---
abstract: 'Massive stars ejected from their parent cluster and supersonically sailing away through the interstellar medium (ISM) are classified as exiled. They generate circumstellar bow shock nebulae that can be observed. We present two-dimensional, axisymmetric hydrodynamical simulations of a representative sample of stellar wind bow shocks from Galactic OB stars in an ambient medium of densities ranging from $n_{\rm ISM }=0.01$ up to $10.0\, \rm cm^{-3}$. Independently of their location in the Galaxy, we confirm that the infrared is the most appropriated waveband to search for bow shocks from massive stars. Their spectral energy distribution is the convenient tool to analyze them since their emission does not depend on the temporary effects which could affect unstable, thin-shelled bow shocks. Our numerical models of Galactic bow shocks generated by high-mass ($\approx 40\, \rm M_{\odot}$) runaway stars yield H$\alpha$ fluxes which could be observed by facilities such as the [*SuperCOSMOS H-Alpha Survey*]{}. The brightest bow shock nebulae are produced in the denser regions of the ISM. We predict that bow shocks [*in the field*]{} observed at H$\alpha$ by means of Rayleigh-sensitive facilities are formed around stars of initial mass larger than about $20\, \rm M_{\odot}$.'
date: 'Received January 18 2015; accepted Month day, 2015'
title: On the observability of bow shocks of Galactic runaway OB stars
---

\[firstpage\]

methods: numerical – circumstellar matter – stars: massive.

Introduction {#sect:introduction}
============

The estimate of massive star feedback is a crucial question in the understanding of the Galaxy’s functioning [@langer_araa_50_2012]. Throughout their short lives, they release strong winds [@holzer_araa_8_1970] and ionising radiation [@diazmiller_apj_501_1998] which modify their ambient medium. This results in diaphanous [H[ii]{} ]{}regions [@dyson_ass_35_1975], parsec-scale bubbles of stellar wind [@weaver_apj_218_1977], inflated [@petrovic_aa_450_2006] or shed [@woosley_rvmp_74_2002; @garciasegura_1996_aa_305] stellar envelopes that impact their close surroundings and which can alter the propagation of their subsequent supernova shock wave [@vanveelen_phd; @meyer_mnras_450_2015]. Understanding the formation processes of these circumstellar structures allows us to constrain the impact of massive stars, e.g. on the energetics or the chemical evolution of the interstellar medium (ISM). Moreover, it links studies devoted to the dynamical evolution of supernova remnants expanding into the ISM [@rozyczka_mnras_261_1993] with works focusing on the physics of the star forming ISM [@peters_apj_711_2010].

These arc-like structures of swept-up stellar wind material and ISM gas the distortion of their stellar wind bubble by the bulk motion of their central star [@weaver_apj_218_1977]. Their size and their morphology are governed by their stellar wind mass loss, the bulk motion of the runaway star and their local ambient medium properties [@comeron_aa_338_1998]. These distorted wind bubbles have been first noticed in optical \[O[iii]{}\] $\lambda 
\, 5007$ spectral emission line around the Earth’s closest runaway star, the OB star $\zeta$ Ophiuchi [@gull_apj_230_1979]. Other noticeable fast-moving massive stars producing a stellar wind bow shock are, e.g. the blue supergiant Vela-X1 [@kaper_apj_475_1997], the red supergiant Betelgeuse [@noriegacrespo_aj_114_1997] and the very massive star BD+43${\ensuremath{^\circ}}$365 running away from Cygnus OB2 [@comeron_aa_467_2007].

   $M_{\star}\, (\rm M_{\odot})$   $t_{\mathrm{ start}}\, (\rm Myr)$   $\textcolor{black}{\log(L_{\star}/\rm L_{\odot})}$   $\log(\dot{M}/\rm M_{\odot}\, \rm yr^{-1})$   $v_{\rm w}\, (\mathrm{km}\, \mathrm{s}^{-1})$   $T_{\rm eff}\, (\mathrm{K})$   $S_{\star} (\mathrm{photon}\, \mathrm{s}^{-1})$   $t_{\rm MS} (\mathrm{Myr})$
  ------------------------------- ----------------------------------- ---------------------------------------------------- --------------------------------------------- ----------------------------------------------- ------------------------------ ------------------------------------------------- -----------------------------
               $10$                              $5.0$                                       $3.80$                                           $-9.52$                                        $1082$                                 $25200$                                 $10^{45}$                                $22.5$
               $20$                              $3.0$                                       $4.74$                                           $-7.38$                                        $1167$                                 $33900$                                 $10^{48}$                              $\,\,\,8.0$
               $40$                              $0.0$                                       $5.34$                                           $-6.29$                                        $1451$                                 $42500$                                 $10^{49}$                              $\,\,\,4.0$

\[tab:lum\_stars\]

Analysis of data from the [*Infrared Astronomical Satellite*]{} facility [[*IRAS*]{}, @neugebauer_278_apj_1984] later extended to measures taken with the [*Wide-Field Infrared Satellite Explorer*]{} [[ *WISE*]{}, @wright_aj_140_2010] led to the compilation of bow shock records, see e.g. [@buren_apj_329_1988]. Soon arose the speculation that those isolated nebulae can serve physics of these stars, to constrain the still highly debated mass loss of massive stars [@gull_apj_230_1979] and/or their ambient medium density [@huthoff_aa_383_2002]. This also raised questions related to the ejection mechanisms of OB stars from young stellar clusters [@hoogerwerf_aa_365_2001]. More recently, multi-wavelengths data led to the publication of the E-BOSS catalog of stellar wind bow shocks [@peri_aa_538_2012; @2015arXiv150404264P].

Early simulations discussed the general morphology of the bow shocks around OB stars [@brighenti_mnras_277_1995, and references therein], their (in)stability [@blondin_na_57_1998] and the general uncompatibility of the shape of stellar wind bow shocks with analytical approximations such as the one of @wilkin_459_apj_1996, see in @comeron_aa_338_1998. However, observing massive star bow shocks remains difficult and they are mostly serendipitously noticed in infrared observations of the neighbourhood of stellar clusters [@gvaramadze_aa_490_2008]. Moreover, their optical emission may be screened by the [H[ii]{} ]{}region which surrounds the driving star and this may affect their H$\alpha$ observations [@brown_aa_439_2005]. We are particularly interested in the prediction of the easiest bow shocks to observe, their optical emission properties and their location in the Galaxy.

In the present study, we extend our numerical investigation of the circumstellar medium of runaway massive stars [@meyer hereafter Paper I]. explores the effects of the ambient medium density on the emission properties of the bow-like nebulae around the most common runaway stars, in the spirit of works on bow shocks generated by low-mass stars [@villaver_apj_748_2012 and references therein].

Our paper is organised as follows. In Section \[sect:method\] we present the numerical methods and the microphysics that is included in our models. The resulting numerical simulations are presented and discussed in Section \[sect:results\]. We then analyze and discuss the emission properties of our bow shock models in Section \[sect:emission\]. Finally, we formulate our conclusions in Section \[section:cc\].

Method {#sect:method}
======

Governing equations {#subsect:goveq}
-------------------

Hydrodynamical simulations {#subsect:hydrosim}
--------------------------

We run two-dimensional, axisymmetric, hydrodynamical numerical simulations using the [ pluto]{} code [@mignone_apj_170_2007; @migmone_apjs_198_2012] in axisymmetric, cylindrical coordinates on a uniform grid $[z_{\rm min},z_{\rm 
max}]\times[O,R_{\rm max}]$ of spatial resolution $\Delta=2.25\times 10^{-4}\, 
\rm{pc}\, \rm{cell}^{-1}$ minimum. The stellar wind is injected into the computational domain filling a circle of radius 20 cells centered onto the origin $O$ [see e.g., @comeron_aa_338_1998; @meyer_mnras_2013 and references therein]. The interaction with the ISM is calculated in the reference frame of the moving star [@vanmarle_aa_469_2007; @vanmarle_apj_734_2011; @vanmarle_aa_561_2014]. Inflowing ISM gas mimicing the stellar motion is set at the $z=z_{\rm max}$ boundary whereas semi-permeable boundary conditions are set at $z=z_{\rm min}$ and at $R=R_{\rm max}$. Wind material is distinguished from the ISM using a passive tracer $Q$ that is advected with the gas and initially set to $Q=1$ in the stellar wind and to $Q=0$ in the ISM. The ISM composition is assumed to be solar [@asplund_araa_47_2009].

  ${\rm {Model}}$    $M_{\star}\, (\rm M_{\odot})$   $v_{\star}\, (\mathrm{km}\, \mathrm{s}^{-1})$   $n_{\rm ISM}\, (\mathrm{cm}^{-3})$     
  ----------------- ------------------------------- ----------------------------------------------- ------------------------------------ -- --
  MS1020n0.01                    $10$                                    $20$                                      $0.01$                   
  MS1040n0.01                    $10$                                    $40$                                      $0.01$                   
  MS1070n0.01                    $10$                                    $70$                                      $0.01$                   
  MS2040n0.01                    $20$                                    $40$                                      $0.01$                   
  MS2070n0.01                    $20$                                    $70$                                      $0.01$                   
  MS1020n0.1                     $10$                                    $20$                                      $0.10$                   
  MS1040n0.1                     $10$                                    $40$                                      $0.10$                   
  MS1070n0.1                     $10$                                    $70$                                      $0.10$                   
  MS2020n0.1                     $20$                                    $20$                                      $0.10$                   
  MS2040n0.1                     $20$                                    $40$                                      $0.10$                   
  MS2070n0.1                     $20$                                    $70$                                      $0.10$                   
  MS4070n0.1                     $40$                                    $70$                                      $0.10$                   
  MS1020n10                      $10$                                    $20$                                      $10.0$                   
  MS1040n10                      $10$                                    $40$                                      $10.0$                   
  MS1070n10                      $10$                                    $70$                                      $10.0$                   
  MS2020n10                      $20$                                    $20$                                      $10.0$                   
  MS2040n10                      $20$                                    $40$                                      $10.0$                   
  MS2070n10                      $20$                                    $70$                                      $10.0$                   
  MS4020n10                      $40$                                    $20$                                      $10.0$                   
  MS4040n10                      $40$                                    $40$                                      $10.0$                   
  MS4070n10                      $40$                                    $70$                                      $10.0$                   

  : The hydrodynamical models. Parameters $M_{\star}$ (in $\rm M_{\odot}$), $v_{\star}$ (in $\mathrm{km}\, \mathrm{s}^{-1}$) and $n_{\rm ISM}$ (in $\mathrm{cm}^{-3}$) are the initial mass of the considered moving star, its space velocity and its local ISM density, respectively.

\[tab:models\]

Microphysics {#subsect:phys}
------------

In order to proceed on our previous bow shock studies [Paper I, @meyer_mnras_450_2015], we include the same microphysics in our simulations of the circumstellar medium of runaway, massive stars, i.e. we take into account losses and gain of internal energy by optically-thin cooling and heating together with electronic thermal conduction. Optically-thin radiative processes are included into the model using the cooling and heating laws established for a fully ionized medium in Paper I. It mainly consist of cooling contributions from hydrogen and helium for temperatures $T<10^{6}\, \rm K$ whereas it is principally due to metals for temperatures $T \ge 10^{6}\, \rm K$ [@wiersma_mnras_393_2009]. A term representing the cooling from collisionally excited forbidden lines [@henney_mnras_398_2009] incorporates the effects of, among other, the \[O[iii]{}\] $\lambda \, 5007$ line emission. The heating contribution includes the reionisation of recombining hydrogen atoms by the starlight [@osterbrock_1989; @hummer_mnras_268_1994]. All our models include electronic thermal conduction [@cowie_apj_211_1977].

![ Stellar wind bow shocks from the main sequence phase of the $20\,  \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n001_legend.eps){width="100.00000%"}

\

![ Stellar wind bow shocks from the main sequence phase of the $20\,  \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n01_legend.eps){width="100.00000%"}

\

![ Stellar wind bow shocks from the main sequence phase of the $20\,  \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n10_legend.eps){width="100.00000%"}

\

![ Stellar wind bow shocks from the main sequence phase of the $20\,  \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n100_legend.eps){width="100.00000%"}

Parameter range {#subsect:para}
---------------

This work consists of a parameter study extending our previous investigation of stellar wind bow shock (Paper I) to regions of the Galaxy where the ISM has either lower or higher densities. The boundary conditions are unchanged, i.e. we consider runaway stars of $10$, $20$ and $40\, \rm M_{\odot}$ star moving with velocity $v_{\star}=20$, $40$ and $70\, \rm km\, \rm s^{-1}$, respectively. Differences come from the chosen ISM number density that ranges from $n_{\rm ISM}=0.01$ to $10.0\, \rm 
cm^{-3}$ whereas our preceeding work exclusively focused on bow shocks models with $n_{\rm ISM}=0.79\, \rm cm^{-3}$.

Bow shocks morphology {#sect:results}
=====================

Bow shocks structure {#subsect:structure}
--------------------

In Fig. \[fig:grid\_density\] we show the density fields in our hydrodynamical simulations of our $20\, \rm M_{\odot}$ star moving with velocity $v_{\star}=70\, \rm 
km\, \rm s^{-1}$ in a medium of number density $n_{\rm ISM}=0.01$ (panel a, model MS2070n0.01), $0.1$ (panel b, model MS2070n0.1), $0.79$ (panel c, model MS2070) and $10.0\, \rm cm^{-3}$ (panel d, model MS2070n10), respectively.

Bow shocks size {#subsect:scaling}
---------------

The bow shocks have a stand-off distance $R(0)$, i.e. the distance separating them from the star along the direction of motion predicted by @wilkin_459_apj_1996. It decreases as a function of (i) $v_{\star}$, (ii) $\dot{M}$ (c.f. Paper I) and (iii) $n_{\rm ISM}$ since $R(0) \propto n_{\rm ISM}^{-1/2}$. A dense ambient medium produces a large ISM ram pressure $n_{\rm ISM}v_{\star}^{2}$ which results in a compression of the whole bow shock and consequently in a reduction of $R(0)$. As an example, our simulations involving a $20\, \rm M_{\odot}$ star with $v_{\star}=70\, 
\mathrm{km}\, \mathrm{s}^{-1}$ has $R(0)\approx 3.80$, $1.14$, $0.38$ and $0.07\rm pc$ when the driving star moves in $n_{\rm ISM}=0.01$, $0.1$, $0.79$ and $10\, \rm cm^{-3}$, respectively (Fig. \[fig:grid\_density\]a-d), which is reasonably in accordance with @wilkin_459_apj_1996. All our measures of $R(0)$ are taken at the contact discontinuity, because it is appropriate measure to compare models with Wilkin’s analytical solution [@mohamed_aa_541_2012].

![ Axis ratio $R(0)/R(90)$ of our bow shock models. The figure shows the ratio $R(0)/R(90)$ measured in the density field of our models measured at their contact discontinuity, as a function of their stand-off distance $R(0)$ (in $\rm pc$). Symbols distinguish models as a function of (i) the ISM ambient medium with $n_{\rm ISM}=0.01$ (triangles), $0.1$ (diamonds), $0.79$ (circles) and $10.0\, \rm cm^{-3}$ (squares) and (ii) of the initial mass of the star with $10\, \rm M_{\odot}$ (blue dots), $20\, \rm M_{\odot}$ (blue plus signs) and $40\, \rm M_{\odot}$ (dark green crosses), respectively. The thin horizontal black line corresponds to the analytic solution $R(0)/R(90)= 1/\sqrt{3}\approx 0.58$ of @wilkin_459_apj_1996. []{data-label="fig:axis_ratio"}](./R0_vs_R90.eps){width="100.00000%"}

Non-linear instabilities and mixing of material {#subsect:stability}
-----------------------------------------------

In Fig. \[fig:grid\_velocity\] we show a time sequence evolution of the density field in hydrodynamical simulations of $40\, \rm M_{\odot}$ zero-age main-sequence star moving with velocity $v_{\star}=70\, \rm km\, \rm s^{-1}$ in a medium of number density $n=10.0\, \rm cm^{-3}$ (model MS4070n10). The figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d), respectively. After $0.02\, \rm Myr$ the whole shell is sparsed with small size clumps which are the seeds of non-linear instabilities (Fig. \[fig:grid\_velocity\]b). The fast stellar motion ($v_{\star}=70\, \rm km\, \rm s^{-1}$) provokes a distortion of the bubble into an ovoid shape [see fig. 7 of @weaver_apj_218_1977] and the high ambient medium density ($n=10.0\, \rm cm^{-3}$) induces rapidly a thin shell after only about $0.01\, \rm Myr$.

![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time2_legend.eps){width="100.00000%"}

\

![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time3_legend.eps){width="100.00000%"}

\

![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time4_legend.eps){width="100.00000%"}

\

![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time5_legend.eps){width="100.00000%"}

![ Bow shock volume ($z\ge0$) in our model (see Fig. \[fig:grid\_velocity\]a-e). The figure shows the volume of perturbed material (in $\rm pc^{3}$) in the computational domain (thick solid blue line), together with the volume of shocked ISM gas (thin solid red line) and shocked stellar wind (thick dotted orange line), respectively, as function of time (in $\rm Myr$). The large dotted black line represents the volume of the thin shell of shocked ISM. []{data-label="fig:volume"}](./volume.eps){width="100.00000%"}

\

The bow shock then experiences a series of cycles in which small scaled eddies grow in the shell (Fig. \[fig:grid\_velocity\]b) and further distort its apex into wing-like structures (Fig. \[fig:grid\_velocity\]c) which are pushed sidewards because of the transverse component of the stellar wind acceleration (Fig. \[fig:grid\_velocity\]d). Our model MS4070n10 has both characteristics from the models E “ High ambient density” and G “ Instantaneous cooling” of @comeron_aa_338_1998. Thin-shelled stellar wind bow shocks develop non-linear instabilities, in addition to the Kelvin-Helmholtz instabilities that typically affect interfaces between shearing flows of opposite directions, i.e. the outflowing stellar wind and the ISM gas penetrating the bow shocks [@vishniac_apj_428_1994; @garciasegura_1996_aa_305; @vanmarle_aa_469_2007]. A detailed discussion of the development of such non-linearities affecting bow shocks generated by OB runaway stars is in @comeron_aa_338_1998.

In Fig. \[fig:volume\] we plot the evolution of the volume of the bow shock in our model MS4070n10 (thick solid blue line), separating the volume of shocked ISM gas (thin dotted red line) from the volume of shocked stellar wind (thick dotted orange line) in the apex ($z\ge0$) of the bow shock. Such a discrimination of the volume of wind and ISM gas is possible because a passive scalar tracer is is numerically advected simultaneously with the flow. The figure further illustrates the preponderance of the volume of shocked ISM in the bow shock compared to the stellar wind material, regardless the growth of eddies. Interestingly, the volume of dense shocked ISM gas (large dotted black line) does not have large time variations (see Section \[sect:emission\]).

Bow shock energetics and emission signatures {#sect:emission}
============================================

![ Bow shocks luminosities. The panels correspond to models with an ISM density The simulations labels are indicated under the corresponding values. []{data-label="fig:lum1"}](./luminosity_grid_001.eps "fig:"){width="44.00000%"} ![ Bow shocks luminosities. The panels correspond to models with an ISM density The simulations labels are indicated under the corresponding values. []{data-label="fig:lum1"}](./luminosity_grid_01.eps "fig:"){width="44.00000%"} ![ Bow shocks luminosities. The panels correspond to models with an ISM density The simulations labels are indicated under the corresponding values. []{data-label="fig:lum1"}](./luminosity_grid_100.eps "fig:"){width="44.00000%"}

Methods {#subsect:methods}
-------

In Fig. \[fig:lum1\] the total bow shock luminosity $L_{\rm total}$ (pale green diamonds) is calculated integrating the losses by optically-thin radiation in the $z \ge 0$ region of the computational domain [@mohamed_aa_541_2012 Paper I]. Shocked wind emission $L_{\rm wind}$ (orange dots) is discriminated from $L_{\rm total}$ with the help of the passive scalar $Q$ that is advected with the gas, Additionaly, we compute $L_{\rm H\alpha}$ (blue crosses) and $L_{[\rm O{\sc III}]}$ (dark green triangles) which stand for the bow shock luminosities at H$\alpha$ and at \[O[iii]{}\] $\lambda \, 5007$ spectral line emission using the prescriptions for the emission coefficients in @dopita_aa_29_1973 and @osterbrock_1989, respectively. The overall X-ray luminosity $L_{\rm X}$ (black right crosses) is computed with emission coefficients generated with the [xspec]{} program [@arnaud_aspc_101_1996] with solar metalicity and chemical abundances from @asplund_araa_47_2009.

Results {#subsect:results}
-------

### Optical luminosities {#subsect:luminosities}

![ Luminosities of our bow shock simulation of a $40\, \rm M_{\odot}$ star moving with velocity $v_{\star}=70\, \rm km\, \rm s^{-1}$ through a medium with $n_{\rm ISM}=10\, \rm cm^{-3}$ (see corresponding time-sequence evolution of its density field in Fig. \[fig:grid\_velocity\]a-e). Plotted quantities and color-coding are similar to Fig. \[fig:lum1\] and are shown as function of time (in $\rm Myr$). []{data-label="fig:lum3"}](./luminosity.eps){width="100.00000%"}

\

![image](./observability.eps){width="100.00000%"}

![image](./feedback_energy_2.eps){width="100.00000%"}

In Fig. \[fig:lum1\] we display the bow shocks luminosities as a function of the initial mass of the runaway star, its space velocity $v_{\star}$ and its ambient medium density $n_{\rm ISM}$. At a given density of the ISM, all of our models have luminosities from optically-thin gas radiation which with respect to the stellar mass loss are as described in Paper I for the simulations with $n_{\rm ISM}\approx 0.79\, \rm 
cm^{-3}$.

The behaviour of the optically-thin emission originating from the shocked stellar wind $L_{\rm wind}$, the \[O[iii]{}\] $\lambda \, 5007$ spectral line emission and the H$\alpha$ emission at fixed $n_{\rm ISM}$ are similar as described in @meyer_mnras_2013. The contribution of $L_{\rm wind}$ is smaller than $L_{\rm total}$ by several orders of magnitude for all models, e.g. our model MS1020n0.1 has $L_{\rm wind}/L_{\rm total} \approx 10^{-5}$. All our models have $L_{\rm H\alpha} < L_{[\rm O{\sc III}]} < L_{\rm
total}$ and the H$\alpha$ emission, the $[\rm O{\sc III}]$ spectral line emission and $L_{\rm wind}$ have variations which are similar to $L_{\rm ISM}$ with respect to $M_{\star}$, $v_{\star}$ and $n_{\rm ISM}$.

Fig. \[fig:lum3\] shows the lightcurve of our model MS4070n10 computed over the whole simulation and plotted as a function of time with the color coding from Fig. \[fig:lum1\]. Very little variations of the emission are present at the beginning of the calculation up to a time of about $0.004\, \rm Myr$ and it remains almost constant at larger times. This is in accordance with the volume of the dense ISM gas trapped into the nebula (see large dotted black line in Fig. \[fig:volume\]). The independence of $L_{\rm IR}$ with respect to the strong volume fluctuations of thin-shelled nebulae (Fig. \[fig:lum3\]) indicates that their spectral energy distributions is likely to be the appropriate tool to analyze them since it constitutes an observable which is not reliable to temporary effects.

### Infrared and X-rays luminosities {#subsect:thermalisation}

reprocessed starlight on dust grains penetrating the bow shocks, $L_{\rm IR}$, is larger than $L_{\rm total}$ by about $1-2$ orders of magnitude. This is possible because the reemission of starlight by dust grains is not taken into account in our simulations. for the models MS2040n0.01 and MS2040n10, respectively. $L_{\rm IR}$ increases with $M_{\star}$ (Figs. \[fig:lum1\]a-d). Particularly, we find that $L_{\rm IR} \gg L_{\rm H\alpha}$ and $L_{\rm IR} \gg L_{[\rm O{\sc III}]}$, and therefore we conclude that the infrared waveband is the best way to detect and observe bow shocks from massive main-sequence runaway stars regardless of $n_{\rm ISM}$ (see section \[sect:observability\]).

Several current and/or planned facilities are designed to observe at these wavelengths and may be able to detect bow shocks from runaway stars:

1.  First, the [*James Webb Space Telescope*]{} (JWST) which [*Mid-Infrared Instrument*]{} [MIRI, @swinyard_2004] observes in the infrared ($5$$-$$28\, \mu \rm m$) that roughly corresponds to our predicted waveband of dust continuum emission from stellar wind bow shocks of runaway OB stars.

2.  Secondly, the [*Stratospheric Observatory for Infrared Astronomy*]{} (SOFIA) airborne facility which [*Faint Object infraRed CAmera for the SOFIA Telescope*]{} [FORCAST, @adams_2008] instrument detects photons in the $5.4$$-$$37\, \mu \rm m$ waveband.

3.  Then, the proposed [*Space Infrared Telescope for Cosmology and Astrophysics*]{} [SPICA, @kaneda_2004] satellite would be the ideal tool the observe stellar wind bow shock, since it is planed to be mounted with (i) a far-infrared imaging spectrometer ($30$$-$$210\, \mu \rm m$), (ii) a mid-infrared coronograph ($3.5/5$$-$$27\, \mu \rm m$) and (iii) a mid-infrared camera/spectrometer ($5$$-$$38\, \mu \rm m$).

4.  Finally, we should mention the proposed [*The Mid-infrared E-ELT Imager and Spectrograph*]{} (METIS) on the planned [*European Extremely Large Telescope*]{} [E-ELT, @brandl_2006], that will be able to scan the sky in the $3$$-$$19\, \mu \rm m$ waveband.

Exploitation of the associated archives of these instruments in regions surroundings young stellar clusters and/or at the locations of previously detected bow-like nebulae [@buren_apj_329_1988; @vanburen_aj_110_1995; @noriegacrespo_aj_113_1997; @peri_aa_538_2012; @2015arXiv150404264P] are research avenues to be explored.

Finally, we notice that the X-rays emission are much smaller than any other emission lines or bands, e.g. the model MS2070 has $L_{\rm X}/L_{\rm H\alpha} \approx 10^{-5}$, and it is consequently not a relevant waveband to observe our bow shocks.

### Feedback {#subsect:feedback}

The ratio $\dot{E}_{\rm motion}/L_{\rm total}$ is shown as a function of the bow shock volume in Fig. \[fig:obser\]b.

![image](./Ha_M10.eps){width="100.00000%"}

![image](./Ha_M20.eps){width="100.00000%"}

![image](./Ha_M40.eps){width="100.00000%"}

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![image](./IR_M10.eps){width="100.00000%"}

![image](./IR_M20.eps){width="100.00000%"}

![image](./IR_M40.eps){width="100.00000%"}

Discussion {#sect:discussion}
----------

### The appropriated waveband to observe stellar wind bow shocks in the Galaxy {#sect:observability}

In Fig. \[fig:paving\_lum\] we show the H$\alpha$ surface brightness (in $\rm erg\,\rm 
s^{-1}\,\rm  cm^{-2}\,\rm  arcsec^{-2}$, panels a-c) and the infrared luminosity (in $\rm erg\, \rm s^{-1}$, panels d-f) for models with $M_{\star}=10\, \rm M_{\odot}$ (left panels), $20\, \rm M_{\odot}$ (middle panels) and $40\, \rm M_{\odot}$ (right panels). The surface brightness $\Sigma^{\rm max}_{\rm H \alpha}$ scales with $n^{2}$, see Appendix A of Paper I, therefore the lower the ISM background density of the star, i.e. the higher its Galactic latitude, the fainter the projected emission of the bow shocks and the lower the probability to observe them. The brightest bow shocks are generated both in infrared and H$\alpha$ by our most massive stars running in the denser regions of the ISM ($\rm n_{\rm ISM} = 
10.0\, \rm cm^{-3}$). The estimate of the infrared luminosity confirms our result relative to bow shock models with $n_{\rm ISM}=0.79\, \rm 
cm^{-3}$ in the sense that the brightest bow shocks are produced by high-mass, stars (Paper I) moving in a relatively dense ambient medium, i.e. within the Galactic plane (Fig. \[fig:paving\_lum\]d-f). At H$\alpha$, these bow shocks are associated to fast-moving stars ($v_{\star}=70\, \rm km\, \rm s^{-1}$) producing the strongest shocks, whereas in infrared they are associated to slowly-moving stars ($v_{\star}=20\, \rm km\, \rm s^{-1}$) generating the largest nebulae.

![image](./map_rotation_legend_one.eps){width="100.00000%"}

![image](./map_rotation1_legend_1.eps){width="100.00000%"}

![image](./map_rotation_legend_a.eps){width="100.00000%"}

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![image](./map_rotation_legend_two.eps){width="100.00000%"}

![image](./map_rotation_legend_2.eps){width="100.00000%"}

![image](./map_rotation_legend_b.eps){width="100.00000%"}

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![image](./map_rotation_legend_three.eps){width="100.00000%"}

![image](./map_rotation_legend_3.eps){width="100.00000%"}

![image](./map_rotation_legend_c.eps){width="100.00000%"}

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![image](./map_rotation_legend_four.eps){width="100.00000%"}

![image](./map_rotation_legend_4.eps){width="100.00000%"}

![image](./map_rotation_legend_d.eps){width="100.00000%"}

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### Synthetic optical emission maps {#sect:maps}

In Fig. \[fig:maps1\] we plot synthetic H$\alpha$ and \[O[iii]{}\] $\lambda \, 5007$ emission maps of the bow shocks generated by our $20\, \rm M_{\odot}$ star moving with velocity $70\, 
\mathrm{km}\, \mathrm{s}^{-1}$ moving through a medium with $n_{\rm ISM}=0.1$ (left column of panels), $0.79$ (middle column of panels) and $10.0\, \rm 
cm^{-3}$ (right column of panels). The region of maximum H$\alpha$ emission of the gas is located close to the apex of the bow shock and extended to its trail ($z\, 
\le 0$). This broadening of the emitting region is due to the high space velocity of the star, see Paper I. Neither the shocked stellar wind nor the hot shocked ISM of the bow shock contributes significantly to these emission since the $\rm H\alpha$ emission coefficient $j_{\rm H\alpha} \propto T^{-0.9}$ and the contact discontinuity is the brightest part of the whole structure (Fig. \[fig:maps1\]a). The \[O[iii]{}\] $\lambda \, 5007$ emission is maximum at the same location but, however, slightly different dependence on the temperature of the corresponding emission coefficient $j_{\rm [OIII]} \propto 
\exp(-1/T)/T^{1/2}$ [@dopita_aa_29_1973] induces a weaker extension of the emission to the tail of the structure (Fig. \[fig:maps1\]a). The unstable simulations with $v_{\star}\, \ge 40\, \mathrm{km}\, \mathrm{s}^{-1}$ and $n_{\rm ISM} \simeq 
10\, \rm cm^{-3}$ have ring-like artefacts which dominate the emission (see Fig. \[fig:maps1\]e-h and Fig. \[fig:maps1\]i-l). They are artificially generated by the over-dense regions of the shell that are rotated and mapped onto the Cartesian grid. A tri-dimensional unstable bow shock would have brighter clumps of matters sparsed around its layer of cold shocked ISM rather than regular rings [@mohamed_aa_541_2012]. Regardless of the properties of their driving star, our bow shocks are brighter in large ambient medium, e.g. the model MS2070n0.1 has $\Sigma^{\rm max}_{[\rm H\alpha]} \approx 10^{-18} 
\, \mathrm{erg}\, \mathrm{s}^{-1}\, \mathrm{cm}^{-2}\, \mathrm{arcsec}^{-2}$ whereas the model MS2070n10 has $\Sigma^{\rm max}_{[\rm H\alpha]} \approx 
3\times 10^{-15} \, \mathrm{erg}\, \mathrm{s}^{-1}\, \mathrm{cm}^{-2}\, 
\mathrm{arcsec}^{-2}$. The projected \[O[iii]{}\] $\lambda \, 5007$ emission behaves similarly.

![ Cross-sections taken along the direction of motion of our $20\, \rm M_{\odot}$ star moving with velocity $70\, \rm km\, \rm s^{-1}$ in an ambient medium of number density $n_{\rm ISM}=0.1\, \rm cm^{-3}$. The data are plotted for inclination angles $\phi=30{\ensuremath{^\circ}}$ (thin solid red line), $\phi=45{\ensuremath{^\circ}}$ (thin dotted blue line), $\phi=60{\ensuremath{^\circ}}$ (thick solid orange line) and $\phi=90{\ensuremath{^\circ}}$ (thick dotted dark green line) through their H$\alpha$ surface brightness (see Fig. \[fig:maps1\]a-d). The position of the star is located at the origin. []{data-label="fig:profiles"}](./cut_profile_2070n01_Ha.eps){width="45.00000%"}

![ Bow shock H$\alpha$ surface brightness (a) and ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/
             \Sigma^{\rm max}_{[\rm H\alpha]}$ (b) as a function of its volume $R(0)^{3}$ (in $\rm pc^{3}$). Upper panel shows the H$\alpha$ surface brightness as a function of the detection threshold of the SuperCOSMOS H$\alpha$ Survey (SHS) of $\Sigma_{\rm SHS}\approx 1.1-2.8 \times 10^{-17}\, 
             \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ [@parker_mnras_362_2005]. Lower panel plots the ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm max}_{[\rm H\alpha]}$ of the same models. []{data-label="fig:vol_vs_S_Ha"}](./vol_vs_S_Ha.eps){width="100.00000%"}

![ Bow shock H$\alpha$ surface brightness (a) and ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/
             \Sigma^{\rm max}_{[\rm H\alpha]}$ (b) as a function of its volume $R(0)^{3}$ (in $\rm pc^{3}$). Upper panel shows the H$\alpha$ surface brightness as a function of the detection threshold of the SuperCOSMOS H$\alpha$ Survey (SHS) of $\Sigma_{\rm SHS}\approx 1.1-2.8 \times 10^{-17}\, 
             \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ [@parker_mnras_362_2005]. Lower panel plots the ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm max}_{[\rm H\alpha]}$ of the same models. []{data-label="fig:vol_vs_S_Ha"}](./vol_vs_OIII_over_Ha.eps){width="100.00000%"}

\

In Fig. \[fig:profiles\] we show cross-sections of the H$\alpha$ surface brightness of the model MS2070n0.1. The cuts are taken along the symmetry axis of the figures and plotted as a function of the inclination angle $\phi$ with respect to the plane of the sky. The emission rises slightly as $\phi$ increases from for $\phi=30{\ensuremath{^\circ}}$ (thin red solid line) to $\phi=60{\ensuremath{^\circ}}$ (thick solid orange line) since $\Sigma^{\rm max}_{[\rm H\alpha]}$ peaks at about $6\times 10^{-19}$ and about $10^{-18}\, \mathrm{erg}\, \mathrm{s}^{-1}\, 
\mathrm{cm}^{-2}\, \mathrm{arcsec}^{-2}$, respectively. The case with $\phi=90{\ensuremath{^\circ}}$ is different since the emission decreases to about $\approx 
2\times 10^{-19}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ (see thick dotted green line in Fig. \[fig:profiles\]). The same is true for the \[O[iii]{}\] emission since its dependence on the post-shock density is similar. Large angles of inclination make the opening of the bow shocks larger (Fig. \[fig:profiles\]a-c, e-g, i-k) and the stand-off distance appears smaller (Fig. \[fig:profiles\]a-c). Note that bow shocks observed with a viewing angle of $\phi=90{\ensuremath{^\circ}}$ do not resemble an arc-like shape but rather an overlapping of iso-emitting concentric circles (Fig. \[fig:profiles\]d,h,l).

### Bow shocks observability at H$\alpha$ and comparison with observations {#sect:comp}

In Fig. \[fig:vol\_vs\_S\_Ha\] we show our bow shocks’ H$\alpha$ surface brightness (a) and their $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm 
max}_{[\rm H\alpha]}$ ratio (b), both as a function of the volume of emitting gas ($z\ge0$). The color coding of both panels takes over the definitions adopted in Fig. \[fig:axis\_ratio\]. The models with a $10\, \rm 
M_{\odot}$ have a volume smaller than about a few $\rm pc^{3}$ and have emission smaller than about $10^{-15}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm 
arcsec^{-2}$. The models with $M_{\star}=20\, \rm M_{\odot}$ have larger volume at equal $n_{\rm ISM}$ and can reach surface brightness of about a few $10^{-14}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ if $n_{\rm 
ISM}=10\, \rm cm^{-3}$. Note that all models with $n_{\rm ISM} \ge 10.0\, \rm 
cm^{-3}$ produce emission larger than the diffuse emission sensitivity threshold of the [*SuperCOSMOS H-Alpha Survey*]{} (SHS) of $\Sigma_{\rm SHS} \approx 
1.1$-$2.8 \times 10^{−17}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm 
arcsec^{-2}$ [@parker_mnras_362_2005] and such bow shocks should consequently be observed by this survey (see horizontal black line in Fig. \[fig:vol\_vs\_S\_Ha\]a).

As discussed above, a significant fraction of our sample of bow shocks models have a H$\alpha$ surface brightness larger than the sensitivity limit of the SHS survey [@parker_mnras_362_2005]. This remark can be extended to other (all-sky) H$\alpha$ observations campaigns, especially if their detection threshold is lower than the SHS. This is the case of, e.g. the [*Virginia Tech Spectral-Line Survey*]{} survey [VTSS, @dennison_aas_195_1999] and the [*Wisconsin H-Alpha Mapper*]{} [WHAM, @reynolds_pasa_15_1998] which provide us with images of diffuse sensitivity detection limit that allow the revelation of structures associated with sub-Rayleigh intensity. Consequently, one can expect to find optical traces of stellar wind bow shocks from OB stars in these data. According to our study, their driving stars are more likely to be of initial mass $M_{\star} \ge 20\, \rm M_{\odot}$ (Fig. \[fig:vol\_vs\_S\_Ha\]a). This also implies that bow shocks [*in the field*]{} that are observed with such facilities are necessary produced by runaway stars of initial mass larger than $M_{\star} \ge 20\, \rm M_{\odot}$. Moreover, we find that the models involving an $10\, \rm M_{\odot}$ star and with $v_{\star}\, \ge 40\, \mathrm{km}\, 
\mathrm{s}^{-1}$ have $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm 
max}_{[\rm H\alpha]}>10$, whereas almost all of the other simulations do not satisfy this criterion (Fig. \[fig:vol\_vs\_S\_Ha\]b).

, we find a similarity between some of the cross-sections taken along the symmetry axis of the H$\alpha$ surface brightness of our bow shock models (Fig. \[fig:profiles\]) and the measure of the radial brightness in emission measure of the bow shock generated by the runaway O star HD 57061 [see fig. 5 of @brown_aa_439_2005]. This observable and our model authorize a comparison since H$\alpha$ emission and emission measures have the same quadratic dependence on the gas number density. The emission measure profile of HD 57061 slightly increases from the star to the bow shock and steeply peaks in the region close to the contact discontinuity, before to decrease close to the forward shock of the bow shock and reach the ISM background emission. Our H$\alpha$ profile with $\phi=60{\ensuremath{^\circ}}$ is consistent with (i) the above described variations and (ii) with the estimate of the inclination of the symmetry axis of HD 57061 with respect to the plane of the sky of about $75{\ensuremath{^\circ}}$, see table 3 of @brown_aa_439_2005. Note that according to our simulations, the emission peaks in the region separating the hot from the cold shocked ISM gas.

### Implication for the evolution of supernova remnants generated by massive runaway stars {#sect:pre_sn}

Massive stars evolve and die supernovae, a sudden and strong release of matter, energy and momentum taking place inside the ISM pre-shaped by their past stellar evolution [@langer_araa_50_2012]. In the case of a runaway progenitor, the circumstellar medium at the pre-supernova phase can be a bow shock nebula with which the shock wave interacts into the unperturbed ISM [@brighenti_mnras_270_1994]. The subsequent growing supernova remnant develops asymmetries since it is braked by the mass at the apex of the bow shock but expands freely in the cavity driven by the star in the opposite direction [@borkowski_apj_400_1992]. If the progenitor is slightly supersonic, the bow shock is mainly shaped during the main-sequence phase of the star; whereas if the progenitor is a fast-moving star then the bow shock is essentially made of material from the last pre-supernova evolutionary phase. In the Galactic plane ($n_{\rm ISM}=0.79\, \rm cm^{-3}$) such asymmetries arise if the apex of the bow shock accumulates at least $1.5\, \rm M_{\odot}$ of shocked material [@meyer_mnras_450_2015].

In Fig. \[fig:mass\] we present the mass trapped into the $z\ge0$ region of our bow shock models as a function of their volume. As in Fig. \[fig:vol\_vs\_S\_Ha\] the figure distinguishes the initial mass and the ambient medium density of each models. Amongst our bow shock simulations, 9 models have $M_{\rm bow}  \gtrsim  
1.5\, \rm M_{\odot}$ and 4 of them are generated by the runaway stars which asymmetric supernova remnant studied in detail in @meyer_mnras_450_2015. The other models with $v_{\star} \le 40\, \rm km\, \rm s^{-1}$ may produce asymmetric remnants because they will explode inside their main-sequence wind bubble. The model MS4070n0.1 has $v_{\star}=70\, \rm km\, \rm s^{-1}$ which indicates that the main-sequence bow shock will be advected downstream by the rapid stellar motion and the surroundings of the progenitor at the pre-supernova phase is made of, e.g. red supergiant material. Consequently, its shock wave may be unaffected by the presence of the circumstellar medium. We leave the examination via hydrodynamical simulations of this conjecture for future works. Interestingly, we notice that most of the potential progenitors of asymmetric supernova remnants are moving in a low density medium $n_{\rm ISM} 
\le 0.1\, \rm cm^{-3}$, i. e. in the rather high latitude regions of the Milky Way. This is consistent with the interpretation of the elongated shape of, e.g. Kepler’s supernova remnant as the consequence of the presence of a massive bow shock at the time of the explosion [@velazquez_apj_649_2006; @toledoray_mnras_442_2014].

### The influence of the interstellar magnetic field on the shape of supernovae remnants

![ Bow shocks mass as a function of the bow shock volume. The figure shows the mass $M_{\rm bow}$ (in $M_{\odot}$) trapped in the $z \ge 0$ region of the bow shock as a function of its volume $R(0)^{3}$ (in $\rm pc^{3}$). The dots distinguish between models (i) as a function of the ISM ambient medium with $n_{\rm ISM}=0.01$ (triangles), $0.1$ (diamonds), $0.79$ (circles) and $10\, \rm cm^{-3}$ (squares), and (ii) as a function of the initial mass of the star with $10$ (blue dots), $20$ (red plus signs) and $40\, \rm M_{\odot}$ (green crosses). The thin horizontal black line corresponds to $M_{\rm bow} = 1.5\, \rm M_{\odot}$, i.e. the condition to produce an asymmetric supernova remnant if $n_{\rm ISM}=0.79\, \rm cm^{-3}$ [@meyer_mnras_450_2015]. []{data-label="fig:mass"}](./vol_vs_mass.eps){width="100.00000%"}

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Conclusion {#section:cc}
==========

Our bow shock simulations indicate that no structural difference arise when changing the density of the background ISM in which the stars move, i.e. their internal organisation is similar as described in @comeron_aa_338_1998 and Paper I. The same is true for their radiative properties, governed by line cooling such as the \[O[iii]{}\] $\lambda \, 5007$ line and showing faint H$\alpha$ emission, both principally originating from outer region of shocked ISM gas. We also find that their X-rays signature is fainter by several orders of magnitude than their H$\alpha$ emission, and, consequently, it is not a good waveband to search for such structures.

The best way to observe bow shocks remains their infrared emission of starlight reprocessed by shocked ISM dust [@meyer_mnras_2013]. We find that the brightest infrared bow shocks, i.e. the most easily observable ones, are produced by high-mass ($M_{\star} \approx 40\, \rm M_{\odot}$) stars moving with a slow velocity ($v_{\star}\approx 20\, \rm km\, \rm s^{-1}$) in the relatively dense regions ($n_{\rm ISM}\approx 10\, \rm cm^{-3}$) of the ISM, whereas the brightest H$\alpha$ structures are produced by these stars when moving rapidly ($v_{\star}\approx 70\, \rm km\, \rm s^{-1}$). Thin-shelled bow shocks have mid-infrared luminosities which does not report the time-variations of their unstable structures. This indicates that spectral energy distributions of stellar wind bow shocks are the appropriate tool to analyze them since they do not depend on the temporary effects that affect their density field. .

A detailed analysis of our grid of simulations indicates that the H$\alpha$ surface brightness of Galactic stellar wind bow shocks increases if their angle of inclination with respect to the plane of the sky increases up to $\phi = 60{\ensuremath{^\circ}}$, however, edge-on viewed bow shocks are particularly faint. We find that all bow shocks generated by a $40\, \rm M_{\odot}$ runaway star could be observed with Rayleigh-sensitive H$\alpha$ facilities and that bow shocks observed [*in the field*]{} by means of these facilities should have an initial mass larger than about $20\, \rm M_{\odot}$. Furthermore, all of our bow shocks generated by a $10\, \rm M_{\odot}$ star moving with $v_{\star}\ge 40\, \rm km\, 
\rm s^{-1}$ have a line ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm max}_{[\rm H\alpha]}>10$. Our study suggests that slowly-moving stars of ZAMS mass $M_{\star} \ge 20\, \rm 
M_{\odot}$ moving in a medium of $n_{\rm ISM }\ge 0.1\, \rm cm^{-3}$ generate massive bow shocks, i.e. are susceptible to induce asymmetries in their subsequent supernova shock wave. This study will be enlarged, e.g. estimating observability of red supergiant stars.

Acknowledgements {#acknowledgements .unnumbered}
================

D. M.-A. Meyer thanks P. F. Velazquez, F. Brighenti and L. Kaper for their advices, and F. P. Wilkin for useful comments on stellar wind bow shocks which partly motivated this work. This study was 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. A.-J. van Marle acknowledges support from FWO, grant G.0227.08, KU Leuven GOA/2008, 04 and GOA/2009/09. The authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JUROPA at J" ulich Supercomputing Centre (JSC).
