a potential alternative to explain this absence of asymmetry is to entertain the idea that wind clumping reduces the X-ray wind opacity, allowing radiation to escape in a porous medium and still allow for higher canonical mass-loss rates. Future multiwavelength mass-loss studies should determine whether the lower opacity is due to an intrinsic mass-loss rate reduction or a porous medium. A complementary way to study wind clumping in X-rays is in high-mass X-ray binaries, HMXB’s, where one can effectively utilize the accreting compact object to probe the clumpy donor wind without the concern of density squared diagnostics.
3 point 1 point 5. Bow-shock method.
A newer method to measure mass-loss rates that is independent of wind clumping is that of the bow-shock method, as it principally only depends on the ram pressure between the stellar wind and the ambient I S M. Following Kobulnicky et al, equation eleven relates:
Mass loss rate equals four pi R zero squared v “A” squared rho “A”, over v w.
Where R zero is the angular size of the standoff distance where the momentum fluxes of the wind and I S M are equal, v “A” and rho “A” are the velocity and density of the ambient Interstellar Medium in the stellar rest-frame, and vw is the velocity of the wind. R zero can be measured from IR images, whereas vw can be taken to be the terminal wind speed. Although there could be systematic issues with this type of analysis, for example, due to the difference between the IR radii and the actual termination shock, or assumptions of adiabatic versus isothermal shocks, the strength of this new bow-shock method is that it does not depend on wind clumping.
In other words, the method has the potential to provide key insights into wind regimes in which there are challenges with traditional diagnostics, such as in the weak-wind regime and on the cool side of the BS-jump.
3 point 2. Wind Clumping.
Due to the variability of spectral lines, as well as the presence of linear polarization, we have known for decades that stellar winds are not stationary but time-dependent, and that this leads to inhomogeneous, clumpy media. H alpha and long-wavelength continuum diagnostics depend on the density squared and are, thus, sensitive to clumping, whereas UV P Cygni lines such as Phosphorus Five are insensitive to clumping, as they depend linearly on density.

In the optically thin micro clumping limit, the wind is divided into a portion of the wind that contains all the material with a volume filling factor f while the remainder of the wind is assumed to be void. In reality, however, clumped winds are porous with a range of clump sizes, masses, and optical depths, and a macro clumping approach is needed.
3 point 2 point 1. Optically thin clumping, micro clumping.
The general concept of optically thin microclumping assumes that the wind is made up of large numbers of small clumps. Motivated by the results from hydrodynamic simulations including the LDI, the interclump medium is usually assumed to be void. The average density is rho equals f rho C, where rho C is the density inside the overdense clumps, and the clumping factor D equals f minus 1 is a measure for this overdensity.
As the interclump space may be assumed to be void, matter is only present inside the clumps, with density rho C, and with its opacity given by kappa equals kappa C (Of D rho). This micro formalism is only correct as long as the clumps are optically thin, and optical depths may be expressed by a mean opacity
Mean kappa equals kappa C (Of D rho) times f equals one over D times kappa C (Of D rho).
Hence, for processes that are linearly dependent on density, the mean opacity of a clumped medium is exactly the same as for