Bow shock nebulae of hot massive stars in a magnetized medium

Bow shock nebulae of hot massive stars in a magnetized medium
MNRAS 464, 3229–3248 (2017)
doi:10.1093/mnras/stw2537
Advance Access publication 2016 October 5
Bow shock nebulae of hot massive stars in a magnetized medium
D. M.-A. Meyer,1‹ A. Mignone,2 R. Kuiper,1 A. C. Raga3 and W. Kley1
1 Institut
für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
di Fisica Generale Facoltà di Scienze M.F.N., Università degli Studi di Torino, Via Pietro Giuria 1, I-10125 Torino, Italy
3 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ap. 70-543, 04510 D.F., México
2 Dipartimento
Accepted 2016 October 2. Received 2016 October 2; in original form 2015 September 30
ABSTRACT
Key words: MHD – methods: numerical – circumstellar matter – stars: massive.
1 I N T RO D U C T I O N
Massive star formation is a rare event that strongly impacts the
whole Galactic machinery. These stars can release strong winds
and ionizing radiation which shape their close surroundings into
beautiful billows of swept-up and irradiated interstellar gas, which
, in the case of a static or a slowly moving star, can produce structures such as the Bubble Nebula (NGC 7635) in the constellation
of Orion (Moore et al. 2002). The detailed study of the circumstellar medium of these massive stars provides us an insight into
their internal physics (Langer 2012); it provides information on their
intrinsic rotation (Langer, Garcı́a-Segura & Mac Low 1999), their
envelope’s (in)stability (Yoon & Cantiello 2010) and allows us to
understand the properties of their close surroundings throughout
their evolution (van Marle et al. 2006; Chita et al. 2008) and after
their death (Orlando et al. 2008; Chiotellis, Schure & Vink 2012).
This information is relevant for evaluating their feedback, i.e. the
amount of energy, momentum and metals that massive stars inject
into the interstellar medium (ISM) of the Galaxy (Vink 2006).
In particular, the bow shocks that develop around some fastmoving massive stars ejected from their parent stellar clusters
E-mail: dominique.meyer@uni-tuebingen.de
provide an opportunity to constrain both their wind and local ISM
properties (Huthoff & Kaper 2002; Meyer et al. 2014a). Over the
past decades, stellar wind bow shocks have first been serendipitously noticed as bright [O III] λ 5007 spectral line arc-like shapes
and/or distorted bubbles surrounding some massive stars having
a particularly large space velocity with respect to their ambient
medium. As a textbook example of such a bow shock, we refer
the reader, e.g. to ζ Ophiuchi (Gull & Sofia 1979, see Fig. 13
below). Further infrared observations, e.g. with the Infrared Astronomical Satellite (IRAS; Neugebauer et al. 1984) and the WideField Infrared Satellite Explorer (WISE; Wright et al. 2010) facilities have made possible the compilation of catalogues of dozens
of these bow shock nebulae (van Buren & McCray 1988a; van
Buren, Noriega-Crespo & Dgani 1995; Noriega-Crespo, van
Buren & Dgani 1997a) and have motivated early numerical simulations devoted to the parsec-scale circumstellar medium of moving
stars (Brighenti & D’Ercole 1995a,b). Recently, modern facilities
led to the construction of multi-wavelengths data bases, see e.g. the
E-BOSS catalogue (Peri et al. 2012; Peri, Benaglia & Isequilla
2015) or the recent study of Kobulnicky et al. (2016). Moreover,
a connection with high-energy astrophysics has been established,
showing that stellar wind bow shocks produce cosmic rays in the
same way as the expanding shock waves of growing supernova
remnants do (del Valle, Romero & Santos-Lima 2015).
C 2016 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Downloaded from http://mnras.oxfordjournals.org/ at University Tuebingen on November 14, 2016
A significant fraction of OB-type, main-sequence massive stars are classified as runaway and
move supersonically through the interstellar medium (ISM). Their strong stellar winds interact
with their surroundings, where the typical strength of the local ISM magnetic field is about
3.5–7 μG, which can result in the formation of bow shock nebulae. We investigate the effects
of such magnetic fields, aligned with the motion of the flow, on the formation and emission
properties of these circumstellar structures. Our axisymmetric, magneto-hydrodynamical simulations with optically thin radiative cooling, heating and anisotropic thermal conduction show
that the presence of the background ISM magnetic field affects the projected optical emission
of our bow shocks at Hα and [O III] λ 5007 which become fainter by about 1–2 orders of
magnitude, respectively. Radiative transfer calculations against dust opacity indicate that the
magnetic field slightly diminishes their projected infrared emission and that our bow shocks
emit brightly at 60 μm. This may explain why the bow shocks generated by ionizing runaway
massive stars are often difficult to identify. Finally, we discuss our results in the context of the
bow shock of ζ Ophiuchi and we support the interpretation of its imperfect morphology as an
evidence of the presence of an ISM magnetic field not aligned with the motion of its driving
star.
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D. M.-A. Meyer et al.
MNRAS 464, 3229–3248 (2017)
magnetic field leads to anisotropic heat conduction (HC; see e.g.
Balsara, Tilley & Howk 2008) and (ii) our study does not concentrate on the secular stellar wind evolution of our bow-shockproducing stars. Note that our study introduces a reduced number
of representative models due to the high numerical cost of the MHD
simulations. Following Acreman, Stevens & Harries (2016), we additionally appreciate the effects of the ISM magnetic field on the
bow shocks with the help of radiative transfer calculations of dust
continuum emission.
This paper is organized as follows. We start in Section 2 with a
review of the physics included in our models for both the stellar
wind and the ISM. We also recall the adopted numerical methods.
Our models of bow shocks generated by main-sequence, runaway
massive stars moving in a magnetized medium are presented together with a discussion of their morphology and internal structure
in Section 3. We detail the emission properties of our bow shocks
and discuss their observational implications in Section 4. Finally,
we formulate our conclusions in Section 5.
2 METHOD
In the present section, we briefly summarize the numerical methods
and microphysics utilized to produce MHD bow shock models of
the circumstellar medium surrounding hot, runaway massive stars.
2.1 Governing equations
We consider a magnetized flow past a source of hot, ionized and
magnetized stellar wind. The dynamics are described by the ideal
equations of magneto-hydrodynamics and the dissipative character
of the thermodynamics originates from the treatment of the gas
with heating and losses by optically thin radiation together with
electronic HC. These equations are
∂ρ
+ ∇ · (ρv) = 0,
∂t
(1)
∂m
+ ∇ · (m⊗v + B⊗B + Îpt ) = 0,
∂t
(2)
∂E
+ ∇ · ((E + pt )v − B(v · B)) = ζ (T , ρ, μ),
∂t
(3)
and,
∂B
+ ∇ · (v⊗B − B⊗v) = 0,
∂t
(4)
where ρ and v are the mass density and the velocity of the plasma.
In the relation of momentum conservation equation (2), the quantity
m = ρv is the linear momentum of a gas element, B is the magnetic
field, Î is the identity matrix, and
B·B
(5)
2
is the total pressure of the gas, i.e. the sum of its thermal component p and its magnetic contribution (B · B)/2, respectively. Equation (3) describes the conservation of the total energy of the gas,
pt = p +
E=
m·m
B·B
p
+
+
,
(γ − 1)
2ρ
2
(6)
where γ is the adiabatic index, which is taken to be 5/3, i.e. we
assume an ideal gas. The right-hand source term ζ (T, ρ, μ) in
equation (3) represents (i) the heating and the losses by optically
thin radiative processes and (ii) the heat transfers by anisotropic
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It is the discovery of bow shocks around the historical stars Betelgeuse (Noriega-Crespo et al. 1997b) and Vela-X1 (Kaper et al. 1997)
that revived the interest of the scientific community regarding such
circumstellar structures generated by massive stars. The fundamental study of Comerón & Kaper (1998) demonstrates that complex
morphologies can arise from massive stars’ wind–ISM interactions.
Bow shocks are subject to a wide range of shear-like and non-linear
instabilities (Blondin & Koerwer 1998) producing severe distortions
of their overall forms, which can only be analytically approximated
(Wilkin 1996) in the particular situations of either a star moving in
a relatively dense ISM (Comerón & Kaper 1998) or a high-mass
star hypersonically moving through the Galactic plane (Meyer et al.
2014b, hereafter Paper I). Tailoring numerical models to runaway
red supergiant stars allows us to constrain the mass loss and local
ISM properties of Betelgeuse (van Marle et al. 2011; Cox et al.
2012; Mackey et al. 2012) or IRC-10414 (Gvaramadze et al. 2014;
Meyer et al. 2014a). For the sake of simplicity, these models neglect
the magnetization of the ISM.
However, magnetic fields are an essential component of the
ISM of the Galaxy, e.g. its large-scale component has a tendency
to be aligned with the galactic spiral arms (Gaensler 1998). If
the strength of the ISM magnetic field can reach up to several
tenths of Gauss in the centre of our Galaxy (see Rand & Kulkarni
1989; Ohno & Shibata 1993; Opher et al. 2009; Shabala, Mead &
Alexander 2010), it can be even stronger in the cold phase of the ISM
(Crutcher et al. 1999). In particular, radio polarization measures of
the magnetic field in the context of Galactic ionized supershells
are reported to be 2–6 μG in Harvey-Smith, Madsen & Gaensler
(2011). This value is in accordance with previous estimates of the
field strength in the warm phase of the ISM (Troland & Heiles
1986) and was supported by hydrodynamical (HD) simulations
(Fiedler & Mouschovias 1993). Such a background magnetic field
should therefore be included in realistic models of circumstellar
nebulae around massive stars.
Numerical studies of magneto-hydrodynamical (MHD) flows
around an obstacle are approximated in the plane-parallel approach
in de Sterck, Low & Poedts (1998) and de Sterck & Poedts (1999).
A significant number of circumstellar structures, such as the vicinity of our Sun (Pogorelov & Semenov 1997), planetary nebulae
developing in the vicinity of intermediate-mass stars (Heiligman
1980) or supernova remnants (Rozyczka & Tenorio-Tagle 1995),
have been studied in such a two-dimensional approach (see also
Soker & Dgani 1997; Pogorelov & Matsuda 2000). The presence
of a weak magnetic field can inhibit the growth rate of shear instabilities in the bow shocks around cool stars such as the runaway
red supergiant Betelgeuse in the constellation of Orion (van Marle,
Decin & Meliani 2014). We place our work in this context, focusing on bow shocks generated by hot, fast winds of main-sequence
massive stars.
In this study, we continue our investigation of the circumstellar medium of runaway massive stars moving within the plane
of the Milky Way (Paper I; Meyer et al. 2015, 2016). As a logical extension of them, we present MHD models of a sample
of some of the most common main-sequence, runaway massive
stars (Kroupa 2001) moving at the most probable space velocities
(Eldridge, Langer & Tout 2011). We ignore any intrinsic inhomogeneity or turbulence in the ISM. Particularly, we assume an axisymmetric magnetization of the ISM surrounding the bow shocks
in the spirit of van Marle et al. (2014). We concentrate our efforts on
an initially 20 M star; however, we also consider bow shocks generated by lower and higher initial mass stars. This project principally
differs from Paper I because (i) the inclusion of an ISM background
MHD bow shocks of hot massive stars
electronic thermal conduction (see Section 2.3). Finally, equation (4) is the induction equation and governs the time evolution
of the vector magnetic field B. The relation
γp
(7)
cs =
ρ
closes the system (equations 1−4), where cs denotes the adiabatic
speed of sound.
2.2 Boundary conditions and numerical scheme
2.3 Gas microphysics
(8)
where (T , ρ) is a function that stands for the processes by optically
thin radiation where
mH p
(9)
T =μ
kB ρ
is the gas temperature, with μ = 0.61 the mean molecular weight
of the gas, kB the Boltzmann constant and mH the proton mass, respectively. The gain and losses by optically thin radiative processes
are taken into account via the following law:
(T , ρ) = nH (T ) − n2H (T ),
(10)
where (T ) and (T ) are the rate of change of the gas internal
energy induced by heating and cooling as a function of T, respectively, and where nH = ρ/μ(1 + χ He, Z )mH is the hydrogen number
1
http://plutocode.ph.unito.it/
Fc = κ|| b̂( b̂ · ∇T ) + κ⊥ (∇T − b̂ · ∇T ),
(11)
where b̂ = B/||B|| is the magnetic field unit vector. It is calculated
through the interface of the nearest neighbouring cells in the whole
computational domain according to the temperature difference T
and the local field orientation b̂ (see appendix of Mignone et al.
2012). The coefficients κ || and κ ⊥ are the heat coefficients along the
directions parallel and normal to the local magnetic field streamline,
respectively. Along the direction of the local magnetic field,
κ|| = K|| T 5/2 ,
(12)
with
1.84 × 10−5
(13)
erg s−1 K−1 cm−1 ,
ln(L)
√
where ln(L) = 29.7 + ln(T /106 n) is the Coulomb logarithm,
with n the gas total number density (Spitzer 1962). The HC coefficients satisfy κ ⊥ /κ || ≈ 10−16 1 for the densities that we
consider (Parker 1963; Velázquez et al. 2004; Balsara et al. 2008;
Orlando et al. 2008). The value of Fc varies between the classical
flux in equation (11) and the saturated conduction regime (Balsara
et al. 2008) which limits the heat flux to
K|| =
3
,
Fsat = 5φρciso
(14)
for very large temperature gradients (≥ 106 K pc−1 ), with ciso = p/ρ
the isothermal speed of sound and φ < 1 a free parameter that we
set to the typical value of 0.3 (Cowie & McKee 1977).
2.4 Setting up the stellar wind
We impose the stellar wind at the surface of a sphere of radius
20z pc centred into the origin O with wind material. Its density
is
ρw =
The source term ζ (T, ρ, μ) in equation (3) represents the non-ideal
thermodynamics processes that we take into account, and reads
ζ (T , ρ, μ) = (T , ρ) + ∇ · Fc
density with χ He,Z the mass fraction of the coolants heavier than hydrogen. Details about the processes included into the cooling (T)
and heating (T) laws are given in section 2 of Paper I.
The divergence term in the source function in equation (8) represents the anisotropic heat flux,
Ṁ
,
4π r 2 vw
(15)
where Ṁ is the star’s mass-loss rate and r is the distance to the
origin O. We interpolate the wind parameters from stellar evolution
models of non-rotating massive stars with solar metallicity that we
used for previous studies (see Paper I). Our stellar wind models
are have been generated with the stellar evolution code described
in Heger, Woosley & Spruit (2005) and subsequently updated by
Yoon & Langer (2005), Petrovic et al. (2005) and Brott et al. (2011).
It utilizes the mass-loss prescriptions of Kudritzki et al. (1989) for
the main-sequence phase of our massive stars and of de Jager,
Nieuwenhuijzen & van der Hucht (1988) for the red supergiant
phase. Despite the fact that our wind models report the marginal
evolution of main-sequence winds (see Paper I), they remain quasiconstant during the part of the stellar evolution that we follow.
We refer the reader interested in a graphical representation of the
utilized wind models to fig. 3 of Paper I, while we report the wind
properties at the beginning of our simulations in our Table 1. Note
that our adopted values for the stellar wind velocity belong to the
lower limit of the range of validity for stellar winds of OB stars (see
below in Section 3.1.3).
MNRAS 464, 3229–3248 (2017)
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We solve the above described system of equations (equations 1−7)
using the open-source PLUTO code1 (Mignone et al. 2007, 2012) on a
uniform two-dimensional grid covering a rectangular computational
domain in a cylindrical frame of reference (O; R, z) of origin O and
symmetry axis about R = 0. The grid [O; Rmax ] × [ − zmin ; zmax ]
where Rmax , −zmin and zmax are the upper and lower limits of the
OR and Oz directions, respectively, which are discretized with NR
= 2Nz = 1000 cells such that the grid resolution is R = z =
Rmax /NR . Learning from previous bow shock models (Comerón &
Kaper 1998; van Marle et al. 2006), we impose inflow boundary
conditions corresponding to the stellar motion at z = zmax whereas
outflow boundaries are set at R = Rmax and z = −zmin . Moreover, the
stellar wind is modelled setting inflow boundary conditions centred
around the origin (see Section 2.4).
We integrate the system of partial differential equations within
the eight-wave formulation of the MHD equations (1)−(7), using
a cell-centred representation consisting in evaluating ρ, m, E and
B using the barycentre of the cells (see section 2 of Paper I). This
formulation, used together with the Harten–Lax–van Leer approximate Riemann solver (Harten, Lax & van Leer 1983), conserves
the divergence-free condition ∇ · B = 0. The method is a secondorder, unsplit, time-marching algorithm scheme controlled by the
Courant–Friedrich–Levy parameter initially set to Ccfl = 0.1. The
gas cooling and heating rates are linearly interpolated from tabulated cooling curves (see Section 2.3) and the corresponding rate
of change is subtracted from the total energy E. The parabolic
term of HC is integrated with the Super-Time-Stepping algorithm
(Alexiades, Amiez & Gremaud 1996).
3231
3232
D. M.-A. Meyer et al.
Table 1. Stellar wind parameters at the beginning of the simulations, at
a time tstart after the beginning of the zero-age main-sequences of the star.
Parameter M (in M ) is the initial mass of the star, L the stellar luminosity
(in L ), Ṁ its mass loss and vw the wind velocity (see also table 1 of Meyer
et al. 2016).
M
(M )
10
20
40
tstart
(Myr)
log(L /L )
log(Ṁ/M yr−1 )
vw (km s−1 )
Teff (K)
5.0
3.0
0.0
3.80
4.74
5.34
−9.52
−7.38
−6.29
1082
1167
1451
25 200
33 900
42 500
where B and R are the stellar surface magnetic field and the stellar
radius, respectively, and of a toroidal component, which, in the
case of a non-rotating star, this reduces to Bφ = 0. The ∝1/r2
radial dependence of equation (16) makes the strength of the stellar
magnetic field almost negligible at the wind termination shock that
is typically about a few tenths of pc from the star that we study
(Paper I). However, imposing a null magnetic field in the stellar
wind region would let the direction of the heat flux Fc undetermined
in the region of (un)shocked wind material of the bow shock (see
magnetic field unit vector b̂ in the right-hand side of equation 11).
Note that, given their analogous radial dependence on r, stellar
wind and stellar magnetic field are similarly implemented into our
axisymmetric simulations. In these simulations, the stellar surface
magnetic field is set to B 1.0 kG (Donati et al. 2002) at R =
3.66 R (Brott et al. 2011) where R is the solar radius.
We first focus on a baseline bow shock generated by an initially
20 M star moving with a velocity v = 40 km s−1 in the Galactic
plane of the Milky Way whose magnetic field is assumed to be
BISM = 7 μG (Draine 2011). Then, we consider models with velocity v = 20 to 70 km s−1 , explore the effects of a magnetization of
BISM = 3.5 μG, and carry out simulations of initially 10 and 40 M
stars moving at velocities v = 40 and 70 km s−1 , respectively. We
investigate the effects of the ISM magnetic field carrying out a couple of additional purely HD simulations, as comparison runs. All
our simulations are started at a time about 4.5 Myr after the zero-age
main-sequence phase of our stars and are run at least four crossing
times |zmax − zmin |/v of the gas through the computational domain,
such that the system reaches a steady or quasi-stationary state in the
case of a stable or unstable bow shock, respectively.
We label our MHD simulations concatenating the values of the
initial mass M of the moving star (in M ), its bulk motion v (in
km s−1 ) and the included physics ‘Ideal’ for dissipativeless simulations, ‘Cool’ if the model includes heating and losses by optically
thin radiative processes, ‘Heat’ for HC and ‘All’ if cooling, heating
and HC are taken into account together. Finally, the labels inform
about the strength of the ISM magnetic field. We distinguish our
MHD runs from our previously published HD studies (Paper I)
adding the prefix ‘HD’ and ‘MHD’ to the simulation labels of our
HD and MHD simulations, respectively. All the information relative
to our models is summarized in Table 2.
3 R E S U LT S A N D D I S C U S S I O N
This section presents the MHD simulations carried out in the context
of our Galactic, ionizing, runaway massive stars. We detail the
effects of the included microphysics on a baseline bow shock model,
discuss the morphological differences between our HD and MHD
simulations and consider the effects of the adopted stellar wind
models. Finally, we review the limitations of the model.
2.5 Setting up the ISM
Our runaway stars are moving through the warm ionized phase
of the ISM, i.e. we assume that they run in their own H II region
inside which the gas is considered as homogeneous, laminar and
fully ionized fluid. The ISM composition assumes solar metallicity
(Lodders 2003), with nH = 0.57 cm−3 (Wolfire et al. 2003) and with
TISM ≈ 8000 K, initially. The model is a moving star within an ISM
at rest. We solve the equations of motion in the frame in which the
star is at rest and, hence, the ISM moves with vISM = −v , where v
is the bulk motion of the star. The gas in the computational domain
is evaluated with the cooling curve for photoionized gas described
in fig. 4(a) of Paper I. In particular, our initial conditions neglect
the possibility that a bow shock might trap the ionizing front of the
H II region (see section 2.4 of Paper I for an extended discussion of
the assumptions underlying our method for modelling bow shocks
from hot massive stars). Additionally, an axisymmetric magnetic
field B = −BISM ẑ field is imposed over the whole computational
domain, with BISM > 0 its strength and ẑ the unit vector along the Oz
direction. Finally, our simulations trace the respective proportions
of ISM gas with respect to the wind material using a passive scalar
tracer according to the advection equation,
∂(ρQ)
+ ∇ · (vρQ) = 0,
(17)
∂t
where Q is a passive tracer with initial value Q(r) = 1 for the wind
material and Q(r) = 0 for the ISM gas, respectively.
MNRAS 464, 3229–3248 (2017)
3.1 Bow shock thermodynamics
3.1.1 Effects of the included physics: hydrodynamics
In Fig. 1, we show the gas density field in a series of bow shock
models of our initially 20 M star moving with velocity 40 km s−1
through a medium of ISM background density nH = 0.59 cm−3 and
of magnetic field strength BISM = 7 μG. The crosses indicate the
position of the moving star. The figures correspond to a time about
5 Myr after the beginning of the main-sequence phase of the star.
The stellar wind and ISM properties are the same for all figures,
only the included physics is different for each model (our Table 2).
Left-hand panels are HD simulations whereas right-hand panels are
MHD simulations, respectively. From top to bottom, the included
thermodynamic processes are adiabatic (a), take into account optically thin radiative processes of the gas (b), heat transfers (c)
or both (d). The black dotted lines are the contours Q(r) = 1/2,
which trace the discontinuity between the stellar wind and the ISM
gas. The streamlines (a–c) and vector velocity field (d) highlight
the penetration of the ISM gas into the different layers of the bow
shock.
The internal structure of the bow shocks can be understood by
comparing the time-scales associated with the different physical
processes at work. The dynamical time-scale represents the time
interval it takes the gas to advect through a given layer of our bow
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Since we assume a spherically symmetric stellar wind density,
thermal pressure and velocity profiles, we use the Parker prescription (Parker 1958) to model the magnetic field in the stellar wind.
It consists of a radial component of the field
2
R
,
(16)
Br = B
r
2.6 Simulation ranges
MHD bow shocks of hot massive stars
3233
Table 2. Nomenclature and grid parameters used in our (magneto-)hydrodynamical simulations. The quantities M (in M ) and v (in
km s−1 ) are the initial mass of the stars and their space velocity, respectively, whereas BISM (in µG) is the strength of the ISM magnetic
field. Parameters , zmin and Rmax are the resolution of the uniform grid (in pc cell−1 ) and the lower and upper limits of the domain
along the R-axis and z-axis (in pc), respectively. The last column contains the physics included in each simulation. HC refers to isotropic
thermal conduction in the case of a HD simulation and to anisotropic thermal conduction in the case of a MHD simulation, respectively.
Model
v (km s−1 )
BISM (µG)
(10−3 pc cell−1 )
zmin (pc)
Rmax (pc)
Included microphysics
20
20
20
20
20
20
20
20
10
20
20
20
40
40
40
40
40
40
40
40
40
40
20
40
70
70
−
−
−
−
7.0
7.0
7.0
7.0
7.0
7.0
3.5
7.0
7.0
8.0
8.0
8.0
8.0
8.0
8.0
8.0
8.0
3.0
6.0
8.0
1.2
1.6
−2.0
−2.0
−2.0
−2.0
−2.0
−2.0
−2.0
−2.0
−2.0
−3.0
−2.0
−1.0
−4.0
8.0
8.0
8.0
8.0
8.0
8.0
8.0
8.0
6.0
12.0
8.0
3.0
16.0
HD, adiabatic
HD, cooling, heating
HD, HC
HD, cooling, heating, HC
MHD
MHD, cooling, heating
MHD, HC
MHD, cooling, heating, HC
MHD, cooling, heating, HC
MHD, cooling, heating, HC
MHD, cooling, heating, HC
MHD, cooling, heating, HC
MHD, cooling, heating, HC
HD2040Ideal
HD2040Cool
HD2040Heat
HD2040All
MHD2040IdealB7
MHD2040CoolB7
MHD2040HeatB7
MHD2040AllB7
MHD1040AllB7
MHD2020AllB7
MHD2040AllB3.5
MHD2070AllB7
MHD4070AllB7
shocks, i.e. the region of shocked ISM or the layer of shocked wind.
It is defined as
l
tdyn =
,
(18)
v
where l is the characteristic lengthscale of the region of the bow
shock measured along the Oz direction and where v is the gas velocity in the post-shock region of the considered layers. According
to the Rankine–Hugoniot relations and taking into account the nonideal character of our model, we should have v v /4 in the shocked
ISM and v vw /4 in the post-region at the forward shock (FS) and
at the reverse shock (RS), respectively.
The cooling time-scale is defined as
tcool =
Eint
p
=
,
(γ − 1)(T )n2H
E˙int
(19)
where E˙int is the rate of change of internal energy Eint (Orlando et al.
2005). The HC time-scale measures the rapidity of heat transfer into
the bow shock, and is given by
theat =
7pl 2
,
2(γ − 1)κ(T )T
(20)
where l is a characteristic length of the bow shock along which heat
transfers take place. Measuring the density, pressure and velocity
fields in our simulations, we evaluate and compare those quantities
defined in equations (18)–(20) at both the post-shock regions at the
forward and RSs. Results for both the layers of shocked wind and
shocked ISM material are given in Table 3.
Our HD, dissipation-free bow shock model HD2040Ideal has a
morphology governed by the gas dynamics only (Fig. 1a). It has a
contact discontinuity separating the outer region of cold shocked
ISM from the inner region of hot shocked stellar wind, which are
themselves bordered by the forward and RSs, respectively. There is
no advection of ISM material into the wind region (see the ISM gas
streamlines in Fig. 1a). The model HD2040Cool including cooling by optically thin radiation has a considerably reduced layer
of dense, shocked ISM gas caused by the rapid losses of internal
energy (tdyn tcool ; see time-scales in our Table 3). Its thinness
favours the growth of Kelvin–Helmholtz instabilities and allows
large eddies to develop in the shocked regions (Fig. 1b). The layer
of hot gas is isothermal because the regular wind momentum input
at the RS prevents it from cooling and it therefore conserves its hot
temperature (tcool tdyn ) whereas the distance between the star and
the contact discontinuity,
Ṁvw
R(0) =
,
(21)
4πρISM v2
does not evolve (Wilkin 1996).
The model HD2040Heat takes into account thermal conduction
which is isotropic in the case of the absence of magnetic field. The
heat flux reads
Fc = κ∇T ,
(22)
and transports internal energy from the RS to the contact discontinuity (tdyn theat ) which in its turn splits the dense region into
a hot (tdyn theat ) and a cold layer of shocked ISM gas (tdyn theat ), respectively. This modifies the penetration of ISM gas into
the bow shock and causes the region of shocked wind to shrink to a
narrow layer of material close to the RS (Fig. 1c). Not surprisingly,
the model with both cooling and conduction HD2040All (Fig. 1d)
presents both the thermally split region of shocked ISM (tdyn theat , tdyn tcool , tcool theat ) and a reduced layer of shocked wind
material (tdyn theat , tdyn tcool , tcool theat ) that reorganizes the
internal structure of the bow shock together with a dense shell of
cool ISM gas (see also discussion in Paper I). For the sake of clarity, Fig. 1(d) overplots the gas velocity fields as white arrows which
illustrate the penetration of ISM gas into the hot layer of the bow
shock.
3.1.2 Effects of the included physics: magneto-hydrodynamics
We plot in the right-hand panels of Fig. 1 the ideal MHD simulation
of our initially 20 M star moving with v = 40 km s−1 through a
medium where the strength of the magnetic field is BISM = 7 μG
(e) together with models including cooling and heating by optically
thin radiation (f), anisotropic HC (g) and both (h). Despite the
fact that the overall morphology of our MHD bow shock models is
globally similar to the models with BISM = 0 μG, a given number of
significant changes relative to both their shape and internal structure
arise. Note that in the context of our MHD models, theat represents
the heat transfer time-scale normal to the fields lines.
Our ideal MHD model has the typical structure of a stellar
wind bow shock, with a region of shocked ISM gas surrounding
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M (M )
3234
D. M.-A. Meyer et al.
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Figure 1. Changes in the morphology of a stellar wind bow shock with variation of the included physics. Figures show gas number density plotted with a
density range from 10−5 to 5 cm−3 in the logarithmic scale for an initially 20 M star moving with velocity 40 km s−1 . Left-hand panels are the HD models
whereas right-hand panels are the MHD models with BISM = 7 µG. The first line of panels shows adiabatic (a) and ideal MHD (e) models, respectively. The
second line of panels plots models with optically thin radiative processes (b,f); the third line shows models including thermal (an-)isotropic conduction (c,g)
and the last line plots models including cooling, heating and (an-)isotropic thermal conduction (d,h). The nomenclature of the models follows our Table 2. For
each figure, the dotted thick line traces the material discontinuity, i.e. the interface of the wind/ISM regions, Q(r) = 1/2. The right part of each figure overplots
ISM flow streamlines, except panel (d), which explicitly plots the velocity field as white arrows over the whole computational domain. The crosses mark the
position of the star. The R-axis represents the radial direction and the z-axis the direction of stellar motion (in pc). Only a fraction of the computational domain
is shown.
the one of shocked wind gas. The contact discontinuity acts as a
border between the two kinds of material (Fig. 1e). The model
with cooling MHD2040CoolB7 has smaller but denser layer of
ISM gas (Fig. 1f) due to the rapid cooling time (tcool tdyn ).
The MHD model with thermal conduction is similar to our model
MHD2040IdealB7 since, due to its anisotropic character, heat transport is cancelled across the magnetic field lines (theat ≫tdyn ). Note
MNRAS 464, 3229–3248 (2017)
the boundary effect close to the apex along the Oz direction as a
result of the HC along the direction of the ISM magnetic field lines
(Fig. 1g). Finally, our model with both processes has its dynamics governed by the cooling in the region of shocked ISM (theat
≫ tdyn , theat ≫ tcool , tcool tdyn ) and by the wind momentum
in the region of shocked wind (theat ≫ tdyn , theat ≫ tcool , tdyn tcool ).
MHD bow shocks of hot massive stars
3235
Table 3. Characteristics dynamical time-scale tdyn , cooling time-scale tcool
and thermal conduction time-scale theat (in Myr) measured along the Oz
direction from our simulations of our initially 20 M star moving velocity
40 km s−1 (see Figs 1a–h). We estimate the various time-scales in both
the post-shock region at the FS and the RS of our bow shocks. The black
hyphen indicates that the corresponding physical process is not included in
the models (our Table 2).
tdyn (Myr)
tcool (Myr)
theat (Myr)
HD2040Ideal (FS)
HD2040Ideal (RS)
HD2040Cool (FS)
HD2040Cool (RS)
HD2040Heat (FS)
HD2040Heat (RS)
HD2020All (FS)
HD2020All (RS)
MHD2040IdealB7 (FS)
MHD2040IdealB7 (RS)
MHD2040CoolB7 (FS)
MHD2040CoolB7 (RS)
MHD2040HeatB7 (FS)
MHD2040HeatB7 (RS)
MHD2040AllB7 (FS)
MHD2040AllB7 (RS)
2.5 × 10−2
4.7 × 10−3
1.0 × 10−2
3.7 × 10−3
6.5 × 10−2
1.3 × 10−2
9.0 × 10−3
1.1 × 10−2
1.8 × 10−2
4.3 × 10−3
1.1 × 10−1
2.0 × 10−3
1.0 × 10−2
6.7 × 10−3
3.0 × 10−1
2.3 × 10−3
−
−
4.5 × 10−3
3.5 × 10+3
−
−
5.1 × 10−3
2.5 × 10+1
−
−
4.3 × 10−2
1.0 × 10+3
−
−
3.4 × 10−3
3.0 × 10+4
−
−
−
−
1.2 × 10+3
1.2 × 10−4
8.7 × 10+5
4.0 × 10−3
−
−
−
−
1.1 × 10+25
5.5 × 10+9
1.4 × 10+18
5.4 × 10+7
3.1.3 Effects of the boundary conditions: stellar wind models
The shape of the bow shock generated around a runaway massive
star in the warm phase of the ISM is a function of the respective
strength of both the ISM ram pressure ρISM v2 and the stellar wind
ram pressure ρw vw2 , as seen in the frame of reference of the moving
object (see explanations in Mohamed, Mackey & Langer 2012).
According to equation (15), ρw = Ṁ/4π r 2 vw which implies that
ρw vw2 ∝ Ṁvw . In other words, in a given ambient medium and at
a given peculiar velocity, the governing quantity in the shaping of
such
bow shock is Ṁvw and its stand-off distance R(0) goes as
Ṁvw (see equation 21). Nevertheless, if the production of stellar
evolution models depends on specific prescriptions relative to Ṁ
that are consistently used through the calculations (in our case the
recipe of Kudritzki et al. 1989), the estimate of the wind velocity
is posterior to the calculation of the stellar structure and it does not
influence Ṁ, Teff or L .
The manner to calculate vw is not unique (Castor, Abbott & Klein
1975; Kudritzki et al. 1989; Kudritzki & Puls 2000; Eldridge et al.
2006) and it can also be assumed to characteristic values for the concerned stars (Comerón & Kaper 1998; van Marle, Decin & Meliani
2014; van Marle, Meliani & Marcowith 2015; Acreman et al. 2016).
In our study, the wind velocities are in the lower limit of the range
of validity for the main-sequence massive stars that we consider;
nonetheless, they still remain within the order of magnitude of, e.g.
late O stars (Martins et al. 2007) or weak-winded stars (Comerón &
Kaper 1998). Furthermore, the evolution of massive stars is governed by physical mechanisms strongly influencing their feedback
such as the presence of low-mass companions (Sana et al. 2012),
which are neglected in our stellar evolution models. Produced before
their zero-age main-sequence phase, e.g. by fragmentation of the
accretion disc that surrounds massive protostars (Meyer et al. 2017),
those dwarf stars entirely modify the evolution of massive stars and
consequently affect their wind properties (de Mink, Pols & Hilditch
2007; de Mink et al. 2009; Paxton et al. 2011; Marchant et al. 2016).
Using wind velocities faster by a factor of α would enlarge the
√
bow shocks by a factor of α and, eventually, in the HD case, favour
Figure 2. Comparison of the quantity Ṁvw between our weak-winded stars
and the non-rotating Galactic models of Brott et al. (2011). The grey zone
of the plot corresponds to the mass regime of massive stars (M ≥ 8 M ).
Solid and dotted lines are lines of constant Ṁvw and M , respectively.
the growth of instabilities (cf. Fig. 1b). However, the results of our
numerical study would be similar in the sense that the presence of
the field essentially stabilizes the nebulae and inhibits the effects
of HC (cf. Figs 1a and h), reduces their size (Section 3.2.1) and
modifies, e.g. their infrared emission accordingly (see Section 4.3).
In Fig. 2, we compare our values of Ṁvw (Table 1) with the nonrotating stellar evolutionary models published in Brott et al. (2011).
We conclude that the bow shocks generated with our initially 10,
20 and 40 M weak-winded stellar models correspond to nebulae
produced by initially ≈10, ≈18 and ≈32 M standard massive
stars at Galactic metallicity, respectively. Therefore, our models
have full validity for this study of magnetized bow shock nebulae,
albeit of lower zero-age main-sequence mass in the case of our
heaviest runaway star.
3.2 Hydrodynamics versus magneto-hydrodynamics
3.2.1 The effects of the magnetic pressure
The ISM magnetic pressure, proportional to B 2ISM , dynamically
compresses the region of shocked ISM gas such that the density in the post-shock region at the FS slightly increases. Similarly, the shape of the bow shock’s wings of shocked ISM is displaced sidewards compared to our model with BISM = 0 μG (Figs 1a
and e). The size of the layer of ISM gas diminishes along the direction of motion of the moving star and the position of the termination
shock sets at a distance from the star where the wind ram pressure
equals the ISM total pressure decreases as measured along the Oz
axis. The effects of the cooling are standard in the sense that it makes
the region of shocked ISM thinner and denser, i.e. the position of
the FS decreases, together with the bow shock volume. The effects
of HC are cancelled (tdyn theat ) in the direction perpendicular to
the field lines, i.e. in the direction perpendicular to the streamline
collinear to both the RS and the contact discontinuity.
3.2.2 Stagnation point morphology and discussion in the context
of plasma physics studies
The topology at the apex of our MHD bow shock (Fig. 1h) is
different from the traditional single-front bow shock morphology
(Fig. 1d). This can be discussed at the light of plasma physics
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Model
3236
D. M.-A. Meyer et al.
studies (de Sterck et al. 1998; de Sterck & Poedts 1999). These
works explore the formation of exotic shocks and discontinuities
that affect the particularly dimpled apex of bow shocks generated
by field-aligned flows around a conducting cylinder (de Sterck et al.
1998). They extended this result to bow shocks produced around
a conducting sphere and showed that the inflow parameter space
leading to such structures is similar to plasma β and Alfvénic Mach
number values allowing the formation of so-called switch-on shocks
(de Sterck & Poedts 1999).
Switch-on shocks are allowed when plasma β of the inflowing
material, i.e. the ratio of the gas and magnetic pressures, which
read
β=
8π nkB T
,
B ISM · B ISM
(24)
where
|B ISM |
vA = √
4π nmH
(25)
is the Alfvénic velocity, satisfy some particular conditions. Note that
in equation (24) the velocities are taken along the shock normal. On
the one hand, the plasma beta must be such that
β<
2
,
γ
(26)
whereas, on the other hand, the Alfvénic Mach number verifies the
following order relation,
γ (1 − β) + 1
,
(27)
1 < MA <
γ −1
where γ is the adiabatic index (see equation 1 in Pogorelov &
Matsuda 2000). Numbers from our simulations indicate that the
ISM thermal pressure nISM kB TISM ≈ 8.62 × 10−13 dyne s−2 , therefore we find β > 2/γ ≈ 1.2 for BISM ≤ 3.5 μG (see bow shocks
with normal morphologies in Figs 3a and b) but β ≈ 0.44 < 2/γ
≈ 1.2 for BISM = 7 μG (see dimpled bow shock in Fig. 3c). The
Alfvénic Mach number MA = v/vA ≈ 40.0 km s−1 /17.2 km s−1 ≈
2.33 which is outside the range 1 < MA < ((γ (1 − β) + 1)/(γ
− 1))1/2 ≈ 1.70. Similarly, the model with v = 70 km s−1 is such
that MA > ((γ (1 − β) + 1)/(γ − 1))1/2 whereas our slower model
with v = 20 km s−1 gives 1 < MA ≈ 1.16 < ((γ (1 − β) + 1)/(γ −
1))1/2 , which is inside the range in equation (27). We conclude that
the upstream ISM conditions in our MHD simulations producing
dimpled bow shocks have values consistent with the existence of
switch-on shocks [see also sketch of the (β, MA ) plane in fig. 3 of
de Sterck & Poedts (1999)].
However, we cannot affirm that the dimpled apex topology of
our MHD bow shock models is of similar origin to the ones in
de Sterck et al. (1998) and de Sterck & Poedts (1999). Only their
particular concave-inward form that differs from the classical shape
of HD bow shocks (Fig. 1e) authorizes a comparison between the
two studies. Nevertheless, we notice that our bow shocks are also
generated in an ambient medium in which the plasma beta and the
Alfvénic Mach number have parameter values consistent with the
formation of switch-on shocks, which has been showed to be similar to the parameter values producing dimpled bow shocks around
charged obstacles (see de Sterck et al. 1998; de Sterck & Poedts
1999, and references therein). Additional investigations, left for
MNRAS 464, 3229–3248 (2017)
Figure 3. Models of stellar wind bow shocks of our initially 20 M star
moving with velocity v = 40 km s−1 represented as a function of its ISM
magnetic field strength, with BISM = 0 (a), 3.5 (b) and 7.0 µG (c).
future studies, are required to assess the question of the exact nature
the various discontinuities affecting MHD bow shocks of OB stars.
3.2.3 Effects of the magnetic field strength
Fig. 3 is similar to Fig. 1 and displays the effects of the ISM magnetic field strength BISM = 0 (a), 3.5 (b) and 7.0 μG (c) on the shape
of the bow shocks produced by our initially 20 M star moving with
velocity v = 40 km s−1 . In Fig. 4 we show density (solid lines) and
temperature (dotted lines) profiles from our HD simulation (thick
blue lines) and MHD model (thin red lines) of the bow shocks in
Fig. 3. The profiles are taken along the symmetry axis of the computational domain. The global structure of the bow shock is similar
for both simulations, i.e. it consists of a hot bubble (T ≈ 107 K)
surrounded by a shell of dense (n ≈ 10 cm−3 ) shocked ISM gas.
The profiles in Fig. 4 highlight the progressive compression of the
bow shocks by the ISM total pressure which magnetic component
increases as BISM is larger. Several mechanisms at work might be
responsible for such discrepancy:
(i) The magnetic pressure in the ISM. If one neglects the thermal
pressures nkB T in both the supersonic stellar wind and the inflowing ISM, and omits the magnetic pressure ∝ B2 ∝ r −4 at the wind
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and its Alfvénic Mach number
v
,
MA =
vA
(23)
MHD bow shocks of hot massive stars
termination shock, then the pressure balance between ISM and stellar wind gas reads
2
BISM
,
(28)
8π
from which one can derive the bow shock stand-off distance in a
planar-aligned field bow shock,
⎛
⎞1/2
2
Ṁv
w
⎠ ,
R(0) = ⎝ 2
(29)
BISM + 8πρISM v2
ρw vw2 = ρISM v2 +
which is slightly smaller from the one derived in a purely HD context
(Wilkin 1996).
(ii) The cooling by optically thin radiative processes. Changes in
the density at the post-shock region at the FS influence the temperature in the shocked ISM gas, which in their turn modify the cooling
rate of the gas, itself affecting its thermal pressure. This results in
not only an increase of the density of the shell of ISM gas but also
a decrease of the temperature in the hot region of shocked stellar
wind material that shrinks in order to maintain its total pressure
2
/8π .
equal to ρISM v2 + BISM
(iii) The magnetic field lines inside the bow shock. The compression of the layer of shocked ISM gas modifies the arrangement of the
2
/8π
field lines in the post-shock region at the FS. Thus, the term BISM
corresponding to the magnetic pressure increases and modifies the
effects of radiative cooling in the simulations (see above).
(iv) Symmetry effects. The solution may also be affected by the
intrinsic two-dimensional nature of our simulations, which may
develop numerical artifices close to the symmetry axis. In the case
of MHD simulations of objects moving supersonically along the
direction of the ISM magnetic field, such effects are more complex
than a simple accumulation of material at the apex of the nebula,
but might present artificial shocks (see also Section 3.4).
Appreciating in detail which of the above cited processes dominates the solution would require three-dimensional numerical simulations which are beyond the scope of this work. Moreover, establishing an analytic theory of the position of the contact discontinuity
of a magnetized bow shock is a non-trivial task since the thin-shell
limit (Wilkin 1996) is not applicable. In particular, the hot bubble
loses about three quarter of its size along the Oz direction when the
ISM magnetic field strength increases up to BISM = 7 μG (Figs 3a
and c). This modifies the volume of hot shocked ISM gas advected
thanks to heat transfers towards the inner part of the bow shock
of our model HD2040All, reducing it to a narrow layer made of
shocked wind material since anisotropic thermal conduction forbids the penetration of ISM gas in the hot region. The effects of
the ISM magnetization on our optical and infrared bow shocks’
emission properties are further discussed in Section 4.
All of our MHD simulations have a stable density field (Figs
1e–h). The simulations with cooling but without heat transfer
(Fig. 1b) show that the presence of the magnetic field inhibits the
growth of Kelvin–Helmholtz instabilities (Fig. 1f) that typically
develops within the contact discontinuity of the bow shocks because
they are the interface of two plasma moving in opposite directions
(Comerón & Kaper 1998; van Marle, Langer & Garcı́a-Segura 2007,
Paper I). The solution does not change performing the simulation
MHD2040AllB7 at double and quadruple spatial resolution, and
conclude that our results are consistent with both numerical studies
devoted to the growth and saturation of these instabilities in the presence of a planar magnetic field (see e.g. Keppens et al. 1999) and
with results obtained for slow-winded, cool runaway stars moving
in a planar-aligned magnetic field (van Marle et al. 2014). Note that
detailed numerical studies demonstrating the suppression of shear
instabilities by the presence of a background magnetic field also
exist in the context of jets from protostars (Viallet & Baty 2007).
3.3 Effects of the star’s bulk motion
Fig. 5 is similar to Fig. 3 and plots a grid of density field of our
initially 20 M star moving with velocity v = 20 (a), 40 (b) and
70 km s−1 (c). The scaling effect of the bulk motion of the star on
the bow shocks morphology is similar to our HD study (Paper I).
At a given strength of the ISM magnetic field, the compression of
the FS increases as the spatial motion of the star increases because
the ambient medium ram pressure is larger. The relative thickness
of the layers of ISM and wind behaves similarly as described in
Paper I. Our model with v = 20 km s−1 has a layer of shocked
ISM larger than the layer of shocked wind because the relatively
small ISM ram pressure induces a weak FS (Fig. 5a). The shell
of shocked ISM is thinner in our simulation with v = 70 km s−1
because the strong FS has a high post-shock temperature which
allows an efficient cooling of the plasma (Fig. 5c).
The density field in our models with ISM inflow velocity
similar to the Alfvénic speed (v = 20 vA ≈ 17.2 km s−1 ) has
the dimpled shape of its apex of the bow shock (Fig. 5a). The
model with v = 70 km s−1 has inflow ISM velocity larger than
the Alfvénic speed and presents the classical single-front morphology (Fig. 5c) typically produced by stellar wind bow shocks
(Brighenti & D’Ercole 1995a,b; Comerón & Kaper 1998; Meyer
et al. 2016). A similar effect of the Alfvénic speed is discussed in,
e.g. fig. 4 of de Sterck & Poedts (1999). Again, exploring in detail
whether the formation mechanisms of our dimpled bow shocks are
identical to the ones obtained in calculations of bow shock flow
over a conducting sphere is far beyond the scope of this work. Note
the absence of instabilities in our MHD bow shock simulations
compared to our HD models.
3.4 Model limitation
First and above, our models suffer from their two-dimensional nature. If carrying out axisymmetric models is advantageous in order
to decrease the amount of computational resources necessary to
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Figure 4. Number density (solid lines, in cm−3 ) and temperature (dotted
lines, in K) profiles in our HD (thick blue lines) and MHD (thin red lines)
bow shock models of an initial 20 M star moving with velocity v =
40 km s−1 . The profiles are measured along the symmetry axis Oz.
3237
3238
D. M.-A. Meyer et al.
Figure 6. Temperature field (in K) in the models MS2040 and
MHD2040AllB7. The cross-like structure in the central region of freely
expanding stellar wind is a boundary effect caused by the pressure.
Figure 5. Grid of stellar wind bow shocks from our initially 20 M star
represented as a function of its space velocity with respect to the ISM, with
velocity v = 20 (a), 40 (b) and 70 km s−1 (c). The nomenclature of the
models follows Table 2. The bow shocks are shown at about 5 Myr after the
beginning of the main-sequence phase of the central star’s evolution. All our
MHD models assume a strength of the ISM magnetic field BISM = 7 µG.
The gas number density is shown with a density range from 10−5 to 5.0 cm−3
in the logarithmic scale. The crosses mark the position of the star. The
solid black contour traces the boundary between wind and ISM material
Q1 (r) = 1/2. The R-axis represents the radial direction and the z-axis the
direction of stellar motion (in pc). Only part of the computational domain is
shown in the figures.
perform the simulations, however, it forbids the bow shocks from
generating a structure which apex would be totally unaffected by
symmetry-axis-related phenomena, common in this case of calculations (Meyer et al. 2016). This prevents our simulations from being
able to assess, e.g. the question of the relation between the seeds
of the non-linear thin-shell instability at the tip of the structure and
the growth of Kelvin–Helmholtz instabilities occurring later in the
wings of the bow shocks. Only full 3D models of the same bow
shocks could fix such problems and allow us to further discuss in
detail the instability of bow shocks from OB stars. We refer the
reader to van Marle et al. (2015) for a discussion of the dimension
dependence of numerical solutions concerning the interaction of
magnetic fields with HD instabilities.
In particular, the selection of admissible shocks, which is generally treated using artificial viscosity in purely HD simulations, is
more complex in our MHD context (see discussion in Pogorelov &
MNRAS 464, 3229–3248 (2017)
4 C O M PA R I S O N W I T H O B S E RVAT I O N S A N D
I M P L I C AT I O N S O F O U R R E S U LT S
In this section, we extract observables from our simulations, compare them to observations and discuss their astrophysical implications. We first recall the used post-processing methods and then
compare the emission by optically thin radiation of our MHD bow
shocks with HD models of the same star moving at the same velocity. Given the high temperature generated by collisional heating
(Fig. 6), we particularly focus on the Hα and [O III] λ 5007 optical
emission. Moreover, stellar wind bow shocks from massive stars
have been first detected at these spectral lines and hence constitute
a natural observable. We complete our analysis with infrared radiative transfer calculations and comment on the observability of our
bow shock nebulae. Finally, we discuss our findings in the context
of the runaway massive star ζ Ophiuchi.
4.1 Post-processing methods
Fig. 7 plots the projected optical emission of our model of an initially 20 M star moving at 40 km s−1 in Hα (a) and [O III] λ 5007
(b) in erg s−1 cm−2 arcsec−2 . The left-hand part of the panels corresponds to the star moving into an ISM with no background magnetic
field (HD model MS2040, Paper I) whereas right-hand parts correspond to BISM = 7 μG (MHD model MHD2040AllB7). We take
into account the rotational symmetry about R = 0 of our models
and integrate the emission rate assuming that our bow shocks lay
in the plane of the sky, i.e. the star moves perpendicular to the
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Matsuda 2000). This can lead to additional fragilities of the solution, especially close to the symmetry axis of our cylindrically
symmetric models. Although the stability of these kinds of shocks
is still under debate (de Sterck & Poedts 2000, 2001), we will
try to address these issues in future three-dimensional simulations.
Moreover, such models would (i) allow us to explore the effects
of a non-aligned ISM magnetic field on the morphology of the
bow shocks and (ii) will make subsequent radiative transfer calculations meaningful, e.g. considering polarization maps using full
anisotropic scattering of the photons on the dust particles in the bow
shocks. The space of parameters investigated in our study is also
limited, especially in terms of the explored range of space velocity v and ISM density nISM and will be extended in a follow-up
project. Finally, other physical processes such as the presence of a
surrounding H II region or the intrinsic viscous, granulous and turbulent character of the ISM are also neglected and deserve additional
investigations.
MHD bow shocks of hot massive stars
3239
et al. (2009). Moreover, the total infrared emission LIR is estimated
as a fraction of the starlight bolometric flux L (Brott et al. 2011)
intercepted by the ISM silicate dust grains in the bow shock,
dust =
L
nd σd (1 − A) erg s−1 cm−3 ,
4π d 2
(33)
plus the collisional heating,
dust
(T ) =
coll,photo
25/2 f Qnnd σd
(kB T )3/2 erg s−1 cm−3 ,
√
π mp
(34)
where a = 5.0 nm is the dust grains radius,
σd = π a 2 cm2
observers’ line-of-sight. The spectral line emission coefficients are
evaluated using the prescriptions for optical spectral line emission
from Dopita (1973) and Osterbrock & Bochkarev (1989), which
read
j[Hα] (T ) ≈ 1.21 × 10−22 T −0.9 n2p erg s−1 cm−3 sr−1 ,
(30)
where np is the number of proton in the plasma, and,
j[O III] (T ) ≈ 3.23 × 10−21
e−
28737
T
√ n2p erg s−1 cm−3 sr−1 ,
4π T
(31)
for the Hα and [O III] λ 5007 spectral lines, respectively. Additionally, we assume solar oxygen abundances (Lodders 2003) and cease
to consider the oxygen as triply ionized at temperatures larger than
106 K (cf. Cox, Gull & Green 1991).
The bow shock luminosities L are estimated integrating the emission rate,
“
L = 2π
D
(T )n2H RdRdz,
(32)
where D represents its volume in the z > 0 part of the computational
domain (Mohamed et al. 2012, Paper I). Similarly, we calculate the
momentum deposited by the bow shock by subtracting the stellar
motion from the ISM gas velocity field. We compute LHα and L[O III] ,
the bow shock luminosity in [O III] λ 5007 and Hα, respectively.
Furthermore, we discriminate the total bow shock luminosity Ltotal
from the shocked wind emission Lwind . For distinguishing the two
kinds of material, we make use of a passive scalar Q that is advected
with the gas. We estimate the overall X-ray luminosity LX with
emission coefficients generated with the XSPEC program (Arnaud
1996) with solar metallicity and chemical abundances from Asplund
is their geometrical cross-section, d their distance from the star
and A = 1/2 their albedo. Additionally, nd is the dust number
density whereas Q 1 represents the grains electrical properties.
More details regarding to the estimate of the bow shock infrared
luminosity are given in appendix B of Paper I.
Last, infrared images are computed performing dust continuum
calculations against dust opacity for the bow shock generated by
our 20 M star moving with velocity 40 km s−1 , using the radiative
transfer code RADMC-3D2 (Dullemond 2012). We map the dust mass
density fields in our models on to a uniform spherical grid [0; Rsph ]
2
2
+ zmax
)1/2 and θmax = 180◦ . We
× [0; θ max ], where Rsph = (Rmax
assume a dust-to-gas mass ratio of 1/200. The dust density field is
computed with the help of the passive scalar tracer Q that allows
us to separate the dust-free stellar wind of our hot OB stars with
respect to the dust-enriched regions of the bow shock, made of
shocked ISM gas. Additionally, we exclude the regions of ISM
material that are strongly heated by the shocks or by electronic
thermal conduction (Paper I), and which are defined as much hotter
than about a few 104 K. RADMC-3D then self-consistently determines
the dust temperature using the Monte Carlo method of Bjorkman &
Wood (2001) and Lucy (1999) that we use as input to the calculations
of our synthetic observations.
The code solves the transfer equation by ray-tracing photons
packages from the stellar atmosphere that we model as a blackbody
point source of temperature Teff (see our Table 1) that is located at
the origin of the spherical grid. The dust is assumed to be composed
of silicates (Draine & Lee 1984) of mass density 3.3 g cm−3 that follow the canonical power-law distribution n(a) ∝ a−q with q = −3.3
(Mathis, Rumpl & Nordsieck 1977) and where amin = 0.005 μm and
amax = 0.25 μm, the minimal and maximal dust sizes (van Marle
et al. 2011). We generate the corresponding RADMC-3D input files
containing the dust scattering κ scat and absorption κ abs opacities
such that the total opacity κ tot = κ scat + κ abs (see Fig. 9a) on the
basis of a run of the Mie code of Bohren and Huffman (Bohren &
Huffman 1983), which is available as a module of the HYPERION3
package (Robitaille 2011). Our radiative transfer calculations produce spectral energy distributions (SEDs) and isophotal images of
the bow shocks at a desired wavelength, which we choose to be λ =
24 and 60 μm because they correspond to the wavelengths at which
stellar wind bow shocks are typically observed [see Sexton et al.
(2015) and van Buren & McCray (1988a), van Buren et al. (1995),
Noriega-Crespo et al. (1997a), respectively]. Our SEDs and images
are calibrated as such that we consider that the objects are located
at a distance 1 pc from the observer.
2
3
http://www.ita.uni-heidelberg.de/dullemond/software/radmc-3d/
http://www.hyperion-rt.org/
MNRAS 464, 3229–3248 (2017)
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Figure 7. Surface brightness maps of Hα (a), [O III] (b) surface brightness
(in erg s−1 cm−2 arcsec−2 ), respectively, of our bow shock model generated
by our initially 20 M star moving with velocity v = 40 km s−1 . Quantities
are calculated excluding the undisturbed ISM and plotted in the linear scale.
The left-hand part of the panels refers to the HD model MS2040, and the
right-hand part to the MHD model MHD2040AllB7. The crosses mark the
position of the star. For the sake of comparison, these optical maps are
presented as in Paper I.
(35)
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D. M.-A. Meyer et al.
Table 4. Maximum optical surface brightness of our MHD simulations with
max and
BISM = 7 µG. The second and third columns are the quantities [Hα]
max representing the maximum projected emission in [O III] λ 5007 and
[O
III]
Hα (in erg cm−2 s−1 arcsec−2 ), respectively. Models consisting of the HD
counterpart of our bow shock models have their labels in italic in the first
column (see description in table 1 in Paper I). The surface brightnesses are
measured along the direction of motion of the star at the apex of our bow
shocks, close to the symmetry axis Oz.
Model
2.5 × 10−19
1.0 × 10−18
1.7 × 10−17
6.0 × 10−17
2.9 × 10−17
1.0 × 10−16
1.2 × 10−16
8.0 × 10−18
1.5 × 10−16
1.2 × 10−17
4.0 × 10−16
7.0 × 10−18
2.5 × 10−17
6.8 × 10−17
7.2 × 10−17
1.6 × 10−16
3.2 × 10−16
2.5 × 10−16
2.0 × 10−16
8.5 × 10−16
5.5 × 10−16
1.0 × 10−15
28.0
25.0
4.0
1.2
5.5
3.2
2.1
25.0
5.7
45.8
2.5
4.2 Results: optically thin emission
In Table 4 we report the maximum surface brightness measured
along the direction of motion of the stars in the synthetic emission
maps build from our models at both the Hα and [O III] λ 5007 spectral line emission. We find that the presence of an ISM magnetic
field makes the Hα signatures fainter by about 1–2 orders of magnitudes whereas the [O III] λ 5007 emission maps are about 1 order of
magnitude fainter, respectively. The luminosity of stellar wind bow
shocks is a volume integral (Paper I) and this volume decreases when
a large ISM magnetic pressure compresses the nebula (Figs 1d and
h). Thus, their surface brightness is fainter despite the fact that the
density and temperature of their shocked regions are similar (Fig. 4).
The ratio of our bow shock models’ maximum [O III] and Hα
maximum surface brightness increases in the presence of the magmax
max
/[Hα]
≈ 2.1
netic field, e.g. the HD model MS2040 has [O
III]
max
max
/
≈
5.5
if
whereas our model MHD2040AllB7 has [O
[Hα]
III]
max
max
/
BISM = 7 μG. We notice that the spectral line ratio [O
[Hα]
III]
augments with the increasing space velocity of the star, e.g. our
models MHD2020B7, MHD2040AllB7 and MHD2070B7 have
max
max
/[Hα]
≈ 4.0, 5.5 and 25.0, respectively. This difference be[O
III]
tween [O III] λ 5007 and Hα emission is more pronounced in our
MHD simulations. As for our HD study, the region of maximum
emission peaks close to the contact discontinuity in the layer
of shocked ISM material, in the region of the stagnation shock
(Paper I, see also Figs 7a and b).
The ISM magnetic field does not change the order relations we
previously established with HD bow shocks generated by mainsequence stars (fig. 13 a in Paper I), i.e. Lwind < LHα < Ltotal < LIR
(see orange dots, blue crosses of Saint-Andrew, dark green triangles and black squares in Fig. 8a, respectively). Additionally, as
discussed above in the context of projected emission maps, we find
that the optical spectral line emission that we consider is such that
L[O III] > LHα . This confirms and extends to MHD bow shocks, a
result previously obtained by integrating the optically thin emission
in the range 8000 ≤ T ≤ 106 K (Paper I). Our MHD bow shock
models have Hα and [O III] emission originating from the shocked
ISM gas and their emission from the wind material is negligible
(Ltotal /Lwind ≈ 10−6 ). Moreover, we find that the bow shock X-rays
MNRAS 464, 3229–3248 (2017)
Figure 8. Bow shock luminosities and feedback of our MHD models. We
separate the infrared reprocessed starlight (red squares, in erg s−1 ) and distinguish the total emission by optically thin radiation from the bow shock
(dark-green triangles, in erg s−1 ) from the emission from the shocked wind
material only (orange dots, in erg s−1 ). Additionally, we show the luminosity from [O III] λ 5007 emission (green losanges, in erg s−1 ), the luminosity
from Hα emission (blue crosses, in erg s−1 ) and the X-rays luminosity in
both the soft and hard energy bands E > 0.5 eV (T > 5.8 × 106 K). For the
sake of comparison, we add the feedback of the HD model HD2040 corresponding to BISM = 0 µG (originally published in Paper I). The simulation
labels are indicated under the corresponding values.
emission is very small in all our simulations (LX /Lwind ≈ 10−1 , see
black crosses in Fig. 8a).
4.3 Results: dust continuum infrared emission
4.3.1 Spectral energy distribution
Fig. 9(b) plots a comparison between the SEDs of two bow
shock models generated by our 20 M star moving with velocity 40 km s−1 , either through an unmagnetized ISM (model
MS2040, solid blue line) or in a medium with BISM = 7 μG (model
MHD2040AllB7, dotted red line) for a viewing angle of the nebulae of φ = 0◦ . The figure represents the flux density Fλ (in Jy)
as a function of the wavelength λ (in μm) for the waveband including the 0.01 ≤ λ ≤ 2000 μm. The star is responsible for the
component in the range 0.01 ≤ λ ≤ 10 μm that corresponds to a
blackbody spectrum of temperature Teff = 33 900 K (see Table 1)
while the circumstellar dust produces the feature in the waveband
10 ≤ λ ≤ 2000 μm. The bow shock’s component is in the waveband including the wavelengths at which stellar wind bow shock
are typically recorded, e.g. at 60 μm (van Buren & McCray 1988a;
van Buren et al. 1995; Noriega-Crespo et al. 1997a).
The SED of the magnetized bow shock has a slightly larger flux
than the SED of the HD bow shock in the waveband 10 ≤ λ ≤
30 μm, because its smaller size makes the shell of dense ISM gas
closer to the star, increasing therefore the dust temperature (Fig. 9b).
At λ ≈ 30 μm, the HD bow shock emits by slightly more than half an
order of magnitude than the magnetized nebula, e.g. at λ ≈ 60 μm
our model MS2040 has a density flux Fλ ≈ 3 × 10−1 Jy whereas our
model MHD2040AllB7 shows Fλ ≈ 1 × 10−1 Jy, respectively. This
is consistent with the previously discussed reduction of the projected
optical emission of our bow shocks. This relates to the changes in
size of the nebulae induced by the inclusion of the magnetic field
in our simulations, which reduces the mass of dust in the structure
responsible for the reprocessing of the starlight, e.g. our models
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MHD1040AllB7
MS1040
MHD2020AllB7
MS2020
MHD2040AllB7
MHD2040AllB3.5
MS2040
MHD2070AllB7
MS2070
MHD4070AllB7
MS4070
max
max
max /
[Hα]
[O
[Hα]
III]
max
−2
−1
−2
−2
−1
−2
(erg cm s arcsec ) (erg cm s arcsec ) [Hα]
MHD bow shocks of hot massive stars
MS2040 and MHD2040AllB7 contain about Md ≈ 3 × 10−2 M
and Md ≈ 2 × 10−3 M , respectively, where Md is the dust mass
trapped into the nebulae. The reduced mass of dust into the magnetized bow shock absorbs a lesser amount of the stellar radiation
and therefore re-emits a smaller quantity of energy, reducing Fλ in
the waveband λ ≥ 30 μm (Fig. 9b). Note that the infrared surface
brightness of a bow shock is also sensible to the density of its ambient medium, i.e. Fλ is much larger in the situation of a runaway
star moving in a medium with nISM 1000 cm−3 (Acreman et al.
2016).
4.3.2 Synthetic infrared emission maps
Our Fig. 10 plots a series of synthetic infrared emission maps of
our bow shock models produced by an initially 20 M star moving with velocity 40 km s−1 in its purely HD (MS2040) or MHD
configuration (MHD2040AllB7) at the wavelengths corresponding
to the central wavelengths of the IRAS facility’s main broad-band
images (van Buren & McCray 1988b), i.e. λ = 25 μm (left column
of panels), 60 μm (middle column of panels) and 100 μm (right
column of panels). The maps are represented with an inclination
angle of φ = 30◦ (Figs 10a, e and i), 45◦ (Figs 10b, f and j), 60◦
(Figs 10c, g and k) and 90◦ (Figs 10d, h and l) with respect to
the plane of the sky and the projected flux is plotted in units of
erg s−1 cm−2 arcsec−2 . As in the context of their optical emission
(Fig. 7), the overall size of the infrared magnetized bow shocks is
smaller than in the HD case because of the reduction of their standoff distance R(0) (see e.g. Figs 10a, e and j). The global morphology
of our infrared bow shock nebulae does not change significantly. It
remains a single, bright arc at the front of an ovoid structure that
is symmetric with respect to the direction of motion of the runaway star and extended to the trail (z ≤ 0) of the bow shocks due
to the supersonic motion of the star (Acreman et al. 2016). In the
HD case, the region of maximum emission is the region containing
the ISM dust, the temperature of which is less than a few 104 K,
i.e. between the contact discontinuity and the FS of the bow shock
(Paper I; Acreman et al. 2016) whereas in the magnetized case, the
maximum emission is reduced to a thin region close to the discontinuity between hot stellar wind and colder ISM. Both the shocked
stellar wind and the shocked ISM of the bow shock do not contribute
to these emission because the material is too hot.
Fig. 11 reports cross-sections taken along the direction of
motion of the bow shock and comparing their surface brightnesses at several wavebands λ and viewing angles φ. It illustrates that, as in the case of the optical emission, the presence
of the ISM magnetic field makes the bow shocks slightly dimmer, e.g. for φ = 45◦ our model has a maximal surface brightness
max
max
−17
erg s−1 cm−2 arcsec−2 whereas 100
of 100
µ m ≈ 4.3 × 10
µm ≈
−17
−1
−2
−2
2.6 × 10 erg s cm arcsec for BISM = 0 and 7 μG, respectively. Fig. 12 shows different cross-sections of the projected infrared emission of the magnetized bow shock of our initially
20 M star moving with velocity v = 40 km s−1 . The emission at
λ = 60 μm is more important that at λ = 25 μm and at λ = 100 μm,
max
−17
erg s−1 cm−2 arcsec−2 whereas
e.g. it peaks at 60
µ m ≈ 8.2 × 10
max
max
−17
−1
−2
−2
≈
2.6
×
10
erg
s
cm
arcsec
and 100
25
µ m ≈ 3.0 ×
µm
10−17 erg s−1 cm−2 arcsec−2 , respectively, at a distance of 0.55 pc
from the star and assuming an inclination angle of the bow shock of
φ = 45◦ (Fig. 12a). All our models have similar behaviour of their
infrared surface brightness as a function of λ and φ. Note also that
the evolution of the position of the stand-off distance of the bow
shock is consistent with the study of Acreman et al. (2016) in the
sense that it increases at larger φ (Fig. 12b).
4.4 Implications of our results and discussion
4.4.1 Bow shocks Hα and [OIII] observability
The surface brightnesses at Hα and [O III] λ 5007 spectral line emission of our stellar wind bow shocks reported in Section 4.4.4 indicate that (i) the presence of the ISM magnetic field makes their
projected emission Hα and [O III] fainter by 2 and 1–2 orders
of magnitude and (ii) that this reduction of the nebulae’s emission is more important as the strength of the B field is larger.
Consequently, the emission signature of a purely HD bow shock
model that is above the diffuse emission sensitivity threshold of,
e.g. the SuperCOSMOS H-Alpha Survey (SHS) of SHS ≈ 1.12.8 × 10−17 erg s−1 cm−2 arcsec−2 can drop down below it once the
ISM magnetic field is switched-on. As an example, our HD model
of a 20 M star moving with velocity v = 70 km s−1 (Paper I)
could be observed since it has Hα ≈ 1.5 × 10−16 ≥ SHS whereas
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Figure 9. Top panel: dust opacities used in this study, inspired from Acreman et al. (2016). The figure shows the total opacity κ tot (blue solid thick
line), the absorption opacity κ abs (red dotted thin line) and the scattering
opacity κ scat (green dashed thin line). Bottom panel: SEDs of our model
involving a 20 M star moving with a velocity of 40 km s−1 , considered in
the HD (model MS2040 with BISM = 0 µG, solid blue line) and in the MHD
contexts (model MHD2040AllB7 with BISM = 7 µG, dotted red line). The
plot shows the flux density Fλ (in Jy) as a function of the wavelength λ (in
µm) for an inclination angle φ = 0◦ of the bow shock.
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D. M.-A. Meyer et al.
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Figure 10. Isophotal infrared emission maps of our bow shock models MS2040 and MHD2040AllB7. It represents our initially 20 M star moving with
velocity 40 km s−1 as seen at wavelengths λ = 25 (a–d), 60 µm (e–h) and 100 µm (i–l). The projected flux is in units of erg s−1 cm−2 arcsec−2 . The maps
are generated with an inclination angle of φ = 30 (a,e,i), 45 (b,f,j), 60 (c,g,k) and 90◦ (d,h,l) with respect to the plane of the sky. For each panel, the surface
brightness is plotted in the linear scale and its maximum corresponds to the maximum of the HD (left) and MHD bow shock models (right).
MNRAS 464, 3229–3248 (2017)
MHD bow shocks of hot massive stars
3243
4.4.2 Surrounding H II region and dust composition
our MHD model of the same runaway star has Hα ≈ 8.0 × 10−18
SHS and would be invisible with regard to the SHS facility (our
Table 4).
This may explain why not so many stellar wind bow shocks are
discovered at Hα around isolated, hot massive stars, despite the fact
the ionization of their circumstellar medium must produce such
emission (Brown & Bomans 2005). Since Hα ∝ n2 (see appendix
A of Paper I), it implies that the more diluted the ISM constituting
the surrounding of an exiled star, i.e. the higher the runaway star’s
Galactic latitude, the smaller the probability to observe its bow
shock at Hα. In other words, the search for bow shocks at this
wavelength may work well within the Galactic plane or in relatively
dense regions of the ISM. Note also that in the presence of the
magnetic field, all models have [OIII] /Hα > 1, which is consistent
with the discovery of the first bow-shock-producing massive stars
ζ Ophiuchi in [O III] λ 5007 emission.
Figure 12. Cross-sections taken through the direction of motion of the bow shock of our initially 20 M star moving in a medium with BISM = 7 µG with
velocity v = 40 km s−1 . The emission is shown for the principal broad-band images of the IRAS telescope for a viewing angle φ = 45◦ (a) and for the
wavelength λ = 60 µm as a function of different viewing angle φ (b). The surface brightness (in erg s−1 cm−2 arcsec−2 ) is plotted as a function of the distance
to the star (in pc). The position of the star is located at the origin.
MNRAS 464, 3229–3248 (2017)
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Figure 11. Cross-sections taken through the direction of motion of the
bow shock of our initially 20 M star moving in a medium with velocity
v = 40 km s−1 , both in a medium with BISM = 0 and 7 µG. The emission
are shown for a viewing angle of φ = 45◦ and at the waveband λ = 60 µm
(dotted red curves) and for φ = 60◦ at λ = 100 µm (solid blue curves). The
surface brightness (in erg s−1 cm−2 arcsec−2 ) is plotted as a function of the
distance to the star (in pc). The position of the star is located at the origin.
Massive stars release huge amount of ultraviolet photons (DiazMiller, Franco & Shore 1998) that ionize the hydrogen constituting
their surroundings (Dyson 1975), giving birth to an H II region
overwhelming the stellar wind bubble of the star (Weaver et al.
1977; van Marle 2006). In the case of a runaway star, the stellar
motion produces a bow shock surrounded by a cometary H II region
(Raga 1986; Mac Low et al. 1991; Raga et al. 1997; Arthur & Hoare
2006; Zhu et al. 2015), the presence of which in our study is simply
taken into account assuming that the ambient medium of the star is
fully ionized; however, we neglect its turbulent internal structure.
The gas that is between the FS of the bow shock and the outer part
of the H II region is filled by ISM dust that emits infrared thermal
emission by efficiently reprocessing the stellar radiation, i.e. it is
brighter than the emission by gas cooling (Paper I).
While our study shows that our nebulae are brighter at 60 μm
(Fig. 12), i.e. at the waveband at which catalogues of bow shocks
from OB stars have been compiled (van Buren & McCray 1988a;
van Buren et al. 1995; Noriega-Crespo et al. 1997a), the study
of Mackey et al. (2016) compared the respective brightnesses of
the front of a distorted circumstellar bubble with the outer edge
of its surrounding H II region and find the 24 μm waveband to be
ideal to observe the structure generated by the stellar wind. However, the presence of the ISM background magnetic field makes
our infrared arc smaller and slightly dimmer, i.e. more difficult to
detect in the case of a distant runaway star which could explain
why a large proportion of observed H II regions do not contain
dust-free cavities encircled with bright mid-infrared arcs (Sharpless 1959; Churchwell et al. 2006; Wachter et al. 2010; Simpson et al. 2012). Further radiation magneto-hydrodynamic simulations are required to fully assess the question of the infrared
screening of stellar wind bow shocks by their own H II regions,
particularly for an ambient medium corresponding to the Galactic
plane (nISM 1 cm−3 ).
Following Pavlyuchenkov, Kirsanova & Wiebe (2013), we consider that the dust filling the H II region and penetrating into the
bow shock is similar to that of the ISM. Our radiative transfer
calculations nevertheless suffer from uncertainties regarding to the
composition of this ISM dust. Our mixture is made of silicates
(Draine & Lee 1984), which could be modified, e.g. changing the
3244
D. M.-A. Meyer et al.
slope of the dust size distribution. Particularly, the inclusion of very
small grains such as polycyclic aromatic hydrocarbon (PAHs; see
Wood et al. 2008) may be an appropriate update of the dust mixture, as it has been shown to be necessary to fit observations of
mid-infrared bow shocks around O stars in dense medium in M17
and RCW 49 (Povich et al. 2008). Enlarging our work in a wider
study, e.g. scanning the parameter space of the quantities governing the formation of Galactic stellar wind bow shocks (v , nISM ,
M ) in order to discuss both their SEDs and infrared images, will
be considered in a follow-up paper, e.g. performing a systematic
post-processing of the grid of bow shock simulations of Meyer
et al. (2016) with RADMC-3D. Then, thorough comparison of numerical simulations with, e.g. the IRAS observations of van Buren &
McCray (1988a), van Buren et al. (1995), Noriega-Crespo et al.
(1997a) would be achievable.
(see in particular appendix A of Meyer et al. 2015, and references
therein).
Our study shows that the presence of background ISM magnetic field aligned with the direction of motion of a main-sequence
runaway star inhibits the growth of both shear instabilities that typically affect these circumstellar structures (Fig. 1). Consequently, a
planar-aligned magnetic field would further shape the RS of moving stars’ bow shocks as a smooth tube in which shocks waves
could be channelled as a jet-like extension, e.g. as in Cox et al.
(1991). Additionally, the shock wave colliding with the FS of circumstellar structures of runaway stars that are sufficiently dense to
make their subsequent supernova remnant asymmetric (Meyer et al.
2015) would be more collimated along the direction of motion of
its progenitor and/or ambient magnetic field. This may produce additional asymmetries to the elongated shape of supernova remnants
exploding in a magnetized ISM (Rozyczka & Tenorio-Tagle 1995).
4.4.3 Shaping of the circumstellar medium of runaway massive
stars at the pre-supernova phase
4.4.4 The case of the hot runaway star ζ Ophiuchi
It has been shown in the context of Galactic, high-mass runaway
stars that the pre-shaped circumstellar medium in which these stars
die and explode as a Type II supernova is principally constituted of
its own main-sequence wind bubble, distorted by the stellar motion.
Further evolutionary phase(s) produces additional bubble(s) and/or
shell(s) whose evolution is contained inside the initial bow shock
(Brighenti & D’Ercole 1994, 1995a). The expansion of the subsequent supernova shock wave is strongly impacted by the progenitor’s pre-shaped circumstellar medium inside which it develops initially (see e.g. Cox et al. 1991). Particularly, the more well-defined
and stable the walls of the tunnel formed by the RS of the bow
shock are, the easier the channelling the supernovae ejecta inside it
MNRAS 464, 3229–3248 (2017)
The O9.5 V star ζ Ophiuchi is the Earth’s closest massive, mainsequence runaway star. Infrared observations, e.g. with the WISE
3.4 μm facility (band W1, Wright et al. 2010, see Fig. 134 )
highlighted the complex topology of its stellar wind bow shock,
originally discovered in [O III] λ 5007 spectral line (Gull &
Sofia 1979) and further observed in the infrared waveband (van
Buren & McCray 1988b). The properties of the particular, nonaxisymmetric shape of its circumstellar nebula which moves in
the H II region Sh 2-27 (Sharpless 1959) are studied in a relatively large literature (see Mackey, Langer & Gvaramadze 2014a,
4
http://wise2.ipac.caltech.edu/docs/release/allsky/
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Figure 13. WISE 3.4 µm (band W1, Wright et al. 2010) of the stellar wind bow shock surrounding the massive runaway O9.5 V star ζ Ophiuchi. The image
represents about 35 arcmin in the horizontal direction, which at a distance of 112 pc corresponds to about 1.12 pc.
MHD bow shocks of hot massive stars
4.4.5 The case of runaway cool stars
Our results apply to bow shocks generated by hot, main-sequence
OB stars that move through the hot ionized gas of their own H II region (Raga et al. 1997) and archetype of which is the nebulae
surrounding ζ Ophiuchi (see above discussion). Externally photoionized cool runaway stars that move rapidly in the H II region
produced by an other source of ionizing radiation have particularly
bright optical emission [see e.g. the cases of the red supergiant Betelgeuse (Mohamed et al. 2012; Mackey et al. 2014b) and IRC-10414
(Meyer et al. 2014a)]. These circumstellar structures are themselves
sensitive to the presence of even a weak ISM background magnetic
field of a few μG (van Marle et al. 2014). Consequently, one can expect that the inclusion of such a field in numerical models tailored
to these objects would affect their associated synthetic emission
maps and update the current estimate of their driving star’s mass
loss and/or ambient medium density (Meyer et al. 2016).
According to the fact that the warm phase of the ISM is typically magnetized, the reduction of both optical and infrared surface
brightnesses of circumstellar structures generated by massive stars
should be a rather common phenomenon. In particular, it should also
concern bow shocks of OB runaway stars once they have evolved
through the red supergiant phase (Paper I). However, the proportion of red supergiant stars amongst the population of all runaway
massive stars should be similar to the proportion of red supergiant
with respect to the population of static OB stars, which is, to the
best of our knowledge, contradicted by observations. The recent
study of van Marle et al. (2014) shows that a background ISM magnetic field can inhibit the growth of shear instabilities, i.e. forbids
the development of potentially bright infrared knots, in the bow
shock of Betelgeuse, and this may help in explaining why the scientific literature only reports four known runaway red supergiant
stars, amongst which only three have a detected bow shock, i.e.
Betelgeuse (Noriega-Crespo et al. 1997b), IRC-10414 (Meyer et al.
2014a) and μ Cep (Cox et al. 2012). The extragalactic, hyperveloce
red supergiant star J004330.06+405258.4 in M31 has all kinematic
characteristics to generate a bow shock but it has not been observed
so far (Evans & Massey 2015). This remark is also valid for bow
shocks generated by runaway massive stars experiencing other evolutionary stages such as the so-called blue supergiant phase (see e.g.
Kaper et al. 1997).
4.4.6 Comparison with the bow shock around the Sun
The Sun is moving into the warm phase of the ISM (McComas et al.
2015) and the properties of its ambient surrounding, the so-called
local interstellar medium (LISM) are similar to the ISM in which
our runaway stars move, especially in terms of Alfvénic Mach number and plasma β (Florinski et al. 2004; Burlaga, Florinski & Ness
2015). The study of the interaction between our Sun and the LISM
led to a large literature, including, amongst other, numerical investigations of the bow shock formed by the solar wind (see e.g.
Baranov & Malama 1993; Pogorelov & Matsuda 1998; Zank 2015,
and references therein). If obvious similitudes between the bow
shock of the Sun and those of our massive stars indicate that the
physical processes governing the formation of circumstellar nebulae around OB stars such as electronic thermal conduction or the
influence of the background local magnetic field have to be included
in the modelling of those structures (Zank et al. 2009), nevertheless,
the bow shock of the Sun is, partly due to the differences in terms of
effective temperature and wind velocity, on a totally different scale.
Further resemblances with bow-like nebulae from massive stars are
therefore mostly morphological.
As a low-mass star (< 8 M ), the Sun is much cooler (Teff ≈
6000 K) than the runaway OB stars considered in the present
work (Teff > 20 000 K) and its mass loss (Ṁ ≈ 10−14 M yr−1 ) is
much smaller than that of a main-sequence star with M ≥ 20 M
(our Table 1), which makes its stellar luminosity fainter by several
orders of magnitude (L /L ≥ 103 ). Moreover, the solar wind velocity at 1 au is about 350 km s−1 (Golub & Pasachoff 1997) whereas
our OB stars have larger wind velocities (> 1000 km s−1 ; see
Table 1). Stellar winds from solar-like stars consequently develop a
smaller ram pressure and expel less linear momentum than massive
stars such as our 20 M star and their associated corresponding
circumstellar structures, i.e. wind bubbles or bow shocks are scaled
down to a few tens or hundreds of au. Note also that the Sun is
too cool to produce ionizing radiations and generated an H II region that is susceptible screen its optical/infrared wind bubble. In
other words, if the numerical methods developed to study the bow
shock surrounding the Sun are similar to the ones utilized in our
study, the solar solutions are more appropriated to investigate the
surroundings of cool, low-mass stars such as asymptotic giant stars
(AGB) (see Wareing, Zijlstra & O’Brien 2007b,a; Raga et al. 2008;
Esquivel et al. 2010; Villaver, Manchado & Garcı́a-Segura 2012;
Chiotellis et al. 2016), or the trails let by planetesimals moving
in stellar systems presenting a common envelope (see Thun et al.
2016, and references therein).
Early two-dimensional numerical models of the solar neighbourhood were carried out assuming that the respective directions of
both the Sun’s motion and the LISM magnetic field are considered
as parallel, as we hereby do with our massive stars (Pogorelov &
Matsuda 1998). More sophisticated simulations have produced
three-dimensional models in which the Sun moves obliquely
through the LISM (see e.g. Baranov, Barmin & Pushkar’ 1996;
Boley, Morris & Desch 2013). Such investigation is observationally
motivated by the perturbated and non-uniform appearance of the
heliopause, e.g. the boundary between the interplanetary and ISM
MNRAS 464, 3229–3248 (2017)
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and references therein). The mass loss of ζ Ophiuchi has been
estimated in the range Ṁζ ≈ 1.58 × 10−9 –1.43 × 10−7 M yr−1
(Gvaramadze, Langer & Mackey 2012), which, according to equation (21), taking R(0) ≈ 0.16 pc (Gvaramadze et al. 2012), adopting
v ≈ 26.5 km s−1 and considering a typical OB star wind velocity
of vw ≈ 1500 km s−1 , constrains its ambient medium density to
nζ ≈ 3-4 cm−3 (cf. Gull & Sofia 1979).
Assuming (i) the magnetization of the close surrounding of
ζ Ophiuchi to be BISM = 7 μG (Mackey et al. 2014a), (ii) that
the conditions for switch-on shocks to be permitted are fulfilled,
i.e. plasma and Alfvénic velocities are normal to the shock, and
(iii) considering that its ISM properties are, in addition to the above
presented quantities, such that γ = 1.67, TISM = 8000 K, it comes
that β ≥ 2/γ and MA < 1. This indicates that, under our hypotheses, the ambient medium of ζ Ophiuchi does not allow the existence of switch-on shocks. Consequently, the imperfect shape of its
bow shock (Fig. 13) may not be explained invoking the particular
double-front topology of bow shocks that can be produced in such
regime, but rather by the presence of a background ISM magnetic
field whose direction is not aligned with respect to the motion of
the star. Further tri-dimensional MHD models are needed in order
to assess the question of ζ Ophiuchi’s background ISM magnetic
field direction, the position of its contact discontinuity and a more
precise estimate of its stellar wind mass loss.
3245
3246
D. M.-A. Meyer et al.
5 CONCLUSION
In this study, we presented MHD models of the circumstellar
medium of runaway, main-sequence, massive stars moving supersonically through the plane of the Galaxy. Our two-dimensional
simulations first investigated the conjugated effects of optically
thin radiative cooling and heating together with anisotropic thermal transfers on a field-aligned, MHD bow shock flow around an
OB-type, fast-moving star. We then explored the effects of the stellar motion with respect to the bow shocks, focusing on an initially
20 M star moving with velocities v = 20, 40 and 70 km s−1 . We
presented additional models of an initially 10 M star moving with
velocities v = 40 km s−1 and of an initially 40 M star moving
with velocities v = 70 km s−1 . The ISM magnetic field strength is
set to BISM = 7 μG. We also considered bow shock nebulae produced within a weaker ISM magnetic field (BISM = 3.5 μG). The
other ISM properties are unchanged for each models.
Our models show that although the magnetization of the ISM
does not radically change the global aspect of our bow shock nebulae, it slightly modifies their internal organization. Anisotropic
thermal transfers do not split the region of shocked ISM gas as
5
http://voyager.jpl.nasa.gov/mission/interstellar.html
MNRAS 464, 3229–3248 (2017)
in our HD models (Paper I), since the presence of the magnetic
field in the regions of shocked material forbids HC perpendicular
to the magnetic field lines. The field lines, initially parallel to the
direction of stellar motion, are bent round by the bow shock into
a sheath around the fast stellar wind bubble. As shown in Heitsch
et al. (2007), the presence of the magnetic field stabilizes the contact discontinuities inhibiting the growth of the Kelvin–Helmholtz
instabilities that typically occur in pure HD models at the interface
between shocked ISM and shocked stellar wind.
As in our previous HD study (Paper I), bow shocks are brighter
in infrared reprocessed starlight. Their emission by optically thin
radiation mostly originates from the shocked ISM and their [O III]
λ 5007 spectral line emission is higher than their Hα emission. Notably, their X-rays emission is negligible compared to their optical
luminosity and therefore it does not constitute the best waveband
to search for hot massive stars’ stellar wind bow shocks. We find
that the presence of an ISM background magnetic field has the effect of reducing the optical synthetic emission maps of our models,
making them fainter by one and two orders of magnitude at [O III]
λ 5007 and Hα, respectively. This may explain why not so many of
them are observed at these spectral lines. We confirm that, under
our assumptions and even in the presence of a magnetic field, circumstellar structures produced by high-mass, slowly moving stars
are the easiest observable bow shock nebulae in the warm neutral
phase of the Milky Way.
We performed dust continuum radiative transfer calculations of
our bow shock models (cf. Acreman et al. 2016) and generated
SEDs and isophotal emission maps for different wavelengths 25 ≤
λ ≤ 100 μm and viewing angles 0◦ ≤ φ ≤ 90◦ . Consistently with
the observation of van Buren & McCray (1988a), van Buren et al.
(1995), Noriega-Crespo et al. (1997a), the calculations show that our
bow shocks are brighter at 60 μm. The projected infrared emission
can also be diminished in the presence of the ISM magnetic field,
in particular at wavelengths λ ≥ 60 μm, since the amount of dust
trapped into the bow shock is smaller. We also notice that the change
in surface brightness of our emission maps as a function of the
viewing angle of the bow shock is similar as in the optical waveband,
i.e. it is brighter if φ = 0◦ and fainter if φ = 90◦ (see Meyer et al.
2016).
In future models, we would like to extend this pioneering study
of massive stars’ bow shocks within the magnetized ISM towards
three-dimensional models in which the ISM magnetic field is unaligned with respect to the motion of the star, as it has been done
in order to appreciate its influence on the morphology of the global
heliopause (Pogorelov & Matsuda 1998). Such simulations will
help to better understand the structure of the circumstellar nebulae forming around hot, ionizing, massive runaway stars and allow
us to predict more accurately, e.g. the optical emission signatures
of these bow shocks. Thorough comparison with particular hot,
bow-shock-producing massive stars, e.g. ζ Ophiuchi, might then be
feasible.
AC K N OW L E D G E M E N T S
The authors thank Tom Hartquist for his kind help and very helpful
advices when reviewing this paper and Richard Stancliffe for numerous grammatical comments when carefully reading the manuscript.
DM-A Meyer gratefully thanks T. Robitaille and C. Dullemond for
their support with the HYPERION and the RADMC-3D radiative transfer codes, respectively, as well as D. Thun for fruitful discussion
concerning his master thesis. Are also acknowledged Professor R.
Jalabert and G. Weick from the Institute of Physics and Chemistry
Downloaded from http://mnras.oxfordjournals.org/ at University Tuebingen on November 14, 2016
(Kawamura, Heerikhuisen & Pogorelov 2010) which revealed the
need for 3D calculations, able to report the non-stationary character of the trail of the bow shock of the Sun (Washimi & Tanaka
1996; Linde et al. 1998; Ratkiewicz et al. 1998). Those models are
more complex than our simplistic two-dimensional simulations and
investigate, e.g. the charge exchanges arising between the stellar
wind and the LISM (Fitzenreiter, Scudder & Klimas 1990). These
studies also highlighted the complexity and fragility of such models, e.g. regarding the variety of instable MHD discontinuities that
affect shock waves propagating through a magnetized flow and differentiating the shocks from purely HD discontinuities described by
the Rankine–Hugoniot (see also de Sterck et al. 1998; de Sterck &
Poedts 1999). Additionally, those solutions are affected by the spatial resolution of the calculations and the inclusion of numerical
viscosity in the models (Lopez, Merkin & Lyon 2011; Wang et al.
2014, and references therein).
Finally, let us mention an other obvious difference between bow
shock of the Sun and the nebulae generated by the runaway OB stars
that we model. The proximity of the Earth with the Sun makes it
easier to be studied and analysed by means of, e.g. radio observations (Baranov, Krasnobaev & Onishchenko 1975) while its innermost substructures are directly reachable with spacecrafts such as
Voyager 1 and Voyager 2.5 Their missions partly consisted in leaving
the neighbourhood of our Sun in order to explore the heliosheath,
i.e. the layer corresponding to the region of shocked solar wind that
is between the contact discontinuity (the heliopause) and the RS
of the solar bow shock (the wind termination shock). The Voyager
engines crossed the outermost edge of the Solar system between
2004 and 2007 at the expected distance of 94 and 84 au from the
Earth (Linde et al. 1998), giving the first experimental data on the
physics of the ISM (Chalov et al. 2016). Those measures proved
the existence of the solar bow shock, and also highlighted the particular conditions of the outer space in terms of magnetic phenomenon
(Richardson 2016) and effects of cosmic rays (Webber 2016). In order to make our models more realistic, those physical processes
should be taken into account in future simulations of bow shocks
from runaway high-mass stars.
MHD bow shocks of hot massive stars
of Materials of Strasbourg (IPCMS) for their kind help concerning the Mie theory. This study was in parts conducted within the
Emmy Noether research group on ‘Accretion Flows and Feedback in
Realistic Models of Massive Star Formation’ funded by the German Research Foundation under grant no. KU 2849/3-1. The authors gratefully acknowledge the computing time provided on the
supercomputer JUROPA at Jülich Supercomputing Centre (JSC)
and on the bwGrid cluster Tübingen. This research has made use of
‘Aladin sky atlas’ and ‘VizieR catalogue access tool’ both developed
at CDS, Strasbourg Observatory, France.
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