density of the surrounding gas. The large number of walkaways formed per each
runaway $\mathcal{R}$ suggests that walkaway stars might change massive stars
feedback more significantly than the faster runaways, although they cannot
reach as far out as the latter because of their lower velocity.

As a consequence of the typical evolutionary scenario described in Sec. 3, the
first CC event in a typical binary system will result either in a stripped SN
or a failed SN with little or no electromagnetic signature. The second CC will
instead be from the accretor star, which could again result in a SNIb/Ic (if
the star becomes a Wolf-Rayet because of its own stellar wind), but, because
of the IMF and the possibly non-fully conservative accretion rate, the
secondary is more likely to remain H-rich until its CC. As already pointed out
by Eldridge et al. (2011), this suggests that stripped SNe should
preferentially happen in star forming regions, while type II SN should show a
larger spatial spread (because of the motion of the ejected star).

In this study, we did not consider the effects of the background cluster
and/or Galactic potential on the motion of the ejected secondaries. These can
in principle be used as test particles to probe the potential itself, and
consequently allow for testing the local and/or Galactic dark matter
distribution (e.g., Rossi et al., 2014; Marchetti et al., 2017).

While runaways have peculiar kinematic characteristics that make them
recognizable, walkaway stars might more easily be mistaken for genuine single
stars. Especially for masses larger than $7.5\,M_{\odot}$ and velocities
larger than $v_{\mathrm{dis}}\gtrsim 20\,\mathrm{km\ s^{-1}}$, the walkaways
and runaways accrete mass before being ejected. Therefore, the disruption of
binaries can pollute observed samples of present-day single stars with binary
evolution by-products populating the field of the Galaxy (Gvaramadze et al.,
2012; de Mink et al., 2014).

As an example, the runaway star $\zeta$ Puppis has been considered to be a
“canonical” O-type star (e.g. Ramiaramanantsoa et al., 2018), and it was used
to calibrate the free parameters in the Castor et al. (1975) (CAK) theory of
stellar winds (Pauldrach et al., 1994) widely applied to model single stars.
However, this object is likely to be a accretor ejected from a binary, (van
Rensbergen et al., 1996; Ramiaramanantsoa et al., 2018).

Massive walkaway stars ejected from a binary might also appear in isolation.
The origin of massive stars observed in isolation is an open problem: did they
form in isolation? or could they possibly have reached that location, if
ejected from a binary? Bestenlehner et al. (2011) identify an isolated $\sim
150\,M_{\odot}$ star (VFTS682) in the 30 Doradus field. Such massive object is
unlikely to be the result of a binary disruption, but might have been ejected
from the massive young cluster R136. Bestenlehner et al. (2011) suggest that
if this star has not been ejected from this cluster, it might be a direct
indication of isolated star formation. Walkaway massive stars can pose similar
problems if not recognized as rejuvenated binary products.

This might also connect to the evolutionary path of luminous blue variable
(LBV) stars. These are massive stars which experience extreme and yet
unexplained outbursts of mass loss. In the classical picture of massive
stellar evolution (so-called “Conti scenario”), LBVs are an intermediate
evolutionary stage between O-type and Wolf-Rayet stars, during which the bulk
of the H-rich envelope is lost (e.g. Conti, 1975; Maeder & Conti, 1994;
Maeder, 1996). However, Smith & Tombleson (2015) pointed out that LBV and LBV-
candidates are farther from the nearest O-type star than Wolf-Rayet stars, and
suggested that LBVs might be rejuvenated products of binary
evolution101010This idea has also been questioned by Humphreys et al. (2016);
Davidson et al. (2016), see also Smith (2016). (accretors or merger products).
The followup study of Aghakhanloo et al. (2017) suggested that partial
rejuvenation during mass transfer coupled to a walkaway (or runaway) velocity
might be sufficient to explain the isolation observed by Smith & Tombleson
(2015).

##  9 Summary and conclusion

We have carried out a suite of numerical simulations to provide predictions
for the kinematics and properties of massive stars ejected from binary systems
disrupted by the supernova explosion of the companion star. Our aim with this
study is three-fold: (i) identify which theoretical predictions are robust
against model uncertainties and discuss their astrophysical implications (ii)
investigate theoretically which uncertain physical processes affect the
kinematics of the population of unbound stars in such a way that their
imprints would be observable and (iii) provide a framework of models that can
be used for comparison against observations or as input for other simulations.

We summarize our main findings below. When we quote percentages or fractions,
the error bars given indicate the maximum variations that we encountered in
our model variations for different physical assumptions (excluding the
unphysical variation with zero natal kicks for all newly born compact
objects).

  * $\bullet$

Nearly a quarter, $22_{-8}^{+26}$%, of all binary systems with at least one
star more massive initially than $7.5{\mathrm{M}_{\odot}}$ merges prior to the
explosion of the first star (see Fig. 4 and Table 1).

  * $\bullet$

The large majority of binary systems is disrupted at the moment of core
collapse of the first star, $\mathcal{D}=86_{-9}^{+11}\%$ in our simulations,
consistent with earlier studies. This prediction is robust against variations
in the treatment of the evolutionary and interaction processes, but it is
sensitive to the choices concerning the natal kicks of neutron stars and black
holes.

  * $\bullet$

The disruption of the binary system produces an unbound stellar companion.
Remarkably, we find that these events only rarely produce runaway stars (i.e.
stars with peculiar motions larger than $30\,{\mathrm{km\ s^{-1}}}$). The
velocity distribution peaks at $\sim 6\,\mathrm{km\ s^{-1}}$, with a
$90^{\mathrm{th}}$ percentile around $20{\mathrm{km\ s^{-1}}}$ well below the
typical threshold for runaway stars.

  * $\bullet$

The slow-moving unbound companions, to which we refer as walkaway stars,
outnumber the runaway stars by at least an order of magnitude in nearly all
our models. This is a robust outcome of the models and is primarily due to the
reversal of the mass ratio prior to the explosion as we show numerically and
analytically. This finding has been noted in earlier studies (e.g., De Donder
et al., 1997; Eldridge et al., 2011), but we believe this result (and its
potential implications) is under appreciated.

  * $\bullet$

We also produce runaway stars but we rarely find them to be faster than
$60\,{\mathrm{km\ s^{-1}}}$, which corresponds to $98.9^{\mathrm{th}}$
percentile of our fiducial population. This is much lower than the theoretical
maximum set by the finite size of the stars and their orbit, which is in the
range 400–1000$\,{\mathrm{km\ s^{-1}}}$ depending on their masses and
evolutionary state. We expect that stars with larger peculiar velocities
result from dynamical ejections from clusters, capture from another Galaxy, or
most exotically, interaction with a intermediate mass or supermassive black
hole.

  * $\bullet$

Isolated binary evolution produces a runaway fraction among stars more massive
than $15\,M_{\odot}$ between $0.2-2.6\%$. None of the combination of
parameters that we consider produces a runaway fraction approaching the
observed value of $\sim$$10\%$ for O-type stars. This result is consistent
with previously published theoretical studies (see also Appendix A), and might
indicate that (i) the observed fraction of O-type stars that are runaways is
overestimated, possibly because of biases favoring the observation of fast and
isolated massive stars or because of the difficulties in defining the frame of
reference in which to measure the velocity, or (ii) contrary to previous
claims, it is not correct that the majority of observed O-type runaways come
from the disruption of binaries, or (iii) massive binary evolution models are
lacking some physical process allowing to produce short pre-collapse orbits
which would result in faster ejection velocities if disrupted.

  * $\bullet$

We find a mild trend for runaway stars to be faster and more common at lower
metallicity, caused by the fact that stars are on average smaller and the
reduced effects of stellar winds. _Gaia_ should in principle be able to test
these predictions by identifying the fastest runaway stars in the Magellanic
clouds. In our most metal poor simulations for $Z=0.0002$, which may have
relevance for the earliest stellar populations, we find 5 times more (massive)
runaway stars, if we keep all other assumptions for the initial distributions
the same. The sensitivity of our results to the assumed metallicity also
suggests that binary evolution might create systematic trends in the stellar
feedback with metallicity and galaxy size.

  * $\bullet$

Both runaways and walkaways typically accrete mass from their companion prior
to the disruption, This is especially true for those more massive than
$7.5\,M_{\odot}$ at the time they are ejected. They typically also gain
angular momentum and nuclearly processed material in the process. These
features might make walkaway stars recognizable as binary products even if
they do not stand out from a kinematic point of view. Their further evolution
and structure may also differ from that of non-rotating single stars, as has
been speculated in the context of the LBV phenomenon.

  * $\bullet$

We suggest that the high mass tail of the runaway mass function could provide
insight on the formation mechanism of black holes from the core-collapse of
the primary star, and in particular on whether these receive a “velocity” or a
“momentum” kick. So far, studies have focussed on the surviving X-ray
binaries, but we propose to use their unbound counter parts since they are
more common and should be identified by _Gaia_.

  * $\bullet$

By ejecting secondaries, the disruption of binaries can enhance the role of
massive stars as feedback engines. The average distance they travel,
calculated neglecting any external potential, is of order $\gtrsim 100$ pc,
and up to $\gtrsim 500$ pc if considering only the faster runaways. These
suggests that both runaways and walkaways can exit the dust cloud embedding
their birth location, thus changing the escape fraction for their ionizing
radiation. If they are massive enough, the ejected star will explode far from
other stars and gas overdensities, with potential implications for the driving
of turbulence and star formation.

Our simulations will be made available upon final publication of this study.
Early requests for access can be addressed to the lead author.

###### Acknowledgements.

We acknowledge for the many helpful discussions F. Broekgaarden, A .de Koter,
D. Hendriks, L. Kaper, N. Langer, I. Mandel, C. J. Neijssel, Ph.
Podsiadlowski, K. Sakar, F. N. Schneider, S. N. Shore, E. van den Heuvel, A.
Vigna-Gomez. MR is grateful to the KITP for the visiting graduate fellowship
during which a large portion of this study has been carried out. This research
was supported in part by the National Science Foundation under Grant No. NSF
PHY11-25915. SdM has received funding under the European Unions Horizon 2020
research and innovation programme from the European Research Council (ERC)
(Grant agreement No. 715063). RGI thanks the Science and Technology Facilities
Council (STFC) for funding his Rutherford fellowship under grant ST/L003910/1.

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##  Appendix A Comparison to selected previous population studies

Various earlier studies have simulated the production of binaries containing a
compact object and/or ejecting runaway stars. We briefly discuss a selection
that focuses on estimates for unbound companions and comment briefly on how
our findings compare.

De Donder et al. (1997) presented an extensive binary evolution study
discussing predictions for the O-type runaways as well as the systems that
remain bound. They consider a variety of assumptions for the initial
distributions and the uncertain physical parameters concerning the efficiency
of mass transfer, angular momentum loss and the common envelope ejection.
Generally, we find that our results agree well. They find that 16–23% of
systems remain bound after the SN explosion of the primary, which is
consistent with our findings $1-\mathcal{D}=14^{+22}_{-10}\%$. They find that
between 6% and 27% of the O-type are unbound former companions that are now
single, which is also agreement with our walkaway fraction (one but last
column in their table 1).

The find that 2–7% of the O-type stars have velocities large than
$30\,{\mathrm{km\ s^{-1}}}$ (final column in their Table 1), which is slightly
larger but still consistent with what we find. We expect that this difference
may be in part the result of a difference the treatment of mass transfer and
angular momentum loss, although the combined effect of further differences in
our assumptions will also contribute. Their default assumption is that a fixed
fraction $\beta_{\mathrm{RLOF}}=0.5$ of the material transferred during Roche-
lobe overflow leaves the system through the outer L2 Lagrangian point forming
a ring around the binary system, which leads to larger angular momentum loss,
shrinking the orbit further. This is somewhat similar to our model variation
where we assumed $\gamma_{\rm RLOF}=\gamma_{\rm disk}$ although our results
cannot be compared one to one, since we adopt a different, physically
motivate, assumption for the mass transfer efficiency. In our simulations, we
find that this assumption is not increasing the number of runaway stars,
because the fraction of system that merges increases too.

Eldridge et al. (2011) perform population synthesis simulations with a
detailed stellar evolutionary evolutionary code to investigate O and early B
type runaway stars. Their primary aim is the predict the spatial distribution
of different types of CCSN and gamma-ray bursts. They find a disruption
fraction, $\mathcal{D}=80\%$, in good agreement with our results,
$\mathcal{D}=86^{+10}_{-22}\%$. In their study, they use the term runaway to
refer to all unbound companions with velocities larger than $5{\mathrm{km\
s^{-1}}}$, which encloses the large majority of what we refer to as walkaway
stars, but they also quote estimates for stars faster than $30{\mathrm{km\
s^{-1}}}$ which corresponds to our definition of runaway stars. They predict a
walkaway (runaway) fraction of $2.2\%$ ($0.5\%$) for O type stars in their
simulations for $Z=0.02$. For comparison, for stars more massive than
$15\,{\mathrm{M}_{\odot}}$, which roughly corresponds to O type stars, we
estimate a walkaway (runaway) fraction of $10^{+4.7}_{-8.5}$%
($0.5^{+1.0}_{-0.3}$%). Our runaway fraction agrees very well. We find a
somewhat larger walkaway fraction, but we consider this a fairly good
agreement, given the uncertainties and differences in definitions that we have
adopted. Also the distance they estimate that ejected stars can reach is in
good agreement with our predictions and they also note that the majority
unbound companions accrete mass from their companions before the first core-
collapse. Our predictions for the velocities of bound systems are slightly
lower compared to Eldridge et al. (2011): the vast majority of our bound post-
core-collapse systems is slower than $30\,\mathrm{km\ s^{-1}}$, and there is a
high-velocity tail barely extending beyond about $100\,\mathrm{km\ s^{-1}}$.

Very recently, Boubert & Evans (2018) published a study investigating the
hypothesis that Be stars are products of mass transfer in binary systems (e.g.
Pols et al. 1991; de Mink et al. 2013). They compare the kinematics of a flux-
limited sample of Galactic Be stars with binary population synthesis
simulations. These simulations were obtained with a different version of the
binary evolutionary code that we use. Generally, our findings are in
agreement, despite the minor differences in the model assumptions. They also
find a large fraction of unbound companions, many of which are rapidly
rotating and moving at velocities slower than $30\,{\mathrm{km\ s^{-1}}}$
(e.g., their Fig. 6 and 7b), in good agreement with our Fig. 7. They also find
that the natal kick distribution does not greatly affect the resulting
velocity distribution and/or the runaway fraction and that the maximum
distance traveled by ejected Be stars is likely to be smaller than the
vertical scale height of the thin disk, in agreement with our findings.

We find some disagreement in the provided explanations of the theoretical
results. For example, the authors state in their section 3.2 that that mass
transfer shrinks the orbit and accelerates the secondary star. While this the
case initially, upon the onset of mass transfer, we find that the orbit
generally widens after the reversal of the mass ratio. The widening and the
inversion of the mass ratio both slow down the orbital velocity as we verified
both analytically Sec. 4 and by detailed inspection of representative example
systems, e.g. Sec. 3. In their Section 3.3, they state that “whether a binary
is disrupted by a supernova is principally determined by whether the primary
loses more than half its mass (Blaauw 1961), and the kick on the compact
object is only a second order effect”. We find instead that mass loss during
the explosion is rarely responsible for unbinding of a binary system that can
produce a runaway star. The amount of mass lost needs to exceed half of the
total mass of the system (and not half of the primary star), which is rarely
achieved in our simulations. This is because the CC progenitor loses most of
its mass during the preceding mass transfer phase. We find that the Blaauw
kick due to rapid mass loss is only important of initially very wide binaries,
in which the two stars have not exchanged mass prior to CC. This difference
may be in part due to the differences in the assumptions for the range of
initial orbital periods. The authors consider systems with initial orbital
periods up to $10^{10}$ days in their simulations, which means that the
majority of their progenitors should effectively evolve as single stars. We
consider instead systems up to $10^{5.5}$ days, which is more appropriate for
the more massive progenitors that we are interested in.

Many further studies investigated the populations of binaries in the context
of the formation of X-ray binaries and binary neutron stars and black holes.
Providing a complete overview and detailed comparison is beyond our present
scope. We discuss below a comparison to a limited set of studies.

Brandt & Podsiadlowski (1995) focus on binary systems remaining bound at the
first CC, with the aim of understanding the effects of natal kicks on X-ray
binaries. They estimate a disruption fraction in their calculations of
$\mathcal{D}\simeq 73-81\%$ (depending on the companion mass), in good
agreement with our results.

Kalogera (1996) presented a similar analytic study of the effects of natal
kicks on the systemic velocities of X-ray binaries. Our population synthesis
results agree in predicting systemic velocities of bound post-CC systems
generally lower than the pre-explosion orbital velocity, except with
$\sigma_{\mathrm{kick}}\gg v_{\mathrm{orb}}^{\mathrm{pre-CC}}$ (corresponding
to large $\xi$ in the notation used by Kalogera 1996).

Fryer et al. (1998) investigated the impact of NS natal kicks on the formation
of NS X-ray binaries and NS-NS binaries, but also present results for the
unbound companions. Assuming a bimodal kick distribution, they find that most
O/B-type ejected companion move slower than about $50{\mathrm{km\ s^{-1}}}$,
consistent with our findings.

Dray et al. (2005) focus on high-mass runaways, which become Wolf-Rayet stars
during or before their post-disruption evolution. They argue in favor of
significant BH kick amplitudes to explain the rarity of BH-WR binaries and the
observed velocity distribution of WR runaways. We reach similar conclusions
based on our simulations for different assumptions for the BH kicks. The
systemic velocities we find for bound systems are also in reasonable agreement
with those found by Dray et al. (2005)

Recently, Tauris et al. (2017) presented a detailed study of the evolutionary
processes leading to the formation of NS NS binary systems. They also find
that the majority of binaries hosting a NS after the first CC have systemic
velocities smaller than $\sim 30\,\mathrm{km\ s^{-1}}$, in good agreement with
the results shown in the left panel of Fig. 11.

##  Appendix B Output files

The outcome of our population synthesis calculations will be made available
upon publication. For early inquiries, please contact the lead author. Each
file corresponds to one parameter variation (see Sec. 2 and Sec. 6), and logs
the following information for each binary system where a star goes CC111111We
stop our computations at the first CC event..

  * •

primary ZAMS, pre-CC, and post-CC (corresponding to the NS or BH mass if the
primary is the star collapsing) masses in $M_{\odot}$ units: M1zams, M1preCC,
M1postCC;

  * •

secondary ZAMS, pre-CC, and post-CC masses in $M_{\odot}$ units: M2zams,
M2preCC, M2postCC;

  * •

fallback fraction $f_{b}$ for each star (set to zero for the star that is not
collapsing): fb1, fb2

  * •

evolutionary stage before and after CC (stellar types listed according to
Hurley et al. 2000): type1preCC, type1postCC, type2preCC, type2postCC

  * •

post-CC velocities of the 2 stars in the original frame, in $\mathrm{km\
s^{-1}}$: v1postCC, v2postCC;

  * •

pre-CC and post-CC eccentricity (the latter is -1 for mergers and disrupted
systems): e_preCC, e_postCC;

  * •

pre-CC and post-CC separation in $R_{\odot}$ units (the latter is set to 0 for
mergers and disrupted systems): a_preCC, a_postCC;

  * •

ZAMS, pre-CC, and post-CC periods in days: Pzams, PpreCC, PpostCC;

  * •

kick amplitude in $\mathrm{km\ s^{-1}}$: v_kick;

  * •

kick direction, with $\theta$ angle between the collapsing star orbital
velocity and the kick (see also TT98 for notation): theta, phi;

  * •

systemic velocity in $\mathrm{km\ s^{-1}}$: v_sys;

  * •

age of the system at the time of CC in Myr: t_explosion;

  * •

time left in the current evolutionary stage in Myr: t_remaining;

  * •

time spent by the system with at least one star more massive than 15
$M_{\odot}$: duration_*;

  * •

system probability (see below): Prob.

The system probability corresponds to the hyper-volume of the initial
parameter space ($M_{1}^{\mathrm{ZAMS}}$, $q^{\mathrm{ZAMS}}$,
$P^{\mathrm{ZAMS}}$) represented by each binary system in our model grid. In
other words, the probability of each system is the statistical weight of the
system seen as sampling point for the initial distributions. To construct
distributions of the output quantities (e.g. those presented in Fig. 5-11),
the properties of each system should be weigh with the corresponding
probability. Similarly, the mean value of a quantity $\langle x\rangle$ (e.g.
$\langle v\rangle$ in Table 1) should be calculated using:

$\langle x\rangle=\frac{\int xP(x)\,dx}{\int
P(x)\,dx}\equiv\frac{\sum_{i}x_{i}P_{i}}{\sum_{i}P_{i}}\ \ ,$ (7)

where $P$ is the probability, and the index $i$ runs over all the binary
systems in a population.

To calculate the average distance traveled by stars ejected by the binary
disruption, we use:

$\displaystyle\langle L\rangle\equiv\langle v\times\Delta t\rangle=$ (8)
$\displaystyle\frac{\sum_{i}\texttt{v2postCC}\times[\texttt{t\\_remaining}+0.1\times\tau_{\mathrm{MS}}(M_{2})]\times
P_{i}}{\sum_{i}P_{i}}\ \ ,$

where the second term in squared brackets accounts, albeit in a simplified
way, for the helium core burning duration of the rejuvenated star.

##  Appendix C Observable velocity distribution

Figure 13 shows the velocity distribution of MS stars ejected by the
disruption of binaries that can be directly compared to observations provided
that (i) the contribution of dynamical ejection can be separated in the
observed sample and (ii) the effects of the Galactic potential can be
neglected, i.e. effectively each ejected star moves in a straight line at
constant velocity for the remaining duration of its MS. This is the same
information of Fig. 5, but each bin is populated considering also the
remaining MS lifetime of the ejected star ($\tau_{\mathrm{MS}}$).

Figure 13: Velocity distribution of ejected stars, including the finite MS
lifetime to populate the bins (see also Fig. 5).  Figure 14: Same as Fig. 5,
but using a logarithmic scale for the velocity. The use of a logarithmic scale
allows for the display of a wider range of velocities. A minor peak in the
grey histogram can be seen between $100\lesssim v_{\mathrm{dis}}/\mathrm{km\
s^{-1}}\lesssim 400$, but is absent in the histograms for massive ejected
stars. Such high ejection velocities are reached through a common envelope
evolution without accreting mass.

##  Appendix D Pre-collapse distributions

We present in this appendix the pre-CC distribution in separation, mass of the
collapsing star, and mass of the companions for all the binaries with a MS
companion to the collapsing star in our fiducial simulation. Similar
distributions can be derived for all our parameter variations from the data
files that will be made available.These distributions can inform studies of
the interaction of the SN shock with the companion star (e.g. Wheeler et al.
1975; Liu et al. 2015; Rimoldi et al. 2016; Hirai et al. 2018).

We show in Fig. 15 the pre-CC separation distribution. The colors indicate
indicate the minimum mass of the MS companion (i.e., not of the collapsing
star). Roughly speaking, the two peaks shown in Fig. 15 correspond to the
orbital widening due to conservative (case A and early case B) and non-
conservative (late case B and case C) mass transfer phase. For all pre-CC
separations shorter than $10^{3}\,R_{\odot}$, the dashed and solid histograms
coincide, indicating that all these binaries have experienced a direct
interaction previously during the evolution.

Figure 15: Pre-CC separation distribution for binaries with a collapsing star
and a main sequence companion. Colors indicate the pre-CC mass of the MS
companion according to the legend. Dashed histograms indicate post-interaction
(RLOF or common envelope) binaries. The top panel shows the corresponding
cumulative distributions.

Figure 16 shows the mass distribution for the exploding star and the MS
companion ($M_{\mathrm{CC}}$, and $M_{2}$, respectively), at the pre-CC stage.
The combination of the distributions shown in Fig. 16 and Fig. 15, together
with the effects of the natal kick distribution results in the ejection
velocities in Fig. 5 which is our main result.

Figure 16: Pre-CC mass distribution for the exploding star ($M_{\mathrm{CC}}$)
and the companion ($M_{2}$). We plot all systems where the companion is a MS
star at the time of the explosion, regardless of whether the binary is
disrupted or not. Dashed histograms indicate post-interaction (RLOF or common
envelope) binaries. The top panel shows the corresponding cumulative
distributions.