- Maximum mass-loss rates of line-driven winds of massive stars
- 1 Introduction
- 2 Description of the physical model
- 3 Illustration of the method
- 4 The influence of the physical effects on the maximum mass-loss rate
- 5 Maximum mass-loss rates as a function of stellar parameters
- 6 Discussion and conclusions
- References

A&A 403, 625-635 (2003)

DOI: 10.1051/0004-6361:20030349

**C. Aerts ^{1} - H. J. G. L. M. Lamers^{2,3}**

1 - Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001
Leuven, Belgium

2 -
Astronomical Institute, Utrecht University, PO Box 80000, 3508 TA Utrecht, The Netherlands

3 -
SRON Laboratory, for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands

Received 20 December 2002 / Accepted 3 March 2003

**Abstract**

We develop a theoretical treatment that allows us to determine the
*maximum* mass-loss rate of a hot rotating star with a wind that is
accelerated by radiation pressure due to spectral lines, taking into account
finite disk correction as well as the effect of photon tiring but neglecting
multiple scattering. The maximum mass-loss rate of a star is obtained by
subsequent numerical integrations of the momentum equation from an assumed
position of the sonic point onwards for increasing values
of the mass loss, until the wind can no longer escape. For stars rotating
below 80% of the critical velocity the decrease in the velocity far out in the
wind due to the maximisation of the mass loss is negligible. Stars rotating at
>
of the critical speed have a kinked velocity law connected with the
highest possible mass-loss rate. In such cases the wind velocity increases up to
typically a few stellar radii, and decreases subsequently almost ballistically
outwards. In these cases the terminal wind velocity is much smaller than the
maximum wind velocity. For O-type main-sequence stars, the maximum mass-loss
rates derived from our formalism are somewhat smaller than those derived for
self-regulated line-driven winds including multiple scattering. For B-type
supergiants, however, the maximum mass-loss rate is higher by about a factor 1.5-2.
Including rotation, but without gravity darkening, results in a maximum
mass-loss rate that is twice as high as for a non-rotating star.

**Key words: **stars: early-type - stars: mass-loss - stars: winds, outflows - stars: evolution

The subject of this paper is to derive theoretical upper limits for the mass-loss rates that can be expelled from non-rotating and rotating massive stars due to radiation pressure in spectral lines. This upper limit is important, because evolutionary tracks of rotating massive stars show that the stars may reach a critical limit, the so-called -limit, after or during their main sequence phase. This -limit is the Eddington-limit modified by rotation, which is the same as the critical rotation limit, modified by radiation pressure. Evolutionary calculations of e.g. Langer (1998), Heger et al. (2000) and Maeder & Meynet (2000a,b) show that stars at the -limit have to eject a large amount of mass in a relatively short time. The question that we want to answer is: can such a high amount of mass be expelled from the star by radiation pressure in spectral lines?

The theory of a radiation-driven stellar wind in a hot non-rotating star was developed with success for the first time by Castor et al. (CAK) in a pioneering paper published in 1975. CAK used the Sobolov approximation to parametrize the radiative acceleration by large numbers of spectral lines as a simple powerlaw of an optical depth parameter. With this parametrisation they were able to solve the momentum equation that describes the hydrodynamics of an isothermal stellar wind. The CAK-theory forms the basis of many more sophisticated studies of radiation-driven winds by several different authors up to the present day. Pauldrach et al. (1986) have shown the CAK-parametrisation to be very appropriate, by comparing it with the actual line driving due to a huge number of spectral lines.

Vink et al. (2000, 2001) have recently provided new theoretical mass-loss rates of O and B stars based on generalisations of the CAK-theory. They succeeded for the first time to find overall good agreement between theoretical calculations and observations of for a wide range of effective temperatures, K, by taking into account the effects of multiple scattering and different metallicities. These results are nowadays used for the calculation of evolutionary tracks of massive stars. In all these models radiation pressure on spectral lines determines both the mass-loss rate and the acceleration of the wind.

In the current paper we take a different approach: we investigate what the highest mass-loss rate is that can be accelerated and removed out of the gravitational potential well of the star by means of radiation pressure on spectral lines. In our calculations the mass-loss rate is a free parameter that may be set by some mechanism other than radiation pressure. This method is analogous to the mass-loss studies of dust-driven winds, where the acceleration is due to radiation pressure on dust, but the mass-loss rate is due to the effect of the pulsations, e.g. Wilson & Bowen (1984) and Lamers & Cassinelli (1999, Chap. 7). The mass-loss rates of hot stars have hardly been studied from this viewpoint.

In our work, we investigate to what extent the conservation of momentum and energy allow the occurrence of extremely high mass loss. In order to do so, we calculate the maximum mass loss by means of a different approach compared to the classical derivation of in the framework of radiation-driven wind theory. Traditionally, is derived from the momentum equation, to which specific conditions - the so-called regularity and singularity condition - are added in order to find one unique value for the mass loss of an isothermal wind. Here, we do not make use of such specific conditions but we rather calculate the upper limit to the mass loss that an optically-thin stellar wind with a temperature gradient can drive, i.e. we investigate how large can become such that the wind velocity fulfills the momentum equation and remains positive at all distances, although its gradient may become negative. We do this by simply expressing conservation of momentum and energy.

Poe et al. (1990) were the first to discuss mass-loss rates that exceed the CAK value at the expense of a non-monotonic wind velocity that declines to small terminal values (see their Fig. 4). They have found such solutions by studying in detail the steady state solution topology of line-driven winds without relying on the Sobolev approximation for the calculation of the line force. Owocki et al. (1994) performed two-dimensional hydrodynamical calculations to simulate a radiation-driven stellar wind for a rapidly rotating Be star, neglecting gravity darkening and non-radial line forces. These authors find an increase in mass-loss with a factor of about 2.8 compared to a non-rotating star when the rotation speed amounts to 80% of the critical velocity (see their Table 1). Moreover, they find the line-driving to be inadequate in the dense equatorial region, resulting in a stagnation and even reaccretion of wind material. Gayley (2000) has also considered the case where the mass loss is treated as an external parameter instead of being self-regulated by the wind, i.e. instead of being determined by the critical-point conditions of CAK theory. He proposes a scaling law for the wind velocity in the simplification of neglecting the gravity far out in the wind, which is in general a valid approximation in the situation where does not differ too much from the CAK value. Gayley (2000) also finds that any process leading to mass-loss enhancement must reduce the terminal speed of the wind, conform with the findings of Poe et al. (1990).

In a recent series of papers, Feldmeier & Shlosman (2000, 2002) and Feldmeier et al. (2002) have considered runaway of line-driven winds towards critical and overloaded solutions. Their physical model is a generalisation of the CAK model, in which they have assumed zero sound speed and for all examples shown in these papers. A perturbation at a fixed height is fed into the wind such that the inner wind is lifted towards higher mass-loss rates. These solutions are accompanied by flow deceleration and kinks in the velocity at the point in the wind where the deceleration starts to occur. The increase in the mass loss these authors find amounts to only a few percent compared to the classical CAK mass loss.

It is important to have accurate predictions of the (maximum) mass-loss rates as a function of stellar parameters in order to derive reliable evolutionary models of massive stars. Indeed, the mass loss significantly affects the evolution of stars with initial masses above some 20 . Maeder & Meynet (2000a,b) have recently reviewed the effects of the interplay between rotation and mass loss on stellar evolution of massive stars. They come to the conclusion that the effects of rotation on the mass-loss rates remain moderate for stars sufficiently far from the Eddington limit. On the other hand, when the rotation-corrected Eddington factor is considerable, (with is the ratio between the radiative acceleration due to electron scattering and the accelaration of gravity corrected for centrifugal acceleration), even a moderate rotation rate can lead to extremely high mass loss. We will briefly discuss the effect of rotation on the maximum mass-loss rate in this paper. A subsequent paper (Aerts et al., in preparation) will be devoted completely to the mass loss of rotating hot stars based on our formalism but with the inclusion of gravity darkening. In that future paper we also provide a comparison to the predictions by Maeder & Meynet (2000b) and Langer (1998).

This paper is organised as follows. In Sect. 2 we give a description of the
different physical ingredients of our model. We derive the momentum equation for
a CAK-type radiatively driven wind with a temperature law of the form
.
We explicitly include the effect of photon tiring
^{}. We also formulate the boundary conditions we adopted to solve the
momentum equation. Subsequently, in Sect. 3, we illustrate our method by
applying it to the stellar parameters of an O5IV star. In Sect. 4 we study the
influence of different physical effects on the maximum mass-loss rate: photon
tiring, the finite disk correction, the position of the sonic point and
rotation. With that purpose, we apply our formalism to an O-type main-sequence
star and a B-type supergiant. In Sect. 5 we provide maximum mass-loss rates for
a grid of stellar parameters. Finally, we end this work with some concluding
remarks.

The theory of radiation-driven winds was reviewed extensively before by many authors. We refer to Chap. 8 of the book by Lamers & Cassinelli (1999) and to the review paper by Kudritzki & Puls (2000), and references therein, for a basic extensive overview and limit ourselves here by giving the main concepts needed to understand our formalism. A concise but very clear short description of basic CAK theory can be found in e.g. Owocki & Puls (2002, Sect. 2.3).

The basic expression for the line acceleration of a spherically symmetric
stationary stellar wind is:

where the first right-hand factor is the radiative acceleration by electron scattering for radiation from a point source, with being the electron scattering opacity per gram, the second factor is the force multiplier and the third takes into account the non-radial direction of the radiation. The force multiplier describes the radiative acceleration in terms of that by electron scattering where the scaling factor

with a reference value of 0.325 cm

where

(4) |

(Pauldrach et al. 1986; see also Lamers & Cassinelli 1999, Chap. 8 for a derivation). This finite disk correction factor takes into account that the radiation has a significant non-radial component close to the star. It was included in a very clever way in the description of radiation-driven winds of hot stars by Kudritzki et al. (1989).

For the wind from a rotating star, the radial component of the equation of
motion in the equatorial plane of a line-driven wind is written as follows:

(5) |

where stands for the equatorial rotation velocity and is the acceleration due to both continuum and line radiation. The fourth term is the centrifugal force in the equatorial plane in case of conservation of angular momentum. For higher stellar latitudes the centrifugal force is smaller by a factor and vanishes in the polar direction. For this study we only consider the mass flux in the equatorial plane. We neglect the deformation of the star due to rotation. This is justified as flattening is only important for stars rotating close to their critical rate (Pelupessy et al. 2000).

Besides this equation, we consider also the equation of mass continuity:

where is the mass flux (in g cm

The equation of state for a perfect gas is:

with

where and stand for the stellar radius and effective temperature respectively. We assumed a temperature structure with

Substitution of the adopted temperature law into the equation of state, and
subsequently into the equation of motion by making use of Eq. (6),
multiplication by *r*^{2}, and substitution of the expression for the line-driving
(1) leads to the following version of the equation of motion:

with the mass loss and

We point out that we have omitted the factor
,
with
and *W* respectively the electron density and the dilution factor,
in our description of the radiative acceleration by lines, Eq. (1).
This factor is connected to the reduction of the line force due to the change in
ionisation in the wind. Its inclusion will only lower the upper limit of the
maximum mass loss, so it is justified to neglect this factor for our purpose,
which we have done in order to simplify finding solutions of Eq. (9).
In our model we also neglect the instability of the radiative line force, which
is a good approximation if the goal is to study the large-scale wind behaviour
(for a review of such wind instabilities, see Owocki 1994).

The concept of *photon tiring* was first put forward by Owocki & Gayley
(1997). The authors used this term to indicate that, for a star with a given
luminosity, only a limited mass-loss rate can be accelerated before the energy
expended in accelerating the outflow becomes a considerable fraction of the
original stellar luminosity. Owocki & Gayley took into account the effect of
photon tiring by reducing the stellar luminosity according to the gained kinetic
and potential energy of the flow at each distance in the wind. They performed
some preliminary calculations of acceptable mass-loss rates for a specific form
of the Eddington factor. This form was such that it allowed to determine the
effect of photon tiring on the mass loss analytically, for illustrative
purposes. In this paper, we include the effect of photon tiring with a more
realistic description.

In order to take into account the effect of photon tiring, we express the
luminosity *L*(*r*) in Eq. (9) as

where is the radiative luminosity from the photosphere. The right-hand side of this equation is obtained by expressing conservation of kinetic and potential energy, as well as enthalpy, from the stellar radius onwards throughout the wind for the temperature law adopted in (8).

Evaluation of Eq. (10) shows that the fractional change of the stellar luminosity always is very small for all stellar models considered in this paper, suggesting the effect of photon tiring to be negligible. We have determined explicitly the effect of photon-tiring by using Eq. (10) when integrating the momentum equation. For none of the mass-loss rates and velocity laws given in the paper foton tiring plays any role. We therefore omit further discussion of this effect.

The traditional approach to solve the momentum Eq. (9) is to
search for the critical point of the equivalent of this equation for an
isothermal wind, by imposing a singularity and a regularity condition, which
both are a function of the mass loss .
These conditions are
constructed in such a way that a smooth transsonic solution is obtained, from
which the mass loss
is determined for a self-regulating or perturbed
wind. For the isothermal analogue of Eq. (9), the critical point is
typically located around
(Kudritzki et al. 1989) and is not
necessarily equal to the sonic point
(Castor et al. 1975; Lamers &
Cassinelli 1999, Chap. 8). From here on, we use the term *classical*
mass-loss rates for such determination of .

In the current work, we take a different attitude. We are interested in solving
the question: *what is the maximum mass loss for which the momentum Eq. (9) still has an outflowing wind that escapes to infinity?* In
other words, we search for the maximum
,
termed
,
for which Eq. (9) still has a solution ánd for
which this solution corresponds to an outflow velocity law at all distances.
The velocity does not have to be monotonically increasing outwards, but it has
to remain positive. We determine
by solving
Eq. (9) for increasing values of
until the
equation is no longer solvable because the line force has become too weak. Out
of this pool of solutions, we select
by picking out
the largest one for which the condition of a positive wind velocity at large
distance of the stellar surface still holds.

As boundary condition, we impose the velocity gradient at the sonic point and we
search for solutions from the sonic point up to
.
We
assume the sonic point to be located somewhere in the interval
.
The velocity gradient in
is
determined numerically by solving the equation:

which is the momentum equation for a fixed location of the sonic point. Since all factors are known for any adopted value of , the value of can be found.

Imposing the position of the sonic point allows one to find solutions of the momentum equation without encountering delicate mathematical complexities compared to making use of the singularity and regularity condition. It also admits inclusion of additional effects, such as photon tiring and non-isothermal decelerated flows. The objective of our study is to investigate how high the mass-loss rate can become in solving the momentum equation under these conditions.

In the following we solve the equation of motion (9) as described in Sect. 2.2 and determine for one particular stellar model assuming that the sonic point is situated at . This choice of hardly affects the resulting mass-loss rate, because we only solve for the supersonic part of the wind. We will illustrate this further in the paper.

We consider the stellar parameters corresponding to an O5IV star - see
Table 1. Accurate values of the CAK line parameters were determined
from NLTE calculations by Pauldrach et al. (1986). For the effective
temperature of an O5IV star they find
.
Further, we have
used
cm^{2} g^{-1} for the O5IV star.

We solve the momentum equation numerically by means of a code written in *Mathematica*^{} with the boundary
condition of
of Eq. (11) described above. We
perform the integrations of the equation for increasing values of
in steps of
yr^{-1} and search for
as described in the previous section.

Figure 1:
Different solutions of the momentum equation for the stellar parameters of an
O5IV star (see Table 1) rotating at
km s^{-1}. The solutions for increasing mass-loss rates are indicated as full
lines. The dashed-dot line indicates graphically that the momentum equation is
no longer solvable for the indicated and higher mass-loss rates. |

In a first example we have considered the O 5 IV star to have an equatorial
rotation velocity of
km s^{-1}, which is about half the
critical velocity. We have determined solutions to the momentum equation for
increasing values of the equatorial mass loss. The courses of the accompanying
wind velocities *v*(*r*) in the equatorial plane are graphically depicted in
Fig. 1. We find a maximum equatorial mass-loss rate of 2.7
yr^{-1} for this case. Increasing the mass-loss rate
does no longer allow to find solutions of the momentum equation. The terminal
wind velocity amounts to some 450 km s^{-1} in this example. For comparison,
we mention that the classical CAK solution for this stellar configuration is
yr^{-1} and
km s^{-1} (this CAK velocity law is not shown in Fig. 1
as it would hinder the visibility of the different non-CAK curves). The fact
that the CAK mass loss is higher than the maximum mass loss is entirely due to
the neglect of the finite disk correction in the classical CAK model. Including
the finite disk correction gives a modified CAK mass loss of
yr^{-1} and
km s^{-1}.
The velocity law corresponding to the maximum mass-loss rate has a much lower
terminal speed than both these CAK solutions, as our formalism is such that all
effort is done to drive as high a mass loss as possible, at the expense of a
high velocity (see also Fig. 4 in Poe et al. 1990).

Figure 2:
Different solutions of the momentum equation for the stellar
parameters of an O5IV star (see Table 1) rotating at
km s^{-1}. Left panel: solutions with increasing velocities towards
infinity are indicated as full lines; the dashed lines are solutions for which
the line force is exhausted at a certain distance. The dashed-dot line
indicates graphically that the momentum equation is no longer solvable for the
indicated and higher mass-loss rates. Right: selected solutions giving the
transition from infinitely increasing to decreasing velocity laws. The thick
vertical line segment indicates the position at which line driving is no longer
effective. |

For all rotation velocities between 0 and 450 km s^{-1}, i.e. between 0%
and 80% of the critical velocity, we find similar types of results as the one
shown in Fig. 1.

In a subsequent example we adopted an equatorial rotation velocity of
km s^{-1}, which is 90% of the critical velocity. The variation of
the radial velocity *v*(*r*) in the equatorial plane is shown in Fig. 2.
In this case of near-critical rotation, we find two types of physically relevant
solutions:

- 1.
- solutions for which the line force is able to drive the mass towards an infinite distance from the star (full lines);
- 2.
- solutions for which the line force is only able to drive the mass to a limited distance (dashed lines).

In this example, solutions of type 1 are found for mass-loss rates up to
yr^{-1}. After having found the highest mass-loss rate
accompanied by an overall increasing velocity law, we can still find solutions
with higher mass-loss rates in such a way that the stellar wind material is
lifted only to a certain limited distance from the star. Such solutions of
type 2 are also valid, provided that the wind material has already accumulated
a sufficient amount of energy at that distance to escape from the star. This
requires that the wind velocity at the point where the radiative acceleration is
no longer effective is higher than the local escape velocity, or rather that the
sum of the kinetic and potential energy and the enthalpy per unit mass is
positive.

Increasing the mass loss to
yr^{-1} implies that the
line driving is only able to lift the wind material to a distance of some
12 ,
after which it can no longer gain momentum.
From that point onwards the velocity starts to decrease according to
the transfer of kinetic energy into potential energy and enthalpy (see
Eq. (10) above). For a mass loss up to
yr^{-1} the wind material has sufficient energy to escape from the star, while
increasing the mass loss to
yr^{-1} results in wind
material falling back to the star. In this example, we thus obtain
yr^{-1} and the *maximum wind velocity*
amounts to some 400 km s^{-1}. This
maximum velocity is no longer the terminal velocity, which will approach zero at
the maximum mass-loss rate.

A velocity kink in the wind speed similar the one we find for the models of the O5IV star near critical rotation (>80%) was recently also found by Porter & Skouza (1999) and Krticka & Kubát (2000), who studied velocity deceleration by considering respectively ionization cutoff and decoupling of the gas and the radiation field in low-density radiately-driven winds. However, these authors did not connect these kinked velocity laws to a detailed study of the mass loss. The latter was done in the already cited recent papers by Feldmeier & Shlosman (2000, 2002), resulting in a maximum of 9% increase of the classical mass-loss rate. This is comptabile with the difference we find between the maximum equatorial mass loss as we have determined it and the modified CAK mass loss for our first example.

Figure 3:
Same as Fig. 2, but for a B5II star rotating at
km s^{-1}. |

star | ) | k |
|||||||||||

O5IV | 4.60 | 5.7 | 40 | 18 | 535 | 19.8 | 525 | 0.326 | 0.640 | 0.124 | 0.34 |
3.72 10^{-6} |
1257 |

B5II | 4.30 | 5.0 | 30 | 30 | 435 | 33.0 | 430 | 0.016 | 0.565 | 0.320 | 0.31 |
0.65 10^{-6} |
1074 |

In order to investigate the effects of the different physical ingredients of our
model, we have determined the maximum mass-loss rates for a typical O5IV and
B5II star of which the relevant stellar parameters are listed in
Table 1. We have done this for different equatorial rotation
velocities, ranging from 0 km s^{-1} up to a value close to the *limiting
rotation velocity*. The latter is defined in this paper as the highest
possible equatorial rotation velocity for which the velocity gradient in the
sonic point derived from Eq. (11) is still positive, i.e. it is the
critical rotation velocity at the sonic point. This limiting velocity is
determined numerically; we find
5 km s^{-1} for
the O5IV star and 430
5 km s^{-1} for the B5II star. We note
that, classically, the critical velocity at the stellar photosphere is usually
defined as

(12) |

where stands for the Eddington ratio corresponding to the electron scattering opacity . This definition applies when the brightness of the star over its surface is uniform, as we are assuming here. We compared and for all the cases calculated in this work and it turns out that their values are very similar, differing less than 5% from each other. We therefore always list any result with respect to , as this quantity is consistently connected to our calculations.

As in the illustrative example explained in the previous section we determine
the solution of the momentum Eq. (9) for increasing values of
,
in steps of
yr^{-1}. A similar plot as
Fig. 2, but now for the B5II star rotating with 400 km s^{-1}, which
is 93% of the critical velocity, is shown in Fig. 3. From the right
panel of Fig. 3 we see that the highest possible mass loss associated
with an increasing velocity law amounts to
yr^{-1} for
the B 5 supergiant. The conclusion from all the different testcases is that
increasing velocity laws are found for rotational velocities below 80% of
the critical velocity. For higher rotation velocities the situation is always
such that, once we have derived the highest mass loss with an overall increasing
velocity law, only slightly higher mass-loss rates can be driven having a kinked
velocity law with a still outflowing wind.

The velocity laws that are found by integrating the momentum equation have low
terminal velocities. Models with mass-loss rates very close to the maximum
value, i.e. models where radiative acceleration is no longer able to provide
acceleration, have an outward decreasing wind velocity at high distance as
illustrated in the right panels of Figs. 2 and 3. All models
with mass-loss rates not too close to the maximum value show an outward
increasing velocity law which can be approximated reasonably well with a
-law of the type
.
This is
shown in Fig. 4 for the O5IV star rotating at 400 km s^{-1}, which
is 72% of the critical velocity, for which we still found an increasing
velocity towards infinity, i.e. for which the line driving is effective all the
way through the wind. On the same plot we show different -type velocity
laws with the same terminal velocity. It can be seen that the derived velocity
law can be approximated by a -law with
or 0.5 close to
the star at
.
However, for larger distances the velocity law
approaches that of a higher value of
.
The velocity laws of
classical line-driven CAK-winds but with the finite disk effect taken into
account have a velocity law that can be approximated with a
law (Kudritzki et al. 1989; Lamers & Cassinelli 1999).

The observed terminal velocities of OB stars (see Prinja et al. 1990; Lamers et al. 1995; Puls et al. 1996) are about 4-7 times higher than the -values we find. This implies that most OB stars for which can be determined are not in the very outermost regime of maximum mass loss.

Conservation of energy determines the velocity behaviour at large distance for the highest possible mass-loss rates. This is illustrated in Fig. 5, which shows the predicted velocity laws of the O5IV star (left) and the B5II star (right).

One important question is in how far the finite disk correction factor given by Exp. (3) influences the maximum mass-loss rates. As a check of the calculations, this factor was in each case determined a posteriori, i.e. after having solved the momentum equation. We found that is smaller than 1 at small distances and larger than 1 at large distances from the star. The values of are due to the non-radial direction of the radiation close to the star, whereas is due to the fact that the line optical depth is smaller in the non-radial direction than in the radial direction (e.g. Kudritzki et al. 1989; Lamers & Cassinelli 1999).

Figure 4:
A comparison between a velocity law found from
Eq. (9) (dashed line) and different
velocity laws
(full lines). The latter are drawn for
(upper curve)
1.9 (lower curve) in steps of 0.2. |

In order to assess the influence of
,
we have determined
from Eq. (9) in which we have set
and compared this to the solutions taking into account the true
behaviour of the finite disk correction factor. The results of such a comparison
are graphically illustrated in Fig. 5 (dashed line: without
;
full line: with
). We find that the inclusion of finite disk
correction leads in general to slightly *higher* mass-loss rates, contrary
to the situation in self-regulating CAK winds. The explanation for this is that
in only a very limited regime very close to the star (in a
self-regulated wind this is where the value of the mass loss is determined),
while the maximum mass loss is mainly dependent on the velocity behaviour
further out in the wind, at a few stellar radii, where
.
Hence,
the finite disk correction helps in a slightly positive way in driving the wind
in the region
.
Although increases in the maximum mass-loss rates
occur by including
,
they always remain small, e.g. below
yr^{-1} in the two examples mentioned in Table 1.

Up to now the sonic point was always assumed to be positioned at 1.1 . Subsequently, we have investigated the effect of changing the position of the sonic point by varying its position between 1.01 and 1.1 . We illustrate our findings for the velocity law corresponding to the highest mass loss for the O 5 II star in Fig. 6. It can be seen from this plot that our results are quite robust against the choice of the position of the sonic point within reasonable limits. The further out the sonic point, the more difficult it is to drive a high mass loss as the velocity gradient is less steep. Changing the sonic point within the indicated limits does not change the velocity law for the B-supergiant. With the goal to find the maximum mass loss we therefore assume the sonic point to be located at 1.01 in all the calculations done in Sect. 5.

The temperature structure of winds from hot stars has been discussed by e.g. de Koter et al. (1993), who found that the temperature does not decrease very
steeply very far out in the wind (see their Fig. 1). Presumably, the precise
temperature law does not matter too much for our study, as the effect of the
gass pressure is very limited compared to the one of the radiation pressure. In
order to assess this statement more quantitatively, we have also performed some
calculations in which we have taken *x*=0.5 instead of *x*=1 in the temperature
law (8). We have adjusted Eqs. (9), (11), and (12) accordingly and
have determined again
and
for this
less steep temperature law. This indeed does not change the values of these two
quantities so that our proposed formalism is sufficiently accurate as a general
description of maximum mass loss.

Figure 6:
The velocity law for the O5IV star having
km s^{-1} and
yr^{-1} with
the inclusion of finite disk correction. The sonic point was
fixed at
(full line) and at
(dashed line). |

An important effect that also influences the mass loss of the star is its rotational velocity. Pioneering studies of the effect of rotation on radiatively-driven wind models were done by Poe & Friend (1986) and Friend & Abbott (1986). Poe & Friend (1986) found that the mass-loss rates depend very strongly on the equatorial rotation velocity, with typically a factor 5 difference in mass loss between slow and near-critical rotation, for magnetic Be star models. Friend & Abbott (1986) showed convincingly that the terminal wind speed decreases due to the inclusion of the centrifugal force, while the mass loss increases with at most a factor 5, in agreement with Poe & Friend's result. The question is what kind of factor between the maximum mass-loss rates of non-rotating and of near-critical rotation we find from our formalism.

In the present paper we discuss the effect of rotation on the
maximum equatorial mass flux only. We have investigated the behaviour of the
maximum mass loss derived with the method explained above as a function of the
equatorial rotation velocity. We stress that we do not take into account
gravitational darkening of the fast rotating star, which reduces the radiative
flux at the equator compared to the pole (Von Zeipel effect), nor the effect of
the increased equatorial radius by rotational flattening. This means that our
calculated maximum equatorial mass flux should be compared with that of a
non-rotating star *with the same gravity, radius and *
* as in the
rotating star at the equator.* The effect of rotation on the maximum overall,
i.e. surface integrated, mass loss, with gravity darkening taken into account,
will be studied in a separate paper (Aerts et al., in preparation).

The results for
with the inclusion of foton tiring
and finite disk correction are listed in Table 2 as a function of
equatorial rotation velocity for the case where the sonic point is situated at
1.1.

O5IV, | B5II, | ||||||||

(km s^{-1}) |
(
yr^{-1}) |
(km s^{-1}) |
(
yr^{-1}) |
||||||

0 | 0.00 | 2.20 | 1.00 | 0.10 | 0 | 0.00 | 0.40 | 1.00 | 0.05 |

100 | 0.18 | 2.27 | 1.03 | 0.10 | 100 | 0.23 | 0.44 | 1.10 | 0.06 |

150 | 0.27 | 2.34 | 1.06 | 0.10 | 150 | 0.35 | 0.48 | 1.20 | 0.07 |

200 | 0.36 | 2.40 | 1.09 | 0.10 | 200 | 0.47 | 0.54 | 1.35 | 0.07 |

250 | 0.45 | 2.56 | 1.16 | 0.11 | 250 | 0.58 | 0.60 | 1.50 | 0.08 |

300 | 0.54 | 2.68 | 1.22 | 0.12 | 300 | 0.70 | 0.70 | 1.75 | 0.09 |

350 | 0.63 | 3.00 | 1.36 | 0.13 | 350 | 0.81 | 0.78 | 1.95 | 0.11 |

400 | 0.72 | 3.40 | 1.55 | 0.15 | 400 | 0.93 | 0.80 | 2.00 | 0.11 |

450 | 0.81 | 3.90 | 1.77 | 0.17 | 420 | 0.98 | 0.80 | 2.00 | 0.11 |

500 | 0.90 | 4.00 | 1.82 | 0.17 | -- | -- | -- | -- | -- |

550 | 0.99 | 4.00 | 1.82 | 0.17 | -- | -- | -- | -- | -- |

We find that the maximum mass-loss rates for the O 5 main-sequence star range from yr

Since the maximum mass-loss rate of a non-rotating star is not much higher than the value predicted by classical radiation driven wind models, within about a factor two (see below), we conclude that it is not possible to drive significantly higher mass-loss rates by line radiation pressure than predicted with the CAK theory. This is essentially also in accord with the findings of Gayley (2000) who has considered a much simpler treatment than we have done. We point out that Maeder & Meynet (2000b) do find very high classical mass-loss rates for rotating stars with a high Eddington factor of . We did not consider this regime in the present work.

The predicted maximum wind velocity turns out to be independent of
.
One can understand this, as with our formalism, we concentrate entirely on
increasing the mass loss as much as possible, irrespective of the consequences
for the velocity law, provided that the wind material can still escape from the
star. We have also calculated the equatorial momentum transfer efficiency
,
which
we define in this paper as

This expression differs from that normally used because in our models differs strongly from , which is zero in the maximum mass loss models. The values are listed in Table 2. As the maximum wind velocity is independent of , 's dependence on is the same as the one of . So increases by about a factor two or three from non-rotating to fast rotating stars with the same equatorial values of and effective gravity. The values of of our models are smaller than those of the classical line-driven wind models, because is not much higher than whereas is about a factor two smaller than .

A most important conclusion from Table 2 is that we find a *finite* maximum mass-loss rate at the limiting velocity from our
formalism. The dependence of the mass-loss rate on the rotation of the star is
the subject of an intensive, still ongoing debate (see, e.g. Maeder & Meynet
2000b). We hence devote a special paper to this topic, in which we study the
dependence of
on
in much more detail,
including also gravity darkening, for all the stellar models considered in the
next section (Aerts et al., in preparation) and compare our results with Maeder
& Meynet's.

In order to provide more systematically determined maximum mass-loss rates, we
determined
in the same way as the ones given in the
previous section for a grid of nine stellar models typical for massive stars in
the upper part of the HR diagram. The characteristics of these models are
given in Table 3. The corresponding mass-loss rates were this time
calculated for
,
which is not only the most appropriate
value (see, e.g. Pauldrach et al. 1986) but leads at the same time to the
highest possible mass loss (see Fig. 6). We included finite disk
correction in the calculations. The results can be found in
Table 4^{}.

Model number | ) | k |
|||||||||

Model 1 | 6.0 | 50 000 K | 60 | 15 | 0.434 | 650 | 0.64 | 0.124 | 0.34 | 2.0 | 2.1 |

Model 2 | 6.0 | 40 000 K | 50 | 23 | 0.521 | 435 | 0.64 | 0.124 | 0.34 | 1.9 | 1.9 |

Model 3 | 6.0 | 30 000 K | 40 | 42 | 0.613 | 260 | 0.59 | 0.17 | 0.32 | 1.7 | 0.7 |

Model 4 | 5.5 | 40 000 K | 40 | 10 | 0.209 | 780 | 0.64 | 0.124 | 0.34 | 2.0 | 1.5 |

Model 5 | 5.5 | 30 000 K | 35 | 18 | 0.221 | 535 | 0.59 | 0.17 | 0.32 | 1.9 | 0.8 |

Model 6 | 5.5 | 20 000 K | 30 | 40 | 0.250 | 325 | 0.565 | 0.32 | 0.31 | 1.8 | 1.5 |

Model 7 | 5.0 | 40 000 K | 30 | 8 | 0.087 | 805 | 0.64 | 0.124 | 0.34 | 2.1 | 1.0 |

Model 8 | 5.0 | 30 000 K | 25 | 14 | 0.089 | 550 | 0.59 | 0.17 | 0.32 | 2.0 | 0.6 |

Model 9 | 5.0 | 20 000 K | 20 | 32 | 0.119 | 320 | 0.565 | 0.32 | 0.31 | 1.8 | 1.3 |

Model 1, | Model 4, | Model 7, | ||||||

0 | 4.30 | 0.10 | 0 | 0.81 | 0.07 | 0 | 0.14 | 0.04 |

250 | 4.80 | 0.11 | 250 | 0.87 | 0.07 | 250 | 0.15 | 0.04 |

500 | 7.30 | 0.17 | 500 | 1.11 | 0.10 | 500 | 0.19 | 0.06 |

650 | 9.00 | 0.21 | 750 | 1.84 | 0.16 | 750 | 0.32 | 0.09 |

Model 2, | Model 5, | Model 8, | ||||||

0 | 6.10 | 0.10 | 0 | 1.01 | 0.06 | 0 | 0.16 | 0.03 |

150 | 6.60 | 0.11 | 200 | 1.13 | 0.07 | 200 | 0.18 | 0.03 |

300 | 8.90 | 0.15 | 400 | 1.82 | 0.11 | 400 | 0.27 | 0.05 |

400 | 12.60 | 0.21 | 500 | 2.17 | 0.13 | 500 | 0.35 | 0.06 |

Model 3, | Model 6, | Model 9, | ||||||

0 | 11.80 | 0.11 | 0 | 3.40 | 0.12 | 0 | 0.53 | 0.06 |

100 | 13.30 | 0.12 | 100 | 3.70 | 0.13 | 100 | 0.58 | 0.06 |

200 | 21.00 | 0.19 | 200 | 5.00 | 0.18 | 200 | 0.79 | 0.09 |

250 | 23.10 | 0.21 | 300 | 7.20 | 0.25 | 300 | 1.13 | 0.13 |

We compare our maximum mass-loss rates for non-rotating stars (
)
with those predicted for two other treatments:

**1)** Classical CAK
self-regulated line-driven winds without multiple scattering and not taking into
account finite disk correction (second but last column of
Table 3). These mass-loss rates were calculated as in Eq. (27) of
Kudritzki et al. (1989). We find that
is roughly
twice as high as
.
This is no surprise, as it is
well known that the neglect of finite disk correction leads to mass-loss rates
that are too high for a self-regulated wind (Kudritzki et al. 1989).

**2)** Line-driven winds with multiple scattering (Vink et al. 2000). In their
models Vink et al. adopted a velocity law of
and the observed ratio
for O-stars and 1.3 for B-stars, as has been derived
from the observations (Lamers et al. 1995). They find that the inclusion of
multiple scattering increases the mass-loss rates of O-stars by about a factor
of 1.4-3.2. Vink et al. have shown that their predicted mass-loss rates
agree very well with the observed ones. In our study we did not include the
effect of multiple scattering. However, we do not expect the inclusion of
multiple scattering to increase the *maximum* mass-loss rate
significantly, because this maximum is not determined by the efficiency, but
rather the overall ability of the momentum transfer from the photons to the gas
throughout the wind. The ratios of the mass-loss rate derived from the recipe
provided by Vink et al. (2000) and our maximum mass-loss rate for the nine
considered stellar models are listed in the last column of
Table 3. In order to make the comparison consistent, we have used
for models 1, 2, 3, 4, 5, 7, 8 and 1.3 for models 6 and 9
in Vink et al.'s recipe. We see that the values for the mass-loss rate derived
from Vink et al.'s work are generally higher, except for Models 3, 5, 8, i.e. for
hot giants and supergiants. It is precisely in these phases that stars have to
lose a large amount of material in a relatively short time according to
evolutionary calculations.

In this paper we studied the maximum mass-loss rate that can be driven by
radiation pressure due to spectral lines in winds of hot stars. The models
include the effects of gravity, radiation pressure by spectral lines including
the finite disk correction, gas pressure, photon tiring and, for rotating
models, centrifugal forces in the case of angular momentum conservation. The
radiative acceleration by spectral lines is described by the CAK-formalism in
terms of the force multiplier parameters, *k* and
(see Sect. 2). We
have solved the momentum equation by integration upward from an imposed sonic
point position close to the star. The weak density dependence of the
ionization, described by the force multiplier parameter
was ignored in
this study; it would reduce
only slightly.

The results of our study can be summarised as follows.

(**1**) Line-driven wind models at the maximum mass-loss rate for stars with a
moderate rotation (<80% of the critical velocity) have a wind velocity that
does not decrease significantly with distance. Stars rotating near critical
velocity (>80%), however, have a velocity that initially increases with
distance, more or less as a -law. From a certain distance onwards, when
radiation pressure is no longer efficient, the velocity decreases outwards
almost ballistically, approaching zero at infinity. Such a kinked velocity law
implies that the maximum velocity is reached rather close to the star, typically
within a few stellar radii. The values of
for maximum mass-loss
rates are much smaller than the values of
derived from the classical treatment of line-driven winds.

(**2**) The maximum mass-loss rates of non-rotating stars are a factor 1-2
lower compared to the mass-loss rates predicted for line-driven winds with
multiple scattering (Vink et al. 2000), except for hot giants and
supergiants. For the latter stars we find the maximum mass-loss rates to be
about 1.2-1.7 times *higher* than those based upon multiple scattering.
Since the predicted mass-loss rates with multiple scattering agree quite well
with the observed values, we conclude that the same is true for the maximum
mass-loss rates derived from our formalism.

(**3**) We have also calculated the maximum mass-loss rates of line-driven
winds of rotating stars. In these calculations we have adopted a *sectorial* model, i.e. the wind is assumed to move outwards in a plane of
constant stellar latitude. This implies that we neglect the tendency of the
gas in a rotating wind to move to the equatorial plane, i.e. the effect of wind
compression (Bjorkman & Cassinelli 1993) or the tendency of the gas to move
away from the equator by radiation pressure in the lateral direction, i.e. the
effect of wind inhibition (Owocki et al. 1996). We also neglected rotational
flattening and gravitational darkening of the star. This means that our results
of the maximum equatorial mass flux of a rotating star should be compared with
those of a non-rotating star with the same values of
and as in the equator of the rapidly rotating star. We find that the equatorial
mass-loss rate, defined as
,
where
is the mass flux at the equator, increases with increasing
rotational velocity and reaches a maximum when
approaches
.
The effect of rotation will be described in detail in Aerts et al.
(in preparation).

Our findings imply that, if additional effects play a role in setting the mass loss (e.g. pulsation or other instabilities), the maximum mass-loss rates of line-driven winds can be higher than predicted for self-regulated line-driven winds. However, the maximum mass-loss rates that can be accelerated and lifted out of the potential well of the star by line driving is at most 2-3 times that of a self-regulated optically-thin line-driven wind. Our conclusion therefore is that much higher mass-loss rates, e.g. those required by stellar evolution calculations close to the -limit (Heger et al. 2000; Maeder & Meynet 2000b), can only be accelerated by either radiation pressure of optically thick continuum-driven winds (e.g. as proposed for Wolf-Rayet stars by Nugis & Lamers 2002), or by a significant increase in, and redistribution of, the line-opacity during such evolutionary stages.

We devote a separate paper (Aerts et al., in preparation) to the detailed study of the effects of rotation on the integrated maximum mass-loss rates as derived from our formalism but including gravity darkening and the oblateness of the star.

This research was started during CA's stay as guest lecturer at the Julius Institute of the University of Utrecht in the period January - February 2001. The authors are indebted to the referee, prof. Stan Owocki, for his numerous and pertinent remarks which helped us to improve our paper. CA is grateful to I. Pelupessy for sending her one of hisMathematicacodes and to prof. Norbert Langer for valuable discussions and helpful comments.

- Abbott, D. C. 1982, ApJ, 259, 282 NASA ADS
- Bjorkman, J. E., & Cassinelli, J. P. 1993, ApJ, 409, 429 NASA ADS
- Castor, J. I., Abbott, D. C., & Klein, R. I. 1975, ApJ, 195, 157 (CAK) NASA ADS
- de Koter, A., Schmutz, W., & Lamers, H. J. G. L. M. 1993, A&A, 277, 561 NASA ADS
- Feldmeier, A., & Shlosman, I. 2000, ApJ, 532, L125 NASA ADS
- Feldmeier, A., & Shlosman, I. 2002, ApJ, 564, 385 NASA ADS
- Feldmeier, A., Shlosman, I., & Hamann, W.-R. 2002, ApJ, 566, 392 NASA ADS
- Friend, D. B., & Abbott, D. C. 1986, ApJ, 311, 701 NASA ADS
- Gayley, K. G. 2000, ApJ, 529, 1019 NASA ADS
- Heger, A., Langer, N., & Woosley, S. E. 2000, ApJ, 528, 368 NASA ADS
- Krticka, J., & Kubát, J. 2000, A&A, 359, 983 NASA ADS
- Kudritzki, R. P., Pauldrach, A., Puls, J., & Abbott, D. C. 1989, A&A, 219, 205 NASA ADS
- Kudritzki, R. P., & Puls, J. 2000, ARA&A, 38, 613 NASA ADS
- Lamers, H. J. G. L. M., & Cassinelli, J. P. 1999, Introduction to stellar winds (Cambridge University Press)
- Lamers, H. J. G. L. M., Snow, T. P., & Lindholm, D. M. 1995, ApJ, 455, 269 NASA ADS
- Maeder, A., & Meynet, G. 2000a, ARA&A, 38, 143 NASA ADS
- Maeder, A., & Meynet, G. 2000b, A&A, 361, 159 NASA ADS
- Nugis, T., & Lamers, H. J. G. L. M. 2002, A&A, 389, 162 NASA ADS
- Owocki, S. P. 1994, Astrophys. Space Sci., 221, 3 NASA ADS
- Owocki, S. P., Cranmer, S. R., & Blondin, J. M. 1994, ApJ, 424, 887 NASA ADS
- Owocki, S. P., Cranmer, S. R., & Gayley, K. G. 1996, ApJ, 472, 115O
- Owocki, S. P., & Gayley, K. G. 1997, in Luminous Blue Variables: Massive Stars in Transition, ed. A. Nota, & H. J. G. L. M. Lamers, ASP Conf. Ser., 120, 121
- Owocki, S. P., & Puls, J. 2002, ApJ, 568, 965 NASA ADS
- Pauldrach, A. W. A., Kudritzki, R. P., Puls, J., Butler, K., & Hunsinger, J. 1994, A&A, 283, 525 NASA ADS
- Pauldrach, A. W. A., Puls, J., & Kudritzki, R. P. 1986, A&A, 164, 86 NASA ADS
- Pelupessy, I., Lamers, H. J. G. L. M., & Vink, J. S. 2000, A&A, 359, 695 NASA ADS
- Poe, C. H., & Friend, D. B. 1986, ApJ, 311, 317 NASA ADS
- Poe, C. H., Owocki, S. P., & Castor, J. I. 1990, ApJ, 358, 199 NASA ADS
- Porter, J. M., & Skouza, B. A. 1999, A&A, 344, 205 NASA ADS
- Prinja, R. K., Barlow, M. J., & Howarth, I. D. 1990, ApJ, 361, 607 NASA ADS
- Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, A&A, 362, 295 NASA ADS
- Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, 574 NASA ADS
- Wilson, L. A., & Bowen, G. H. 1984, Nature, 312, 429 NASA ADS

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