# The structure of blue supergiant winds and the accretion in supergiant High Mass X–ray Binaries

###### Abstract

We have developed a stellar wind model for OB supergiants to investigate the effects of accretion from a clumpy wind on the luminosity and variability properties of High Mass X–ray Binaries. Assuming that the clumps are confined by ram pressure of the ambient gas and exploring different distributions for their mass and radii, we computed the expected X–ray light curves in the framework of the Bondi-Hoyle accretion theory, modified to take into account the presence of clumps. The resulting variability properties are found to depend not only on the assumed orbital parameters but also on the wind characteristics. We have then applied this model to reproduce the X-ray light curves of three representative High Mass X-ray Binaries: two persistent supergiant systems (Vela X-1 and 4U 1700–377) and the Supergiant Fast X-ray Transient IGR J112155952. The model can reproduce well the observed light curves, but requiring in all cases an overall mass loss from the supergiant about a factor 310 smaller than the values inferred from UV lines studies that assume a homogeneous wind.

###### keywords:

X–ray: individuals: Vela X1/4U 190040, 4U 1700377, IGR J112155952. stars: supergiants.^{†}

^{†}pagerange: The structure of blue supergiant winds and the accretion in supergiant High Mass X–ray Binaries–References

^{†}

^{†}pubyear: 2009

## 1 Introduction

A new class of massive X-ray binaries has been recognized in the
last few years, mainly thanks to observations carried out with
the *INTEGRAL* satellite. They are transient X-ray sources
associated to O or B supergiant stars and characterized by short
outbursts. These *Supergiant Fast Xray Transients*
(SFXTs) [Sguera et al. (2005); Negueruela et al. (2006)]
are remarkably different from the classical
High Mass X-ray Binaries (HMXBs) with supergiant
companions, that are bright persistent sources, and also differ
from the Be transients for their optical companions and shorter
outbursts.

The outbursts of SFXTs involve a high dynamic range, spanning 3 to 5 orders of magnitudes, from a quiescent luminosity of erg s up to the peak luminosity of erg s. The outbursts typically last a few days and are composed of many short flares with duration of a few hours. Besides these bright outbursts, the SFXTs can display a fainter flaring activity with luminosity erg s (Sidoli et al., 2008). Different mechanisms have been proposed to explain the SFXTs properties (see Sidoli (2009) and references therein for a recent review).

Sidoli et al. (2007), proposed that SFXTs outbursts are due to the presence of an equatorial wind component, denser and slower than the spherically symmetric wind from the supergiant, and possibly inclined with respect to the orbital plane of the system. The enhanced accretion rate occurring when the neutron star crosses this wind component can explain SFXTs showing periodic outbursts, such as IGR J112155952 and also other SFXTs, assuming different geometries for the outflowing equatorial wind.

Another possibility involves the gated mechanisms due to transitions across the centrifugal barrier (Grebenev & Sunyaev, 2007). Bozzo et al. (2008) showed that a centrifugal or a magnetic barrier can explain the SFXTs properties only if the supergiant wind is inhomogeneous, and the accreting neutron star has a strong magnetic field ( G) and a long spin period ( s).

in’t
Zand (2005) proposed that the SFXTs flares are produced
by accretion of clumps of matter from the companion wind.
In the framework of the clumpy wind model proposed by
Oskinova et al. (2007),
Walter &
Zurita Heras (2007) and
Negueruela et al. (2008) proposed that what distinguishes
the SFXTs from the persistent HMXBs with supergiant companions is
their different orbital separation:
in persistently bright sources the compact object orbits the
companion at a small distance (2 stellar radii) where there is
a high number density of clumps, while the transient emission in SFXTs is
produced by accretion of much rarer clumps present at larger
distances.
On the other hand, the monitoring with *Swift* of a sample
of four SFXTs (Sidoli et al., 2008) has demonstrated that these
sources accrete matter also outside the bright outbursts, so any
model should also account for this important observational
property.

The winds of O and B type stars are driven by the momentum transfer of the radiation field and Lucy & Solomon (1970) showed that the dominant mechanism is the line scattering. The analytical formulation for line-driven winds has been developed by Castor et al. (1975) (CAK). In the CAK theory the mass lost by the star is smoothly accelerated by the momentum transferred from the stellar continuum radiation, and forms a stationary and homogeneous wind. However, both observational evidence and theoretical considerations indicate that the stellar winds are variable and non homogeneous. Changes in the UV line profiles, revealing wind variability, have been observed on time scales shorter than a day (Kudritzki & Puls, 2000). The X-ray variability observed in 4U 1700377, Vela X1 and other HMXBs can be explained in terms of wind inhomogeneity [White et al. (1983); Kreykenbohm et al. (2008)] and further indications for the presence of clumps come from Xray spectroscopy. For example, the Xray spectrum of Vela X1 during the eclipse phase shows recombination lines produced by a hot ionized gas and fluorescent K-shell lines produced by cool and dense gas of near-neutral ions (Sako et al., 1999). These authors proposed that the coexistence of highly ionized and near neutral ions can be explained with an inhomogeneous wind, where cool, dense clumps are embedded in a lower density, highly ionized medium.

Lucy & White (1980) suggested that the wind acceleration is subject to a strong instability since small perturbations in the velocity or density distribution grow with time producing a variable and strongly structured wind. The first time-dependent hydrodynamical simulations of unstable line-driven winds were performed by Owocki et al. (1988) and, more recently, by Runacres & Owocki (2002) and Runacres & Owocki (2005). All these simulations show that the line-driven instability produces a highly structured wind, with reverse and forward shocks that compress the gas into clumps. Moreover, the shock heating can generate a hot inter-clump medium into which the colder clumps are immersed (Carlberg, 1980).

Based on these considerations, we study in this work the expected variability and Xray luminosity properties of neutron stars accreting from a clumpy wind. In the next two Sections we describe our clumpy wind model and the assumptions for the mass accretion. In Section 4 we study the dependence of the parameters introduced in the previous sections. In Sections 5, 6 and 7 we compare the X-ray light curves predicted with our model to the observations of three high mass X-ray binaries, showing that it is possible to reproduce well the observed flaring behaviors.

## 2 Clumpy stellar wind properties

In our model, where the dynamical problem has not been treated, we assume that a fraction of the stellar wind is in form of clumps with a power law mass distribution

(1) |

in the mass range - . The rate of clumps produced by the supergiant is related to the total mass loss rate by:

(2) |

where is the fraction of mass lost in clumps and is the average clump mass, which can be computed from Equation (1).

Clumps are driven radially outward by absorption of UV spectral lines (Castor et al., 1975). From spectroscopic observations of O stars, Lépine & Moffat (2008) suggest that clumps follow on average the same velocity law of a smooth stellar wind. We can then assume the following clump velocity profile without solving the dynamical problem:

(3) |

where is the terminal wind velocity, is the radius of the supergiant, is a parameter which ensures that km s, and is a constant in the range – [Lamers & Cassinelli (1999); Kudritzki et al. (1989)].

Assuming that the clumps are confined by the ram pressure of the ambient gas, their size can be derived by the balance pressure equation. Following Lucy & White (1980) and Howk et al. (2000), the average density of a clump is:

(4) |

where is the density profile of the homogeneous
(inter-clump) wind, and are the inter-clump wind
and the clump thermal velocity, respectively: and .
is the Boltzmann constant, and are the
temperatures of the inter-clump wind and of the clumps,
respectively, and is the mean atomic weight. The constant
accounts for the confining effect of the bow shock
produced by the ram pressure around the clump
(Lucy &
White, 1980). is the relative velocity
between the wind and the clump (). Adopting cm s,
K, K and
, we obtain^{1}^{1}1This value is just an estimate of a typical value
of ratio of clump to ambient density, and will not adopted throughout.:

(5) |

Since the density radial profile of the homogeneous inter-clump wind (obtained from the continuity equation ) is:

(6) |

where is a generic distance from the supergiant, from Equations (4), (5), (6), we obtain:

(7) |

where . Bouret et al. (2005) analyzed the far-ultraviolet spectrum of O-type supergiants and found that clumping starts deep in the wind, just above the sonic point , at velocity km s. In the CAK model the sonic point is defined as the point where the wind velocity is equal to the sound speed ():

(8) |

Adopting typical parameters for O supergiants, from Equation (8) we obtain that the clumping phenomenon starts close to the photosphere (). Assuming spherical geometry for the clumps and that the mass of each clump is conserved, it is possible to obtain the expansion law of the clumps from Equation (7), with :

(9) |

From Equation (9) we find that the clump size increases with the distance from the supergiant star.

For the initial clump dimensions we tried two different distributions, a power law

(10) |

and a truncated gaussian function:

(11) |

where , and is a free parameter.

For any given mass clump we derived the minimum and maximum values for the initial radii as follows. The minimum radius is that below which the clump is optically thick for the UV resonance lines. In this case gravity dominates over the radiative force causing the clump to fall back onto the supergiant. The momentum equation of a radiatively driven clump is:

(12) |

where is the mass of the supergiant, is the radiative acceleration due to the continuum opacity by electron scattering, is the radiative acceleration due to line scattering. While Equation (12) is an approximation that ignores the pressure gradient, the solution of the momentum equation differs only slightly from the accurate solution derived by Kudritzki et al. (1989); another assumption is that the photosphere has been treated as a point source (Lamers & Cassinelli, 1999). The radiative acceleration due to electron scattering is:

(13) |

where is the opacity for electron scattering, and is given by , where is the Thomson cross section, is the number density of electrons, is the density of the clump and is the luminosity of the OB supergiant. Lamers & Cassinelli (1999) found . Assuming a constant degree of ionization in the wind both and are constant.

The radiative acceleration due to the line scattering is:

(14) |

where cm g, , , are the force multiplier parameters, which are obtained with the calculation of the line radiative force (Abbott, 1982). is the dimensionless optical depth parameter (Castor et al., 1975). According to the model of Howk et al. (2000), we assume no velocity gradient inside the clump (of size ), then we utilize the dimensionless optical depth parameter for a static atmosphere . is the geometrical dilution factor (Lamers & Cassinelli, 1999), given by:

(15) |

According to the Equation (7), we find that the number density of electrons in each clump is given by:

(16) |

where such that:

(17) |

From Equation (12) we obtain that the minimum radius of the clump is given by:

(18) |

where is the acceleration due to gravity. Approximating as

(19) |

and assuming the force multiplier parameters calculated by Shimada et al. (1994), with km s and , we finally obtain the lower-limit for the clump radius:

(20) |

where:

(21) | |||||

(22) | |||||

(23) |

We found that for the interesting range of the clump masses, the drag force (Lucy & White, 1980) values are less than 3% of the forces resulting from Equations (13) and (14). Thus, in the determination of the minimum clump radius, we can neglect this contribution.

The upper-limit to the clump radius is obtained from the definition of a clump as an over-density with respect to the inter-clump smooth wind:

(24) |

where is the mass loss rate of the homogeneous wind component (inter-clump). From Equation (24) we obtain the upper-limit for the clump radius

(25) |

The lower and the upper limits for the clump radius (Equations 20 and 24) depend on the supergiant parameter , , , , , . Thus supergiants of different spectral type have, for any given mass of the clump, different minimum and maximum values for the initial radii distribution. However, these differences are smaller than a factor 10 for OB supergiants, and the intersection of the upper-limit and lower-limit functions ranges from g to g.

## 3 Luminosity computation

The Bondi-Hoyle-Lyttleton accretion theory
[Hoyle &
Lyttleton (1939); Bondi &
Hoyle (1944)]
is usually applied to the HMXBs where
a OB supergiant loses mass in the form of a fast stellar wind,
(terminal velocity, km s),
that is assumed to be homogeneous and spherically symmetric.
Only matter within a distance
smaller than the *accretion radius* () is
accreted:

(26) |

where is the neutron star mass and is the relative velocity between the neutron star and the wind. The fraction of stellar wind gravitationally captured by the neutron star is given by:

(27) |

where is the density of the wind (Davidson & Ostriker, 1973).

The Bondi-Hoyle-Lyttleton accretion theory, that is based on a homogeneous wind, requires an important modification to properly take into account the presence of inhomogeneity in a clumpy wind. In the homogeneous case the wind particles are deflected by the gravitational field of the neutron star, and collide with the particles having the symmetric trajectory in a cylindrical region with axis along the relative wind direction. The collisions dissipate the kinetic energy perpendicular to this axis, and only the particles with a parallel kinetic energy component lower than the gravitational potential energy are accreted. The application of this accretion mechanism to an inhomogeneus wind can lead to a partial accretion of the clump: when the distance between the neutron star and the projection of the centre of the clump on the accretion cross section is smaller than the clump radius and , only a fraction of the mass of the clump will be accreted (see Figure 1). In particular, if an incoming clump that crosses the accretion cross-section , is smaller than the accretion radius () and , the accretion cross-section in Equation (27) must be replaced by

(28) |

and the mass accretion rate is (see Figure 2,A). If we have three cases:

The number density of clumps obeys the equation of continuity , where is the rate of clumps emitted by the OB supergiant (see Equation 1). Thus:

(31) |

The rate of clumps accreted by the neutron star is given by:

(32) |

When the neutron star accretes only the inter-clump wind, the Xray luminosity variations are due to changes in its distance from the OB companion due to orbital eccentricity. The corresponding luminosity (see Equations 26 and 27) is:

(33) |

where is the orbital phase, and is given by the Equation (6).

When the neutron star accretes a clump its Xray luminosity is given by:

(34) |

where is given by the Equation (9), and by the Equations (28) – (30). If two or more clumps are accreted at the same time, the Xray luminosity at the peak of the flare produced by the accretion is given by the sum of the luminosities (Equation 34) produced by the accretion of each clump.

## 4 Application of the clumpy wind model

In this section we investigate how the X-ray luminosity and variability properties depend on the different clumpy wind parameters and orbital configurations.

As an example, we consider a binary consisting of an O8.5I star with M, R (Vacca et al., 1996), and a neutron star with M, km. The parameters for the supergiant wind are the following: M yr (Puls et al., 1996), km s, , km s, and the force multiplier parameters are , , and (Shimada et al., 1994). The corresponding upper and lower limits to the clump radius, derived from Equations (20) and (25), are shown in Figure 3.

### 4.1 The effect of the mass distribution

To study the effects of the clump masses we computed the distributions of the flares luminosity and durations for different values of , , and considering for simplicity a circular orbit with d. We first neglected the clump radii distribution assuming that clumps which start from the sonic radius have radii given by Equation (20), and follow the expansion law (Equation 9). We found that when and/or increase, the number of clumps produced by the supergiant decreases (see Equation 2), resulting in a smaller number of Xray flares. The average flare luminosity, the average flare duration, the number of flares and the shapes of the luminosity and flare duration distributions do not change much for different values of and .

In Figure 4 we show the dependence of the distributions of flare luminosities and durations on and (for g, g). If increases, the number of clumps and their density increases (see Equations 2 and 20), implying a higher number of flares, a shift to higher luminosities and to shorter flare durations.

Figure 5 shows the effect of changing the fraction of wind mass in the form of clumps for different values of . When increases, the supergiant produces more clumps (see Equations 2), thus the number of flares and their average luminosity increase, while their average duration remains unchanged.

### 4.2 The effect of the radii distribution

In Section 2 we showed that, for any given mass, the clump dimensions are constrained within the limits given by Equations (20) and (25). Here we show the effect of different assumptions on the radii distributions laws. We considered both a power law (Equation 10) and a truncated normal distribution (Equation 11), described by the parameters (or ).

When increases, the number of clumps with larger density decreases, thus the average luminosity of the flares decreases and their average duration increases (see Figure 6). We also found that when increases, the flare distributions with a positive follow a different behaviour than those with a negative value: for , the flare distributions behave as described above (i.e. the number of flares increases with ), while this does not happen for . This is due to the fact that in this case the clumps are larger, thus there is a high probability that two or more clumps overlap thus reducing the number of flares.

For the case of a normal distribution of clump radii, we calculated the distributions of flare luminosities and durations for different values of (see Figure 7). When increases, the number of clumps with larger and smaller radii is reduced, resulting in a narrower flare luminosity distribution. When increases, the flare distributions have the same behaviour described above for the case .

We then tried another test, increasing the orbital period (e.g. from 10 days to 100 days), finding that the shape of the integral distributions in Figure 6 remains similar, except for the number of Xray flares, which decreases.

### 4.3 The effect of the mass-loss rate

The effects of different wind mass-loss rates are shown in Figure 8. The mass-loss rate is usually derived observationally from the strength of the H emission line, since this gives smaller uncertainties than the method based on UV P-Cygni lines (Kudritzki & Puls, 2000).

Since the H line opacity depends on , the presence of wind inhomogeneities leads to an over-estimate of the mass-loss rate. In particular, the mass-loss rates from O stars derived from smooth-wind models measurements with the H method need to be reduced by a factor 3 to 10 if the wind is clumpy [see Lépine & Moffat (2008) and Hamann et al. (2008)]. In Figure 8 we show that if decreases, also the number of flares decreases due to the reduced number of clumps. Also the average luminosity of the flares is reduced because the number density of clumps decreases resulting in a smaller probability that two or more clumps overlap.

### 4.4 The effect of the orbital parameters

In Figure 9 we show the effect of changing the orbital period and the eccentricity . We assumed , , .

When the orbital period increases, the number of flares emitted by the neutron star decreases (see Equation 31) and the neutron star accretes clumps with a smaller density (see Equation 9), implying a shift to lower luminosities and higher flare durations (see Figure 9). When the eccentricity increases, the neutron star accretes clumps with a higher density range (in general clumps are denser when they are closer to the supergiant), thus the luminosity range of the flares increases, as shown in Figure 9.

## 5 Comparison with the HMXB Vela X1

Vela X1 (4U 190040) is a bright eclipsing X-ray binary (= d, ) formed by the B0.5 Ib supergiant HD 77581 (Brucato & Kristian, 1972) (M = M, R= R, (van Kerkwijk et al., 1995)) and a pulsar with spin period s and mass M. This source shows significant X–ray variability on short time-scales, with flares lasting from s to s [Haberl (1994); Kreykenbohm et al. (2008)].

Recently Kreykenbohm
et al. (2008) analyzed *INTEGRAL*
observations of Vela X1 obtained during a phase of high flaring
activity, finding two kinds of flares: brief and bright flares
softer than longer flares. They also found several off-states,
during which the source is not detected (at least by *INTEGRAL*).
Kreykenbohm
et al. (2008) proposed that the short flares are
caused by the flip-flop instability, while the long ones are due
to the accretion of clumps ejected by the supergiant. The
off-states are explained as due to the onset of the propeller
effect when the neutron star crosses the lower density inter-clump
medium.

In this Section we apply our wind model to Vela X-1, assuming the wind parameters derived by Searle et al. (2008) and reported in Table 1.

Vela X1 | 4U 1700377 | IGR J112155952 | |
---|---|---|---|

Type | SGXB | SGXB | SFXT |

Spectral Type | B0.5 Ib | O6.5 Iaf | B0.7 Ia |

M | M | M () | |

R | R | R () | |

K () | K | K () | |

() | () | ||

M yr () | M yr | M yr | |

km s () | km s | km s () () | |

() | |||

M | M | M | |

d | d | d | |

– | |||

s | – | s | |

distance | kpc | kpc | kpc |

*RXTE*(lower panel), with that calculated with our clumpy wind model for the parameters reported in Table 1, and M yr, , km s, , g, g, , (middle panel), and (upper panel), , , . Orbital phase corresponds to 53750 MJD. Note that Vela X1 is not continuously observed with ASM/

*RXTE*. Therefore it is possible that some flares have been missed.

In Figure 10 we compare the light
curve measured with the ASM/*RXTE* instrument with that
calculated with our clumpy wind model assuming a spherical
symmetry for the outflowing wind. The
ASM/*RXTE* count rate, measured in the keV range, has
been converted to the keV luminosity using the average
spectral parameters obtained by Orlandini
et al. (1998)
and the distance of kpc (Sadakane
et al., 1985).
The observed light curve of Vela X1 is well reproduced by our
clumpy wind model for
M yr,
g and g,
, and .
Acceptable light curves were also obtained with
and M yr,
and with a normal distribution law for the clump radii, with .
We point out that the average luminosity
observed by ASM/*RXTE* out of the flares is erg s. This luminosity is obtained in our
model with the accretion of numerous clumps with low density.

## 6 Comparison with the HMXB 4U 1700377

4U 1700377 (Jones et al., 1973) is a day eclipsing HMXB
composed of a compact object, (a neutron star or a black hole),
and the O6.5 Iaf star HD 153919, located at a distance of
kpc (Ankay et al., 2001). Despite extensive searches,
no Xray pulsations have been found in this system. Therefore
the Xray mass function cannot be determined and the system
parameters (, , ,) cannot be derived
directly. They have been estimated from the radial velocity curve
of the supergiant and from the duration of the Xray eclipse, by
making assumptions about possible values of the radius of the O
star and the orbital inclination. Several studies indicate that
the mass of the compact object is larger than M
[Rubin
et al. (1996); Clark
et al. (2002)]. The
similarity of the Xray spectrum to other pulsars suggest that
the compact object of 4U 1700377 is a neutron star
(White
et al., 1983), but the presence of a low-mass black
hole cannot be excluded (Brown
et al., 1996).
Reynolds et al. (1999) reported the
presence of a possible cyclotron feature at keV observed
with *BeppoSAX*. If confirmed, this would demonstrate that
4U 1700377 hosts a neutron star with a magnetic field of about
G.

The Xray light curve of 4U 1700377 is characterized by a strong flaring activity with variations as large as a factor of on time scales from minutes to hours [Haberl et al. (1989); White et al. (1983); Rubin et al. (1996)]. We assumed the most recent set of system parameters of 4U 1700377, obtained by means of a detailed NLTE (Non-Local Thermal Equilibrium) line-driven wind model analysis of HD 153919 and a Monte Carlo simulation for the determination of the masses of both components (Clark et al., 2002). These authors found that the supergiant has a luminosity , an effective temperature K, radius R, mass M, mass loss rate M yr, and a mass for the compact object M (Table 1).

The Xray spectrum of 4U 1700377 is well described by an
absorbed power law with high-energy cutoff
(van der Meer
et al., 2005). The spectrum above keV was
studied using different satellites (e.g. BATSE detector on board
the *CGRO*, *INTEGRAL*) and can be modelled using a
thermal bremsstrahlung model with keV
(Rubin
et al., 1996) or with a thermal Comptonization model
(Orr
et al., 2004).

The analysis of the keV spectrum with XMM-*Newton*
during the eclipse, the egress, and a low-flux interval led
van der Meer
et al. (2005) to suggest that the low-flux
interval is probably due to a lack of accretion such as expected
in a structured and inhomogeneous wind. Moreover,
van der Meer
et al. (2005) proposed that the fluorescence
line from near-neutral iron detected in all spectra is produced by
dense clumps. They also observed recombination lines during the
eclipse which indicate the presence of ionized zone around the
compact object.

We analyzed the public archival *INTEGRAL* data of
4U 1700377, using all the IBIS/ISGRI observations obtained
from 2003 March 12 to 2003 April 22, and from 2004 February 2 to
2004 March 1. These data correspond to a net exposure time of
days (excluding the eclipse phase). We reduced the data
using OSA 7.0, and extracted the light curve in the energy range
keV, finding a total of flares. For each flare we
extracted the spectrum in the range keV. All the spectra
could be well fit by a thermal Comptonization model (comptt
in xspec). Based on the spectral results, we computed the
keV luminosity of each flare. All of them have a
luminosity greater than erg s (for
lower luminosities it is difficult to evaluate the flare duration
and then to distinguish the flares from the average level of the
Xray emission). For each flare we have measured two parameters:
the peak luminosity and the flare duration. We then applied our
clumpy wind model to the *INTEGRAL* observations of
4U 1700377.

We first compared the observed distributions of the flare luminosities and durations with those computed adopting the system parameters reported in Table 1. We assumed for the computed distribution a time interval equal to the exposure time of the 4U 1700377 observations considered here. As shown in Figure 11 the flare properties are well reproduced with our clumpy wind model for M yr, g and g, , and . We found that the numbers of observed (123) and calculated flares (116) are in good agreement.

The light curve of 4U 1700377 calculated with our clumpy wind model is shown in Figure 12.

## 7 Comparison with the SFXT IGR J112155952

The SFXT IGR J112155952 was discovered in April 2005 with
*INTEGRAL* (Lubinski et al., 2005). It is associated
with HD 306414, a B0.7 Ia star located at an estimated
distance of kpc (Negueruela et al., 2007).
*RXTE* observations showed a pulse period s
[Smith
et al. (2006); Swank
et al. (2007)]. This is
the first SFXT for which a periodicity in the outbursts
recurrence time was discovered (Sidoli
et al., 2006).
Subsequent observations (Romano et al., 2009) showed that the
true periodicity is about 164.5 days, i.e. half of the originally
proposed value. This periodicity is very likely due to the orbital
period of the system.

For a distance of kpc the peak fluxes of the outbursts
correspond to a luminosity of erg s
[ keV, Romano et al. (2007)].
*Swift* monitoring of this source revealed that the outburst
(lasting a few days) is composed by many flares (lasting from minutes to a few hours),
and before and after the whole outburst the source is fainter than
erg s (Sidoli
et al., 2007).

Sidoli
et al. (2007) showed that
accretion from a spherically symmetric homogeneous wind
could not reproduce the observed light curve and therefore
proposed a model based on an anisotropic wind characterized by a
denser and slower equatorial component, periodically
crossed by the neutron star along its orbit.
However this result was based on the old determination
of the orbital period (329 days).
Therefore, before applying our clumpy wind model, we checked the
spherically symmetric homogeneous wind with d, different eccentricity values, and the set of
stellar parameters derived by Searle et al. (2008) and
Lefever
et al. (2007) and reported in Table 1.
We assumed a terminal velocity ranging from
km s to km s, and ranging
from M yr to M yr.
In all cases we found that the duration of the Xray outburst
observed with *Swift* is shorter than that of the calculated
light curves.

A better agreement with the observations could be obtained with our clumpy wind model, especially for what concerns the flaring variability during the outburst phase. However, also in this case the calculated light curve always produces an outburst longer than the observed one. This is shown in Fig. 13, where the green symbols corresponds to the accretion of a dense clump (producing a flare), while the blue symbols indicate the lower luminosity level produced by the accretion of the inter-clump matter. We found that the probability to observe a flare, rather than the inter-clump luminosity level, is 90%.

In order to improve the agreement between the observed and the calculated light curve, we introduced an anisotropic outflow similar to that proposed by Sidoli et al. (2007). In this modified model we introduce a denser clumpy wind component in the equatorial plane, with a thickness , a terminal velocity and a mass loss rate , together with a spherically symmetric clumpy wind component (polar wind) with terminal velocity and a mass loss rate . We linked the mass loss rate from the equatorial outflow with the mass loss rate from the polar wind, by means of the factor :

(35) |

Both wind components are clumpy and obey laws described in Section 2. We assume that the second wind component has a Gaussian density profile perpendicular to the equatorial plane of the supergiant. In this framework we have considered an orbital period d and a high eccentricity in order to produce only one outburst per orbit.

The comparison between the *Swift* light curve and that
predicted with this model is shown in Figure
14. A good agreement with the
data is obtained with the parameters reported in Table
2.

Parameter | Value |
---|---|

M yr | |

km s | |

cm | |

g | |

g | |

km s |

## 8 Discussion and Conclusions

We have developed a clumpy wind model (where the dynamical problem is not treated) and explored the resulting effects on an accreting compact object in order to explain the observed behavior of the SFXTs and the SGXBs. Compared to previous attempts to explain the SFXTs outbursts in the context of clumpy winds [Walter & Zurita Heras (2007); Negueruela et al. (2008)], we introduced a distribution for the masses and initial dimensions of the clumps. We described the subsequent expansion of the clumps (Equation 9) taking into account realistic upper and lower limits for their radius (Equations 20 and 25).

This model, together with the theory of wind accretion modified because of the presence of clumps, allow us a comparison with the observed properties of both the light curves and luminosity distributions of the flares in SGXBs and SFXTs.

From the calculated integral distributions (Section 4), we found that the observable characteristics of the flares, such as luminosity, duration, number of flares produced, depend mainly on the orbital period (Figure 9), the scaling parameter of the power-law distribution for the clump formation rate (Equation 1), and the fraction of wind mass in the form of clumps (), as shown in Figures 4 and 5. Thus the variability properties of the different systems do not depend only on the orbital parameters, but are also significantly affected by the properties of the clumps (in particular by the parameters , , , , ).

We successfully applied our clumpy wind model to three different high mass X-ray binaries: Vela X1, 4U 1700377 and IGR J112155952. For the latter source, however, we had to introduce a denser equatorial component (still with a clumpy structure) in order to reproduce the flare duration.

In Figure 15 we show an example of a light curve of a generic SFXT calculated in the case of anisotropic clumpy wind. We assume for the generic SFXT properties similar to IGR J112155952, with an orbital period of d and an eccentricity . The orbital plane intersects the equatorial wind component at two phases (, ) producing two outbursts. The third outburst is produced at the periastron passage (). This implies that, if this explanation is correct, up to 3 outbursts per orbit are possible.

Source | Type | Supergiant | Compact | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Object | (M yr) | km s | (g) | (g) | ||||||

Vela X1 | SGXB | B0.5 Ib | NS | |||||||

4U 1700377 | SGXB | O6.5 Iaf | NS? | |||||||

IGR J112155952 | SFXT | B0.7 Ia | NS |

In Table 3 we have summarized the wind parameters obtained for the 3 sources. This table shows the typical differences in the mass-loss rate and terminal velocity expected from the CAK theory, similar values for and , and different values for , , , which seem to be proportional to the inverse of the effective temperature of the supergiant. Moreover, we found that the values of that best reproduce the observed light curves of the 3 HMXBs studied, are in agreement with the hypothesis that the mass-loss rate derived by the H emission is overestimated by a factor 2–10, in agreement with recent studies (see Section 4.3). We can exclude that the observed light curves of the three HMXBs studied can be reproduced with the same set of wind parameters.

In conclusion, the different values of , , obtained for the 3 HMXBs studied in this paper reveal that, in the framework of our clumpy wind model, the properties of the clumps of these 3 X–ray binary systems are slightly different, independently of the orbital period. This discrepancy could be due to the different spectral type of the 3 supergiants, which could eject structurally inhomogeneus winds with slightly different properties. We suggest as a possible cause of this behaviour that the values of , , could be related to the sonic radius where the clumps start (Bouret et al., 2005), which depends on the supergiant properties (Equation 8).

## Acknowledgments

L.D. thanks Prof. A. Treves for very helpful discussions. We thank the referee (L.B. Lucy) for useful comments which helped significantly improving our paper. This work was supported in Italy by ASI contracts I/023/05/0, I/088/06/0 and I/008/07/0.

## References

- Abbott (1982) Abbott D. C., 1982, ApJ, 259, 282
- Ankay et al. (2001) Ankay A., Kaper L., de Bruijne J. H. J., Dewi J., Hoogerwerf R., Savonije G. J., 2001, A&A, 370, 170
- Bondi & Hoyle (1944) Bondi H., Hoyle F., 1944, MNRAS, 104, 273
- Bouret et al. (2005) Bouret J.-C., Lanz T., Hillier D. J., 2005, A&A, 438, 301
- Bozzo et al. (2008) Bozzo E., Falanga M., Stella L., 2008, ApJ, 683, 1031
- Brown et al. (1996) Brown G. E., Weingartner J. C., Wijers R. A. M. J., 1996, ApJ, 463, 297
- Brucato & Kristian (1972) Brucato R. J., Kristian J., 1972, ApJL, 173, L105+
- Carlberg (1980) Carlberg R. G., 1980, ApJ, 241, 1131
- Castor et al. (1975) Castor J. I., Abbott D. C., Klein R. I., 1975, ApJ, 195, 157
- Clark et al. (2002) Clark J. S., Goodwin S. P., Crowther P. A., Kaper L., Fairbairn M., Langer N., Brocksopp C., 2002, A&A, 392, 909
- Davidson & Ostriker (1973) Davidson K., Ostriker J. P., 1973, ApJ, 179, 585
- Grebenev & Sunyaev (2007) Grebenev S. A., Sunyaev R. A., 2007, Astronomy Letters, 33, 149
- Haberl (1994) Haberl F., 1994, A&A, 288, 791
- Haberl et al. (1989) Haberl F., White N. E., Kallman T. R., 1989, ApJ, 343, 409
- Hamann et al. (2008) Hamann W.-R., Feldmeier A., Oskinova L. M., eds, 2008, Clumping in hot-star winds
- Howk et al. (2000) Howk J. C., Cassinelli J. P., Bjorkman J. E., Lamers H. J. G. L. M., 2000, ApJ, 534, 348
- Hoyle & Lyttleton (1939) Hoyle F., Lyttleton R. A., 1939, in Proceedings of the Cambridge Philosophical Society Vol. 34 of Proceedings of the Cambridge Philosophical Society, The effect of interstellar matter on climatic variation. p. 405
- in’t Zand (2005) in’t Zand J. J. M., 2005, A&A, 441, L1
- Jones et al. (1973) Jones C., Forman W., Tananbaum H., Schreier E., Gursky H., Kellogg E., Giacconi R., 1973, ApJL, 181, L43+
- Kreykenbohm et al. (2008) Kreykenbohm I., Wilms J., Kretschmar P., Torrejón J. M., Pottschmidt K., Hanke M., Santangelo A., et al. 2008, A&A, 492, 511
- Kudritzki et al. (1989) Kudritzki R. P., Pauldrach A., Puls J., Abbott D. C., 1989, A&A, 219, 205
- Kudritzki & Puls (2000) Kudritzki R.-P., Puls J., 2000, Annu. Rev. Astro. Astrophys., 38, 613
- Lamers & Cassinelli (1999) Lamers H. J. G. L. M., Cassinelli J. P., 1999, Introduction to Stellar Winds. Introduction to Stellar Winds, by Henny J. G. L. M. Lamers and Joseph P. Cassinelli, pp. 452. ISBN 0521593980. Cambridge, UK: Cambridge University Press, June 1999.
- Lefever et al. (2007) Lefever K., Puls J., Aerts C., 2007, A&A, 463, 1093
- Lépine & Moffat (2008) Lépine S., Moffat A. F. J., 2008, Astronomical Journal, 136, 548
- Lubinski et al. (2005) Lubinski P., Bel M. G., von Kienlin A., Budtz-Jorgensen C., McBreen B., Kretschmar P., Hermsen W., Shtykovsky P., 2005, The Astronomer’s Telegram, 469, 1
- Lucy & Solomon (1970) Lucy L. B., Solomon P. M., 1970, ApJ, 159, 879
- Lucy & White (1980) Lucy L. B., White R. L., 1980, ApJ, 241, 300
- Negueruela et al. (2006) Negueruela I., Smith D. M., Reig P., Chaty S., Torrejón J. M., 2006, in Wilson A., ed., The X-ray Universe 2005 Vol. 604 of ESA Special Publication, Supergiant Fast X-ray Transients: A New Class of High Mass X-ray Binaries Unveiled by INTEGRAL. p. 165
- Negueruela et al. (2007) Negueruela I., Smith D. M., Torrejon J. M., Reig P., 2007, ArXiv e-prints
- Negueruela et al. (2008) Negueruela I., Torrejón J. M., Reig P., Ribó M., Smith D. M., 2008, in Bandyopadhyay R. M., Wachter S., Gelino D., Gelino C. R., eds, A Population Explosion: The Nature & Evolution of X-ray Binaries in Diverse Environments Vol. 1010 of American Institute of Physics Conference Series, Supergiant Fast X-ray Transients and Other Wind Accretors. pp 252–256
- Orlandini et al. (1998) Orlandini M., dal Fiume D., Frontera F., Cusumano G., del Sordo S., Giarrusso S., Piraino S., et al. 1998, A&A, 332, 121
- Orr et al. (2004) Orr A., Torrejón J. M., Parmar A. N., 2004, in Schoenfelder V., Lichti G., Winkler C., eds, 5th INTEGRAL Workshop on the INTEGRAL Universe Vol. 552 of ESA Special Publication, An INTEGRAL Open Time Observation of the HMXRB 4U 1700-377. p. 361
- Oskinova et al. (2007) Oskinova L. M., Hamann W.-R., Feldmeier A., 2007, A&A, 476, 1331
- Owocki et al. (1988) Owocki S. P., Castor J. I., Rybicki G. B., 1988, ApJ, 335, 914
- Puls et al. (1996) Puls J., Kudritzki R.-P., Herrero A., Pauldrach A. W. A., Haser S. M., Lennon D. J., Gabler R., et al. 1996, A&A, 305, 171
- Reynolds et al. (1999) Reynolds A. P., Owens A., Kaper L., Parmar A. N., Segreto A., 1999, A&A, 349, 873
- Romano et al. (2009) Romano P., Sidoli L., Cusumano G., Vercellone S., Mangano V., Krimm H. A., 2009, ArXiv e-prints
- Romano et al. (2007) Romano P., Sidoli L., Mangano V., Mereghetti S., Cusumano G., 2007, A&A, 469, L5
- Rubin et al. (1996) Rubin B. C., Finger M. H., Harmon B. A., Paciesas W. S., Fishman G. J., Wilson R. B., Wilson C. A., et al. 1996, ApJ, 459, 259
- Runacres & Owocki (2002) Runacres M. C., Owocki S. P., 2002, A&A, 381, 1015
- Runacres & Owocki (2005) Runacres M. C., Owocki S. P., 2005, A&A, 429, 323
- Sadakane et al. (1985) Sadakane K., Hirata R., Jugaku J., Kondo Y., Matsuoka M., Tanaka Y., Hammerschlag-Hensberge G., 1985, ApJ, 288, 284
- Sako et al. (1999) Sako M., Liedahl D. A., Kahn S. M., Paerels F., 1999, ApJ, 525, 921
- Searle et al. (2008) Searle S. C., Prinja R. K., Massa D., Ryans R., 2008, A&A, 481, 777
- Sguera et al. (2005) Sguera V., Barlow E. J., Bird A. J., Clark D. J., Dean A. J., Hill A. B., Moran L., et al. 2005, A&A, 444, 221
- Shimada et al. (1994) Shimada M. R., Ito M., Hirata B., Horaguchi T., 1994, in Balona L. A., Henrichs H. F., Le Contel J. M., eds, Pulsation; Rotation; and Mass Loss in Early-Type Stars Vol. 162 of IAU Symposium, Radiatively driven winds of OB stars. p. 487
- Sidoli (2009) Sidoli L., 2009, Advances in Space Research, 43, 1464
- Sidoli et al. (2006) Sidoli L., Paizis A., Mereghetti S., 2006, A&A, 450, L9
- Sidoli et al. (2008) Sidoli L., Romano P., Mangano V., Pellizzoni A., Kennea J. A., Cusumano G., Vercellone S., Paizis A., Burrows D. N., Gehrels N., 2008, ApJ, 687, 1230
- Sidoli et al. (2007) Sidoli L., Romano P., Mereghetti S., Paizis A., Vercellone S., Mangano V., Götz D., 2007, A&A, 476, 1307
- Smith et al. (2006) Smith D. M., Bezayiff N., Negueruela I., 2006, The Astronomer’s Telegram, 773, 1
- Swank et al. (2007) Swank J. H., Smith D. M., Markwardt C. B., 2007, The Astronomer’s Telegram, 999, 1
- Vacca et al. (1996) Vacca W. D., Garmany C. D., Shull J. M., 1996, ApJ, 460, 914
- van der Meer et al. (2005) van der Meer A., Kaper L., di Salvo T., Méndez M., van der Klis M., Barr P., Trams N. R., 2005, A&A, 432, 999
- van Kerkwijk et al. (1995) van Kerkwijk M. H., van Paradijs J., Zuiderwijk E. J., Hammerschlag-Hensberge G., Kaper L., Sterken C., 1995, A&A, 303, 483
- Walter & Zurita Heras (2007) Walter R., Zurita Heras J., 2007, A&A, 476, 335
- White et al. (1983) White N. E., Kallman T. R., Swank J. H., 1983, ApJ, 269, 264