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Trans­sonic propeller stage
M.E. Prokhorov 1,# , S.B. Popov 1,2,##
1 SAI MSU, Moscow, Russia
2 University of Padova, Italy
e­mail: # mike@sai.msu.ru, ## polar@sai.msu.ru
Abstract
We follow the approach used by Davies and Pringle [1] and discuss the
trans­sonic substage of the propeller regime. This substage is intermediate
between the supersonic and subsonic propeller substages. At the trans­sonic
regime an envelope around a magnetosphere of a neutron star passes through
a kind of a reorganization process. The envelope in this regime consists of two
parts. In the bottom one turbulent motions are subsonic. Then at some dis­
tance, r s , the turbulent velocity becomes equal to the sound velocity. During
this substage the boundary r s propagates outwards till it reaches the outer
boundary, and so the subsonic regime starts.
We found that the trans­sonic substage is unstable, so the transition be­
tween supersonic and subsonic substages goes on at the dynamical time scale.
For realistic parameters this time is in the range from weeks to years.
Keywords: compact objects, isolated neutron stars, evolution
1 Introduction
Observational appearances of neutron stars (NSs) are mainly determined by their
interactions with the surrounding plasma. The main stages are: Ejector, Propeller
and Accretor (see [2]). Normally a NS is born at the stage of ejection, then as the
spin period increases the NS passes the propeller and accretor stages. For NSs with
large spatial velocities another stage (Georotator) can appear.
In a simplified model it is possible to define transitions between stages by com­
paring external and internal pressure. The external one can be approximated as
the ram pressure of a flow of the interstellar medium or as the pressure of matter
falling down onto the NS in its gravitational field. The internal one inside r # (the
light cylinder) can be estimated as a pressure of the magnetic field of a NS.
2 Propeller stage with an intermediate regime
In [3, 1] the authors distinguish three substages of the propeller regime. (1) Very
rapid rotator (c s (r in ) # r
in# # v # ). (2) Supersonic propeller (r
in# # c s (r in )). (3)

2 M.E. Prokhorov and S.B. Popov
Subsonic propeller (r
in# # c s (r in ) and v t (r) < c s (r) in r in < r < r out = r G ). Here
r in and r out -- are internal and external radii of the envelope, c s -- a sound velocity,
v t -- a turbulent velocity, v # -- a free­fall velocity, r G = 2GM/v 2 -- gravitational
capture radius.
We will not discuss the stage of the very rapid rotator here. In this note we focus
on the super­ and subsonic substages and on a transition between them. At the
supersonic stage (a classical propeller) accretion is not possible due to a centrifugal
barrier. At the subsonic stage the magnetospheric (Alfven) radius, r M , is smaller
than the corotation radius, r c , but the accretion does not start because temperature
in the envelope is too high. Why does an intermediate (transitional) regime between
the supersonic and subsonic substages should exist?
In general supersonic and subsonic regimes do cover all possible values of the
rotational
velocity,# = 2#/P . The supersonic propeller formally operates till
r super
in# # c s (r super
in
) . Here r super
in
= r 2/9
G r 7/9
M
[1]. The subsonic regime is on when
r sub
in# # c s (r sub
in ) and v t (r) # c s (r) for r sub
in # r # r out = r G , (1)
here r sub
in = r M [1]. As for the subsonic stage v t (r sub
in ) # r sub
in# and v t /c s # r 1/3 ,
then this regime is valid for r sub
in# # c s (r sub
in ) # r sub
in /r G # 1/3
.
It is easy to check, that the end of the supersonic substage and the beginning
of the subsonic one both correspond
to:# 0 = # 2GMr -7/6
M r -1/3 G . (2)
However, the structure of the envelope on two substages is di#erent, and an interme­
diate regime during which the structure of the envelope is reorganized in inevitable.
We call this intermediate stage trans­sonic propeller.
3 Trans­sonic propeller
We assume that for all substages we can write c s (r) # v # [1]. At the trans­sonic
substage the envelope is divided into two parts with a boundary at r s . Processes
in the lower part of the atmosphere are similar to the ones on the subsonic stage:
v t (r in ) # r
in# < c s . In the outer part of the envelope physical conditions are similar
to the ones on the supersonic substage.
If the envelope is adiabatic, then for its bottom part the politropic index is
equal to n = 3/2 and #(r) # r -3/2 , p(r) # r -5/2 . Following [1] we assume that
the rotational energy of the NS is dissipated at the magnetospheric boundary, and
that this energy is transported outwards by a turbulence. For such assumptions we
have: v t (r) # r -1/6 and M t (r) # v t (r)/c s (r) # r 1/3 .
Till M t < 1, i.e. while to r < r s (r in < r s < r G ) the structure of the envelope
is not changed. For large radii turbulence becomes supersonic. Small­scale shock
waves are formed, and they quickly dissipate part of the energy, so that the turbulent
velocity decreases down to the sound velocity. In the range from r s up to r G the
envelope structure is di#erent from the bottom part: M t (r) # 1 , and #(r) #
r -1/2 , p(r) # r -3/2 .

Trans­sonic propeller stage 3
To determine parameters of the whole atmosphere it is necessary to calculate
the position of the boundary between two parts of the envelope, r s , and position of
the inner boundary of the bottom part, r in (during the transition it decreases from
r 2/9
G r 7/9
M to r M ). To do it is necessary to solve the following system of equations:
# # # # # # #
µ 2
8#
1
r 6
in
= 1
2
—
Mv#
4#r 2
G
# r G
r s
# 3/2
# r s
r in
# 5/2
# 2GM
r s
=# r 7/6
in r -1/6 s
(3)
However, the system is degenerate, and each equation can be reduced to: r s # r -7/2 in
.
If the following equation is fulfilled:
µ 2
8# = # 1
2
—
Mv#
4#r 1/2
G
# (2GM)
3/2# 3
(4)
then the system is compatible, i.e. for # r in in r M < r in < r 7/9
M r 2/9
G # r s , that is a
solution of (3) (where r M = (µ 2 / —
M # 2GM ) 2/7 is the Alfven radius).
The compatibility condition is fulfilled at the end of the supersonic
substage(# =
# 0 ). Later (during the transition) at any given moment (for
any# <# 0 ) left­hand
side of eq.(4) is smaller than the right­hand one. It means that the magnetospheric
pressure and the envelope pressure have the same dependences on r in , but the latter
one is always larger (first equation of (3)).
An energy release during the transition stage is negligible. Estimates for realistic
isolated NS parameters give a value #E # 10 30 erg.
4 Conclusions
1. The intermediate trans­sonic propeller substage in unstable.
2. The duration of the transition can be roughly estimated as #rG /v# (from
weeks to years for realistic isolated NSs).
3. The spin frequency is nearly unchanged during this transition.
4. The energy release during the transition is small.
The work was partially supported by grants RBRF 04­02­16720 and 03­02­16068.
References
[1] R.E. Davies, J.E. Pringle, MNRAS, 196, 209 (1981).
[2] V.M. Lipunov, Astrophysics of neutron stars, Springer­Verlag (1992).
[3] R.E. Davies, A.C. Fabian, J.E. Pringle, MNRAS, 186, 779 (1979).
[4] N.R. Ikhsanov, Astron. Astrophys., 399, 1147 (2003).