A source of high--velocity white dwarfs
Melvyn B.\ Davies$^1$
Andrew King$^1$
and Hans Ritter$^2$
$^1$ Department Physics \& Astronomy
Leicester LE1 7RH
UK
$^2$ Max--Planck--Institut f\"ur Astrophysik
Karl--Schwarzschild-Str.~1
D--85740 Garching
Germany
We investigate whether the recently--observed population of high--velocity white dwarfs can be derived from a population of binaries residing initially within the thin disk of the Galaxy. In particular we consider binaries where the primary is sufficiently massive to explode as a type II supernova. A large fraction of such binaries are broken up when the primary then explodes as a supernova owing to the combined effects of the mass loss from the primary and the kick received by the neutron star on its formation. For binaries where the primary evolves to fill its Roche lobe, mass transfer from the primary leads to the onset of a common envelope phase during which the secondary and the core of the primary spiral together as the envelope is ejected. Such binaries are the progenitors of X-ray binaries if they are not broken up when the primary explodes. For those systems which are broken up, a large number of the secondaries receive kick velocities $\sim 100 - 200$ kms$^{-1}$ and subsequently evolve into white dwarfs. We compute trajectories within the Galactic potential for this population of stars and relate the birthrate of these stars over the entire Galaxy to those seen locally with high velocities relative to the LSR. We show that for a reasonable set of assumptions concerning the Galactic supernova rate and the binary population, our model produces a local number density of high--velocity white dwarfs compatible with that inferred from observations. We therefore propose that a population of white dwarfs originating in the thin disk may make a significant contribution to the observed population of high--velocity white dwarfs.
psbox.tex
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%       By Jean Orloff
%       Comments & suggestions by e-mail: ORLOFF@surya11.cern.ch
%       No modification of this file allowed if not e-sent to me.
%
% WHAT IS IT:
% psbox is a set of machine-independent TeX macros to
% 1) allow (Encapsulated) PostScript figure inclusion in all versions
%    of TeX (Plain, LaTeX) on all machines using a PostScript printer
% 2) facilitate the communication (e-mail, ftp, ...) of all the files
%    (text, macros, figs) needed to reproduce a TeX document by grouping
%    them together into a single, TeXable file.
% For more info, get the file pub/TeX/psbox/PSBOXALL.TEX by anonymous
%     ftp from cs.nyu.edu(=128.122.140.24)
%
% History:
%  1.34  \readfilename=final fix for all filename scans; try \psforptips
%  1.33: corrects \psnewinput for LaTeX (still fails if fname=a{b}c)
%  1.32: corrects \psfordvialw and adds .TEX to PSBOXALL(!)
%  1.31: adds \psfordvialw(?)
%  1.30: adds \splitfile & \joinfiles for multi-file management
%  1.24: fix error handling & add \psonlyboxes
%  1.22: makes \drawingBox \global for use in Phyzzx
%  1.21: accepts %%BoundingBox: (atend)
%  1.20: tries to add \psfordvitps for the TeXPS package.
%  1.10: adds \psforoztex, error handling...
%2345678 1 2345678 2 2345678 3 2345678 4 2345678 5 2345678 6 2345678 7 23456789
%
% Checking version no to avoid multiple loadings
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%
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\def\psfordvialw{%   Try for dvialw, a UNIX public domain
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%\def\psfor...  add your own!
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\begin{document}
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\input epsf.sty
\input psbox.tex

\title{A source of high--velocity white dwarfs}
\author[M. B. Davies et al]{Melvyn B.\ Davies$^1$,
Andrew King$^1$, and Hans Ritter$^2$\\
$^1$ Department Physics \& Astronomy, University of Leicester, Leicester LE1
7RH, UK\\
$^2$ Max--Planck--Institut f\"ur Astrophysik,
Karl--Schwarzschild-Str.~1, D--85740 Garching, Germany}

\maketitle            
\begin{abstract}
We investigate whether the recently--observed population of high--velocity
white dwarfs can be derived from a population of binaries residing initially
within the thin disk of the Galaxy. In particular we consider binaries
where the primary is sufficiently massive to explode as a type II supernova.
A large fraction of such binaries are
broken up when the primary then explodes as a supernova owing to the
combined effects of the mass loss from the primary and the kick received
by the neutron star on its formation.
For binaries where the primary evolves 
to fill its Roche lobe, mass transfer from the primary
leads to the onset of a common envelope phase during which
the secondary and the core of the primary spiral together as the
envelope is ejected. Such binaries are the progenitors of 
X-ray binaries if they are not broken up when the primary explodes.
For those systems which are broken up, a large number of the 
secondaries receive kick velocities $\sim 100 - 200$ kms$^{-1}$
and subsequently evolve into white dwarfs. 
We compute trajectories within the Galactic potential for this population
of stars and relate the birthrate of 
these stars over the entire Galaxy to those
seen locally with high velocities relative to the LSR. 
We show that for a reasonable set of assumptions concerning the Galactic
supernova rate and the binary population, our model produces a local
number density of high--velocity white dwarfs compatible with that inferred
from observations. We therefore propose that a population of
white dwarfs originating in the thin disk 
may make a significant contribution to the 
observed population of high--velocity white dwarfs. 
\end{abstract}
\begin{keywords}
accretion, accretion discs --- binaries: close
stars: evolution --- stars: stars.
\end{keywords}

\section{introduction}


Oppenheimer et al (2001) have reported a population of high-velocity wide dwarfs
in the solar neighbourhood
which they claim to be halo objects.
From their observations they infer a local number density of such objects
to be $\sim 10^{-4}$ pc$^{-3}$ much larger than previous estimates
(Gould, Flynn \& Bahcall 1998).
Reid et al (2001) however, claim these stars 
are simply the high--velocity tail of the so-called
thick disk population. This result is controversial; for example
Hansen (2001) notes that the ages of the white dwarfs
appear to have a spread, whereas the thick disk stars have 
similar ages (around 12 Gyr old).



We investigate whether the observed population can be
derived from the binary population within the thin disk. 
Recently, Koopmans \& Blandford (2001) suggested that young white dwarfs
could be ejected from the thin disk to the halo, through orbital
instabilities in triple--star systems. In this paper, we consider 
a simpler scenario, where the secondary of a tight binary is ejected 
when the primary explodes as a supernova, breaking up the binary
through the combined effects of mass loss
in the supernova explosion and the kick received by the
neutron star. To produce the high--velocity
objects seen by Oppenheimer et al, one must impart kicks in the range
100 -- 200 km/s to stars in the thin disk. As the kick velocity the
secondary receives is similar to the star's binary orbital velocity at the
moment of the supernova explosion, we require the stars to be in 
tight binaries, with separations much smaller than the size of a 
pre--supernova red giant. This in turn implies that the binaries
must have passed through a common envelope phase.
Once ejected, such secondaries evolve into white dwarfs in a relatively
short time if they are sufficiently massive.


Our evolutionary scenario is discussed in Section 2 of this paper
and described in more quantitative detail in Appendix 1.
In Section 3, we calculate the distribution of kicks
received by the secondaries. In Section 4, we compute trajectories
of secondaries within the Galaxy to determine whether they 
can produce a population of high--velocity objects similar
to those seen by Oppenheimer et al.
The likely production rate of such objects is discussed in Section 5
and Section 6 gives our conclusions.



\begin{figure}
\begin{center}
\begin{minipage}{74mm}
\psboxto(74mm;0cm){MB604fig1.ps}
\end{minipage}
\end{center}
\caption{An evolutionary scenario for producing high-velocity main-sequence
stars, some of which will later evolve into white dwarfs.
Beginning with two main-sequence stars (phase 1), the primary evolves
into a red giant (phase 2) filling its Roche lobe, leading to the onset
of a common envelope phase (3) where the red-giant envelope engulfs
the main-sequence star secondary and the helium-star core. The 
post-common-envelope system consists of a tight binary  containing the
helium-star core and the main-sequence star (phase 4). The compact
star loses its helium envelope either via a wind or mass transfer,
leaving a carbon-oxygen core (phase 5), which then explodes as a supernova,
with the main-sequence star secondary often being ejected at high speed
(phase 6).}
\end{figure}

\section{Evolutionary scenario}


The evolutionary scenario for producing high-velocity
stars considered in this paper is shown in Figure 1. 
We consider the evolution of
binary stars where convective mass transfer occurs when the
primary evolves off the main sequence and fills its Roche lobe. 
A common envelope phase follows.
The secondary and the core of the primary
spiral together as the envelope is ejected.
After the common envelope phase, further mass transfer occurs
if the primary (now a helium star) evolves to fill its Roche
lobe for a second time. This (conservative) mass transfer 
increases the mass of the secondary but may also lead 
to the merger of the helium star and main--sequence star. Alternatively, if
the primary fails to fill its Roche lobe, mass--loss occurs
via a wind. In either case, the carbon-oxygen core of the primary remains.
Providing the primary was initially sufficiently massive, 
it then explodes as a supernova. The secondary
and the newly-formed neutron star become unbound in a large
fraction of such binaries
through the combined effects of mass--loss during the
supernova explosion and the kick received by the neutron star
on its formation. The kick velocity received by the secondary
will be similar to its orbital velocity at the moment of the 
explosion. Hence in very wide binaries,  where a common envelope phase
is avoided, such velocities will be relatively low; too low to produce
the high-velocity stars we require. However we will see in Section 4
that kick velocities $\go 100$ kms$^{-1}$ are realised in a subset of 
binaries which have passed through a common envelope phase. 
A large fraction of the high-velocity secondaries subsequently
evolve into white dwarfs, especially if a large fraction of secondaries
are relatively massive.
High--velocity stars should also be visible before they evolve into
white dwarfs. For example Hoogerwerf, de Bruijne \& de Zeeuw (2001)
and Magee et al (2001) both observe a population of high--velocity
O and B--type main-sequence stars.



It is possible that the helium star in the wider systems
does not have sufficient time
to lose its entire He--rich envelope via a wind before exploding as
a supernova. There would be two consequences of this. Firstly, the
binary would be closer when the primary explodes as a supernova
hence the secondary would have a higher orbital velocity. 
Secondly, the primary ejects more mass when it explodes, hence
a larger fraction of binaries would be broken up.


There are a number of restrictions on the initial separations of binaries
for the scenario described above. 
The system must initially be wide enough that the primary has evolved off
the main sequence, and be convective, before it fills its Roche lobe.
Yet the binary must not be too wide, otherwise no mass transfer will occur.
We require that the main-sequence-star secondary not
fill its Roche lobe at the end of the common
envelope phase. If it does, we assume that the two stars merge and remove
such binaries from further consideration.


The evolutionary scenario described above is discussed more quantitatively
in Appendix 1.




\section{Population produced}

\begin{figure}
%\psboxto(\hsize;0cm){MB604fig2.ps}
\psboxto(\hsize;0cm){figs/lavslm2.12.0a.ps}
\caption{Plot of initial separation $a_i$, for
$M_1 = 12$ M$_\odot$, as a function of secondary mass
$M_2$(both in solar units) showing constraints for the formation of
a binary as described in Section 3 and Appendix 1.}
\end{figure}




In this section we consider the population of objects produced
via the evolutionary scenario described in the previous section.
If we assume
that all stars of mass 8.3 M$_\odot \leq M_\star \leq$ 40 M$_\odot$
produce type II supernovae, and that the IMF is given by
$dN/dM \propto M^{-2.35}$, the majority of supernovae
originate from stars less massive than 20 M$_\odot$. 
For illustrative purposes, we consider binaries of primary
mass $M_1 = 12$ M$_\odot$. In Figure 2 we plot the initial separations
as a function of secondary mass, $M_2$, showing the various constraints
for the evolution of binaries as described in Appendix 1.
The lowest of the three solid lines represents the minimum 
separation required to avoid mass transfer whilst the primary is still
on the main sequence.
The top two solid lines are the limits represented by equation (A1).
Systems above the uppermost of the three lines
will remain detached throughout the evolution of the primary. 
Those below the uppermost line, but above the middle line, will
undergo mass transfer after the primary has evolved
off the main sequence and become convective (and thus enter a common envelope
phase) while those below the middle line but above the bottom line
will undergo mass transfer while the primary is still
radiative. This will also lead
to a common envelope phase if $M_2 \lo M_1 /3$. In this case, 
the common envelope phase will result in the merger of the
primary core and the secondary.
The dotted line is the limit set by requiring that the 
main-sequence star secondary remains detached immediately after the common
envelope phase (ie equation [A5], where to produce Figure 2 we
have set $\alpha_{\rm CE} \lambda_{\rm CE} = 0.5$).
The dashed line is the minimum initial separation to avoid the 
helium-star primary filling its Roche lobe when it evolves after the common
envelope phase (ie equation [A6]). In the case of systems
of primary mass, $M_1 = 12$ M$_\odot$, mass transfer from the Helium 
star leads to a merger with the main--sequence star secondary
for $M_2 \lo 3$ M$_\odot$. Hence such systems will not produce
high--velocity white dwarfs.
The black dots in Figure 2 each represent systems 
evolving as described in the previous section,
where the secondary is in a tight binary around the primary
which explodes as a supernova.
This region is limited from above by the requirement that the primary
fills its Roche lobe (upper limit in equation [A1]) and from below
by the requirement that the
main-sequence star secondary remains detached immediately after the common
envelope phase (ie equation [A5]).
For systems where  the post-CE binary is sufficiently wide that
the helium star does not fill its Roche lobe, this star
loses its envelope in a wind, with the binary separation increasing
slightly as given by equation (A10).
In addition to the constraints shown in Figure 2, there is a lower
limit on the mass of the secondaries ($M_2 \geq 0.8 M_\odot$)
if the secondaries are to evolve into white dwarfs during the age of
the Galaxy.
The dotted region in Figure 2 will shift slightly for other 
values of $\alpha_{\rm CE} \lambda_{\rm CE}$. However the range of
binary separations at the moment of the supernovae explosion, and thus
the distribution of kicks received by the secondary are similar. 


A similar picture to that shown in Figure 2 emerges for primary
masses  $M_1 \lo 12$ M$_\odot$. For more massive primaries, 
the helium-star primary never fills its Roche lobe in the systems of 
interest (as higher--mass
stars produce higher--mass helium stars which evolve to have smaller radii).

\begin{figure}
%\psboxto(\hsize;0cm){MB604fig3.ps}
\psboxto(\hsize;0cm){figs/vvsm2psn.12.0long.ps}
\caption{Plot of kick velocities received by secondaries (in km/s) 
contained in binaries which are broken up when the primary explodes
as a supernova. The binary population is that shown in Figure 2
where the initial primary mass is 12 M$_\odot$ in all cases.  The kick
velocities are shown as a function of secondary mass $M_2$ (in solar
units).}
\end{figure}


We now consider the effects of the supernova explosion on the binaries.
We allow for the effect of mass--loss from the primary
(we assume the post-supernova mass of the primary is $1.4$ M$_\odot$).
We also include the kick received by the neutron star at formation,
using a kick distribution from Hansen \& Phinney (1997).
Approximately two-thirds of the binaries identified in Figure 2
are broken up.
We show the kick velocity received by the secondaries,
 as a function of secondary mass, in Figure 3.
Here we have taken, as example, a Salpeter IMF for the secondary
masses, ie $dN/dM_2 \propto M_2^{-2.35}$.
We note from this figure that the kick velocity is often above 100 km/s.
The distribution of kick velocities is similar for other values
of primary mass.
We also note that in all cases $M_2 \go 0.9$ M$_\odot$, even though
here we have assumed a Salpeter--like distribution for the secondary
masses. The median mass for the secondaries is $\sim 3$ M$_\odot$ hence
we would expect the vast majority of ejected secondaries to have
evolved into white dwarfs by today.
In Section 4 we use this kick distribution to compute
the resulting spatial distribution of secondaries throughout the
Galaxy. 

\begin{figure}
%\psboxto(\hsize;0cm){MB604fig4.ps}
\psboxto(\hsize;0cm){figs/rvsz.12salp.ps}
\caption{The positions of 1000 secondaries which have received
kicks drawn randomly from the distribution shown in Figure 3 ($M_1 = 
12$ M$_\odot$ and initial secondary masses are drawn from a Salpeter 
distribution).
The trajectories have been followed for
10 Gyr and the positions noted every 50 Myr.
The positions are shown in cylindrical coordinates, with units of kpc.}
\end{figure}


\section{Trajectories of high--velocity white dwarfs}


Using the model for the Galactic potential described in
Appendix 2, we now compute the subsequent trajectories
of the secondaries of the binaries which are broken up when the
primary explodes as a supernova. For illustration
we again consider binaries of initial primary mass $M_1 = 12$ M$_\odot$,
with the secondaries having a Salpeter--like distribution of masses.
Hence we use the kick distribution given in Figure 3. 
We will assume that the initial (circular) orbital radius of the
binary in the Galaxy is drawn uniformly from the range  1 kpc -- 10 kpc.
We then apply to the secondary
a kick drawn randomly from our kick distribution (and directed
in a random direction with respect to the original motion within
the Galaxy) and integrate its motion within the Galactic
potential for 10 Gyr, writing out its position and velocity every 50 Myr.
We consider 1000 trajectories. 
A plot of all the positions (in cylindrical coordinates) is shown in Figure 4.
We note that a large fraction of positions are relatively close
to their birth site. In Figure 5 we plot the tangential and radial velocities
of all stars within the limits 8 kpc $\leq R \leq$ 9 kpc
and $-0.5$ kpc $\leq z \leq 0.5$ kpc.
We see that the distribution of velocities in the 
$U-V$ plane is encouragingly similar to that reported by Oppenheimer
et al. The distribution shown in Figure 5 will clearly depend on
the distribution of kicks received by the secondaries, which will
depend on the distribution of secondary masses as shown in Figure 3.



\begin{figure}
%\psboxto(\hsize;0cm){MB604fig5.ps}
\psboxto(\hsize;0cm){figs/vphivsvr.12salp.ps}
\caption{Plot of velocities (in kms$^{-1}$) in the U--V plane for 
a subset of positions in Figure 4 given by
8.0 kpc $\leq R \leq$ 9.0 kpc and $-0.5$ kpc $\leq z \leq 0.5$ kpc.}
\end{figure}

\section{production rate}


In the previous section we showed that a reasonable set of assumptions
for the kick distribution and the initial positions of the binaries
yields a distribution of stellar velocities
in the solar neighbourhood not unlike that seen by Oppenheimer et al.
However, we have not yet addressed the issue of formation rates of
such systems. The formation rate is 
dependent on a number of uncertain factors: the global supernova
rate within the Galaxy, the distribution of primary masses, 
the binary fraction, the distribution of secondary
masses for a given primary mass, the distribution of initial binary
separations, and the initial 
spatial distribution of binaries within the Galaxy.


Given all these uncertainties, it is
impossible to give a precise figure for the local number density of white
dwarfs produced as described in this paper. Rather we can
obtain a reasonable idea of the order of magnitude of the likely 
population given a reasonable set of assumptions.
Experimentally we note that some 0.3\% of systems plotted in Figure
4 are within the limits 8.4 kpc $\leq R \leq$ 8.6 kpc
and $-0.1$ kpc $\leq z \leq 0.1$ kpc. The volume of this region
is $V \sim 2 \times 10^9$ pc$^3$. If we assume a (slightly high) supernova rate
of $10^{-1}$/yr, the total number of type II supernovae over the entire
life of the Galaxy is $N_{\rm SN} \sim 1.5 \times 10^9$. The local number
density of ejected secondaries is given by $N_{\rm SN} / V \sim 2 \times
10^{-3} f_2$ pc$^{-3}$, where $f_2$ is the fraction of type II supernovae 
occurring in the sorts of binaries discussed in Section 3.
In other words the process described here will produce the population
of objects observed by Oppenheimer et al if $f_2 \go 0.05$. 
It is quite possible that another set of assumptions would
lead to an even higher production rate than the one derived here. 


\section{Conclusions}


We have described a process producing a population of 
high--velocity stars from binaries in the thin disk.
If the primary fills its Roche lobe after leaving the main sequence,
convective mass transfer leads to a
common envelope phase. The separation between the core of the
primary and the secondary decreases significantly as the
envelope is ejected. 
A second phase of mass transfer may occur if
the primary (now a helium star) evolves to fill its Roche lobe.
This phase is conservative and therefore
increases the mass of the secondary.
Alternatively if the helium star fails to fill its Roche lobe, 
mass loss will occur via a wind. In both cases, the carbon-oxygen core of the
primary remains. 



For initial primary masses, $M_{\rm 1,i} \geq
8.3$ M$_\odot$, this carbon-oxygen core then 
explodes as a type II supernova.
Most of the binaries become unbound through
the combined effects of mass-loss in the supernova explosion 
and the kick received by the neutron star at formation. 



We have computed the distribution of kick velocities for these 
secondaries. Considering a population of binaries initially on
circular orbits within the disk of the Galaxy, we apply kick
velocities drawn from the distribution in random directions
and integrate the motion of the ejected secondaries within the
Galaxy. We show that for a reasonable set of assumptions, 
we are able to obtain a population of high-velocity stars
similar to that seen by Oppenheimer et al. There are number
of uncertainties associated with calculating a production rate
of these stars, but for a reasonable set of assumptions, we
show that this population may make a significant contribution 
to the population observed by Oppenheimer et al.


\bigskip

\paragraph*{Acknowledgments}

MBD gratefully acknowledges the support of a URF from the Royal Society.
Theoretical astrophysics research at Leicester is supported by a rolling grant
from the UK Particle Physics \& Astronomy Research Council.


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\bigskip

\appendix{\noindent {\bf APPENDIX 1: EVOLUTIONARY SCENARIO}

\bigskip


 In order for the late
(convective) Case B mass transfer from the primary to occur, leading
to the onset of a common envelope phase, we require that the
primary fill its Roche lobe, but not until it has a radius
larger than the maximum allowed for early (radiative) Case B
mass transfer. This requires

\begin{equation}
{R_{\rm max,rB}(M_{\rm 1,i}) \over f_1(q_{\rm i})}
< a_{\rm i} < {R_{\rm max,B}(M_{\rm 1,i}) \over f_1(q_{\rm i})}
\end{equation}
where $q_{\rm i}= M_{\rm 1,i}/ M_{\rm 2,i}$ and $R_{\rm max,B}(M_{\rm 1,i})$
is the maximum radius allowed for Case B mass transfer for a star
of mass $M_{\rm 1,i}$.
The radius of star 1 when it fills its Roche lobe
is given by $R_{\rm 1,i} = f_1(q_{\rm i}) a_{\rm i}$,
where $f_1(q)= { 0.49 q^{2/3} / (0.6 q^{2/3} + {\rm ln}(1+q^{1/3})) }$
(Eggleton 1983).  


The mass of the primary after the common envelope phase is given by


\begin{equation}
M_{\rm 1,CE} = M_{\rm He}(M_{\rm 1,i})= a M_{\rm 1,i}^b
\end{equation}
where $a=0.125$ and $b=1.4$ (van den Heuvel 1994).


The inspiral during the common envelope phase may be computed
by equating the change in total energy  of the two stars to
the binding energy of the envelope up to an efficiency
$\alpha_{\rm CE}$, ie $E_{\rm env} = \alpha_{\rm CE} \Delta E$.
Here $E_{\rm env}$ can be written in the following form (Webbink 1984)

\begin{equation}
E_{\rm env} = { G M_{\rm 1,i} ( M_{\rm 1,i} - M_{\rm 1,CE})
\over \lambda_{\rm CE} f_1(q_{\rm i}) a_{\rm i} }
\end{equation}


Combining equation (A2) with the expression
for $\Delta E$, after some rearrangement
we arrive at the following expression for the inspiral during the common
envelope phase


\begin{eqnarray}
F_{{\rm i} \rightarrow {\rm CE}} = { a_{\rm CE} \over a_{\rm i} } & = &
( {2 M_{1,{\rm i}} (
M_{1,{\rm i}} - M_{1,{\rm CE}}) \over \alpha_{\rm CE} \lambda_{\rm CE}
f_2(q_{\rm i}) M_{1,{\rm CE}} M_{2,{\rm i}} } + \nonumber \\
&& { M_{1,{\rm i}} \over M_{1,{\rm CE}} } )^{-1}
\end{eqnarray}


We require that the main-sequence-star secondary not 
fill its Roche lobe at the end of the common
envelope phase. If it does, we assume that the two stars merge and remove
such binaries from further consideration.

\begin{equation}
a_{\rm i} >
{R_{\rm MS}(M_{\rm 2,CE}) \over
F_{{\rm i} \rightarrow {\rm CE}} f_2(q_{\rm CE})}
\end{equation}
where $R_{\rm MS}(M_{\rm 2,CE})$ is the radius of the 
main-sequence-star secondary and $q_{\rm CE}= M_{\rm 1,CE}/M_{\rm 2,CE}$.


The primary will now be a helium star. Such stars will expand as they 
evolve and may, in turn, fill their Roche lobe if

\begin{equation}
a_{\rm CE} < {R_{\rm max,He}(M_{\rm 1, CE}) \over
f_1(q_{\rm CE}) }
\end{equation}
Under these conditions,
conservative (Case BB) mass transfer will follow, where the
envelope of the helium star will be transferred to the secondary
leaving the carbon-oxygen core of the primary.
The masses of the two stars
$M_{1,{\rm CO}}$ and $M_{2,{\rm PSN}}$
after such a phase of mass transfer are given by

\begin{equation}
M_{\rm 1,CO} = M_{\rm CO}(M_{\rm 1,He})
\end{equation}


\begin{equation}
M_{\rm 2,PSN} = M_{\rm 2,i} + (M_{\rm 1,He} - M_{\rm 1,CO})
\end{equation}


The ratio of the separation after this phase to the separation after the
common envelope phase is given by 

\begin{equation}
F_{{\rm CE } \rightarrow {\rm PSN}} = {a_{\rm PSN} \over a_{\rm CE}} =
{M_{1,{\rm CE}}^2 M_{2,{\rm i}}^2 \over  M_{1,{\rm CO}}^2 M_{2,{\rm PSN}}^2}
\end{equation}


If, on the other hand, the He star never fills its Roche lobe,
the envelope will be lost via a wind. Leaving behind the carbon-oxygen core,
while the secondary mass will be unchanged (ie $M_{2,{\rm PSN}} =
M_{\rm 2,CE}$).
The ratio of the separation after this phase to the separation after the
common envelope phase is given by


\begin{equation}
F_{{\rm CE } \rightarrow {\rm PSN}} = {a_{\rm PSN} \over a_{\rm CE}} =
{M_{\rm 1,He} + M_{\rm 2,CE} \over M_{1,{\rm CO}} + M_{2,{\rm PSN}} } 
\end{equation}


The functions
$R_{\rm max,rB}(M)$,
$R_{\rm max,B}(M)$,
$R_{\rm max,He}(M)$,
and $M_{\rm CO}(M)$,
were tabulated from Habets (1986), Paczynski (1971) and Bressan
et al (1993) and linear interpolation was used to
evaluate them for a particular mass, $M$.


In order for the primary to explode as a Type II supernova, we
require that the core mass exceeds the Chandrasekhar limit,
ie $M_{1,{\rm CO}} > M_{\rm CH}$. For the assumptions here
(equations [A2] and [A7]), this is equivalent to $M_{\rm 1,i}
> 8.3$ M$_\odot$. 
The mass-loss (from the supernova explosion)
and the kick received by the neutron star may unbind
the binary. 


If the binary is broken up, the kick velocity of the secondary
will be similar to its orbital velocity at the moment of the 
supernova explosion, which is given by

\begin{equation}
V_{\rm 2,PSN} = { M_{\rm 1,CO} \over M_{\rm 1,CO} + M_{\rm 2,PSN} }
\sqrt{{G ( M_{\rm 1,CO} + M_{\rm 2,PSN} ) \over a_{\rm PSN} }}
\end{equation}

\bigskip

{\noindent {\bf APPENDIX 2: THE GALACTIC POTENTIAL}

\bigskip

The Galactic potential can be modelled as the sum of three
potentials. The spheroid and disc components are given by

\begin{equation}
\Phi_s(R,z) = { G M_s \over \left( R^2 + [a_s + (z^2 + b_s^2)^{1/2}]^2
\right)^{1/2}}
\label{P1}
\end{equation}

\begin{equation}
\Phi_d(R,z) = { G M_d \over \left( R^2 + [a_d + (z^2 + b_d^2)^{1/2}]^2
\right)^{1/2}}
\label{P2}
\end{equation}
where $R^2 = x^2 + y^2$. The component from the Galactic halo
can be derived assuming a halo density distribution, $\rho_{\rm h}$,
given by

\begin{equation}
\rho_h = {\rho_c \over 1 + (r/r_c)^2}
\label{P4}
\end{equation}
where $r^2 = x^2 + y^2 + z^2$. The above density distribution yields
the potential

\begin{equation}
\Phi_h = - {G M_c \over r_c} \left[ {1 \over 2} {\rm ln} \left(
1+ {r^2 \over r_c^2} \right) 
+ {r_c \over r} \ {\rm atan} \left({r \over r_c} \right)
\right]
\end{equation}
where $M_c = 4 \pi \rho_c r_c^3$. The total Galactic potential is
the sum 

\begin{equation}
\Phi=\Phi_s + \Phi_d + \Phi_h	
\end{equation}
Following Paczynski (1990), we use the following choice of parameters:

\begin{equation}
a_s=0, \ b_s = 0.277 \ {\rm kpc}, \  M_s = 1.12 \times 10^{10} M_\odot,  
\end{equation}

\begin{equation}
a_d=3.7 \ {\rm kpc}, \ b_d = 0.20 \ {\rm kpc}, \ 
 M_d = 8.07 \times 10^{10} M_\odot,  
\end{equation}

\begin{equation}
r_c = 6.0 \ {\rm kpc}, \  M_c = 5.0 \times 10^{10} M_\odot, 
\end{equation}
Because of the cylindrical symmetry of the potential, the integration
of the trajectories can be simplified to consider the evolution of
the $z$ and $R$ components only, as given below

\begin{eqnarray}
{dR \over dt} &=& v_R, \ \ {dz \over dt} = v_z, \nonumber \\
{dv_R \over dt} &=& \left( {\partial \Phi \over \partial R} \right)_z
+ { j_z^2 \over R^3}, \ \ {dv_z \over dt} = \left( {\partial \Phi 
\over \partial z } \right)_R
\label{P9}
\end{eqnarray}
where the $z$ component of the angular momentum, $j_z = R v_\phi$.
%The circular velocity for the total
%galactic potential is plotted as a function of galactic radius
%in Fig. 1.

%The total energy $E=v^2/2 - \left( \Phi_s + \Phi_d + \Phi_h \right)$.	

\end{document}



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