In our calculations we assumed wind velocities of the order of the
escape velocity from the companion star of mass
, Mach numbers of order unity, orbital
distances of order 
, and a wind efficiency of
conversion of irradiating energy into outflow kinetic energy 
of
less than a few per cent[5]. We find that for relatively large
irradiating energy fluxes, for example 
, and pulsar luminosities
(typical of systems such as
PSR1957+20), the shock at the inner binary tends to produce a velocity
that forces the evaporated material to escape from the binary to
infinity with a typical two-wing shaped spiral (see
Fig. 1). Material in the 'inner' wing interacts with the
'outer' wing, and additional shocks may occur at large distances.
We find, however, that for smaller irradiating fluxes (for example,
and
) a substantial fraction of
the mass outflow remains bound at large distances even after having
been 'reprocessed' by the pulsar termination shock. The pulsar may in
this case be totally or partially enshrouded and the outflowing
material is shielded from the direct action of the pulsar wind. Figure
2 shows the results of a two-dimensional hydrodynamic model
characterized by a Roche-lobe-filling companion of mass
, pulsar luminosity 
, initial gas temperature at injection on the irradiated half
of the companion's surface 
, Mach number
M=1, and 
. Figure 2 a shows the details
of the mass outflow near the binary, and Fig. 2b gives a
large-scale view of the same mass outflow with the inclusion of a
ballistic trajectory representing the behaviour of non-interacting
mass particles with velocities and initial directions obtained from
the hydrodynamic calculation. We note that the outflowing material has
in this case negative total energy and that the maximum radius reached
is 
times the original orbital radius.
 |
Figure 1: SVP with mass outflow escaping to infinity, 1.6 orbital periods after starting the simulation. The pulsar companion is filling its Roche radius, with  . The initial
temperature of the outflow is  the mass-loss rate is
 , the initial velocity of
the gas is  , the period of the binary orbit is 5h and the
luminosity of the pulsar is  .
The pulsar is at the point (0, 0) and the orbital motion is
counterclockwise. The units of length are in  . The
plot is in the corotating frame. |
In a realistic case, the mass particles cannot spiral inward because
of the interaction with the mass outflow and therefore the outflowing
material in the orbital plane can accumulate in a circumbinary disk
for a relatively long time. For a disk mass 
Earth
masses (from the properties of PSR1257+12) we obtain the estimate of the mass
loss rate from the companion as 
, where 
is the SVP evolutionary timescale (see below). The orbital angular
momentum of the companion in the original orbit is 
, where 
and 
with a the
orbital distance. On the other hand, the total angular momentum of the
two planets orbiting PSR1257+12 is 
, i.e., smaller than the original binary angular momentum by a
factor 
with 
for an
ideal solar composition of the circumbinary material, and with 
for material originating from a He companion. Therefore,
the fraction of the initial angular momentum which needs to
be stored in a circumbinary disk depends on the chemical
composition of the companion. Heavy elements are more likely to be
gravitationally bound and form planets, whereas the remaining H, He,
and C,N,O elements may be evaporated away from the binary system.
The rest of the angular momentum is redistributed by inward binary
motion and outward mass outflow.
Angular momentum transport in the mass outflow may occur
because of turbulent viscosity induced by hydrodynamic shocks and,
possibly, by magnetic instabilities[17]. The timescale for
viscous transport of angular momentum is 
,
where r is the radius and 
the viscosity, and it is expected to
be of order 
yr for the densities, temperatures and
distances involved in outer disks (see also [18], [19]). The
angular momentum transported outwards influences the binary orbit,
which is therefore expected to shrink as in the case of PSR1957+20
[20]. The expanding gas cools because of adiabatic losses and
radiative losses. The upper limit to the temperature at a distance all
obtained from adiabatic losses is 
where 
is the initial temperature
in units of 
, and additional shocks in the outflow will
decrease the temperature even further because of radiative losses. The
pulsar may be completely enshrouded and therefore 'hidden'[14,21,22], and the outflowing gas is exposed only to the high-energy
radiation (X-rays and 
) produced near the
termination shock of the pulsar windy[12,14]. The evaporated gas
at large distance from the binary is expected to be only marginally
affected by high-energy radiation as detailed calculations suggest for
PSR1744-24A [8]. In this case, a gravitationally bound
circum-binary disk is likely to be formed. The heating from the
central source does not influence the large-scale hydrodynamics of the
circumbinary disk in SVP systems with hidden pulsars, and the
outflowing material eventually reaches a radius 
at
which additional molecular cooling and dust formation occurs. At this
distance, grains can coalesce and coagulate rapidly[23,24], and
planetesimal formation is expected on a timescale
[19]. Planet formation therefore
naturally tends to occur at a distance from the central source of
order 1 AU, although one cannot exclude the possibility that planets
form at larger distances.
A complete treatment of viscous and resonant torques acting on the SVP
binary evolution is beyond our scope here. It is plausible, however,
that because of the angular momentum loss, the binary progressively
contracts on a timescale 
as in the case
of PSR1957+20 [20]. This is smaller than the typical spin-down
timescale of millisecond pulsars 
(
for PSR1257+12; [20]) and comparable with 
. In this
case, the companion star is exposed to a larger and larger radiation
flux which enhances both the mass loss from its surface and the
pressure forces at the inner shock. The effect of the termination
shock during this second evolutionary phase, in which the companion
moves closer to the central pulsar, is to increase the velocity of the
outflowing gas. The latest stages of the starvaporization process may
be characterized by relatively large mass-loss rates
(
) and outflow velocities larger than the
binary escape velocity. No material is expected to be gravitationally
bound in a circum-binary disk in this evolutionary phase. The original
companion star can be completely evaporated in a timescale less than
[5], and only the previously coagulated planets or
planetesimals can witness the process of pulsar-driven star
evaporation.
 |
Figure: 2 SVP with a gravitationally bound outflow, 1.6 orbital periods after starting the simulation (compare with Fig. 1). The initial temperature of the outflow is  ,
the mass-loss rate is  , the
initial velocity of the gas is  , the period of the binary
orbit is 5 h and the luminosity of the pulsar is
 . a, Outflowing material is
trapped in the system (compare with Fig. 1). b, Ballistic
trajectory superimposed over the hydrodynamic result of a The
initial conditions for the ballistic trajectory were calculated from
a block of particles at position (O. -5). |
 |