DESCRIPTION OF THE SCENARIO MACHINE
V.M. Lipunov1, K.A. Postnov2, M.E. Prokhorov3, A.I. Bogomazov4
Sternberg astronomical institute, Universitetskij prospect, 13, 119992, Moscow, Russia
ABSTRACT
We present here an updated description of the “Scenario Machine” code. This tool is
used to carry out a population synthesis of binary stars. Previous version of the descrip-
tion can be found at http://xray.sai.msu.ru/∼mystery//articles/review/contents.html; see also
(Lipunov et al. 1996b,c).
Subject headings: binaries: close — binaries: general
1. Basic equations and initial distributions
We use the current scenario of evolution of bi-
nary stellar systems based upon the original ideas
that appeared in the papers by Paczyn’ski (1971);
Tutukov & Yungelson (1973),
van den Heuvel & Heise (1972) (see also re-
view by van den Heuvel (1994)). The sce-
nario for normal star evolution was joined with
the ideas of neutron star evolution (see pi-
oneer works by Shvartsman (1970, 1971a,b);
Illarionov & Sunyaev (1975); Shakura (1975);
Bisnovatyi-Kogan & Komberg (1976),
Lipunov & Shakura (1976),
Savonije & van den Heuvel (1977)). This joint
scenario has allowed to construct a two-dimensional
classification of possible states of binary systems
containing NS (Kornilov & Lipunov (1983a,b);
Lipunov (1992)). According to this classification,
we will distinguish four basic evolutionary stages
for a normal star in a binary system:
I — A main sequence (MS) star inside its Roche
lobe (RL);
II — A post-MS star inside its RL;
III — AMS or post-MS star filling its RL; the mass
is transferred onto the companion.
1E-mail: lipunov@xray.sai.msu.ru
2E-mail: pk@.sai.msu.ru
3E-mail: mystery@xray.sai.msu.ru
4E-mail: a78b@yandex.ru
IV — A helium star left behind the mass-transfer
in case II and III of binary evolution; may
be in the form of a hot white dwarf (for M ≤
2.5M⊙), or a non-degenerate helium star (a
Wolf-Rayet-star in case of initial MS mass
> 10M⊙).
The evolution of single stars can be represented
as a chain of consecutive stages: I→ II→ compact
remnant; the evolution of the most massive single
stars probably looks like I → II → IV → compact
remnant. The component of a binary system can
evolve like I→ II→ III→ IV→ compact remnant.
In our calculations we choose the distributions
of initial binary parameters: mass of the primary
zero age main sequence component (ZAMS), M1,
the binary mass ratio, q = M2/M1 < 1, the orbital
separation a. Zero initial eccentricity is assumed.
The distribution of binaries by orbital separa-
tions can be taken from observations (Krajcheva et al.
1981; Abt 1983),
f(log a) = const,
max(10R⊙,RL[M1]) < a ≤ 106R⊙;
Especially important from the evolutionary point
of view is how different are initial masses of the
components (see e.g. Trimble (1983)). We have
parametrized it by a power-law shape, assuming
the primary mass to obey Salpeter’s power law:
f(M) = M−2.351 , 0.1M⊙ < M1 < 120M⊙, (2)
http://arxiv.org/abs/0704.1387v2
http://xray.sai.msu.ru/~mystery//articles/review/contents.html
f(q) ∼ qαq , q = M2/M1 < 1; (3)
We should note that some apparently reason-
able distributions – such as both the primary and
secondary mass obeying Salpeter’s law, or “hier-
archical” distributions involving the assumption
that the total binary mass and primary’s mass are
distributed according to the Salpeter mass func-
tion – all yield essentially flat-like distributions by
the mass ratio (i.e. with our parameter αq ≃ 0).
We assume that the neutron star is formed
in the core collapse of the pre-supernova star.
Masses of the young neutron stars are randomly
distributed in the range MminNS – M
NS . Initial
NS masses are taken to be in the range MNS =
1.25 − 1.4M⊙. The range of initial masses of the
young NSs was based on the masses of the neu-
tron star in the B1913+16 binary system and of
the radio pulsar in the J0737-3039 binary. In the
B1913+16 system (the Hulse-Taylor pulsar, radio
pulsar + neutron star) the mass of the neutron
star, which is definitely not accreting matter from
the optical donor, is MNS = 1.3873 ± 0.0006M⊙
(Thorsett & Chakrabarty 1999; Wex et al. 2000;
Weisberg & Taylor 2003). The mass of the pul-
sar in the J0737-3039 system (radio pulsar + radio
pulsar), which likewise does not accrete from an
optical companion, is MPSR = 1.250 ± 0.010M⊙
(Lyne et al. 2004). The mass range of stars pro-
ducing neutron stars in the end of their evolution is
assumed to be Mn – Mb; stars with initial masses
M > Mb are assumed to leave behind black holes;
Mn is taken to be equal to 10M⊙ in most cases
(but in general this parameter is free). Note that
according to some stellar models, a very massive
star (≈ 50 − 100M⊙) can leave behind a neutron
star as a remnant due to very strong mass loss
via powerful stellar wind, so we account for this
possibility in the corresponding models.
We take into account that the collapse of mas-
sive star into a neutron star can be asymmetrical,
so that the newborn neutron star can acquire an
additional, presumably randomly oriented in space
kick velocity w (see Section 4 below for more de-
tails).
The magnetic field of rotating compact ob-
jects (neutron stars and white dwarfs) largely
define the evolutionary stage of the compact
object in a binary system (Shvartsman (1970);
Davidson & Ostriker (1973); Illarionov & Sunyaev
(1975)), so we use the general classification of mag-
netic rotating compact objects (see e.g. Lipunov
(1992)) in our calculations. The initial magnetic
dipole moment of the newborn neutron star is
taken according to the distribution
f(logµ) ∝ const,
1028 ≤ µ ≤ 1032G cm3, (4)
The initial rotational period of the newborn neu-
tron star is assumed to be ∼ 10 ms.
It is not definitely clear as yet whether the mag-
netic field of neutron stars decays or not (see for a
comprehensive review Chanmugam (1992)). We
assume that the magnetic fields of neutron stars
decays exponentially on a timescale of td (usually
we take this parameter to be equal to 108, 5 · 107
and 107 years). A radio pulsar is assumed to be
“switched on” until its period P (in seconds) has
reached the “death-line” defined by the relation
µ30/P
d = 0.4, where µ30 is the dipole magnetic
moment in units of 1030 G cm3.
We assumed that magnetic fields of neu-
tron stars decay exponentially to minimal value
Bmin = 8 · 107 G and do not decay further:
B0 exp(−t/td), t < td ln(B0/Bmin),
Bmin, t ≥ td ln(B0/Bmin).
Parameters B0 and td in equation (5) are the ini-
tial field strength and the field decay time.
We also assume that the mass limit for neutron
stars (the Oppenheimer-Volkoff limit) is MOV =
2.0M⊙ (in general, it is a free parameter in the
code; it depends on equation of state of the mate-
rial of the neutron star).
The most massive stars are assumed to leave be-
hind black holes after the collapse, provided that
the progenitor mass before the collapse has a mass
Mcr. The masses of the black holes are calcu-
lated as Mbh = kbhMPreSN , where the parameter
kbh = 0.0 − 1.0, MPreSN is the mass of the pre-
supernova star.
We consider binaries with M1 ≥ 0.8M⊙ with a
constant chemical (solar) composition. The pro-
cess of mass transfer between the components is
treated as conservative when appropriate, that is
the total angular momentum of the binary system
is assumed to be constant. If the accretion rate
from one component to another is sufficiently high
(say, the mass transfer occurs on a timescale few
times shorter than the thermal Kelvin-Helmholtz
time for the normal companion) or a compact ob-
ject is engulfed by a giant companion, the com-
mon envelope stage of binary evolution can begin
(Paczynski 1976; van den Heuvel 1983).
Other cases of non-conservative evolution (for
example, stages with strong stellar wind or those
where the loss of binary angular momentum oc-
curs due to gravitational radiation or magnetic
stellar wind) are treated using the well known
prescriptions (see e.g. Verbunt & Zwaan (1981);
Rappaport et al. (1982); Lipunov & Postnov
(1988)).
2. Evolutionary scenario for binary stars
Significant discoveries in the X-ray astronomy
made during the last decades stimulated the as-
tronomers to search for particular evolutionary
ways of obtaining each type of observational ap-
pearance of white dwarfs, neutron stars and black
holes, the vast majority of which harbours in bi-
naries. Taken as a whole, these ways costitute a
general evolutionary scheme, or the “evolutionary
scenario”. We follow the basic ideas about stel-
lar evolution to describe evolution of binaries both
with normal and compact companions.
To avoid extensive numerical calculations in
the statistical simulations, we treat the continu-
ous evolution of each binary component as a se-
quence of a finite number of basic evolutionary
states (for example, main sequence, red super-
giant, Wolf-Rayet star, hot white dwarf, etc.), at
which stellar parameters significantly differ from
each other. The evolutionary state of the binary
can thus be determined as a combination of the
states of each component, and is changed once the
more rapidly evolving component goes to the next
evolutionary stage.
At each such stage, we assume that the star
does not change its physical parameters (mass, ra-
dius, luminosity, the rate and velocity of stellar
wind, etc.) that have effect on the evolution of
the companion (especially in the case of compact
magnetized stars). Every time the faster evolving
component passes into the next stage, we recalcu-
late its parameters. Depending on the evolution-
ary stage, the state of the slower evolving star is
changed or can remain unchanged. With some ex-
ceptions (such as the common envelope stage and
supernova explosion), states of both components
cannot change simultaneously. Whenever possible
we use analytical approximations for stellar pa-
rameters.
Prior to describing the basic evolutionary states
of the normal component, we note that unlike
single stars, the evolution of a binary compo-
nent is not fully determined by the initial mass
and chemical composition only. The primary star
can fill its Roche lobe either when it is on the
main sequence, or when it has a (degenerate) he-
lium or carbon-oxygen core. This determines the
rate of mass transfer to the secondary compan-
ion and the type of the remnant left behind. We
will follow Webbink (1979) in treating the first
mass exchange modes for normal binary compo-
nents, whose scheme accounts for the physical
state of the star in more detail than the simple
types of mass exchange (A, B, C) introduced by
Kippenhahn & Weigert (1967). We will use both
notations A, B, C and D (for very wide systems
with independently evolving companions) for evo-
lutionary types of binary as a whole, and Web-
bink’s notations for mass exchange modes for each
component separately [Ia], [Ib], [IIa], [IIb], [IIIa],
3. Basic evolutionary states of normal
stars
The evolution of a binary system consisting ini-
tially of two zero-age main sequence stars can be
considered separately for each components until a
more massive (primary) component fills its Roche
lobe. Then the matter exchange between the stars
begins.
The evolutionary states of normal stars will be
denoted by Roman figures (I-IV), whereas those
of compact stars will be marked by capital letters
(E, P, A, SA ...). We divide the evolution of a nor-
mal star into four basic stages, which are signifi-
cant for binary system evolution and bear a clear
physical meaning. We will implicitly express the
mass and radius of the star and the orbital semi-
major axis in solar units (m ≡ M/M⊙, r ≡ R/R⊙,
a ≡ A/R⊙), the time in million years, the lumi-
nosities in units of 1038 erg s−1, the wind velocities
in units of 108 cm s−1 and the accretion rates Ṁ
onto compact objects in units of 10−8M⊙ yr
unless other units are explicitely used.
3.1. Main sequence stars
At this stage, the star is on the zero-age main
sequence (ZAMS) and its size is much smaller than
the Roche lobe radius. The time the star spends
on the main sequence is the core hydrogen burning
time, tH , which depends on the stellar mass only
(Iben & Tutukov 1987):
1.0 + 0.95 79
, m ≥ 79.0,
103.9−3.8 logm+log
2 m, 79 > m ≥ 10,
2400m−2.16, 10 > m ≥ 2.3,
104m−3.5, m < 2.3,
The radius of the ZAMS star is assumed to be
100.66 logm+0.05, m > 1.12,
m, m ≤ 1.2, (7)
and its luminosity is
logL =
−5.032 + 2.65 logm, (α)
−4.253 + 4.8 logm, (β)
−4.462 + 3.8 logm, (γ)
−3.362 + 3.0 logm, (δ)
−3.636 + 2.7 logm, (ǫ)
here we assume the next indication: (α), m < 0.6;
(β), 0.6 ≤ m < 1.0; (γ), 1.0 ≤ m < 10.0; (δ),
10 ≤ m < 48.0; (ǫ), m ≥ 48.0.
The initial mass of the primary and the mode
of the first mass exchange (which is determined by
the initial orbital period and masses of the com-
ponents; see Webbink (1979)) determine the mass
and the type of the core that will be formed during
stage I. For example, for single stars and primaries
in “type C” binaries that fill its Roche lobe having
a degenerate core, we use the expressions
0.1mmax, (α)
0.446 + 0.106mmax, (β)
0.24m0.85max, (γ)
min(0.36m0.55max, 0.44m
max), (δ)
0.44m0.42max, (ǫ)
≈ MCh, (ζ)
0.1m1.4max, (η)
where mmax is the maximum mass the star had
during the preceding evolution. We assume the
next indication in this formula: (α), mmax < 0.8;
(β), 0.8 ≤ mmax < 2.3; (γ), 2.3 ≤ mmax < 4.0;
(δ), 4.0 ≤ mmax < 7.5; (ǫ), 7.5 ≤ mmax < 8.8;
(ζ), 8.8 ≤ mmax < 10.0; (η), mmax ≥ 10.0.
A main-sequience star accreting matter during
the first mass transfer will be treated as a rapidly
rotating “Be-star” with the stellar wind rate dif-
ferent from what is expected from a single star of
the same mass (see below).
3.2. Post main-sequence stars
The star leaves the main sequence and goes to-
ward the red (super)giant region. The star still
does not fill its Roche lobe. The duration of this
stage for a binary component is not any more a
function of the stellar mass only (as in the case
of single stars), but also depends on the initial
binary type (A, B, or C) (see Iben & Tutukov
(1985, 1987)):
tII =
0, (α),
2tKH , (β),
6300m−3.2, (γ),
tHe, (δ),
In type A systems, the primaries fill their Roche
lobes when they belong to the main sequence. We
assume the next indication in this formula: (α),
type A; (β), type B excluding mode [IIIA]; (γ),
types C, D and mode [IIIA], mmax < 5; (δ), types
C, D and mode [IIIA], mmax > 5.
The radius of the post-MS star rapidly increases
(on the thermal time scale) and reaches the char-
acteristic giant values. The star spends the most
time of helium burning with such large radius. In
the framework of our approximate description, we
take the radius of the giant star to be equal to
the maximum value, which depends strongly on
the mass of its core and is calculated according
to Webbink’s mass transfer modes as follows (see
also Iben & Tutukov (1985, 1987)):
rII =
3000m4c, (α)
1050(mc − 0.5)0.68, (β)
10m0.44c , (γ)
We assume the next indication in this formula:
(α), mode [IIIA] or [IIIB] with He core; (β),
mode [IIIB] with CO or ONeMg core; (γ), modes
[I] or [II]. Formula (11) depicts stars with mass
≤ 10M⊙.
This maximum radius can formally exceed the
Roche lobe size; in such cases we put it equal
to 0.9RL during the stage II. The most sensi-
tive to this crude approximation are binaries with
compact companions, which can lead, for exam-
ple, to the overestimation of the number of ac-
creting neutron stars observed as X-ray pulsars.
However, these stages are less important for our
analysis than the stages at which the optical star
fills its Roche lobe. A more detailed treatment
of normal star evolution (given, for example, by
Pols & Marinus (1994)) can reduce such uncer-
tainties.
Luminosities of giants are taken from de Jager
(1980):
log lII =
15.92m6c
(1.0+m4c)(2.512+3.162mc)
, (α)
10−4.462+3.8 logm, (β)
10−3.362+3.0 logm, (γ)
10−3.636+2.7 logm, (δ)
We assume the next indication in this formula:
(α), m < 23.7, mc < 0.7; (β), m < 23.7, mc > 0.7;
(γ), 48 > m > 23.7; (δ), m > 48.
Radii of (super)giants are determined by us-
ing the effective temperature Teff and luminosi-
ties. Typical effective temperatures are taken from
Allen (1973):
logTeff =
4.50, (α)
3.60, (β)
3.70, (γ)
We assume the next indication in this formula:
(α), m > 10.0; (β), m < 10.0, type C or D; (γ),
m < 10.0, type A or B. So, we calculate RII with
mass higher than 10M⊙ calculate using formula
rII = exp {2.3(0.5 log lii − 2 logTeff + 9.7)}.
3.3. Roche lobe overflow
At this stage the star fills its Roche lobe (RL)
and mass transfer onto the companion occurs.
The mass transfer first proceeds on the thermal
time scale (see extensive discussion of this approx-
imaiton in van den Heuvel (1994))
tKH ∼ 30m2r−1∗ (L/L⊙)−1, (15)
The common envelope stage (CE) may be
formed if the Roche lobe overflow occurs in the
type C system (where the primary has a well de-
veloped core) even for q < 1; otherwise (for type
B systems) we use the condition q ≤ qcr = 0.3 for
the CE stage to occur. Radius of the star at the
Roche lobe filling stage is taken to be that of the
equivalent Roche lobe radius (Eggleton 1983):
a(1− e)
0.49q2/3
0.6q2/3 + ln(1 + q1/3)
, (16)
Here a is the binary orbital separation and e is the
orbital eccentricity, q is the arbitrary mass ratio.
For q < 0.6 a more precise approximation can
be used:
= 0.4622
1 + q
. (17)
A star filling its Roche lobe has quite differ-
ent boundary conditions in comparison with sin-
gle stars. The stellar radius at this stage is limited
by the Roche lobe. If the stellar size exceeds the
Roche lobe, the star can lose matter on a time
scale close to the dynamical one until its radius
becomes smaller than the new Roche lobe size.
Now we consider how the RL-filling star loses
matter. Let the star be in equilibrium and
Req(M) = RL(M) at the initial moment of time.
When a fraction of mass δm is transported to the
companion, the mass ratio q and semi-major axis
a of the binary changes depending on the mass
transfer mode assumed (see below for details).
The RL size then becomes equal to RL(M − δm).
On the other hand, mass loss disturbs equi-
librium of the star (hydrodynamical and ther-
mal). The hydrodynamical equilibrium is restored
on the dynamical timescale td ∝
GM/R3
)−1/2
The stellar radius changes to a value Rad(M −
δm) (where “ad” means adiabatic), which can
be bigger or smaller than the equilibrium radius
Req(M − δm) of the star. The thermal equilib-
rium establishes on the thermal time scale TKH ≈
GM2/RL, so after that the stellar radius relaxes
to the equilibrium value Req(M − δm).
Relations RL(M − δm), Rad(M − δm) and
Req(M−δm) determine the mode of mass transfer
during the RL overflow stage. Following Webbink
(1985), one usually introduces the logarithmic
derivative ζ = d lnR/d lnM (R ∝ M ζ). It lo-
cally fits the real dependence R(M). Three values
of ζ are relevant:
d lnRL
d lnM
ζad =
d lnRad
d lnM
, (18)
ζeq =
d lnReq
d lnM
where “L”, “ad” and “eq” correspond to the values
of radii discussed above.
Three possible cases are considered depending
on ζi:
1. If ζad < ζL, the star cannot be inside its RL
regardless of the mass loss rate (dM/dt < 0).
Such stars lose their matter in hydrodynami-
cal time scale. Mass loss rate is limited only
by the speed of sound near the inner La-
grangian point L1. ζeq is unimportant be-
cause the size of the star becomes bigger and
bigger than RL. The equilibrium is impossi-
2. ζeq < ζL < ζad. The star losing mass cannot
be in thermal equilibrium, because otherwise
its size would exceed RL. Nevertheless, in
this case Rad < RL. So the hydrodynamical
equilibrium is established. As a result, the
star loses mass on thermal time scale.
3. ζL < ζad, ζeq . In this case the size of the star
losing mass becomes smaller than its RL.
The evolutionary expansion of the star or
the binary semi-major axis decrease due to
orbital angular momentum loss via magnetic
stellar wind (MSW) or gravitational radia-
tion (GW) support the permanent contact of
the star with RL. The star then loses mass
on a time scale dictated by ots own evolu-
tionary expansion or on a time scale corre-
sponding to the orbital angular momentum
loss.
For non-degenerate stars, Req increases mono-
tonically with M . On the other hand, the expo-
nent ζad is determined by the entropy distribution
over the stellar radius which is different for stars
with radiative and convective envelopes. It can be
shown that stars with radiative envelopes should
shrink in response to mass loss, while those with
convective envelopes should expand 5.
Therefore, stars with convective envelopes in bi-
naries should generally have a higher mass loss
rate than those with radiative envelopes under
other equal conditions. The next important fac-
tor is the dependence RL(M). It can found by
substituting one of the relations a(M) (see below)
into equation (16) or into equation (17) and dif-
ferentiating it with respect to M . For example,
assuming the conservative mass exchange when
the total mass and the orbital angular momen-
tum of the system do not change, one readily gets
that the binary semi-major axis decreases when
the mass transfer occurs from the more massive to
the less massive component; RL decreases of the
primary correspondingly. When the binary mass
ratio reaches unity, the semi-major axis takes on a
minimal value. In contrast, if the less massive star
loses its mass conservatively the system expands.
In that case the mass transfer can be stable.
If more massive component with radiative en-
velope fills its RL, the mass transfer proceeds on
thermal time scale until the masses of the compo-
nents become equal 6. The next stage of the first
mass exchange poceeds in more slower (nuclear)
time scale. If the primary has convective enve-
lope, the mass transfer can proceed much faster
on a time scale intermediate between the thermal
and hydrodynamical one, and probably on the hy-
drodynamical time scale. In that case the fast
stage of the mass exchange ends when the mass
of the donor decreases to ∼ 0.6 mass of the sec-
ondary companion (Tutukov & Yungelson 1973).
Further mass transfer should proceed on the evo-
lutionary time scale.
5The adiabatic convection in stellar envelope can be de-
scribed by the polytropic equation of state P ∝ ρ5/3, simi-
lar to non-relativistically degenerate white dwarfs. For such
equation of state the mass-radius relation becomes inverse:
R ∝ M−1/3. For non-degenerate stars with convective en-
velopes this relation holds approximately.
6The mass exchange can stop earlier if the entire envelope
is lost and the stellar core is stripped (the core has other
values ζad and ζeq).
The process of mass exchange strongly depends
on stellar structure at the moment of the RL over-
flow. The structure of the star in turn depends
on its age and the initial mass. The moment of
the RL overflow is determined by the mass of the
components and by the initial semi-major axis of
the system. to calculate a diagram in the M − a
(or M − Porb) plane which allows us to conclude
when the primary in a binary with given initial pa-
rameters fills its RL, what is its structure at that
moment and what type of the first mass exchange
is expected. We use the diagram calculated by
Webbink (1979) (see also the description of mod-
ern stellar wind scenarios below).
We distinguish different sub-stages of the RL
overflow according to the characteristic timescales
of the mass transfer:
stage III:
This is the most frequent case for the first mass
transfer phase. The primary fills its RL and the
mass transfer proceeds faster than evolutionary
time scale (if outer layers of the star are radia-
tive, then it is thermal time scale, if outer layers
are convective, then time scale is shorter, up to
hydrodynamical time scale). This stage comes to
the end when mass ratio in the system changes
(“role-to-role transition”), i.e. when the mass of
the donor (mass losing) star is equal to mass of
second companion (for radiative envelopes) or 0.6
of the mass of the second companion (for convec-
tive envelopes). This stage also stops if the donor
star totally lost its envelope.
stage IIIe:
This is the slow (evolutionary driven) phase of
mass transfer. We assume it to occur in short-
period binaries of type A. However, it is not ex-
cluded that it may occur after the mass reversal
during the first stage of mass exchange for binaries
of type B (see van den Heuvel (1994)), e.g. as in
wide low-mass X-ray binaries.
stage IIIs
This is the specific to super-accreting com-
pact companions substage of fast mass transfer at
which matter escapes from the secondary compan-
ion carrying away its orbital angular momentum.
Its duration is equal to
tIIIs = tKH
q(1 + q)
2− q − 2q2
, (19)
q = Ma/Md < 0.5,
here and below subscripts “a” and “d” refer to
the accreting and donating mass star, respectively.
For systems with small mass ratios, q < 0.5, this
timescale corresponds to an effective q-time short-
ening of the thermal time for the RL-overflowing
star.
stages IIIm,g
At these stages, the mass transfer is controlled
by additional losses of orbital angular momentum
Jorb caused by magnetic stellar wind or gravita-
tional wave emission. The characteristic time of
the evolution is defined as τJ = −(Jorb/J̇orb),
and in the case of MSW is (see Verbunt & Zwaan
(1981); Iben & Tutukov (1987))
τMSW = 4.42
a5mxλ
(m1 +m2)2m4op
, (20)
Here mop denotes mass of the low-mass optical
star (0.3 < m < 1.5) that is capable of producing
an effective magnetic stellar wind (because only
such stars have outer convective envelopes which
are prerequisit for effecftive MSW), λ is a numer-
ical parameter of order of unity. We have used
the mass-radius relation r ≈ m for main sequence
stars in deriving this formula. The upper limit
of the mass interval and empirical braking law for
main-sequence G-stars are taken from Skumanich
(1972), the lower limit is determined by absence of
cataclysmic variables with orbital periods Porb ≈
3h (Verbunt 1984; Mestel 1952; Kawaler 1988;
Tout & Pringle 1992; Zangrilli et al. 1997). We
use λ = 1 (see for details Kalogera & Webbink
(1998)).
The time scale of the gravitational wave emis-
sion is
τGW = 124.2
m1m2(m1 +m2)
× (21)
(1− e2)−7/2,
Wether the evolution is governed by MSW or
GW is decided by which time scale (τMSW or
τGW ) turns out to be the shortest among all ap-
propriate evolutionary time scales.
stage IIIwd
This is a special case where the white dwarf
overflows its RL. This stage is encountered for very
short period binaries (like Am CVn stars and low-
mass X-ray binaries like 4U 1820-30) whose evolu-
tion is controlled by GW or MSW. The mass trans-
fer is calculated using the appropriate time scale
(GW or MSW). The radius of the white dwarf in-
creases with mass RWD ∝ M−1/3. This fact, how-
ever, does not automatically imply that the mass
transfer is unstable, since the less massive WD
fills its RL first. It can be shown that the mass
exchange is always stable in such systems if the
mass ratio q < 0.8. This condition always holds
in WD+NS and WD+BH systems. WD loses its
matter until its mass decreases to that of a huge
Jupiter-like planet (∼ a few 10−3M⊙), where the
COulomb interaction reverses the mass-radius re-
lation R(M). Such a planet can approach the sec-
ondary companion of the system due to GW emis-
sion until the tidal forces destroy it completely.
The matter of the planet can fall onto the sur-
face of the second companion ir form a long-living
disk around it. If the second star is a neutron star
and its rotation had been spun up by accretion
such that a millisecond radio pulsar appeared, the
planet can be evaporated by relativistic particles
emmited by the pulsar (Paczynski & Sienkiewicz
1983; Joss & Rappaport 1983; Kolb et al. 1998;
Kalogera & Webbink 1998).
The mass loss rate at each of the III-stages is
calculated according to the relation
Ṁ = ∆M/τi, (22)
where ∆M is the a priori known mass to be lost
during the mass exchange phase (e.g. ∆M = M1−
(M1 +M2)/2 in the case of the conservative stage
III, or ∆M = M1−Mcore(M1) in case of III(e,m,g)
or CE) and τi is the appropriate time scale.
The radius of the star at stage III is assumed
to follow the RL radius:
Rdl (Md(t)) = Rd(Md(t)). (23)
3.4. Wolf-Rayet and helium stars
In the process of mass exchange the hydrogen
envelope of the star can be lost almost completely,
so a hot white dwarf (for m ≤ 2.5), or a non-
degenerate helium star (for higher masses) is left
as a remnant. The life-time of the helium star is
determined by the helium burning in the stellar
core (Iben & Tutukov 1985)
tHe =
1658m−2, (α)
1233m−3.8, (β)
0.1tH , (γ)
6913m−3.47, (δ)
≃ 10, (ǫ)
0.1tH , (ζ)
We assume the next indication in this formula:
(α), m < 1.1, modes [IIA-IIF]; (β), m > 1.1,
mmax < 10, modes [IIA-IIF]; (γ), mmax > 10,
modes [IIA-IIF]; (δ), mode [IIIA]; (ǫ), mode [IIIC];
(ζ), modes [IIIB,D,E] and type D.
If the helium (WR) star fills its Roche lobe
(a relatively rare so-called “BB” case of evolu-
tion; Delgado & Thomas (1981); see discussion in
van den Heuvel (1994)), the envelope is lost and
a CO stellar core is left with mass
1.3 + 0.65(m− 2.4),m ≥ 2.5,
0.83m0.36,m < 2.5,
The mass-radius dependence in this case is
(Tutukov & Yungelson 1973)
rWR = 0.2m
0.6. (26)
3.5. Stellar winds from normal stars
The effect of the normal star on the compact
magnetized component is largely determined by
the rate Ṁ and the velocity of stellar wind at in-
finity v∞, which is assumed to be
v∞ = 3vp ≈ 1.85
m/r, (27)
where vp is the escape velocity at the stellar sur-
face.
For “Be-stars” (i.e. those stars at the stage “I”
that increased its mass during the first mass ex-
change), the wind velocity at the infinity is taken
to be equal to the Keplerian velocity at the stellar
surface:
GM/R ≈ 0.44
m/r . (28)
The lower stellar wind velocity leads to an effec-
tive increase of the captured mass rate by the sec-
ondary companion to such “Be-stars”.
The stellar wind mass loss rate at the stage “I”
is calculated as
(de Jager 1980)
ṁ = 52.3αwl/v∞, (29)
Here αw = 0.1 is a numerical coefficient (in gen-
eral, we can treat it as free parameter).
For giant post-MS stars (stage “II”) we assume
v∞ = 3vp and for massive star we take the max-
imum between the stellar wind rate given by de
Jager’s formula and that given by Lamers (1981)
ṁ = max(52.3αw
, 102.33
l1.42r0.61
m0.99
), (30)
M ≥ 10M⊙;
For red super-giants we use Reimers’s formula
(Kudritzki & Reimers 1978):
ṁ = max(52.3αw
, 1.0
), (31)
M ≥ 10M⊙;
For a Wolf-Rayet star the stellar wind loss rate
can significantly increase (up to 10−5M⊙ year
We parametrize it as
ṀWR = kWRMWR/tHe, (32)
where the numerical coefficient is taken to be
kWR = 0.3 (in general, it can be changed if nec-
essary). The mass loss in other stages (MS, (su-
per)giant) is assumed to be limited by 10% of the
mass of the star at the beginning of the stage.
3.6. Change of binary parameters: mass,
semi-major axis and eccentricity
The duration of any evolutionary stage is de-
termined by the more rapidly evolving compo-
nent ∆t = min(∆t1,∆t2). On the other hand,
based on the evolutionary considerations we are
able to calculate how the mass of the faster evolv-
ing star changes (e.g. due to the stellar wind or
RL overflow), that is we can estimate the quantity
∆M = Mi −Mf . Then we set the characteristic
mass loss rate at this stage as
Ṁo = ∆M/∆t, (33)
Next, we should calculate the change of mass for
the slower evolving companion, the orbital semi-
major axis and the eccentricity.
3.7. Mass change
The mass of the star loosing matter is calcu-
lated as
Mf = Mi − Ṁo ×∆t, (34)
Accordingly, the mass of the accreting star is
Mf = Mi + Ṁc ×∆t, (35)
where Ṁc is the accretion rate of the captured
matter.
For stages without RL overflow the accretion
rate of the captured stellar wind matter is
ṁc = 3.8×10−2
a(v2w + 0.19(m1 +m2)/a)
ṁo .
At stages where RL overflow ocurs and both
components are normal (non-degenerate), we will
assume that the accretor can accomodate mass at
the rate determined by its thermal time, i.e.
ṁc = ṁo
tKH(donor)
tKH(accretor)
. (37)
This means that the evolution can not be
fully conservative, especially during the first mass
transfer where the primary component usually has
a shorter thermal time scale.
The mass increase rate by compact accretors
is assumed to be limited by the critical Edding-
ton luminosity (see, however, the possible hyper
accretion stage discussed below):
LEdd =
4πGMmp
≈ 1.3 · 1038 ×m erg/s (38)
(σT is the Thomson cross-section) at the stopping
radius Rstop for the accreted matter (see, e.g., de-
tailed discussion in Lipunov (1992)). This corre-
sponds to the critical accretion rate
Ṁcr = Rstop
. (39)
Thus, the mass of the accreting compact star at
the end of the stage is determined by the relation
Mf = Mi +min(Ṁc, Ṁcr)×∆t . (40)
3.8. Semi-major axis change
The binary separation a changes differently for
various mass exchange modes. First, we introduce
a measure of non-conservativeness of the mass ex-
change as the ratio between the mass change of
the accretor and the donor:
β ≡ −(M iaccr −Mfaccr)/(M idonor −M
donor). (41)
If the mass exchange is conservative (β = 1,
i.e. Ma + Md = const) and one can neglect the
angular momenta of the components, the orbital
momentum conservation implies
M iaM
. (42)
In a more general case of quasi-conservative
mass transfer 0 ≤ β < 1, the orbital separation
changes differently depending on the specific an-
gular momentum carried away from the system by
the escaping matter (see van den Heuvel (1994)
for more detail). We treat the quasi-conservative
mass transfer by assuming the isotropic mass loss
mode in which the matter carries away the spe-
cific orbital angular momentum of the accreting
component (ja)
J̇out = (1− β)Ṁcja . (43)
From here we straightforwardly find
1 + qi
1 + qf
1 + β
1 + β
)3+2/β
. (44)
In this formula q = Maccr/Mdonor and the
non-conservative parameter β is set to be the
minimal value between β = 1 and the ra-
tio TKH(donor)/TKH(accr) (TKH(donor) and
TKH(accr) is the thermal time of donor and ac-
cretor, respectively).
When no matter is captured by the secondary
companion without additional losses of angu-
lar momentum (the so-called “absolutely non-
conservative case”), which relates to the spherical-
symmetric stellar wind from one component, we
use another well-known formula
M i1 +M
. (45)
In this case the orbital separation always increases.
When the orbital angular momentum is carried
away by GW or MSW with no RL overflow, the
following approximate formulas are used:
(1−∆t/τMSW )1/4, for MSW,
(1−∆t/τGW )1/5, for GW,
In a special case of a white dwarf filling its
RL (stage “IIIwd” above), assuming a stable (i.e.
where d lnRwd/d lnM = d lnRRL/d lnM) conser-
vative mass transfer with account for the mass-
radius relation Rwd ∝ M−1/3wd , the orbital separa-
tion must increase according to the equation
)−2/3
, (47)
where mi is the initial mass of the WD donor and
mf is its mass at the end of the mass transfer.
3.9. The change of eccentricity
Tidal interaction between components, as well
as the orbital angular momentum loss due to MSW
or GR decrease the eccentricity of the binary sys-
tem. The tidal interaction is essential in very
close binaries or even during the common envelope
stage. MSW is effective only in systems with low-
mass late-type main sequence stars (see above),
GW losses become significant only in short-period
binaries.
The tidal interaction conserves the orbital an-
gular momentum which implies the relation
a(1− e2) = const. (48)
It seems that the orbit becomes a circle faster
than major semi-axis of the orbit decreases during
common envelope stage. We suppose that tcyr =
1/3tCE. We accept that tcyr for RL-filling stars
is equal to its Kelvin-Helmholtz time. Detached
systems change their eccentricity during the next
character time (Zahn 1975; Press & Teukolsky
1977; Zahn 1989b; Zahn & Bouchet 1989b)
tcyr = tKH
1 + e
)3/2(
. (49)
here tKH ≈ GM2/RL is thermal time of the star,
R is radius of the star, RL is its RL size. For sys-
tems which consist of two normal stars we choose
minimal value of tcyr. Resonances at very high ec-
centricities are not taken into account (Mardling
1995a,b).
Orbits of the systems with MSW become circu-
lar during tcyr = τMSW (see 46).
In case of GW analytical exact solutions were
obtained for a(t) and e(t) (Peters & Mathews
1963; Peters 1964).
4. Special cases: supernova explosion and
common envelope
Supernova explosion in a binary is treated as
an instantaneous mass loss of the exploding star.
The additional kick velocity can be imparted to
the newborn neutron star due the collapse asym-
metry (see below for discussion). In this case the
eccentricity and semi-major axis of the binary af-
ter the explosion can be straightforwardly calcu-
lated (Boersma 1961) (see necessary formulas also
in Grishchuk et al. 2001). Briefly, we use the fol-
lowing scheme.
1. First, velocities and locations of the compo-
nents on the orbit prior to the explosion are
calculated;
2. then the mass of the exploding star Mpr -
Mremnant is changed and the arbitrarily di-
rected kick velocity w is added to its orbital
velocity;
3. after that the transition to the new system’s
barycenter is performed (at this point the
spatial velocity of the new center of mass of
the binary is calculated);
4. in this new reference frame the new total en-
ergyE′tot and the orbital angular momentum
J ′orb are computed; if the new total energy is
negative, the new semi-major axis a′ and ec-
centricity e′ are calculated by using the new
J ′orb end E
tot; if the total energy is positive
(that is, the binary is unbound) spatial ve-
locities of each component are calculated.
The kick velocity w distribution is taken in the
Maxwellian form:
f(w) ∼ w
0 . (50)
We suppose that the absolute value of the ve-
locity that can be added during the formation of
a black hole depends on the mass loss by the col-
lapsing star, the value of the parameter w0 during
the BH formation is defined as
wbh0 = (1− kbh)w0. (51)
An effective spiral-in of the binary compo-
nents occurs during the common envelope (CE)
stage. This complicated process (introduced by
Paczynski (1976)) is not fully understood as yet,
so we use the conventional energy consideration
to find the binary system parameters after the CE
by introducing a parameter αCE = ∆Eb/∆Eorb,
where ∆Eb = Egrav − Ethermal is the binding en-
ergy of the ejected envelope matter and ∆Eorb is
the drop in the orbital energy of the system during
the spiral-in phase (van den Heuvel 1994). This
parameter measures the fraction of the system’s
orbital energy that comes during the spiral-in pro-
cess to the binding energy (gravitational minus
thermal) of the ejected common envelope. Thus
GMaMc
− GMaMd
GMd(Md −Mc)
where Mc is the mass of the core of the mass-
losing star with the initial mass Md and radius Rd
(which is simply a function of the initial separation
ai and the initial mass ratio Ma/Md, where Ma is
the mass of the accreting star).
On the CE stage the luminosity of the accret-
ing star can reach the Eddington limit so that the
further increase of the accretion rate can be pre-
vented by radiation pressure. This usually hap-
pens at accretion rates Ṁ ≃ 10−4−10−5M⊙ yr−1.
However, Chevalier (1993) suggested that when
the accretion rate is higher (Ṁ ≃ 10−2− 10−3M⊙
yr−1), the energy is radiated away not by high-
energy photons only, but also by neutrinos (see
also Zeldovich et al. (1972) and the next section).
On the typical time scale for the hyper accretion
stage of 102 yr, up to ∼ 1M⊙ of matter may be
incident onto the surface of the neutron star.
5. Three regimes of mass accretion by
neutron stars
A considerable fraction of observed neutron
stars have increased their masses in the course of
their evolution, or are still increasing their masses
(e.g., in X-ray sources). But how large can this
mass increase be? It is clear that the only origin
of a mass increase is accretion. It is evident that
the overall change in the mass of a neutron star
is determined not only by the accretion rate, but
also by the duration of the accretion stage:
Ṁdt = ṀTa, (53)
where Ṁ is the mean accretion rate and Ta is the
lifetime of the accretion stage. We emphasize that,
in the case under consideration, the accretion rate
is the amount of matter falling onto the surface
of the neutron star per unit time, and can differ
signifcantly from the values indicated by the clas-
sical Bondi-Hoyle formulas. Three regimes of ac-
cretion are possible in a close binary containing a
neutron star: ordinary accretion, super-accretion,
and hyper-accretion.
5.1. Ordinary accretion
The ordinary accretion regime is realized when
all matter captured by the gravitational field of the
neutron star falls onto its surface. This is possible
only if the radiation pressure and electromagnetic
forces associated with the magnetic field of the star
and its rotation are small compared to the gravita-
tional force. In this case, the increase in the mass
will be precisely determined by the gas dynamics
of the accretion at the gravitational-capture ra-
dius or, if the donor fills its Roche lobe, by the
binary mass ratio and the evolutionary status of
the optical component. In this case, the accretor
is observed as an X-ray source with luminosity
Lx = Ṁ
, (54)
where Mx and R∗ are the mass and radius of the
neutron star. The accretion rate Ṁ is determined
by the Bondi-Hoyle formula
Ṁ = πR2Gρv , (55)
where RG is the gravitational-capture radius of
the neutron star, v is the velocity of the gas flow
relative to the neutron star, and ρ is the density
of the gas.
The X-ray luminosity of the accretor Lx and
its other main parameters can be used to estimate
the mass ∆M accumulated during the accretion
phase:
LxR∗Ta
, (56)
5.2. Super-accretion
Regime of super accretion was considered, for
instance, in paper Lipunov (1982d). Despite the
absence of detailed models for supercritical disk
accretion (supercritical accretion is realized pre-
cisely via an accretion disk), it is possible to esti-
mate the main characteristics of the process – the
accretion rate, magnetosphere radius, and evolu-
tion equations. Accretion is considered to be su-
percritical when the energy released at the radius
where the accretion exceeds the Eddington limit:
Rstop
> LEdd = 1.38×1038(Mx/M⊙) erg s−1,
where Rstop is either the radius of the neutron star
or the magnetosphere radius RA.
For strongly magnetized neutron stars with
magnetic fields B ≫ 108 G, all matter arriving
at the magnetosphere is accreted onto the mag-
netic poles, where the corresponding gravitational
energy is released. If the black body temperature
T , roughly estimated as
SσT 4 = Ṁ
, (58)
is higher than 5 × 109 K (S is the area of the
base of the accretion column), most of the energy
will escape from the neutron star in the form of
neutrinos, and, hence, will not hinder accretion
(Zeldovich et al. 1972; Basko & Sunyaev 1975).
In this case, the rate at which the neutron star
accumulates mass will be
Ṁ ≃ Ṁcrit
≫ Ṁcrit, (59)
For lower temperatures there should be an up-
per limit on the accretion rate equal to the stan-
dard Eddington limit.
5.3. Hyper-accretion
A considerable fraction of neutron stars in bi-
nary systems pass through the common-envelope
stage in the course of their evolution. In this case,
the neutron star is effectively immersed in its opti-
cal companion, and for a short time (102-104 yrs)
spirals-in inside a dense envelope of the compan-
ion. The formal accretion rate estimated using
the Bondi-Hoyle formulas is four to six orders of
magnitude higher than the critical rate and, as
was suggested by Chevalier (1993), this may re-
sult in hyper-accretion, when all the energy is car-
ried away by neutrinos for the reasons described
above. There are currently no detailed theories
for hyper-accretion or the common-envelope stage.
The amount of matter accreted by the neutron
star can be estimated as
∫ Thyper
Ṁdt ≃ (60)
(Mopt −Mcore)
where Thyper is the duration of the hyper-accretion
stage, RG is the gravitational-capture radius of
the neutron star, a is the initial semi-major axis
of the close binary orbit, Mcore is the mass of the
core of the optical star, and Mopt and Mx are the
total masses of the optical star and of the neu-
tron star at the onset of the hyper-accretion stage.
The mass of the neutron star can increase during
the common envelope stage as much as ∼ 1M⊙
(Bogomazov et al. 2005), up to MOV . Such NSs
collapse into black holes.
6. Mass accretion by black holes
If the black hole has formed in the binary sys-
tem, its X-ray luminosity is
Lx = µṀc
2, (61)
where µ = 0.06 for Schwarzschild black hole and
µ = 0.42 (maximum) for extremal Kerr BH.
We use the Bondi-Hoyle formulas to estimate
the accretion rate onto BH.
Powerful X-ray radiation is able to originate
only if an accretion disc has formed around the
black hole (Karpov & Lipunov 2001). For spher-
ically symmetric accretion onto a black hole the
X-ray luminosity is insignificant. A very low stel-
lar wind velocity is necessary to form an accretion
disc (Lipunov 1992)
V < Vcr ≈ (62)
≈ 320(4η)1/4m3/8T−1/410 R
8 (1 + tan
2 β)−1/2,
where η is averaged over the z-coordinate dynamic
viscous coefficient, m = Mx/M⊙, Mx is the rela-
tivistic star mass, T10 = T/10, T is the orbital
period in days, R8 = Rmin/10
8cm, Rmin is the
minimal distance from the compact object up to
which free Keplerian motion is still possible and
β is the accretion disk axis inclination angle with
respect to the radial direction. For black holes
Rmin = 3Rg, where Rg = 2GMbh/c
7. Accretion induced collapse and com-
pact objects merging
WD and NS are degenerate configurations,
which have upper limit of their mass (the Chan-
drasekhar and Oppenheimer-Volkov limits corre-
spondingly). The Chandrasekhar limit depends
on chemical composition of the white dwarf
MCh =
1.44M⊙, He WD,
1.40M⊙, CO WD,
1.38M⊙, ONeMg WD,
If the mass of WD becomes equal to MCh,
the WD loses stability and collapses. The col-
lapse is accompanied by the powerful thermonu-
clear burst observed as a type Ia supernova. Col-
lapses of He and CO WDs leave no remnants
(Nomoto & Kondo 1991). The outcome of the
collapse of a ONeMg WD is not clear. It can lead
to the formation of a neutron star (the accretion
induced collapse, AIC). Some papers come to the
different conclusion about the result of AICs (see
e.g. Garcia-Berro & Iben (1994); Ritossa et al.
(1996)). The question about the NS formation
during the WD collapse remains undecided. Nev-
ertheless, some NSs could have been formed from
AIC WD (van Paradijs 1997). In the “Scenario
Machine” code the possibility of NS formation
during ONeMg WD is optional.
The merging of two compact objects in a binary
WD system (e.g., due to the GW losses) should
likely to be similar to AIC. During the merging
of two helium WDs, one object with a mass of
less than MCh (for example, 0.5M⊙ + 0.6M⊙ →
1.1M⊙) can form. At the same time, if the total
mass of the components exceeds 1.0 − 1.2M⊙, a
thermonuclear burning can happen. It is likely
that the merging of a ONeMg WD with another
WD can form a NS.
For the typical NS mass ≈ 1.4M⊙, the binary
NS+NS merging event can produce a black hole.
Massive (≈ 2.8M⊙) neutron star can be formed
only if the NS equation of state is very hard and
MOV ≃ 2.8− 3.0M⊙.
BH+BH merging should produce a rapidly ro-
tating black hole with the mass equal to the total
mass of the coalescing binary.
8. Additional scenarios of stellar wind
from massive stars
8.1. Evolutionary scenario B
In the end of 1980s and in the beginning of
1990s the series of new evolutionary tracks were
calculated. The authors used new tables of opaci-
ties (Rogers & Iglesias 1991; Kurucz 1991), new
cross-sections in nuclear reactions (Landre et al.
1990) and new parameters of convection in stars
(Stothers & Chin 1991). For stars with M <
10M⊙ those tracks proved to be almost coincident
with previous calculations. More massive stars
had much stronger stellar winds.
A massive star loses up to 90% of its ini-
tial mass in the main-sequence, supergiant, and
Wolf-Rayet stages via stellar wind. There-
fore, the presupernova mass in this case can
be ≈ 8 − 10M⊙, essentially independent of the
initial mass of the star (de Jager et al. 1988;
Nieuwenhuijzen & de Jager 1990; Schaller et al.
1992).
8.2. Evolutionary scenario C
The papers mentioned above were criticised and
in 1998 a new version of the evolutionary scenario
was developed (Vanbevern et al. 1998). The stel-
lar wind loss rates were corrected taking into ac-
count empirical data about OB and WR stars.
Here we list the main equations of this scenario
(all results concern only with the stars with initial
mass M0 > 15M⊙).
log Ṁ =
1.67 logL− 1.55 logTeff − 8.29, (α)
logL+ logR− logM − 7.5, (β)
0.8 logL− 8.7, (γ)
logL− 10, (δ)
We assume the next indication in this formula:
(α), H burning in the core; (β), giant, M0 ≥
40M⊙; (γ), giant, M0 < 40M⊙; (δ), Wolf-Rayet
star.
Note that in this subsection we used the mass
M is in M⊙, the luminosity L is in L⊙ and the ra-
dius R is in R⊙. With these new calculations, the
Webbink diagram described above changed signif-
icantly Vanbevern et al. (1998).
In this scenario, the total mass loss by a star is
calculated using the formula
∆M = (M −Mcore), (65)
where Mcore is the stellar core mass (66). If the
maximum mass of the star (usually it is the initial
mass of a star, but the mass transfer in binary
systems is able to increase its mass above its initial
value) Mmax > 15M⊙, the mass of the core in the
main sequence stage is determined using (66α),
and in giant and supergiant stages using (66β). In
the Wolf-Rayet star stage (helium star), ifMWR <
2.5M⊙ and Mmax ≤ 20M⊙ it is described using
(66γ), if MWR ≥ 2.5M⊙ and Mmax ≤ 20M⊙ as
(66δ), if Mmax > 20M⊙ as (66ǫ)
mcore =
1.62m0.83opt (α)
10−3.051+4.21 lgmopt−0.93(lgmopt)
0.83m0.36WR (γ)
1.3 + 0.65(mWR − 2.4) (δ)
mcore = 3.03m
0.342
opt (ǫ)
These evolutionary scenarios have some pecu-
liar properties with respect to the classical sce-
nario. One of them is that the strong stellar wind
from massive stars leads to a rapid and significant
increase of the system’s orbit and such stars can-
not fill its RL at all.
There are three observational facts that conflict
with the strong stellar wind scenarios:
1. A very high Ṁ is a problem by itself. The
observers calculate this quantity for most
of OB and WR stars using the emission
measure EM ∝
n2edl. The estimate of
Ṁ using EM is maximal for homogenous
wind. However, there are evidences (see
e.g. Cherepashchuk et al. (1984)) that
stellar winds of massive stars are strongly
“clumpy”. In this case the real Ṁ must
be 3-5 times less. This note is especially
important for the scenario with high stellar
wind.
2. Very massive Wolf-Rayet stars do exist.
There are at least three double WR+OB sys-
tems including very massive WR-stars: CQ
Cep 40M⊙, HD 311884 48M⊙ and HD92740
77M⊙ (Cherepashchuk et al. 1996). Such
heavy WR stars are at odds with the as-
sumed high mass loss rate.
3. In the semi-detached binary system RY Sct
(W Ser type) the mass of the primary com-
ponent is ≈ 35M⊙ (Cherepashchuk et al.
1996). This mass is near the limit
(Vanbevern et al. 1998) beyond which the
star, according to the high mass loss sce-
nario, cannot fill its RL.
8.3. Evolutionary Scenario W
The evolutionary scenario W is based on the
stellar evolution calculations by Woosley et al.
(Woosley et al. (2002), Fig. 16), which repre-
sents the relationship between the mass of the
relativistic remnant and the initial mass of the
star. We included into population-synthesis code
two models with W-type stellar winds, which we
label Wb and Wc. In models Wb and Wc, the
mass-loss rates were computed as in scenario B
and scenario C, respectively. The use of these
models to calculate the wind rate in a scenario
based on Woosley’s diagram (Woosley et al.
(2002), Fig. 16) is justified by the fact that
scenarios B and C are based on the same nu-
merical expressions for the mass-loss rates from
Schaller et al. (1992); Vanbevern et al. (1998);
Nieuwenhuijzen & de Jager (1990) that were
used by Woosley et al. (2002).
9. The “Ecology” of Magnetic Rotators
One of the most important achievements in as-
trophysics in the end of the 1960s was the real-
ization that in addition to “ordinary” stars, which
draw energy from nuclear reactions, there are ob-
jects in the Universe whose radiation is caused by a
strong gravitational and magnetic field. The well-
known examples include neutron stars and white
dwarfs. The property that these objects have in
common is that their astrophysical manifestations
are primarily determined by interaction with the
surrounding matter.
In the early 1980s, this approach led to the cre-
ation of a complete classification scheme involv-
ing various regimes of interaction between neu-
tron stars and their environment, as well as to the
first Monte Carlo simulation of the NS evolution
(Lipunov 1984). In addition to NSs, this scheme
has been shown to be applicable to other types of
magnetized rotating stars.
By virtue of the relationship between the grav-
itational and electromagnetic forces, the NS in
various states can manifest itself quite differently
from the astronomical point of view. Accordingly,
this leads to the corresponding classification of NS
types and to the idea of NS evolution as a gradual
changing of regimes of interaction with the envi-
ronment. The nature of the NS itself turns out to
be important also when constructing the classifica-
tion scheme. This indicates that there should be a
whole class of quite different objects which have an
identical physical nature. To develop the theory
describing properties of such objects (in a sense, it
should establish “ecological” links between differ-
ent objects), it proved to be convenient to use sym-
bolic notations elaborated for the particular case
of NS. We start this subsection with recollecting
the magnetic rotator formalism (mainly according
to the paper Lipunov (1987)).
9.1. A Gravimagnetic Rotator
We call any gravitationally-bounded object
having an angular momentum and intrinsic mag-
netic field by the term “gravimagnetic rotator”
or simply, rotator. In order to specify the intrin-
sic properties of the rotator, three parameters are
sufficient – the mass M , the total angular mo-
mentum J = Iω (I is the moment of inertia and ω
is the angular velocity), and the magnetic dipole
moment µ. Given the rotator radius R0, one can
express the magnetic field strength at the poles
B0 by using the dipole moment B0 = 2µ/R
0. The
angle β between the angular moment J and the
magnetic dipole moment µ can also be of impor-
tance: β = arccos(Jµ).
9.2. The Environment of the Rotator
We assume that the rotator is surrounded by
an ideally conductive plasma with a density ρ∞
and a sound velocity a∞ at a sufficiently far dis-
tance from the rotator. The rotator moves rela-
tive to the environment with a velocity v∞. Un-
der the action of gravitational attraction, the sur-
rounding matter should fall onto the rotator. A
rotator without a magnetic field would capture a
stationary flow of matter, Ṁc, which can be es-
timated using the Bondy-Hole-Lyttleton formu-
lae (Bondi & Hole 1944; Bondi 1952; McCrea
1953):
Ṁc = δ
(2GM)2
(a2∞ + v
ρ∞, (67)
where δ is a dimensionless factor of the order of
unity. When one of the velocities, a∞ or v∞, far
exceeds the other, the accretion rate is determined
by the dominating velocity, and can be written in
a convenient form as (55).
In the real astrophysical situation, the param-
eters of the surrounding matter at distances R ≫
RG can be taken as conditions at infinity
As already noted, the matter surrounding a
NS or a WD is almost always in the form of
a high-temperature plasma with a high conduc-
tivity. Such accreting plasma must interact effi-
ciently with the magnetic field of the compact star
(Amnuel’ & Guseinov 1968). Hence, the interac-
tion between the compact star and its surround-
ings cannot be treated as purely gravitational and
therefore the accretion is not a purely gas dynamic
process. In general, such interaction should be
described by the magneto hydrodynamical equa-
tions. This makes the already complicated pic-
ture of interaction of the compact star with the
surrounding medium even more complex.
The following classification of magnetic ro-
tators is based on the essential characteristics
7RG =
of the interaction of the plasma surrounding
them with their electromagnetic field. This
approach was proposed by Shvartsman (1970)
who distinguished three stages of interaction
of magnetic rotators: the ejection stage, the
propeller stage, which was later rediscovered
by Illarionov & Sunyaev (1975) and named as
such, and the accretion stage. Using this ap-
proach, Shvartsman (1971a) was able to pre-
dict the phenomenon of accreting X-ray pulsars
in binary systems. New interaction regimes dis-
covered later have led to a general classifica-
tion of magnetic rotators (Lipunov 1982a, 1984;
Kornilov & Lipunov 1983a).
It should be noted that the interaction of the
magnetic rotator with the surrounding plasma is
not yet understood in detail. However, even the
first approximation reveals a multitude of interac-
tion models. To simplify the analysis, we assume
the electromagnetic part of the interaction to be
independent of the accreting flux parameters, and
vice versa.
Henceforth, we shall assume in almost all cases
that the intrinsic magnetic field of a rotator is a
dipole field (Landau & Lifshiz 1971):
(1 + 3 sin2 θ)1/2, (68)
This is not just a convenient mathematical sim-
plification. We will show that the magnetoplasma
interaction takes place at large distances from the
surface of the magnetic rotator, where the dipole
moment makes the main contribution. Moreover,
the collapse of a star into a NS is known to
“cleanse” the magnetic field. Indeed, the conser-
vation of magnetic flux leads to a decrease of the
ratio of the quadrupole magnetic moment q to the
dipole moment µ in direct proportion to the radius
of the collapsing star, q/µ ∝ R.
It should be emphasized, however, that the con-
tribution of the quadrupole component to the field
strength at the surface remains unchanged.
The light cylinder radius is the first important
characteristric of the rotating magnetic field:
, (69)
where c is the speed of light.
A specific property of the field of the rotating
magnetic dipole in vacuum is the stationarity of
the field inside the light cylinder and formation of
magneto dipole radiation beyond the light cylin-
der. The luminosity of the magnetic dipole radia-
tion is equal to (Landau & Lifshiz 1971):
sin2 β = kt
ω, (70)
where kt =
sin2 β.
This emission exerts a corresponding braking
torque
Km = −
µ2 sin2 β
ω3cω, (71)
leading to a spin down of the rotator. Although
magnetic dipole radiation from pulsars do not ex-
ists, almost all models predict energy loss quantity
near this value. At the same time we do not take
into account possibly complicated angular depen-
dence of such loss.
9.3. The Stopping Radius
Now we consider qualitatively the effect of the
electromagnetic field of a magnetic rotator on the
accreting plasma. Consider a magnetic rotator
with a dipole magnetic moment µ, , rotational fre-
quency ω, and mass M . At distances R ≫ RG
the surrounding plasma is characterized by the
following parameters: density ρ∞, sound veloc-
ity a∞ and/or velocity v∞ relative to the star.
The plasma will tend to accrete on to the star un-
der the action of gravitation. The electromagnetic
field, however, will obstruct this process, and the
accreting matter will come to a stop at a certain
distance.
Basically, two different cases can be considered:
• When the interaction takes place beyond the
light cylinder, Rstop > Rl. This case first
considered by Shvartsman (1970, 1971a). In
this case the magnetic rotator generates a
relativistic wind consisting of a flux of differ-
ent kinds of electromagnetic waves and rel-
ativistic particles. The form in which the
major part of the rotational energy of the
star is ejected is not important at this stage.
What is important is that both relativistic
particles and magnetic dipole radiation will
transfer their momentum and hence exert
pressure on the accreting plasma. Indeed,
random magnetic fields are always present
in the accreting plasma. The Larmor radius
of a particle with energy ≪ 1010 eV mov-
ing in the lowest interstellar magnetic field
∼ 10−6 G is much smaller than the charac-
teristic values of radius of interaction, so the
relativistic wind will be trapped by the mag-
netic field of the accreting plasma and thus
will transfer its momentum to it.
Thus, a relativistic wind can effectively im-
pede the accretion of matter. A cavern is
formed around the magnetic rotator, and the
pressure of the ejected wind Pm at its bound-
ary balances the ram pressure of the accret-
ing plasma Pa:
Pm(Rstop) = Pa(Rstop), (72)
This equality defines a characteristic size
of the stopping radius, which we call the
Shvartzman radius RSh.
• The accreting plasma penetrates the light
cylinder Rstop < Rl. The pressure of the ac-
creting plasma is high enough to permit the
plasma to enter the light cylinder. Since the
magnetic field inside the light cylinder de-
creases as a dipole field, the magnetic pres-
sure is given by
, (73)
Matching this pressure to the ram pressure
of the accreting plasma yields the Alfven ra-
dius RA.
The magnetic pressure and the pressure of
the relativistic wind can be written in the
following convenient form:
, R ≤ Rl,
4πR2c
, R > Rl,
We introduce a dimensionless factor kt such
that the power of the ejected wind is
Lm = kt
ω, (75)
Assuming kt = 1/2 we get for R = Rl a
continuous function Pm = Pm(R).
The accreting pressure of plasma outside the
capture radius is nearly constant, and hence
gravitation does not affect the medium pa-
rameters significantly. In contrast, at dis-
tances inside the gravitational capture ra-
dius RG the matter falls almost freely and
exerts pressure on the “wall” equal to the dy-
namical pressure. For spherically symmetric
accretion we obtain
, R > RG,
, R ≤ RG,
Here we used the continuity equation Ṁc =
4πR2Gρ∞v∞. When presented in this form,
the pressure Pa is a continuous function of
distance.
Summarizing, for the stopping radius we get
Rstop =
Ra, Rstop ≤ Rl,
RSh, Rstop > Rl,
The expressions for the Alfven radius are:
2µ2G2M2
Ṁcv5
, RA > RG,
, RA ≤ RG,
and for the Shvartzman radius:
RSh =
2G2m2ω4
Ṁcv5∞c
, RSh > RG,
9.4. The Stopping Radius in the Super-
critical Case
The estimates presented above for the stopping
radius were obtained under the assumption that
the energy released during accretion does not ex-
ceed the Eddington limit, so we neglected the re-
verse action of radiation on the accretion flux pa-
rameters.
Now, we turn to the situation where one can-
not neglect the radiation pressure. Consider this
effect after Lipunov (1982b). Suppose that the
accretion rate of matter captured by the magnetic
rotator is such that the luminosity at the stopping
radius exceeds the Eddington limit (see equation
We shall assume, following Shakura & Sunyaev
(1973), that the radiation sweeps away exactly
that amount of matter which is needed for the
accretion luminosity of the remaining flux to be
of the order of the Eddington luminosity at any
radius:
Ṁ(R)
= Ledd, (80)
This yields
Ṁ(R) = Ṁc
, Rs =
Ṁc, (81)
where Rs is a spherization radius (where the accre-
tion luminosity first reaches the Eddington limit),
and κ designates the specific opacity of matter.
Using the continuity equation, the ram pressure of
the accreting plasma is now obtained as another
function of the radial distance
Pa ≈ ρv2 ≈
Ṁ(R)
R−3/2, R ≤ Rs,
in contrast to the subcritical regime when Pa ∝
R−5/2. Matching Pa and Pm (see the previous
section) for the supercritical case gives
, (83)
RSh =
, (84)
The critical accretion rate Ṁcr is defined by the
boundary of the inequality
Ṁc ≥ Ṁcr, (85)
and, correspondingly, is
Ṁcr =
Rst. (86)
The dependence of the Alfven radius on the ac-
cretion rate is such that the Alfven radius (be-
yond the capture radius) slightly decreases with
increasing accretion rate as Ṁ−1/6, while it de-
cerases below the capture radius as Ṁ−2/7 and
attains its lowest value for the critical accretion
rate Ṁc > Ṁcr, beyond which it is independent of
the external conditions.
We also note that in the supercritical regime,
the pressure of the accreting plasma increases
more slowly (as R−3/2) when approaching the
magnetic rotator than the pressure of the relativis-
tic wind (as R−2) ejected by it. This means that
in the supercritical case a cavern may exist even
below the capture radius.
The estimates presented here, of course, are
most suitable for the case of disk accretion. In
fact, the supercritical regime seems to emerge
most frequently under these conditions. This can
be simply understood. Indeed, the accretion rate
is proportional to the square of the capture radius
Ṁc ∝ R2G. At the same time, the angular momen-
tum of the captured matter is also proportional to
R2G. Hence, at high accretion rates the formation
of the disk looks natural.
9.5. The Effect of the Magnetic Field
Apparently, the magnetic field of a star be-
comes significant only when the stopping radius
exceeds the radius of the star, Rst > Rx. We take
the Alfven radius RA for Rst, since it is the small-
est of the two quantities RA and RSh. Hence, we
can estimate the lowest value of magnetic field of
a star which will influence the flow of matter
µmin =
4G2M2x
, RA > RG,
, RA ≤ RG
Ṁc ≤ Ṁcr,
2GMxR
, Ṁc > Ṁcr,
The case RA ≥ RG, Ṁc ≥ Ṁcr is considered
most frequently, and for this case we get the fol-
lowing numerical estimations
µmin = 10
x G cm
3, (88)
or, equivalently,
Bmin = 10
x G, (89)
Most presently observed NS have magnetic
fields ∼ 1012 G and dipole moments ∼ 1030 G
cm3, so the magnetic field must necessarily be
taken into account when considering interaction
of matter with these stars.
9.6. The Corotation Radius
The corotation radius is another important
characteristics of a magnetic rotator. Suppose
that an accreting plasma penetrates the light
cylinder and is stopped by the magnetic field at a
certain distance Rst given by the balance between
the static magnetic field pressure and the plasma
pressure. Suppose that the plasma is “frozen” in
the rotator’s magnetic field. This field will drag
the plasma and force it to rotate rigidly with the
angular velocity of the star. The matter will fall
on to the stellar surface only if its rotational ve-
locity is smaller than the Keplerian velocity at the
given distance Rst:
ωRst <
GMx/Rst, (90)
Otherwise, a centrifugal barrier emerges and
the rapidly rotating magnetic field impedes the ac-
cretion of matter (Shvartsman 1970; Pringle & Rees
1972; Davidson & Ostriker 1973; Lamb 1973;
Illarionov & Sunyaev 1975). The latter authors
assumed that if ωRst ≫
GMx/Rst, the mag-
netic field throws the plasma back beyond the
capture radius. They called this effect the “pro-
peller” regime. In fact, matter may not be shed
(Lipunov 1982a), but it is important to note that
a stationary accretion is also not possible.
The corotation radius is thus defined as
Rc = (GMx/ω
2)1/3 ∼ 2.8×108m1/3x (P/1s)2/3 cm,
where P is the rotational period of the star.
If Rst < Rc , rotation influences the accretion
insignificantly. Otherwise, a stationary accretion
is not possible for Rst > Rc.
9.7. Nomenclature
The interaction of a magnetic rotator with the
surrounding plasma to a large extent depends on
the relation between the four characteristic radii:
the stopping radius, Rst, the gravitational capture
radius, RG, the light cylinder radius, Rl, and the
corotation radius, Rc. The difference between the
interaction regimes is so significant that the mag-
netic rotators behave entirely differently in differ-
ent regimes. Hence, the classification of the inter-
action regimes may well mean the classification of
magnetic rotators. The classification notation and
terminology is described below and summarized in
Table 1, based on paper by Lipunov (1987).
Naturally, not all possible combinations of the
characteristic radii can be realized. For example,
the inequality Rl > Rc is not possible in princi-
ple. Furthermore, some combinations require un-
realistically large or small parameters of magnetic
rotators. Under the same intrinsic and external
conditions, the same rotator may gradually pass
through several interaction regimes. Such a pro-
cess will be referred to as the evolution of a mag-
netic rotator.
We describe the classification by considering an
idealized scenario of evolution of magnetic rota-
tors. Suppose the parameters ρ∞, v∞ and Ṁc of
the surrounding medium remain unchanged. We
shall also assume for a while a constancy of the ro-
tator’s magnetic moment µ. Let the potential ac-
cretion rate Ṁc at the beginning be not too high,
so that the reverse effect of radiation pressure can
be neglected, Ṁc ≤ Ṁcr. We also assume that
the star initially rotates at a high enough speed to
provide a powerful relativistic wind.
Ejectors (E). We shall call a magnetic rotator
an ejecting star (or simply an ejector E) if the pres-
sure of the electromagnetic radiation and ejected
relativistic particles is so high that the surround-
ing matter is swept away beyond the capture ra-
dius or radius of the light cylinder (if Rl > RG).
Ejector: RSh > max(Rl, RG), (92)
It follows from here that Pm ∝ R−2 while the
accretion pressure within the capture radius is
Pa ∝ R5/2 i.e. increases more rapidly as we ap-
proach an accreting star. Consequently, the radius
of a stable cavern must exceed the capture radius
(Shvartsman 1970).
It is worth noting that the reverse transition
from the propeller (P) stage to the ejector (E)
stage is non-symmetrical and occurs at a lower
period (see below). This means that to switch a
pulsar on is more difficult than to turn it off. This
is due to the fact that in the case of turning-on of
the pulsar the pressures of plasma and relativis-
tic wind must be matched at the surface of the
light cylinder, not at the gravitational capture ra-
dius. In fact, the reverse transition occurs under
the condition of equality of the Alfven radius to
the radius of the light cylinder (RA = Rl).
It should be emphasized that, as mentioned by
(Shvartsman 1970), relativistic particles can be
formed also at the propeller stage by a rapidly
rotating magnetic field (see also Kundt (1990)).
Propellers (P). After the ejector stage, the
propeller stage sets in under quite general condi-
tions, when accreting matter at the Alfven sur-
face is hampered by a rapidly rotating magnetic
field of the magnetic rotator. In this regime the
Alfven radius is greater than the corotation ra-
dius, RA > Rc. A finite magnetic viscosity causes
the angular momentum to be transferred to the
accreting matter so that the rotator spins down.
Until now, the propeller stage is one of the poorly
investigated phenomena. However, it is clear that
sooner or later the magnetic rotator is spin down
enough for the rotational effects to be of no im-
portance any longer, and the accretion stage sets
Accretors (A). In the accretion stage, the
stopping radius (Alfven radius) must be smaller
than the corotation radius RA < Rc. This is the
most thoroughly investigated regime of interac-
tion of magnetic rotators with accreting plasma.
Examples of such systems span a wide range of
bright observational phenomena from X-ray pul-
sars, X-ray bursters, low-mass X-ray binaries to
most of the cataclysmic variables and X-ray tran-
sient sources.
Georotators (G). Imagine that the star be-
gins rotating so slowly that it cannot impede
the accretion of plasma, i.e. all the conditions
mentioned in the previous paragraph are sat-
isfied. However, matter still can not fall on
to the rotator’s surface if the Alfven radius
is larger than the gravitational capture radius
(Illarionov & Sunyaev 1975; Lipunov 1982c).
This means that the attractive gravitational force
of the star at the Alfven surface is not signifi-
cant. A similar situation occurs in the interaction
of solar wind with Earth’s magnetosphere. The
plasma mainly flows around the Earth’s magne-
tosphere and recedes to infinity. This analogy ex-
plains the term “georotator” used for this stage.
Clearly, a georotator must either have a strong
Table 1: Classification of neutron stars and white dwarfs.
Abbrevi- Type Characteristic Accretion Well known
ation radii relation rate observational appearances
E Ejector Rst > RG Ṁ ≤ Ṁcr Radio pulsars
Rst > Rl
P Propeller Rc > Rst Ṁc ≤ Ṁcr ?
Rst ≤ RG
Rst ≤ Rl
A Accretor Rst ≤ RG Ṁc ≤ Ṁcr X-ray pulsars
Rst ≤ Rl
G Georotator RG ≤ Rst Ṁc ≤ Ṁcr ?
Rst ≤ Rc
M Magnetor Rst > a Ṁc ≤ Ṁcr AM Her, polars,
Rc > a ? soft gamma repeaters,
anomalous X-ray pulsars
SE Super- Rst > Rl Ṁc > Ṁcr ?
ejector
SP Super- Rc < Rst Ṁc > Ṁcr ?
propeller Rst ≤ Rl
SA Super- Rst ≤ Rc Ṁc > Ṁcr ?
accretor Rst ≤ RG
magnetic field or be embedded in a strongly rar-
efied medium.
Magnetors (M). When a rotator enters a bi-
nary system, it may happen that its magneto-
sphere engulfs the secondary star. Such a regime
was considered by Mitrofanov et al. (1977) for
WD in close binary systems called polars due to
their strongly polarized emission. In the case of
NS, magnetors M may be realized only under the
extreme condition of very close binaries with no
matter within the binary separation.
Supercritical interaction regimes. So far,
we have assumed that the luminosity at the stop-
ping surface is lower than the Eddington limit.
This is fully justified for G and M regimes since
gravitation is not important for them. For types
E, P, and especially A, however, this is not always
true. The critical accretion rate for which the Ed-
dington limit is achieved is
Ṁcr = 1.5× 10−6R8M⊙ yr−1, (93)
where R8 ≡ Rst/108 cm is the stopping radius
(Schwartzman radius or Alfven radius, see above).
We stress here that the widely used condition
of supercritical accretion rate
Ṁ & 10−8(M/M⊙)M⊙ yr
is valid only for the case of non-magnetic NS,
where Rst ≈ 10 km coincides with the stellar ra-
dius. In reality, for a NS with a typical magnetic
field of 1011 − 1012 G, the Alfven radius reaches
107 − 108 cm, so much higher accretion rates are
required for the supercritical accretion to set in.
The electromagnetic luminosity released at the NS
surface, however, will be restricted by Ledd, and
most of the liberated energy may be carried away
by neutrinos (Basko & Sunyaev 1975) (see also
section about hyper accretion).
Most of the matter in the dynamic model of
supercritical accretion forms an outflowing flux
covering the magnetic rotator by an opaque shell
(Shakura & Sunyaev 1973). The following three
additional types are distinguished, depending on
the relationship between the characteristic radii:
superejector (SE), superpropeller (SP) and super-
accretor (SA).
9.8. A Universal Diagram for Gravimag-
netic Rotators
The classification given above was based on re-
lations between the characteristic radii, i.e. quan-
tities which cannot be observed directly. This
drawback can be removed if we note that the light
cylinder radius Rl, Shvartzman radius RSh and
corotation radius Rc are functions of the well-
observed quantity, rotational period of the mag-
netor p. Hence, the above classification can be
reformulated in the form of inequalities for the ro-
tational period of a magnetic rotator.
One can introduce two critical periods pE and
pA such that their relationship with period p of a
magnetic rotator specifies the rotator’s type:
p < pE , → E or SE,
pE ≤ p < pA, → P or SP,
p > pA, → A, SA, G or M,
The values of pE and pA can be determined
from Table 2 which defines the basic nomencla-
ture, and are functions of the parameters v∞, Ṁc,
µ and Mx. The parameters p and µ character-
ize the electromagnetic interaction, while Ṁc de-
scribes the gravitational interaction. Instead of
Ṁc we introduce the potential accretion luminos-
ity L
L ≡ Ṁc
, (95)
The physical sense of the potential luminosity
is quite clear: the accreting star would be ob-
served to have this luminosity if the matter for-
mally falling on the gravitational capture cross-
section were to reach its surface.
Approximate expressions for critical periods
(Lipunov 1992) are:
0.42v
38 s, (α)
38 s, (β)
1.4 · 10−2m−1/9µ4/930 s, (γ)
We assume the next indication in this formula:
(α), Ṁc ≤ Ṁcr, p ≤ pGL; (β), Ṁc ≤ Ṁcr, p >
pGL; (γ), Ṁc > Ṁcr.
38 , s, (α)
1.2m−5/7µ
38 , s, (β)
0.17m−2/3µ
30 , s, (γ)
We assume the next indication in this formula:
(α), Ṁc ≤ Ṁcr and RA > RG; (β), Ṁc ≤
Ṁcr and RA ≤ RG; (γ), Ṁc > Ṁcr.
Here a new critical period pGL was introduced
from the condition RG = Rl:
pGL =
4πGMx
≈ 500mxv−27 s, (98)
Treating the rotator’s magnetic dipole moment
µ and Mx as parameters, we find that an over-
whelming majority of the magnetor’s stages can
be shown on a “p-L” diagram (Lipunov 1982a).
The quantity L also proves to be convenient be-
cause it can be observed directly at the accretion
stage.
9.9. The Gravimagnetic Parameter
By expecting the expression for the stopping
radius in the subcritical regime (Ṁc ≤ Ṁcr) one
can note that the magnetic dipole moment µ and
the accretion rate Ṁc always appear in the same
combination,
, (99)
as was noticed by Davies & Pringle (1981). The
parameter y characterizes the ratio between the
gravitational and magnetic “properties” of a star
and will, therefore, be called the gravimagnetic pa-
rameter. Two magnetic rotators having quite dif-
ferent magnetic fields, subjected to different ex-
ternal conditions but with identical gravimagnetic
parameters, have similar magnetospheres, as long
as the accretion rate is quite low (Ṁc ≤ Ṁcr).
Otherwise, the flux of matter near the stopping
radius no longer depends on the accretion rate at
a large distance.
In fact, the number of independent parameters
can be further reduced (see e.g. Lipunov (1992))
by introducing the parameter
Ṁcv∞
, (100)
Table 2: Parameter of the evolution equation of a magnetic rotators.
Parameter Regime
E, SE P, SP A SA G M
Ṁ 0 0 Ṁc Ṁc(RA/Rs) 0 Ṁc
κt ∼ 2/3 . 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3
Rt Rl Rm Rc Rc RA a
Plotting the rotator’s period p versus Y -
parameter we can draw a somewhat less obvious
but more general classification diagram than the
“p-L” diagram discussed above. This permits us
to show on a single plot the rotators with key
parameters Ṁc, µ and v∞ spanning a very wide
range.
In the case of supercritical accretion, another
characteristic combination is found in all the ex-
pressions:
, (101)
In analog to the subcritical “p-Y” diagram, a
supercritical “p-Ys” diagram can be drawn.
10. Evolution of Magnetic Rotators
The evolution of a magnetic rotator, which de-
termines its observational manifestations, involves
the slow changing of the regimes of its interaction
with the surrounding medium. Such an approach
to the evolution was developed in the 1970s by
Shvartsman (1970); Bisnovatyi-Kogan & Komberg
(1976); Illarionov & Sunyaev (1975); Shakura
(1975); Wickramasinghe & Whelan (1975),
Savonije & van den Heuvel (1977) and others.
Three stages were mostly considered in these pa-
pers: ejector, propeller and accretor. All these
stages can be described by a unified evolutionary
equation.
10.1. The evolution equation
Analysis of the nature of interaction of a mag-
netized star with the surrounding plasma allows
us to write an approximate evolution equation for
the angular momentum of a magnetic rotator in
the general form (Lipunov 1982a):
= Ṁksu − κt
, (102)
where ksu is a specific angular momentum applied
by the accretion matter to the rotator. This quan-
tity is given by
ksu =
(GMxRd)
1/2, Keplerian disk accretion,
G, wind accretion in a binary,
∼ 0, a single magnetic rotator.
(103)
where Rd is the radius of the inner disk edge,
Ω is the rotational frequency of the binary sys-
tem, and ηt ≈ 1/4 (Illarionov & Sunyaev 1975).
The values of dimensionless factor κt, characteris-
tic radius Rt and the accretion rate Ṁ in different
regimes are presented in Table 2.
The evolution equation (102) is approximate.
In practice, the situation with propellers and su-
perpropellers is not yet clear. In Table 2 Rm is
the size of a magnetosphere whose value at the
propeller stage is not known accurately and which
may differ significantly from the standard expres-
sions for the Alfven radius.
10.2. The equilibrium period
The evolution equation presented above indi-
cates that an accreting compact star must en-
deavor to attain an equilibrium state in which the
resultant torque vanishes (Davidson & Ostriker
1973; Lipunov & Shakura 1976). This hypothe-
sis is confirmed by observations of X-ray pulsars.
By equating the right-hand side of equation
(103) to zero, we obtain the equilibrium period:
peq ≈ 7.8π
κt/ǫ2(GMx)
−5/7y−3/7 s, (α)
peq =
A/BwL
10 s, (β)
(104)
where A ≈ 5× 10−4(3κt)µ230I−145 m−1x s yr−1, and
Bw ≈ 5.2×10−6R26m
0 /(10
2/3m2x)I
45 Ṁ−6η s yr
L37 = L/10
37, T10 = T/10 days; (α), disk accre-
tion; (β), quasi-spherical accretion.
Alternatively:
peq ≈ 1.0L−3/737 µ
30 s, disk,
peq = 10η
×(m2/30 /(102/3m2x))−1/2×
×L−137 T
10 µ30 s, stellar wind,
(105)
Let us turn to the case of disk accretion. The
above model of the spin-up and spin-down torques
possesses an unexpected property. The equilib-
rium period obtained by setting the torque to zero
is connected with the critical period pA through a
dimensionless factor:
peq(A) = 2
pA, (106)
The parameters κt and ε must be such that
peq > pA. Since κt ≈ ε ≈ 1, the equilibrium period
in the case of disk accretion is close to the critical
period, pA, separating accretion stage A and the
propeller stage P . In the case of the supercritical
accretion the equilibrium period is determined by
formula
(Lipunov 1982b):
peq(SA) ≃ 0.17µ2/330 m−1/9x s, (107)
10.3. Evolutionary Tracks
The evolution of NS in binaries must be stud-
ied in conjunction with the evolution of nor-
mal stars. This problem was discussed qualita-
tively by Bisnovatyi-Kogan & Komberg (1976);
Savonije & van den Heuvel (1977),
Lipunov (1982a) and other. We begin with the
qualitative analysis presented in the latter of these
paper.
The most convenient method of analysis of NS
evolution is using the ”p-L” diagram. It should
be recalled that L is just the potential accretion
luminosity of the NS. This quantity is equal to the
real luminosity only at the accretion stage.
In Figure 1 we show the evolutionary tracks of
a NS. As a rule, a NS in a binary is born when
the companion star belongs to the main sequence
(loop-like track). During the first 105 − 107 years,
the NS is at the ejector stage, and usually it is
not seen as a radiopulsar since its pulse radia-
tion is absorbed in the stellar wind of the normal
star. The period of the NS increases in accordance
with the magnetic dipole losses. After this, the
matter penetrates into the light cylinder and the
NS passes first into the propeller stage and then
into the accretor stage. By this time, the normal
star leaves the main sequence and the stellar wind
strongly increases. This results in the emergence
of a bright X-ray pulsar. The period of the NS
stabilizes around its equilibrium value. Finally,
the normal star fills the Roche lobe and the accre-
tion rate suddenly increases; the NS moves first
to the right and then vertically downward in the
”p-L” diagram. In other words, the NS enters the
supercritical stage SA (superaccretor) and its spin
period tends to a new equilibrium value (see equa-
tion (107)).
After the mass exchange, only the helium core
of the normal star is left (a WR star in the case
of massive stars), the system becomes detached
and the NS returns back to the propeller or ejec-
tor state. Accretion is still hampered by rapid NS
rotation. This is probably the reason underlying
the absence of X-ray pulsars in pairs with Wolf-
Rayet stars (Lipunov 1982c). Since the helium
star evolves on a rather short time-scale (≈ 105
yr), the NS does not have time to spindown con-
siderably: after explosion of the normal star, the
system can be disrupted leaving the old NS as an
ejector, i.e. as a high-velocity radio pulsar.
The “loop-shaped” track discussed above can
be written in the form:
• I+E → I+P → II+P → II+A → III+SA →
IV+P → E+E (recycled pulsar) → . . .
• I+E → I+P → II+A → III+SA → IV+E
→ (recycled ejector) → IV+P → E+E (re-
cycled pulsar) → . . .
• Another version of the evolutionary track of
a NS formed in the process of mass exchange
within a binary system is:
III+SE → III+SP → IV+P → E+E → . . .
The overall lifetime of a NS in a binary system
depends on the lifetime of the normal star and on
the parameters of the binary system. However,
Fig. 1.— Tracks of NS on the period (p) - gravimagnetic parameter (Y) diagram: track of a single NS
(vertical line) and of a NS in a binary system (looped line). For the second track, possible observational
appearances of the NS are indicated.
the number of transitions from one stage to an-
other during the time the NS is in the binary is
proportional to the magnetic field strength of the
Figure 2 demonstrates the effect of NS mag-
netic field decay (track (a) with and (b) without
magnetic field decay). The first track illustrates
the common path which results in the production
of a typical millisecond pulsar.
10.4. Evolution of Magnetic Rotators in
Non-circular Orbits
So far we have considered evolution of a mag-
netic rotator related to single rotators or those
entering binary system with circular orbits. This
approximation was appropriate for the gross anal-
ysis of binary evolutionary scenario performed
by Kornilov & Lipunov (1983a,b). This approx-
imation is further justified by the fact that the
tidal interaction in close binaries leads to or-
bital circularization in a short time (Hut 1981).
However, the more general case of a binary
with eccentric orbit must be considered for fur-
ther analysis. It is especially important because
many of the currently observed X-ray pulsars, as
well as radio pulsars with massive companions
PSR B1259-63 (Johnston et al. 1992) and PSR
B0042-73 (Kaspi et al. 1994), are in highly ec-
centric orbits around massive companions. Pre-
viously, such studies have been performed by
Gnusareva & Lipunov (1985); Prokhorov (1987).
Orbital eccentricity necessarily emerges after
the first supernova explosion and mass expulsion
from the binary system. In massive binaries with
long orbital periods & 10 days, the eccentricity
may be well conserved until the second episode of
mass exchange (Hut 1981). Here, we concentrate
on the evolutionary consequences of eccentricity.
10.5. Mixing types of E-P-A binary sys-
tems with non-zero orbital eccentric-
The impact of eccentricity on the observed
properties of X-ray pulsars has been consid-
ered in many papers (Amnuel’ & Guseinov 1971;
Shakura & Sunyaev 1973; Pacini & Shapiro 1975;
Lipunov & Shakura 1976). The most important
consequence of orbital eccentricity for the evolu-
tion of rotators can be understood without de-
tailed calculations, and suggests the existence of
two different types of binary systems separated by
a critical eccentricity, ecr (Gnusareva & Lipunov
1985).
Consider an ideal situation when a rotator en-
ters a binary system with some eccentricity. The
normal star (no matter how) supplies matter to
the compact magnetized rotator. We assume that
all the parameters of the binary system (binary
separation, eccentricity, masses, accretion rate,
etc.) are stationary and unchanged. Then a crit-
ical eccentricity ecr appears such that at e > ecr
the rotator is not able to reach the accretion state
in principle. Let the rotator be rapid enough ini-
tially to be at the ejector (E) state. With other
parameters constant, the evolution of such a star
is determined only by its spindown. The star will
gradually spindown to such a state that when pass-
ing close to the periastron where the density of
the surrounding matter is higher, the pulsar will
“choke” with plasma and pass into the propeller
regime. Therefore, for a small part of its life the ro-
tator will be in a mixed EP-state, being in the pro-
peller state at periastron and at the ejector state
close to apastron. The subsequent spindown of
the rotator leads most probably to the propeller
state along the entire orbit. This is due to the
fact that the pressure of matter penetrating the
light cylinder Rl increases faster than that caused
by relativistic wind and radiation, as first noted
by Shvartsman (1971a). So it proves to be much
harder for the rotator to pass from the P state to
the E state than from E to P state (see the follow-
ing section).
The rotator will spindown ultimately to some
period, pA, at which accretion will be possible dur-
ing the periastron passage. Accretion, in contrast,
will lead to a spin-up of the rotator, so that it
reaches some average equilibrium state character-
ized by an equilibrium period peq defined by the
balance of accelerating and decelerating torques
averaged over the orbital period. If the eccentric-
ity was zero, the rotator would be in the accretion
state all the time. By increasing the eccentricity
and keeping the periastron separation between the
stars unchanged, we increase the contribution of
the decelerating torque over the orbital period and
thus decrease peq. At some ultimate large enough
eccentricity ecr the equilibrium period will be less
than the critical period pA permitting the transi-
Fig. 2.— The period-gravimagnetic parameter diagram for NS in binary systems. (a) with NS magnetic
field decay (the oblique part of the track corresponds to “movement” of the accreting NS along the so-called
“spin-up” line), (b) a typical track of a NS without field decay in a massive binary system.
tion from the propeller state to the accretion state
at apastron to occur. The rotational torque ap-
plied to the rotator, averaged over orbital period,
vanishes, and in this sense the equilibrium state is
achieved, but the rotator periodically passes from
the propeller state to the accretion state.
Thus, X-ray pulsars with unreachable full-orbit
accretion state must exist. This means that from
the observational point of view such binaries will
be observed as transient X-ray sources with sta-
tionary parameters for the normal component.
Typically, the evolutionary track of a rotator in
an eccentric binary is
• E → PE → P → AP, e > ecr
• E → PE → P → AP → A, e < ecr
this may be the principal formation channel of
transient X-ray sources.
10.6. Ejector-propeller hysteresis
As mentioned earlier, the transition of the ro-
tator from the ejector state to the propeller state
is not symmetrical. Here we consider this effect in
more detail. In terms of our approach, we must
study the dependence of Rst on Ṁc. To find Rst,
we must match the ram pressure of the accreting
plasma with that caused by the relativistic wind or
by the magnetosphere of the rotator. This depen-
dence Rst(Ṁc) will be substantially different for
rapidly (Rl < RG) and slowly (Rl > RG) rotat-
ing stars (see Figure 3). One can see that in the
case of a fast rotator, an interval of Ṁc appears
where three different values of Rst are possible,
the upper value R1 corresponding to the ejector
state and the bottom value R3 to the propeller
state; the intermediate value R2 is unstable. This
means that the rotator’s state is not determined
solely by the value of Ṁc , but also depends on
previous behavior of this value.
Now consider a periodic changing of Ṁc caused,
for example, by the rotator’s motion along an ec-
centric orbit, and large enough for the rotator to
transit from the ejector state to the propeller state
and vice versa. Initially, the rotator is in the ejec-
tor state. By approaching the normal star, the
accretion rate Ṁc increases and reaches a criti-
cal value ṀEP , where the equilibrium points R1
(stable point corresponding to the ejector state)
and R2 (unstable) approach RG (upper kink),
where they merge (see Figure 2). After that only
one equilibrium point remains in the system, the
stopping radius Rst jumps from ≈ RG down to
Fig. 3.— Dependence of the stopping radius Rst on the modified gravimagnetic radius Ry for two possible
relations between the light cylinder radius Rl and the gravitation capture radius RG: Rl > RG (left-hand
panel) and Rl < RG (right-hand panel; here the ejector-propeller hysteresis becomes possible).
R3 < Rl, and the rotator changes to the propeller
state.
As Ṁc decreases further along the orbit and
reaches the critical value ṀEP once again, the
reverse transition from propeller to ejector does
not occur. The transition only occurs when Ṁc
reaches another critical value, ṀPE < ṀEP ,
where the unstable point R2 meets the stable pro-
peller point R3, and the stopping radius Rst jumps
from ∼ Rl up to R1 > RG. It should be noted
that for fast enough rotators, a situation is pos-
sible when the step down from the ejector state
occurs in such a manner that the stopping ra-
dius Rst < Rc and the rotator passes directly to
the accretion state. The reverse transition always
passes through the propeller stage: A → P →
E. In principle, transitions from the ejector state
to supercritical states SP or SA are also possible
(Prokhorov 1987). In the case of slow rotators
(Rl > RB) ), the “E-P” hysteresis is not possible,
and transitions between these states are symmet-
rical.
10.7. E-P transitions for different orbits
Orbital motion of the rotator around the nor-
mal companion in an eccentric binary draws a hor-
izontal line on the ”p-Y” diagram, with the begin-
ning at a point corresponding to Ṁc(ap), and the
end at a point corresponding to Ṁc(aa) (here ap
and aa are the periasrton and apastron distances,
respectively). The length of this segment is de-
termined by the eccentricity. Since Y ∝ Ṁc , the
rotator moves along this segment from left to right
and back as it revolves from the apastron to the
periastron. At each successive orbital period, this
line slowly drifts up to larger periods. The evolu-
tion of this system is thus determined by the or-
der the critical lines on the diagram are crossed by
this “line”. It is seen from the ”p-Y” diagram that
the regions with and without hysteresis are sepa-
rated by a certain value of the parameter Y = Yk.
Since Y ∝ Ṁc ∝ 1/r2 , four different situations are
possible depending on the relationship of the bi-
nary orbital separation a with critical value acr,
corresponding to Yk (see e.g. Lipunov (1992);
Lipunov et al. (1996b)).
1. ap > acr. In this case no hysteresis oc-
curs and transitions E–P and the reverse
take place in symmetrical points of the orbit.
The rotator passes the following sequence of
stages: E → EP → P → . . . Here EP means
a mixed state of the rotator at which it is
in the ejector state during one part of the
orbit, and in the propeller state during the
rest of the orbital cycle.
2. aa > acr > ap. In this case, the hysteresis
occurs at the beginning of the mixed EP-
state (state EPh), but as the rotator slows
down the hysteresis gradually decays and
disappears. E-P transition for this system
is: E → EPh → EP → P → . . .
3. aa < acr. The hysteresis is possible in prin-
ciple, but the shape of the transition depends
on the eccentricity. Suppose a pulsar has
spun down so much that the first transition
from the ejector to the propeller occurred at
the periastron. If the eccentricity were smal
e < ecr, (do not confuse this ecr with the
critical eccentricity introduced in the pre-
vious section) the reverse transition to the
ejector state would not occur even at the
apastron, and the evolutionary path would
be E → P → . . .
4. If e > ecr , the track is E → EPh → P →
. . . It should be noted that just after the first
EP transition (as well as before the last), the
system spends a finite time in the E and P
states at every revolution.
The value of ecr can be expressed through the
orbital parameters as
ecr ≃
cr − a1/7a
cr + a
, (108)
To conclude, we note that the hysteresis during
the ejector-propeller transition may be possible for
single radio pulsars also. For example, when the
pulsar moves through a dense cloud of interstellar
plasma, the pulses can be absorbed. The radio
pulsar turns on again when it comes out from the
cloud. The hysteresis amplitude for single pulsars
can be high enough because of small relative ve-
locities of the interstellar gas and the pulsar, so
that RG ≫ Rl.
11. Summary
The “Scenario Machine” is the numerical code
for theoretical investigations of statistical proper-
ties of binary stars, i.e. this is population synthesis
code (Lipunov et al. 1996b). It includes the evo-
lution of normal stars and the evolution of their
compact remnants. This is especially important
for studies of the neutron stars (Lipunov 1992).
We always include the most important observa-
tional discoveries and theoretical estimations into
our code.
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This 2-column preprint was prepared with the AAS LATEX
macros v5.2.
	Basic equations and initial distributions
	Evolutionary scenario for binary stars
	Basic evolutionary states of normal stars
	Main sequence stars
	Post main-sequence stars
	Roche lobe overflow
	Wolf-Rayet and helium stars
	Stellar winds from normal stars
	Change of binary parameters: mass, semi-major axis and eccentricity
	Mass change
	Semi-major axis change
	The change of eccentricity
	Special cases: supernova explosion and common envelope
	Three regimes of mass accretion by neutron stars
	Ordinary accretion
	Super-accretion
	Hyper-accretion
	Mass accretion by black holes
	Accretion induced collapse and compact objects merging
	Additional scenarios of stellar wind from massive stars
	Evolutionary scenario B
	Evolutionary scenario C
	Evolutionary Scenario W
	The ``Ecology'' of Magnetic Rotators
	A Gravimagnetic Rotator
	The Environment of the Rotator
	The Stopping Radius
	The Stopping Radius in the Supercritical Case
	The Effect of the Magnetic Field
	The Corotation Radius
	Nomenclature
	A Universal Diagram for Gravimagnetic Rotators
	The Gravimagnetic Parameter
	Evolution of Magnetic Rotators
	The evolution equation
	The equilibrium period
	Evolutionary Tracks
	Evolution of Magnetic Rotators in Non-circular Orbits
	Mixing types of E-P-A binary systems with non-zero orbital eccentricity
	Ejector-propeller hysteresis
	E-P transitions for different orbits
	Summary

  We present here an updated description of the "Scenario Machine" code. This
tool is used to carry out a population synthesis of binary stars. Previous
version of the description can be found at
http://xray.sai.msu.ru/~mystery//articles/review/contents.html

DESCRIPTION OF THE SCENARIO MACHINE
V.M. Lipunov1, K.A. Postnov2, M.E. Prokhorov3, A.I. Bogomazov4
Sternberg astronomical institute, Universitetskij prospect, 13, 119992, Moscow, Russia
ABSTRACT
We present here an updated description of the “Scenario Machine” code. This tool is
used to carry out a population synthesis of binary stars. Previous version of the descrip-
tion can be found at http://xray.sai.msu.ru/∼mystery//articles/review/contents.html; see also
(Lipunov et al. 1996b,c).
Subject headings: binaries: close — binaries: general
1. Basic equations and initial distributions
We use the current scenario of evolution of bi-
nary stellar systems based upon the original ideas
that appeared in the papers by Paczyn’ski (1971);
Tutukov & Yungelson (1973),
van den Heuvel & Heise (1972) (see also re-
view by van den Heuvel (1994)). The sce-
nario for normal star evolution was joined with
the ideas of neutron star evolution (see pi-
oneer works by Shvartsman (1970, 1971a,b);
Illarionov & Sunyaev (1975); Shakura (1975);
Bisnovatyi-Kogan & Komberg (1976),
Lipunov & Shakura (1976),
Savonije & van den Heuvel (1977)). This joint
scenario has allowed to construct a two-dimensional
classification of possible states of binary systems
containing NS (Kornilov & Lipunov (1983a,b);
Lipunov (1992)). According to this classification,
we will distinguish four basic evolutionary stages
for a normal star in a binary system:
I — A main sequence (MS) star inside its Roche
lobe (RL);
II — A post-MS star inside its RL;
III — AMS or post-MS star filling its RL; the mass
is transferred onto the companion.
1E-mail: lipunov@xray.sai.msu.ru
2E-mail: pk@.sai.msu.ru
3E-mail: mystery@xray.sai.msu.ru
4E-mail: a78b@yandex.ru
IV — A helium star left behind the mass-transfer
in case II and III of binary evolution; may
be in the form of a hot white dwarf (for M ≤
2.5M⊙), or a non-degenerate helium star (a
Wolf-Rayet-star in case of initial MS mass
> 10M⊙).
The evolution of single stars can be represented
as a chain of consecutive stages: I→ II→ compact
remnant; the evolution of the most massive single
stars probably looks like I → II → IV → compact
remnant. The component of a binary system can
evolve like I→ II→ III→ IV→ compact remnant.
In our calculations we choose the distributions
of initial binary parameters: mass of the primary
zero age main sequence component (ZAMS), M1,
the binary mass ratio, q = M2/M1 < 1, the orbital
separation a. Zero initial eccentricity is assumed.
The distribution of binaries by orbital separa-
tions can be taken from observations (Krajcheva et al.
1981; Abt 1983),
f(log a) = const,
max(10R⊙,RL[M1]) < a ≤ 106R⊙;
Especially important from the evolutionary point
of view is how different are initial masses of the
components (see e.g. Trimble (1983)). We have
parametrized it by a power-law shape, assuming
the primary mass to obey Salpeter’s power law:
f(M) = M−2.351 , 0.1M⊙ < M1 < 120M⊙, (2)
http://arxiv.org/abs/0704.1387v2
http://xray.sai.msu.ru/~mystery//articles/review/contents.html
f(q) ∼ qαq , q = M2/M1 < 1; (3)
We should note that some apparently reason-
able distributions – such as both the primary and
secondary mass obeying Salpeter’s law, or “hier-
archical” distributions involving the assumption
that the total binary mass and primary’s mass are
distributed according to the Salpeter mass func-
tion – all yield essentially flat-like distributions by
the mass ratio (i.e. with our parameter αq ≃ 0).
We assume that the neutron star is formed
in the core collapse of the pre-supernova star.
Masses of the young neutron stars are randomly
distributed in the range MminNS – M
NS . Initial
NS masses are taken to be in the range MNS =
1.25 − 1.4M⊙. The range of initial masses of the
young NSs was based on the masses of the neu-
tron star in the B1913+16 binary system and of
the radio pulsar in the J0737-3039 binary. In the
B1913+16 system (the Hulse-Taylor pulsar, radio
pulsar + neutron star) the mass of the neutron
star, which is definitely not accreting matter from
the optical donor, is MNS = 1.3873 ± 0.0006M⊙
(Thorsett & Chakrabarty 1999; Wex et al. 2000;
Weisberg & Taylor 2003). The mass of the pul-
sar in the J0737-3039 system (radio pulsar + radio
pulsar), which likewise does not accrete from an
optical companion, is MPSR = 1.250 ± 0.010M⊙
(Lyne et al. 2004). The mass range of stars pro-
ducing neutron stars in the end of their evolution is
assumed to be Mn – Mb; stars with initial masses
M > Mb are assumed to leave behind black holes;
Mn is taken to be equal to 10M⊙ in most cases
(but in general this parameter is free). Note that
according to some stellar models, a very massive
star (≈ 50 − 100M⊙) can leave behind a neutron
star as a remnant due to very strong mass loss
via powerful stellar wind, so we account for this
possibility in the corresponding models.
We take into account that the collapse of mas-
sive star into a neutron star can be asymmetrical,
so that the newborn neutron star can acquire an
additional, presumably randomly oriented in space
kick velocity w (see Section 4 below for more de-
tails).
The magnetic field of rotating compact ob-
jects (neutron stars and white dwarfs) largely
define the evolutionary stage of the compact
object in a binary system (Shvartsman (1970);
Davidson & Ostriker (1973); Illarionov & Sunyaev
(1975)), so we use the general classification of mag-
netic rotating compact objects (see e.g. Lipunov
(1992)) in our calculations. The initial magnetic
dipole moment of the newborn neutron star is
taken according to the distribution
f(logµ) ∝ const,
1028 ≤ µ ≤ 1032G cm3, (4)
The initial rotational period of the newborn neu-
tron star is assumed to be ∼ 10 ms.
It is not definitely clear as yet whether the mag-
netic field of neutron stars decays or not (see for a
comprehensive review Chanmugam (1992)). We
assume that the magnetic fields of neutron stars
decays exponentially on a timescale of td (usually
we take this parameter to be equal to 108, 5 · 107
and 107 years). A radio pulsar is assumed to be
“switched on” until its period P (in seconds) has
reached the “death-line” defined by the relation
µ30/P
d = 0.4, where µ30 is the dipole magnetic
moment in units of 1030 G cm3.
We assumed that magnetic fields of neu-
tron stars decay exponentially to minimal value
Bmin = 8 · 107 G and do not decay further:
B0 exp(−t/td), t < td ln(B0/Bmin),
Bmin, t ≥ td ln(B0/Bmin).
Parameters B0 and td in equation (5) are the ini-
tial field strength and the field decay time.
We also assume that the mass limit for neutron
stars (the Oppenheimer-Volkoff limit) is MOV =
2.0M⊙ (in general, it is a free parameter in the
code; it depends on equation of state of the mate-
rial of the neutron star).
The most massive stars are assumed to leave be-
hind black holes after the collapse, provided that
the progenitor mass before the collapse has a mass
Mcr. The masses of the black holes are calcu-
lated as Mbh = kbhMPreSN , where the parameter
kbh = 0.0 − 1.0, MPreSN is the mass of the pre-
supernova star.
We consider binaries with M1 ≥ 0.8M⊙ with a
constant chemical (solar) composition. The pro-
cess of mass transfer between the components is
treated as conservative when appropriate, that is
the total angular momentum of the binary system
is assumed to be constant. If the accretion rate
from one component to another is sufficiently high
(say, the mass transfer occurs on a timescale few
times shorter than the thermal Kelvin-Helmholtz
time for the normal companion) or a compact ob-
ject is engulfed by a giant companion, the com-
mon envelope stage of binary evolution can begin
(Paczynski 1976; van den Heuvel 1983).
Other cases of non-conservative evolution (for
example, stages with strong stellar wind or those
where the loss of binary angular momentum oc-
curs due to gravitational radiation or magnetic
stellar wind) are treated using the well known
prescriptions (see e.g. Verbunt & Zwaan (1981);
Rappaport et al. (1982); Lipunov & Postnov
(1988)).
2. Evolutionary scenario for binary stars
Significant discoveries in the X-ray astronomy
made during the last decades stimulated the as-
tronomers to search for particular evolutionary
ways of obtaining each type of observational ap-
pearance of white dwarfs, neutron stars and black
holes, the vast majority of which harbours in bi-
naries. Taken as a whole, these ways costitute a
general evolutionary scheme, or the “evolutionary
scenario”. We follow the basic ideas about stel-
lar evolution to describe evolution of binaries both
with normal and compact companions.
To avoid extensive numerical calculations in
the statistical simulations, we treat the continu-
ous evolution of each binary component as a se-
quence of a finite number of basic evolutionary
states (for example, main sequence, red super-
giant, Wolf-Rayet star, hot white dwarf, etc.), at
which stellar parameters significantly differ from
each other. The evolutionary state of the binary
can thus be determined as a combination of the
states of each component, and is changed once the
more rapidly evolving component goes to the next
evolutionary stage.
At each such stage, we assume that the star
does not change its physical parameters (mass, ra-
dius, luminosity, the rate and velocity of stellar
wind, etc.) that have effect on the evolution of
the companion (especially in the case of compact
magnetized stars). Every time the faster evolving
component passes into the next stage, we recalcu-
late its parameters. Depending on the evolution-
ary stage, the state of the slower evolving star is
changed or can remain unchanged. With some ex-
ceptions (such as the common envelope stage and
supernova explosion), states of both components
cannot change simultaneously. Whenever possible
we use analytical approximations for stellar pa-
rameters.
Prior to describing the basic evolutionary states
of the normal component, we note that unlike
single stars, the evolution of a binary compo-
nent is not fully determined by the initial mass
and chemical composition only. The primary star
can fill its Roche lobe either when it is on the
main sequence, or when it has a (degenerate) he-
lium or carbon-oxygen core. This determines the
rate of mass transfer to the secondary compan-
ion and the type of the remnant left behind. We
will follow Webbink (1979) in treating the first
mass exchange modes for normal binary compo-
nents, whose scheme accounts for the physical
state of the star in more detail than the simple
types of mass exchange (A, B, C) introduced by
Kippenhahn & Weigert (1967). We will use both
notations A, B, C and D (for very wide systems
with independently evolving companions) for evo-
lutionary types of binary as a whole, and Web-
bink’s notations for mass exchange modes for each
component separately [Ia], [Ib], [IIa], [IIb], [IIIa],
3. Basic evolutionary states of normal
stars
The evolution of a binary system consisting ini-
tially of two zero-age main sequence stars can be
considered separately for each components until a
more massive (primary) component fills its Roche
lobe. Then the matter exchange between the stars
begins.
The evolutionary states of normal stars will be
denoted by Roman figures (I-IV), whereas those
of compact stars will be marked by capital letters
(E, P, A, SA ...). We divide the evolution of a nor-
mal star into four basic stages, which are signifi-
cant for binary system evolution and bear a clear
physical meaning. We will implicitly express the
mass and radius of the star and the orbital semi-
major axis in solar units (m ≡ M/M⊙, r ≡ R/R⊙,
a ≡ A/R⊙), the time in million years, the lumi-
nosities in units of 1038 erg s−1, the wind velocities
in units of 108 cm s−1 and the accretion rates Ṁ
onto compact objects in units of 10−8M⊙ yr
unless other units are explicitely used.
3.1. Main sequence stars
At this stage, the star is on the zero-age main
sequence (ZAMS) and its size is much smaller than
the Roche lobe radius. The time the star spends
on the main sequence is the core hydrogen burning
time, tH , which depends on the stellar mass only
(Iben & Tutukov 1987):
1.0 + 0.95 79
, m ≥ 79.0,
103.9−3.8 logm+log
2 m, 79 > m ≥ 10,
2400m−2.16, 10 > m ≥ 2.3,
104m−3.5, m < 2.3,
The radius of the ZAMS star is assumed to be
100.66 logm+0.05, m > 1.12,
m, m ≤ 1.2, (7)
and its luminosity is
logL =
−5.032 + 2.65 logm, (α)
−4.253 + 4.8 logm, (β)
−4.462 + 3.8 logm, (γ)
−3.362 + 3.0 logm, (δ)
−3.636 + 2.7 logm, (ǫ)
here we assume the next indication: (α), m < 0.6;
(β), 0.6 ≤ m < 1.0; (γ), 1.0 ≤ m < 10.0; (δ),
10 ≤ m < 48.0; (ǫ), m ≥ 48.0.
The initial mass of the primary and the mode
of the first mass exchange (which is determined by
the initial orbital period and masses of the com-
ponents; see Webbink (1979)) determine the mass
and the type of the core that will be formed during
stage I. For example, for single stars and primaries
in “type C” binaries that fill its Roche lobe having
a degenerate core, we use the expressions
0.1mmax, (α)
0.446 + 0.106mmax, (β)
0.24m0.85max, (γ)
min(0.36m0.55max, 0.44m
max), (δ)
0.44m0.42max, (ǫ)
≈ MCh, (ζ)
0.1m1.4max, (η)
where mmax is the maximum mass the star had
during the preceding evolution. We assume the
next indication in this formula: (α), mmax < 0.8;
(β), 0.8 ≤ mmax < 2.3; (γ), 2.3 ≤ mmax < 4.0;
(δ), 4.0 ≤ mmax < 7.5; (ǫ), 7.5 ≤ mmax < 8.8;
(ζ), 8.8 ≤ mmax < 10.0; (η), mmax ≥ 10.0.
A main-sequience star accreting matter during
the first mass transfer will be treated as a rapidly
rotating “Be-star” with the stellar wind rate dif-
ferent from what is expected from a single star of
the same mass (see below).
3.2. Post main-sequence stars
The star leaves the main sequence and goes to-
ward the red (super)giant region. The star still
does not fill its Roche lobe. The duration of this
stage for a binary component is not any more a
function of the stellar mass only (as in the case
of single stars), but also depends on the initial
binary type (A, B, or C) (see Iben & Tutukov
(1985, 1987)):
tII =
0, (α),
2tKH , (β),
6300m−3.2, (γ),
tHe, (δ),
In type A systems, the primaries fill their Roche
lobes when they belong to the main sequence. We
assume the next indication in this formula: (α),
type A; (β), type B excluding mode [IIIA]; (γ),
types C, D and mode [IIIA], mmax < 5; (δ), types
C, D and mode [IIIA], mmax > 5.
The radius of the post-MS star rapidly increases
(on the thermal time scale) and reaches the char-
acteristic giant values. The star spends the most
time of helium burning with such large radius. In
the framework of our approximate description, we
take the radius of the giant star to be equal to
the maximum value, which depends strongly on
the mass of its core and is calculated according
to Webbink’s mass transfer modes as follows (see
also Iben & Tutukov (1985, 1987)):
rII =
3000m4c, (α)
1050(mc − 0.5)0.68, (β)
10m0.44c , (γ)
We assume the next indication in this formula:
(α), mode [IIIA] or [IIIB] with He core; (β),
mode [IIIB] with CO or ONeMg core; (γ), modes
[I] or [II]. Formula (11) depicts stars with mass
≤ 10M⊙.
This maximum radius can formally exceed the
Roche lobe size; in such cases we put it equal
to 0.9RL during the stage II. The most sensi-
tive to this crude approximation are binaries with
compact companions, which can lead, for exam-
ple, to the overestimation of the number of ac-
creting neutron stars observed as X-ray pulsars.
However, these stages are less important for our
analysis than the stages at which the optical star
fills its Roche lobe. A more detailed treatment
of normal star evolution (given, for example, by
Pols & Marinus (1994)) can reduce such uncer-
tainties.
Luminosities of giants are taken from de Jager
(1980):
log lII =
15.92m6c
(1.0+m4c)(2.512+3.162mc)
, (α)
10−4.462+3.8 logm, (β)
10−3.362+3.0 logm, (γ)
10−3.636+2.7 logm, (δ)
We assume the next indication in this formula:
(α), m < 23.7, mc < 0.7; (β), m < 23.7, mc > 0.7;
(γ), 48 > m > 23.7; (δ), m > 48.
Radii of (super)giants are determined by us-
ing the effective temperature Teff and luminosi-
ties. Typical effective temperatures are taken from
Allen (1973):
logTeff =
4.50, (α)
3.60, (β)
3.70, (γ)
We assume the next indication in this formula:
(α), m > 10.0; (β), m < 10.0, type C or D; (γ),
m < 10.0, type A or B. So, we calculate RII with
mass higher than 10M⊙ calculate using formula
rII = exp {2.3(0.5 log lii − 2 logTeff + 9.7)}.
3.3. Roche lobe overflow
At this stage the star fills its Roche lobe (RL)
and mass transfer onto the companion occurs.
The mass transfer first proceeds on the thermal
time scale (see extensive discussion of this approx-
imaiton in van den Heuvel (1994))
tKH ∼ 30m2r−1∗ (L/L⊙)−1, (15)
The common envelope stage (CE) may be
formed if the Roche lobe overflow occurs in the
type C system (where the primary has a well de-
veloped core) even for q < 1; otherwise (for type
B systems) we use the condition q ≤ qcr = 0.3 for
the CE stage to occur. Radius of the star at the
Roche lobe filling stage is taken to be that of the
equivalent Roche lobe radius (Eggleton 1983):
a(1− e)
0.49q2/3
0.6q2/3 + ln(1 + q1/3)
, (16)
Here a is the binary orbital separation and e is the
orbital eccentricity, q is the arbitrary mass ratio.
For q < 0.6 a more precise approximation can
be used:
= 0.4622
1 + q
. (17)
A star filling its Roche lobe has quite differ-
ent boundary conditions in comparison with sin-
gle stars. The stellar radius at this stage is limited
by the Roche lobe. If the stellar size exceeds the
Roche lobe, the star can lose matter on a time
scale close to the dynamical one until its radius
becomes smaller than the new Roche lobe size.
Now we consider how the RL-filling star loses
matter. Let the star be in equilibrium and
Req(M) = RL(M) at the initial moment of time.
When a fraction of mass δm is transported to the
companion, the mass ratio q and semi-major axis
a of the binary changes depending on the mass
transfer mode assumed (see below for details).
The RL size then becomes equal to RL(M − δm).
On the other hand, mass loss disturbs equi-
librium of the star (hydrodynamical and ther-
mal). The hydrodynamical equilibrium is restored
on the dynamical timescale td ∝
GM/R3
)−1/2
The stellar radius changes to a value Rad(M −
δm) (where “ad” means adiabatic), which can
be bigger or smaller than the equilibrium radius
Req(M − δm) of the star. The thermal equilib-
rium establishes on the thermal time scale TKH ≈
GM2/RL, so after that the stellar radius relaxes
to the equilibrium value Req(M − δm).
Relations RL(M − δm), Rad(M − δm) and
Req(M−δm) determine the mode of mass transfer
during the RL overflow stage. Following Webbink
(1985), one usually introduces the logarithmic
derivative ζ = d lnR/d lnM (R ∝ M ζ). It lo-
cally fits the real dependence R(M). Three values
of ζ are relevant:
d lnRL
d lnM
ζad =
d lnRad
d lnM
, (18)
ζeq =
d lnReq
d lnM
where “L”, “ad” and “eq” correspond to the values
of radii discussed above.
Three possible cases are considered depending
on ζi:
1. If ζad < ζL, the star cannot be inside its RL
regardless of the mass loss rate (dM/dt < 0).
Such stars lose their matter in hydrodynami-
cal time scale. Mass loss rate is limited only
by the speed of sound near the inner La-
grangian point L1. ζeq is unimportant be-
cause the size of the star becomes bigger and
bigger than RL. The equilibrium is impossi-
2. ζeq < ζL < ζad. The star losing mass cannot
be in thermal equilibrium, because otherwise
its size would exceed RL. Nevertheless, in
this case Rad < RL. So the hydrodynamical
equilibrium is established. As a result, the
star loses mass on thermal time scale.
3. ζL < ζad, ζeq . In this case the size of the star
losing mass becomes smaller than its RL.
The evolutionary expansion of the star or
the binary semi-major axis decrease due to
orbital angular momentum loss via magnetic
stellar wind (MSW) or gravitational radia-
tion (GW) support the permanent contact of
the star with RL. The star then loses mass
on a time scale dictated by ots own evolu-
tionary expansion or on a time scale corre-
sponding to the orbital angular momentum
loss.
For non-degenerate stars, Req increases mono-
tonically with M . On the other hand, the expo-
nent ζad is determined by the entropy distribution
over the stellar radius which is different for stars
with radiative and convective envelopes. It can be
shown that stars with radiative envelopes should
shrink in response to mass loss, while those with
convective envelopes should expand 5.
Therefore, stars with convective envelopes in bi-
naries should generally have a higher mass loss
rate than those with radiative envelopes under
other equal conditions. The next important fac-
tor is the dependence RL(M). It can found by
substituting one of the relations a(M) (see below)
into equation (16) or into equation (17) and dif-
ferentiating it with respect to M . For example,
assuming the conservative mass exchange when
the total mass and the orbital angular momen-
tum of the system do not change, one readily gets
that the binary semi-major axis decreases when
the mass transfer occurs from the more massive to
the less massive component; RL decreases of the
primary correspondingly. When the binary mass
ratio reaches unity, the semi-major axis takes on a
minimal value. In contrast, if the less massive star
loses its mass conservatively the system expands.
In that case the mass transfer can be stable.
If more massive component with radiative en-
velope fills its RL, the mass transfer proceeds on
thermal time scale until the masses of the compo-
nents become equal 6. The next stage of the first
mass exchange poceeds in more slower (nuclear)
time scale. If the primary has convective enve-
lope, the mass transfer can proceed much faster
on a time scale intermediate between the thermal
and hydrodynamical one, and probably on the hy-
drodynamical time scale. In that case the fast
stage of the mass exchange ends when the mass
of the donor decreases to ∼ 0.6 mass of the sec-
ondary companion (Tutukov & Yungelson 1973).
Further mass transfer should proceed on the evo-
lutionary time scale.
5The adiabatic convection in stellar envelope can be de-
scribed by the polytropic equation of state P ∝ ρ5/3, simi-
lar to non-relativistically degenerate white dwarfs. For such
equation of state the mass-radius relation becomes inverse:
R ∝ M−1/3. For non-degenerate stars with convective en-
velopes this relation holds approximately.
6The mass exchange can stop earlier if the entire envelope
is lost and the stellar core is stripped (the core has other
values ζad and ζeq).
The process of mass exchange strongly depends
on stellar structure at the moment of the RL over-
flow. The structure of the star in turn depends
on its age and the initial mass. The moment of
the RL overflow is determined by the mass of the
components and by the initial semi-major axis of
the system. to calculate a diagram in the M − a
(or M − Porb) plane which allows us to conclude
when the primary in a binary with given initial pa-
rameters fills its RL, what is its structure at that
moment and what type of the first mass exchange
is expected. We use the diagram calculated by
Webbink (1979) (see also the description of mod-
ern stellar wind scenarios below).
We distinguish different sub-stages of the RL
overflow according to the characteristic timescales
of the mass transfer:
stage III:
This is the most frequent case for the first mass
transfer phase. The primary fills its RL and the
mass transfer proceeds faster than evolutionary
time scale (if outer layers of the star are radia-
tive, then it is thermal time scale, if outer layers
are convective, then time scale is shorter, up to
hydrodynamical time scale). This stage comes to
the end when mass ratio in the system changes
(“role-to-role transition”), i.e. when the mass of
the donor (mass losing) star is equal to mass of
second companion (for radiative envelopes) or 0.6
of the mass of the second companion (for convec-
tive envelopes). This stage also stops if the donor
star totally lost its envelope.
stage IIIe:
This is the slow (evolutionary driven) phase of
mass transfer. We assume it to occur in short-
period binaries of type A. However, it is not ex-
cluded that it may occur after the mass reversal
during the first stage of mass exchange for binaries
of type B (see van den Heuvel (1994)), e.g. as in
wide low-mass X-ray binaries.
stage IIIs
This is the specific to super-accreting com-
pact companions substage of fast mass transfer at
which matter escapes from the secondary compan-
ion carrying away its orbital angular momentum.
Its duration is equal to
tIIIs = tKH
q(1 + q)
2− q − 2q2
, (19)
q = Ma/Md < 0.5,
here and below subscripts “a” and “d” refer to
the accreting and donating mass star, respectively.
For systems with small mass ratios, q < 0.5, this
timescale corresponds to an effective q-time short-
ening of the thermal time for the RL-overflowing
star.
stages IIIm,g
At these stages, the mass transfer is controlled
by additional losses of orbital angular momentum
Jorb caused by magnetic stellar wind or gravita-
tional wave emission. The characteristic time of
the evolution is defined as τJ = −(Jorb/J̇orb),
and in the case of MSW is (see Verbunt & Zwaan
(1981); Iben & Tutukov (1987))
τMSW = 4.42
a5mxλ
(m1 +m2)2m4op
, (20)
Here mop denotes mass of the low-mass optical
star (0.3 < m < 1.5) that is capable of producing
an effective magnetic stellar wind (because only
such stars have outer convective envelopes which
are prerequisit for effecftive MSW), λ is a numer-
ical parameter of order of unity. We have used
the mass-radius relation r ≈ m for main sequence
stars in deriving this formula. The upper limit
of the mass interval and empirical braking law for
main-sequence G-stars are taken from Skumanich
(1972), the lower limit is determined by absence of
cataclysmic variables with orbital periods Porb ≈
3h (Verbunt 1984; Mestel 1952; Kawaler 1988;
Tout & Pringle 1992; Zangrilli et al. 1997). We
use λ = 1 (see for details Kalogera & Webbink
(1998)).
The time scale of the gravitational wave emis-
sion is
τGW = 124.2
m1m2(m1 +m2)
× (21)
(1− e2)−7/2,
Wether the evolution is governed by MSW or
GW is decided by which time scale (τMSW or
τGW ) turns out to be the shortest among all ap-
propriate evolutionary time scales.
stage IIIwd
This is a special case where the white dwarf
overflows its RL. This stage is encountered for very
short period binaries (like Am CVn stars and low-
mass X-ray binaries like 4U 1820-30) whose evolu-
tion is controlled by GW or MSW. The mass trans-
fer is calculated using the appropriate time scale
(GW or MSW). The radius of the white dwarf in-
creases with mass RWD ∝ M−1/3. This fact, how-
ever, does not automatically imply that the mass
transfer is unstable, since the less massive WD
fills its RL first. It can be shown that the mass
exchange is always stable in such systems if the
mass ratio q < 0.8. This condition always holds
in WD+NS and WD+BH systems. WD loses its
matter until its mass decreases to that of a huge
Jupiter-like planet (∼ a few 10−3M⊙), where the
COulomb interaction reverses the mass-radius re-
lation R(M). Such a planet can approach the sec-
ondary companion of the system due to GW emis-
sion until the tidal forces destroy it completely.
The matter of the planet can fall onto the sur-
face of the second companion ir form a long-living
disk around it. If the second star is a neutron star
and its rotation had been spun up by accretion
such that a millisecond radio pulsar appeared, the
planet can be evaporated by relativistic particles
emmited by the pulsar (Paczynski & Sienkiewicz
1983; Joss & Rappaport 1983; Kolb et al. 1998;
Kalogera & Webbink 1998).
The mass loss rate at each of the III-stages is
calculated according to the relation
Ṁ = ∆M/τi, (22)
where ∆M is the a priori known mass to be lost
during the mass exchange phase (e.g. ∆M = M1−
(M1 +M2)/2 in the case of the conservative stage
III, or ∆M = M1−Mcore(M1) in case of III(e,m,g)
or CE) and τi is the appropriate time scale.
The radius of the star at stage III is assumed
to follow the RL radius:
Rdl (Md(t)) = Rd(Md(t)). (23)
3.4. Wolf-Rayet and helium stars
In the process of mass exchange the hydrogen
envelope of the star can be lost almost completely,
so a hot white dwarf (for m ≤ 2.5), or a non-
degenerate helium star (for higher masses) is left
as a remnant. The life-time of the helium star is
determined by the helium burning in the stellar
core (Iben & Tutukov 1985)
tHe =
1658m−2, (α)
1233m−3.8, (β)
0.1tH , (γ)
6913m−3.47, (δ)
≃ 10, (ǫ)
0.1tH , (ζ)
We assume the next indication in this formula:
(α), m < 1.1, modes [IIA-IIF]; (β), m > 1.1,
mmax < 10, modes [IIA-IIF]; (γ), mmax > 10,
modes [IIA-IIF]; (δ), mode [IIIA]; (ǫ), mode [IIIC];
(ζ), modes [IIIB,D,E] and type D.
If the helium (WR) star fills its Roche lobe
(a relatively rare so-called “BB” case of evolu-
tion; Delgado & Thomas (1981); see discussion in
van den Heuvel (1994)), the envelope is lost and
a CO stellar core is left with mass
1.3 + 0.65(m− 2.4),m ≥ 2.5,
0.83m0.36,m < 2.5,
The mass-radius dependence in this case is
(Tutukov & Yungelson 1973)
rWR = 0.2m
0.6. (26)
3.5. Stellar winds from normal stars
The effect of the normal star on the compact
magnetized component is largely determined by
the rate Ṁ and the velocity of stellar wind at in-
finity v∞, which is assumed to be
v∞ = 3vp ≈ 1.85
m/r, (27)
where vp is the escape velocity at the stellar sur-
face.
For “Be-stars” (i.e. those stars at the stage “I”
that increased its mass during the first mass ex-
change), the wind velocity at the infinity is taken
to be equal to the Keplerian velocity at the stellar
surface:
GM/R ≈ 0.44
m/r . (28)
The lower stellar wind velocity leads to an effec-
tive increase of the captured mass rate by the sec-
ondary companion to such “Be-stars”.
The stellar wind mass loss rate at the stage “I”
is calculated as
(de Jager 1980)
ṁ = 52.3αwl/v∞, (29)
Here αw = 0.1 is a numerical coefficient (in gen-
eral, we can treat it as free parameter).
For giant post-MS stars (stage “II”) we assume
v∞ = 3vp and for massive star we take the max-
imum between the stellar wind rate given by de
Jager’s formula and that given by Lamers (1981)
ṁ = max(52.3αw
, 102.33
l1.42r0.61
m0.99
), (30)
M ≥ 10M⊙;
For red super-giants we use Reimers’s formula
(Kudritzki & Reimers 1978):
ṁ = max(52.3αw
, 1.0
), (31)
M ≥ 10M⊙;
For a Wolf-Rayet star the stellar wind loss rate
can significantly increase (up to 10−5M⊙ year
We parametrize it as
ṀWR = kWRMWR/tHe, (32)
where the numerical coefficient is taken to be
kWR = 0.3 (in general, it can be changed if nec-
essary). The mass loss in other stages (MS, (su-
per)giant) is assumed to be limited by 10% of the
mass of the star at the beginning of the stage.
3.6. Change of binary parameters: mass,
semi-major axis and eccentricity
The duration of any evolutionary stage is de-
termined by the more rapidly evolving compo-
nent ∆t = min(∆t1,∆t2). On the other hand,
based on the evolutionary considerations we are
able to calculate how the mass of the faster evolv-
ing star changes (e.g. due to the stellar wind or
RL overflow), that is we can estimate the quantity
∆M = Mi −Mf . Then we set the characteristic
mass loss rate at this stage as
Ṁo = ∆M/∆t, (33)
Next, we should calculate the change of mass for
the slower evolving companion, the orbital semi-
major axis and the eccentricity.
3.7. Mass change
The mass of the star loosing matter is calcu-
lated as
Mf = Mi − Ṁo ×∆t, (34)
Accordingly, the mass of the accreting star is
Mf = Mi + Ṁc ×∆t, (35)
where Ṁc is the accretion rate of the captured
matter.
For stages without RL overflow the accretion
rate of the captured stellar wind matter is
ṁc = 3.8×10−2
a(v2w + 0.19(m1 +m2)/a)
ṁo .
At stages where RL overflow ocurs and both
components are normal (non-degenerate), we will
assume that the accretor can accomodate mass at
the rate determined by its thermal time, i.e.
ṁc = ṁo
tKH(donor)
tKH(accretor)
. (37)
This means that the evolution can not be
fully conservative, especially during the first mass
transfer where the primary component usually has
a shorter thermal time scale.
The mass increase rate by compact accretors
is assumed to be limited by the critical Edding-
ton luminosity (see, however, the possible hyper
accretion stage discussed below):
LEdd =
4πGMmp
≈ 1.3 · 1038 ×m erg/s (38)
(σT is the Thomson cross-section) at the stopping
radius Rstop for the accreted matter (see, e.g., de-
tailed discussion in Lipunov (1992)). This corre-
sponds to the critical accretion rate
Ṁcr = Rstop
. (39)
Thus, the mass of the accreting compact star at
the end of the stage is determined by the relation
Mf = Mi +min(Ṁc, Ṁcr)×∆t . (40)
3.8. Semi-major axis change
The binary separation a changes differently for
various mass exchange modes. First, we introduce
a measure of non-conservativeness of the mass ex-
change as the ratio between the mass change of
the accretor and the donor:
β ≡ −(M iaccr −Mfaccr)/(M idonor −M
donor). (41)
If the mass exchange is conservative (β = 1,
i.e. Ma + Md = const) and one can neglect the
angular momenta of the components, the orbital
momentum conservation implies
M iaM
. (42)
In a more general case of quasi-conservative
mass transfer 0 ≤ β < 1, the orbital separation
changes differently depending on the specific an-
gular momentum carried away from the system by
the escaping matter (see van den Heuvel (1994)
for more detail). We treat the quasi-conservative
mass transfer by assuming the isotropic mass loss
mode in which the matter carries away the spe-
cific orbital angular momentum of the accreting
component (ja)
J̇out = (1− β)Ṁcja . (43)
From here we straightforwardly find
1 + qi
1 + qf
1 + β
1 + β
)3+2/β
. (44)
In this formula q = Maccr/Mdonor and the
non-conservative parameter β is set to be the
minimal value between β = 1 and the ra-
tio TKH(donor)/TKH(accr) (TKH(donor) and
TKH(accr) is the thermal time of donor and ac-
cretor, respectively).
When no matter is captured by the secondary
companion without additional losses of angu-
lar momentum (the so-called “absolutely non-
conservative case”), which relates to the spherical-
symmetric stellar wind from one component, we
use another well-known formula
M i1 +M
. (45)
In this case the orbital separation always increases.
When the orbital angular momentum is carried
away by GW or MSW with no RL overflow, the
following approximate formulas are used:
(1−∆t/τMSW )1/4, for MSW,
(1−∆t/τGW )1/5, for GW,
In a special case of a white dwarf filling its
RL (stage “IIIwd” above), assuming a stable (i.e.
where d lnRwd/d lnM = d lnRRL/d lnM) conser-
vative mass transfer with account for the mass-
radius relation Rwd ∝ M−1/3wd , the orbital separa-
tion must increase according to the equation
)−2/3
, (47)
where mi is the initial mass of the WD donor and
mf is its mass at the end of the mass transfer.
3.9. The change of eccentricity
Tidal interaction between components, as well
as the orbital angular momentum loss due to MSW
or GR decrease the eccentricity of the binary sys-
tem. The tidal interaction is essential in very
close binaries or even during the common envelope
stage. MSW is effective only in systems with low-
mass late-type main sequence stars (see above),
GW losses become significant only in short-period
binaries.
The tidal interaction conserves the orbital an-
gular momentum which implies the relation
a(1− e2) = const. (48)
It seems that the orbit becomes a circle faster
than major semi-axis of the orbit decreases during
common envelope stage. We suppose that tcyr =
1/3tCE. We accept that tcyr for RL-filling stars
is equal to its Kelvin-Helmholtz time. Detached
systems change their eccentricity during the next
character time (Zahn 1975; Press & Teukolsky
1977; Zahn 1989b; Zahn & Bouchet 1989b)
tcyr = tKH
1 + e
)3/2(
. (49)
here tKH ≈ GM2/RL is thermal time of the star,
R is radius of the star, RL is its RL size. For sys-
tems which consist of two normal stars we choose
minimal value of tcyr. Resonances at very high ec-
centricities are not taken into account (Mardling
1995a,b).
Orbits of the systems with MSW become circu-
lar during tcyr = τMSW (see 46).
In case of GW analytical exact solutions were
obtained for a(t) and e(t) (Peters & Mathews
1963; Peters 1964).
4. Special cases: supernova explosion and
common envelope
Supernova explosion in a binary is treated as
an instantaneous mass loss of the exploding star.
The additional kick velocity can be imparted to
the newborn neutron star due the collapse asym-
metry (see below for discussion). In this case the
eccentricity and semi-major axis of the binary af-
ter the explosion can be straightforwardly calcu-
lated (Boersma 1961) (see necessary formulas also
in Grishchuk et al. 2001). Briefly, we use the fol-
lowing scheme.
1. First, velocities and locations of the compo-
nents on the orbit prior to the explosion are
calculated;
2. then the mass of the exploding star Mpr -
Mremnant is changed and the arbitrarily di-
rected kick velocity w is added to its orbital
velocity;
3. after that the transition to the new system’s
barycenter is performed (at this point the
spatial velocity of the new center of mass of
the binary is calculated);
4. in this new reference frame the new total en-
ergyE′tot and the orbital angular momentum
J ′orb are computed; if the new total energy is
negative, the new semi-major axis a′ and ec-
centricity e′ are calculated by using the new
J ′orb end E
tot; if the total energy is positive
(that is, the binary is unbound) spatial ve-
locities of each component are calculated.
The kick velocity w distribution is taken in the
Maxwellian form:
f(w) ∼ w
0 . (50)
We suppose that the absolute value of the ve-
locity that can be added during the formation of
a black hole depends on the mass loss by the col-
lapsing star, the value of the parameter w0 during
the BH formation is defined as
wbh0 = (1− kbh)w0. (51)
An effective spiral-in of the binary compo-
nents occurs during the common envelope (CE)
stage. This complicated process (introduced by
Paczynski (1976)) is not fully understood as yet,
so we use the conventional energy consideration
to find the binary system parameters after the CE
by introducing a parameter αCE = ∆Eb/∆Eorb,
where ∆Eb = Egrav − Ethermal is the binding en-
ergy of the ejected envelope matter and ∆Eorb is
the drop in the orbital energy of the system during
the spiral-in phase (van den Heuvel 1994). This
parameter measures the fraction of the system’s
orbital energy that comes during the spiral-in pro-
cess to the binding energy (gravitational minus
thermal) of the ejected common envelope. Thus
GMaMc
− GMaMd
GMd(Md −Mc)
where Mc is the mass of the core of the mass-
losing star with the initial mass Md and radius Rd
(which is simply a function of the initial separation
ai and the initial mass ratio Ma/Md, where Ma is
the mass of the accreting star).
On the CE stage the luminosity of the accret-
ing star can reach the Eddington limit so that the
further increase of the accretion rate can be pre-
vented by radiation pressure. This usually hap-
pens at accretion rates Ṁ ≃ 10−4−10−5M⊙ yr−1.
However, Chevalier (1993) suggested that when
the accretion rate is higher (Ṁ ≃ 10−2− 10−3M⊙
yr−1), the energy is radiated away not by high-
energy photons only, but also by neutrinos (see
also Zeldovich et al. (1972) and the next section).
On the typical time scale for the hyper accretion
stage of 102 yr, up to ∼ 1M⊙ of matter may be
incident onto the surface of the neutron star.
5. Three regimes of mass accretion by
neutron stars
A considerable fraction of observed neutron
stars have increased their masses in the course of
their evolution, or are still increasing their masses
(e.g., in X-ray sources). But how large can this
mass increase be? It is clear that the only origin
of a mass increase is accretion. It is evident that
the overall change in the mass of a neutron star
is determined not only by the accretion rate, but
also by the duration of the accretion stage:
Ṁdt = ṀTa, (53)
where Ṁ is the mean accretion rate and Ta is the
lifetime of the accretion stage. We emphasize that,
in the case under consideration, the accretion rate
is the amount of matter falling onto the surface
of the neutron star per unit time, and can differ
signifcantly from the values indicated by the clas-
sical Bondi-Hoyle formulas. Three regimes of ac-
cretion are possible in a close binary containing a
neutron star: ordinary accretion, super-accretion,
and hyper-accretion.
5.1. Ordinary accretion
The ordinary accretion regime is realized when
all matter captured by the gravitational field of the
neutron star falls onto its surface. This is possible
only if the radiation pressure and electromagnetic
forces associated with the magnetic field of the star
and its rotation are small compared to the gravita-
tional force. In this case, the increase in the mass
will be precisely determined by the gas dynamics
of the accretion at the gravitational-capture ra-
dius or, if the donor fills its Roche lobe, by the
binary mass ratio and the evolutionary status of
the optical component. In this case, the accretor
is observed as an X-ray source with luminosity
Lx = Ṁ
, (54)
where Mx and R∗ are the mass and radius of the
neutron star. The accretion rate Ṁ is determined
by the Bondi-Hoyle formula
Ṁ = πR2Gρv , (55)
where RG is the gravitational-capture radius of
the neutron star, v is the velocity of the gas flow
relative to the neutron star, and ρ is the density
of the gas.
The X-ray luminosity of the accretor Lx and
its other main parameters can be used to estimate
the mass ∆M accumulated during the accretion
phase:
LxR∗Ta
, (56)
5.2. Super-accretion
Regime of super accretion was considered, for
instance, in paper Lipunov (1982d). Despite the
absence of detailed models for supercritical disk
accretion (supercritical accretion is realized pre-
cisely via an accretion disk), it is possible to esti-
mate the main characteristics of the process – the
accretion rate, magnetosphere radius, and evolu-
tion equations. Accretion is considered to be su-
percritical when the energy released at the radius
where the accretion exceeds the Eddington limit:
Rstop
> LEdd = 1.38×1038(Mx/M⊙) erg s−1,
where Rstop is either the radius of the neutron star
or the magnetosphere radius RA.
For strongly magnetized neutron stars with
magnetic fields B ≫ 108 G, all matter arriving
at the magnetosphere is accreted onto the mag-
netic poles, where the corresponding gravitational
energy is released. If the black body temperature
T , roughly estimated as
SσT 4 = Ṁ
, (58)
is higher than 5 × 109 K (S is the area of the
base of the accretion column), most of the energy
will escape from the neutron star in the form of
neutrinos, and, hence, will not hinder accretion
(Zeldovich et al. 1972; Basko & Sunyaev 1975).
In this case, the rate at which the neutron star
accumulates mass will be
Ṁ ≃ Ṁcrit
≫ Ṁcrit, (59)
For lower temperatures there should be an up-
per limit on the accretion rate equal to the stan-
dard Eddington limit.
5.3. Hyper-accretion
A considerable fraction of neutron stars in bi-
nary systems pass through the common-envelope
stage in the course of their evolution. In this case,
the neutron star is effectively immersed in its opti-
cal companion, and for a short time (102-104 yrs)
spirals-in inside a dense envelope of the compan-
ion. The formal accretion rate estimated using
the Bondi-Hoyle formulas is four to six orders of
magnitude higher than the critical rate and, as
was suggested by Chevalier (1993), this may re-
sult in hyper-accretion, when all the energy is car-
ried away by neutrinos for the reasons described
above. There are currently no detailed theories
for hyper-accretion or the common-envelope stage.
The amount of matter accreted by the neutron
star can be estimated as
∫ Thyper
Ṁdt ≃ (60)
(Mopt −Mcore)
where Thyper is the duration of the hyper-accretion
stage, RG is the gravitational-capture radius of
the neutron star, a is the initial semi-major axis
of the close binary orbit, Mcore is the mass of the
core of the optical star, and Mopt and Mx are the
total masses of the optical star and of the neu-
tron star at the onset of the hyper-accretion stage.
The mass of the neutron star can increase during
the common envelope stage as much as ∼ 1M⊙
(Bogomazov et al. 2005), up to MOV . Such NSs
collapse into black holes.
6. Mass accretion by black holes
If the black hole has formed in the binary sys-
tem, its X-ray luminosity is
Lx = µṀc
2, (61)
where µ = 0.06 for Schwarzschild black hole and
µ = 0.42 (maximum) for extremal Kerr BH.
We use the Bondi-Hoyle formulas to estimate
the accretion rate onto BH.
Powerful X-ray radiation is able to originate
only if an accretion disc has formed around the
black hole (Karpov & Lipunov 2001). For spher-
ically symmetric accretion onto a black hole the
X-ray luminosity is insignificant. A very low stel-
lar wind velocity is necessary to form an accretion
disc (Lipunov 1992)
V < Vcr ≈ (62)
≈ 320(4η)1/4m3/8T−1/410 R
8 (1 + tan
2 β)−1/2,
where η is averaged over the z-coordinate dynamic
viscous coefficient, m = Mx/M⊙, Mx is the rela-
tivistic star mass, T10 = T/10, T is the orbital
period in days, R8 = Rmin/10
8cm, Rmin is the
minimal distance from the compact object up to
which free Keplerian motion is still possible and
β is the accretion disk axis inclination angle with
respect to the radial direction. For black holes
Rmin = 3Rg, where Rg = 2GMbh/c
7. Accretion induced collapse and com-
pact objects merging
WD and NS are degenerate configurations,
which have upper limit of their mass (the Chan-
drasekhar and Oppenheimer-Volkov limits corre-
spondingly). The Chandrasekhar limit depends
on chemical composition of the white dwarf
MCh =
1.44M⊙, He WD,
1.40M⊙, CO WD,
1.38M⊙, ONeMg WD,
If the mass of WD becomes equal to MCh,
the WD loses stability and collapses. The col-
lapse is accompanied by the powerful thermonu-
clear burst observed as a type Ia supernova. Col-
lapses of He and CO WDs leave no remnants
(Nomoto & Kondo 1991). The outcome of the
collapse of a ONeMg WD is not clear. It can lead
to the formation of a neutron star (the accretion
induced collapse, AIC). Some papers come to the
different conclusion about the result of AICs (see
e.g. Garcia-Berro & Iben (1994); Ritossa et al.
(1996)). The question about the NS formation
during the WD collapse remains undecided. Nev-
ertheless, some NSs could have been formed from
AIC WD (van Paradijs 1997). In the “Scenario
Machine” code the possibility of NS formation
during ONeMg WD is optional.
The merging of two compact objects in a binary
WD system (e.g., due to the GW losses) should
likely to be similar to AIC. During the merging
of two helium WDs, one object with a mass of
less than MCh (for example, 0.5M⊙ + 0.6M⊙ →
1.1M⊙) can form. At the same time, if the total
mass of the components exceeds 1.0 − 1.2M⊙, a
thermonuclear burning can happen. It is likely
that the merging of a ONeMg WD with another
WD can form a NS.
For the typical NS mass ≈ 1.4M⊙, the binary
NS+NS merging event can produce a black hole.
Massive (≈ 2.8M⊙) neutron star can be formed
only if the NS equation of state is very hard and
MOV ≃ 2.8− 3.0M⊙.
BH+BH merging should produce a rapidly ro-
tating black hole with the mass equal to the total
mass of the coalescing binary.
8. Additional scenarios of stellar wind
from massive stars
8.1. Evolutionary scenario B
In the end of 1980s and in the beginning of
1990s the series of new evolutionary tracks were
calculated. The authors used new tables of opaci-
ties (Rogers & Iglesias 1991; Kurucz 1991), new
cross-sections in nuclear reactions (Landre et al.
1990) and new parameters of convection in stars
(Stothers & Chin 1991). For stars with M <
10M⊙ those tracks proved to be almost coincident
with previous calculations. More massive stars
had much stronger stellar winds.
A massive star loses up to 90% of its ini-
tial mass in the main-sequence, supergiant, and
Wolf-Rayet stages via stellar wind. There-
fore, the presupernova mass in this case can
be ≈ 8 − 10M⊙, essentially independent of the
initial mass of the star (de Jager et al. 1988;
Nieuwenhuijzen & de Jager 1990; Schaller et al.
1992).
8.2. Evolutionary scenario C
The papers mentioned above were criticised and
in 1998 a new version of the evolutionary scenario
was developed (Vanbevern et al. 1998). The stel-
lar wind loss rates were corrected taking into ac-
count empirical data about OB and WR stars.
Here we list the main equations of this scenario
(all results concern only with the stars with initial
mass M0 > 15M⊙).
log Ṁ =
1.67 logL− 1.55 logTeff − 8.29, (α)
logL+ logR− logM − 7.5, (β)
0.8 logL− 8.7, (γ)
logL− 10, (δ)
We assume the next indication in this formula:
(α), H burning in the core; (β), giant, M0 ≥
40M⊙; (γ), giant, M0 < 40M⊙; (δ), Wolf-Rayet
star.
Note that in this subsection we used the mass
M is in M⊙, the luminosity L is in L⊙ and the ra-
dius R is in R⊙. With these new calculations, the
Webbink diagram described above changed signif-
icantly Vanbevern et al. (1998).
In this scenario, the total mass loss by a star is
calculated using the formula
∆M = (M −Mcore), (65)
where Mcore is the stellar core mass (66). If the
maximum mass of the star (usually it is the initial
mass of a star, but the mass transfer in binary
systems is able to increase its mass above its initial
value) Mmax > 15M⊙, the mass of the core in the
main sequence stage is determined using (66α),
and in giant and supergiant stages using (66β). In
the Wolf-Rayet star stage (helium star), ifMWR <
2.5M⊙ and Mmax ≤ 20M⊙ it is described using
(66γ), if MWR ≥ 2.5M⊙ and Mmax ≤ 20M⊙ as
(66δ), if Mmax > 20M⊙ as (66ǫ)
mcore =
1.62m0.83opt (α)
10−3.051+4.21 lgmopt−0.93(lgmopt)
0.83m0.36WR (γ)
1.3 + 0.65(mWR − 2.4) (δ)
mcore = 3.03m
0.342
opt (ǫ)
These evolutionary scenarios have some pecu-
liar properties with respect to the classical sce-
nario. One of them is that the strong stellar wind
from massive stars leads to a rapid and significant
increase of the system’s orbit and such stars can-
not fill its RL at all.
There are three observational facts that conflict
with the strong stellar wind scenarios:
1. A very high Ṁ is a problem by itself. The
observers calculate this quantity for most
of OB and WR stars using the emission
measure EM ∝
n2edl. The estimate of
Ṁ using EM is maximal for homogenous
wind. However, there are evidences (see
e.g. Cherepashchuk et al. (1984)) that
stellar winds of massive stars are strongly
“clumpy”. In this case the real Ṁ must
be 3-5 times less. This note is especially
important for the scenario with high stellar
wind.
2. Very massive Wolf-Rayet stars do exist.
There are at least three double WR+OB sys-
tems including very massive WR-stars: CQ
Cep 40M⊙, HD 311884 48M⊙ and HD92740
77M⊙ (Cherepashchuk et al. 1996). Such
heavy WR stars are at odds with the as-
sumed high mass loss rate.
3. In the semi-detached binary system RY Sct
(W Ser type) the mass of the primary com-
ponent is ≈ 35M⊙ (Cherepashchuk et al.
1996). This mass is near the limit
(Vanbevern et al. 1998) beyond which the
star, according to the high mass loss sce-
nario, cannot fill its RL.
8.3. Evolutionary Scenario W
The evolutionary scenario W is based on the
stellar evolution calculations by Woosley et al.
(Woosley et al. (2002), Fig. 16), which repre-
sents the relationship between the mass of the
relativistic remnant and the initial mass of the
star. We included into population-synthesis code
two models with W-type stellar winds, which we
label Wb and Wc. In models Wb and Wc, the
mass-loss rates were computed as in scenario B
and scenario C, respectively. The use of these
models to calculate the wind rate in a scenario
based on Woosley’s diagram (Woosley et al.
(2002), Fig. 16) is justified by the fact that
scenarios B and C are based on the same nu-
merical expressions for the mass-loss rates from
Schaller et al. (1992); Vanbevern et al. (1998);
Nieuwenhuijzen & de Jager (1990) that were
used by Woosley et al. (2002).
9. The “Ecology” of Magnetic Rotators
One of the most important achievements in as-
trophysics in the end of the 1960s was the real-
ization that in addition to “ordinary” stars, which
draw energy from nuclear reactions, there are ob-
jects in the Universe whose radiation is caused by a
strong gravitational and magnetic field. The well-
known examples include neutron stars and white
dwarfs. The property that these objects have in
common is that their astrophysical manifestations
are primarily determined by interaction with the
surrounding matter.
In the early 1980s, this approach led to the cre-
ation of a complete classification scheme involv-
ing various regimes of interaction between neu-
tron stars and their environment, as well as to the
first Monte Carlo simulation of the NS evolution
(Lipunov 1984). In addition to NSs, this scheme
has been shown to be applicable to other types of
magnetized rotating stars.
By virtue of the relationship between the grav-
itational and electromagnetic forces, the NS in
various states can manifest itself quite differently
from the astronomical point of view. Accordingly,
this leads to the corresponding classification of NS
types and to the idea of NS evolution as a gradual
changing of regimes of interaction with the envi-
ronment. The nature of the NS itself turns out to
be important also when constructing the classifica-
tion scheme. This indicates that there should be a
whole class of quite different objects which have an
identical physical nature. To develop the theory
describing properties of such objects (in a sense, it
should establish “ecological” links between differ-
ent objects), it proved to be convenient to use sym-
bolic notations elaborated for the particular case
of NS. We start this subsection with recollecting
the magnetic rotator formalism (mainly according
to the paper Lipunov (1987)).
9.1. A Gravimagnetic Rotator
We call any gravitationally-bounded object
having an angular momentum and intrinsic mag-
netic field by the term “gravimagnetic rotator”
or simply, rotator. In order to specify the intrin-
sic properties of the rotator, three parameters are
sufficient – the mass M , the total angular mo-
mentum J = Iω (I is the moment of inertia and ω
is the angular velocity), and the magnetic dipole
moment µ. Given the rotator radius R0, one can
express the magnetic field strength at the poles
B0 by using the dipole moment B0 = 2µ/R
0. The
angle β between the angular moment J and the
magnetic dipole moment µ can also be of impor-
tance: β = arccos(Jµ).
9.2. The Environment of the Rotator
We assume that the rotator is surrounded by
an ideally conductive plasma with a density ρ∞
and a sound velocity a∞ at a sufficiently far dis-
tance from the rotator. The rotator moves rela-
tive to the environment with a velocity v∞. Un-
der the action of gravitational attraction, the sur-
rounding matter should fall onto the rotator. A
rotator without a magnetic field would capture a
stationary flow of matter, Ṁc, which can be es-
timated using the Bondy-Hole-Lyttleton formu-
lae (Bondi & Hole 1944; Bondi 1952; McCrea
1953):
Ṁc = δ
(2GM)2
(a2∞ + v
ρ∞, (67)
where δ is a dimensionless factor of the order of
unity. When one of the velocities, a∞ or v∞, far
exceeds the other, the accretion rate is determined
by the dominating velocity, and can be written in
a convenient form as (55).
In the real astrophysical situation, the param-
eters of the surrounding matter at distances R ≫
RG can be taken as conditions at infinity
As already noted, the matter surrounding a
NS or a WD is almost always in the form of
a high-temperature plasma with a high conduc-
tivity. Such accreting plasma must interact effi-
ciently with the magnetic field of the compact star
(Amnuel’ & Guseinov 1968). Hence, the interac-
tion between the compact star and its surround-
ings cannot be treated as purely gravitational and
therefore the accretion is not a purely gas dynamic
process. In general, such interaction should be
described by the magneto hydrodynamical equa-
tions. This makes the already complicated pic-
ture of interaction of the compact star with the
surrounding medium even more complex.
The following classification of magnetic ro-
tators is based on the essential characteristics
7RG =
of the interaction of the plasma surrounding
them with their electromagnetic field. This
approach was proposed by Shvartsman (1970)
who distinguished three stages of interaction
of magnetic rotators: the ejection stage, the
propeller stage, which was later rediscovered
by Illarionov & Sunyaev (1975) and named as
such, and the accretion stage. Using this ap-
proach, Shvartsman (1971a) was able to pre-
dict the phenomenon of accreting X-ray pulsars
in binary systems. New interaction regimes dis-
covered later have led to a general classifica-
tion of magnetic rotators (Lipunov 1982a, 1984;
Kornilov & Lipunov 1983a).
It should be noted that the interaction of the
magnetic rotator with the surrounding plasma is
not yet understood in detail. However, even the
first approximation reveals a multitude of interac-
tion models. To simplify the analysis, we assume
the electromagnetic part of the interaction to be
independent of the accreting flux parameters, and
vice versa.
Henceforth, we shall assume in almost all cases
that the intrinsic magnetic field of a rotator is a
dipole field (Landau & Lifshiz 1971):
(1 + 3 sin2 θ)1/2, (68)
This is not just a convenient mathematical sim-
plification. We will show that the magnetoplasma
interaction takes place at large distances from the
surface of the magnetic rotator, where the dipole
moment makes the main contribution. Moreover,
the collapse of a star into a NS is known to
“cleanse” the magnetic field. Indeed, the conser-
vation of magnetic flux leads to a decrease of the
ratio of the quadrupole magnetic moment q to the
dipole moment µ in direct proportion to the radius
of the collapsing star, q/µ ∝ R.
It should be emphasized, however, that the con-
tribution of the quadrupole component to the field
strength at the surface remains unchanged.
The light cylinder radius is the first important
characteristric of the rotating magnetic field:
, (69)
where c is the speed of light.
A specific property of the field of the rotating
magnetic dipole in vacuum is the stationarity of
the field inside the light cylinder and formation of
magneto dipole radiation beyond the light cylin-
der. The luminosity of the magnetic dipole radia-
tion is equal to (Landau & Lifshiz 1971):
sin2 β = kt
ω, (70)
where kt =
sin2 β.
This emission exerts a corresponding braking
torque
Km = −
µ2 sin2 β
ω3cω, (71)
leading to a spin down of the rotator. Although
magnetic dipole radiation from pulsars do not ex-
ists, almost all models predict energy loss quantity
near this value. At the same time we do not take
into account possibly complicated angular depen-
dence of such loss.
9.3. The Stopping Radius
Now we consider qualitatively the effect of the
electromagnetic field of a magnetic rotator on the
accreting plasma. Consider a magnetic rotator
with a dipole magnetic moment µ, , rotational fre-
quency ω, and mass M . At distances R ≫ RG
the surrounding plasma is characterized by the
following parameters: density ρ∞, sound veloc-
ity a∞ and/or velocity v∞ relative to the star.
The plasma will tend to accrete on to the star un-
der the action of gravitation. The electromagnetic
field, however, will obstruct this process, and the
accreting matter will come to a stop at a certain
distance.
Basically, two different cases can be considered:
• When the interaction takes place beyond the
light cylinder, Rstop > Rl. This case first
considered by Shvartsman (1970, 1971a). In
this case the magnetic rotator generates a
relativistic wind consisting of a flux of differ-
ent kinds of electromagnetic waves and rel-
ativistic particles. The form in which the
major part of the rotational energy of the
star is ejected is not important at this stage.
What is important is that both relativistic
particles and magnetic dipole radiation will
transfer their momentum and hence exert
pressure on the accreting plasma. Indeed,
random magnetic fields are always present
in the accreting plasma. The Larmor radius
of a particle with energy ≪ 1010 eV mov-
ing in the lowest interstellar magnetic field
∼ 10−6 G is much smaller than the charac-
teristic values of radius of interaction, so the
relativistic wind will be trapped by the mag-
netic field of the accreting plasma and thus
will transfer its momentum to it.
Thus, a relativistic wind can effectively im-
pede the accretion of matter. A cavern is
formed around the magnetic rotator, and the
pressure of the ejected wind Pm at its bound-
ary balances the ram pressure of the accret-
ing plasma Pa:
Pm(Rstop) = Pa(Rstop), (72)
This equality defines a characteristic size
of the stopping radius, which we call the
Shvartzman radius RSh.
• The accreting plasma penetrates the light
cylinder Rstop < Rl. The pressure of the ac-
creting plasma is high enough to permit the
plasma to enter the light cylinder. Since the
magnetic field inside the light cylinder de-
creases as a dipole field, the magnetic pres-
sure is given by
, (73)
Matching this pressure to the ram pressure
of the accreting plasma yields the Alfven ra-
dius RA.
The magnetic pressure and the pressure of
the relativistic wind can be written in the
following convenient form:
, R ≤ Rl,
4πR2c
, R > Rl,
We introduce a dimensionless factor kt such
that the power of the ejected wind is
Lm = kt
ω, (75)
Assuming kt = 1/2 we get for R = Rl a
continuous function Pm = Pm(R).
The accreting pressure of plasma outside the
capture radius is nearly constant, and hence
gravitation does not affect the medium pa-
rameters significantly. In contrast, at dis-
tances inside the gravitational capture ra-
dius RG the matter falls almost freely and
exerts pressure on the “wall” equal to the dy-
namical pressure. For spherically symmetric
accretion we obtain
, R > RG,
, R ≤ RG,
Here we used the continuity equation Ṁc =
4πR2Gρ∞v∞. When presented in this form,
the pressure Pa is a continuous function of
distance.
Summarizing, for the stopping radius we get
Rstop =
Ra, Rstop ≤ Rl,
RSh, Rstop > Rl,
The expressions for the Alfven radius are:
2µ2G2M2
Ṁcv5
, RA > RG,
, RA ≤ RG,
and for the Shvartzman radius:
RSh =
2G2m2ω4
Ṁcv5∞c
, RSh > RG,
9.4. The Stopping Radius in the Super-
critical Case
The estimates presented above for the stopping
radius were obtained under the assumption that
the energy released during accretion does not ex-
ceed the Eddington limit, so we neglected the re-
verse action of radiation on the accretion flux pa-
rameters.
Now, we turn to the situation where one can-
not neglect the radiation pressure. Consider this
effect after Lipunov (1982b). Suppose that the
accretion rate of matter captured by the magnetic
rotator is such that the luminosity at the stopping
radius exceeds the Eddington limit (see equation
We shall assume, following Shakura & Sunyaev
(1973), that the radiation sweeps away exactly
that amount of matter which is needed for the
accretion luminosity of the remaining flux to be
of the order of the Eddington luminosity at any
radius:
Ṁ(R)
= Ledd, (80)
This yields
Ṁ(R) = Ṁc
, Rs =
Ṁc, (81)
where Rs is a spherization radius (where the accre-
tion luminosity first reaches the Eddington limit),
and κ designates the specific opacity of matter.
Using the continuity equation, the ram pressure of
the accreting plasma is now obtained as another
function of the radial distance
Pa ≈ ρv2 ≈
Ṁ(R)
R−3/2, R ≤ Rs,
in contrast to the subcritical regime when Pa ∝
R−5/2. Matching Pa and Pm (see the previous
section) for the supercritical case gives
, (83)
RSh =
, (84)
The critical accretion rate Ṁcr is defined by the
boundary of the inequality
Ṁc ≥ Ṁcr, (85)
and, correspondingly, is
Ṁcr =
Rst. (86)
The dependence of the Alfven radius on the ac-
cretion rate is such that the Alfven radius (be-
yond the capture radius) slightly decreases with
increasing accretion rate as Ṁ−1/6, while it de-
cerases below the capture radius as Ṁ−2/7 and
attains its lowest value for the critical accretion
rate Ṁc > Ṁcr, beyond which it is independent of
the external conditions.
We also note that in the supercritical regime,
the pressure of the accreting plasma increases
more slowly (as R−3/2) when approaching the
magnetic rotator than the pressure of the relativis-
tic wind (as R−2) ejected by it. This means that
in the supercritical case a cavern may exist even
below the capture radius.
The estimates presented here, of course, are
most suitable for the case of disk accretion. In
fact, the supercritical regime seems to emerge
most frequently under these conditions. This can
be simply understood. Indeed, the accretion rate
is proportional to the square of the capture radius
Ṁc ∝ R2G. At the same time, the angular momen-
tum of the captured matter is also proportional to
R2G. Hence, at high accretion rates the formation
of the disk looks natural.
9.5. The Effect of the Magnetic Field
Apparently, the magnetic field of a star be-
comes significant only when the stopping radius
exceeds the radius of the star, Rst > Rx. We take
the Alfven radius RA for Rst, since it is the small-
est of the two quantities RA and RSh. Hence, we
can estimate the lowest value of magnetic field of
a star which will influence the flow of matter
µmin =
4G2M2x
, RA > RG,
, RA ≤ RG
Ṁc ≤ Ṁcr,
2GMxR
, Ṁc > Ṁcr,
The case RA ≥ RG, Ṁc ≥ Ṁcr is considered
most frequently, and for this case we get the fol-
lowing numerical estimations
µmin = 10
x G cm
3, (88)
or, equivalently,
Bmin = 10
x G, (89)
Most presently observed NS have magnetic
fields ∼ 1012 G and dipole moments ∼ 1030 G
cm3, so the magnetic field must necessarily be
taken into account when considering interaction
of matter with these stars.
9.6. The Corotation Radius
The corotation radius is another important
characteristics of a magnetic rotator. Suppose
that an accreting plasma penetrates the light
cylinder and is stopped by the magnetic field at a
certain distance Rst given by the balance between
the static magnetic field pressure and the plasma
pressure. Suppose that the plasma is “frozen” in
the rotator’s magnetic field. This field will drag
the plasma and force it to rotate rigidly with the
angular velocity of the star. The matter will fall
on to the stellar surface only if its rotational ve-
locity is smaller than the Keplerian velocity at the
given distance Rst:
ωRst <
GMx/Rst, (90)
Otherwise, a centrifugal barrier emerges and
the rapidly rotating magnetic field impedes the ac-
cretion of matter (Shvartsman 1970; Pringle & Rees
1972; Davidson & Ostriker 1973; Lamb 1973;
Illarionov & Sunyaev 1975). The latter authors
assumed that if ωRst ≫
GMx/Rst, the mag-
netic field throws the plasma back beyond the
capture radius. They called this effect the “pro-
peller” regime. In fact, matter may not be shed
(Lipunov 1982a), but it is important to note that
a stationary accretion is also not possible.
The corotation radius is thus defined as
Rc = (GMx/ω
2)1/3 ∼ 2.8×108m1/3x (P/1s)2/3 cm,
where P is the rotational period of the star.
If Rst < Rc , rotation influences the accretion
insignificantly. Otherwise, a stationary accretion
is not possible for Rst > Rc.
9.7. Nomenclature
The interaction of a magnetic rotator with the
surrounding plasma to a large extent depends on
the relation between the four characteristic radii:
the stopping radius, Rst, the gravitational capture
radius, RG, the light cylinder radius, Rl, and the
corotation radius, Rc. The difference between the
interaction regimes is so significant that the mag-
netic rotators behave entirely differently in differ-
ent regimes. Hence, the classification of the inter-
action regimes may well mean the classification of
magnetic rotators. The classification notation and
terminology is described below and summarized in
Table 1, based on paper by Lipunov (1987).
Naturally, not all possible combinations of the
characteristic radii can be realized. For example,
the inequality Rl > Rc is not possible in princi-
ple. Furthermore, some combinations require un-
realistically large or small parameters of magnetic
rotators. Under the same intrinsic and external
conditions, the same rotator may gradually pass
through several interaction regimes. Such a pro-
cess will be referred to as the evolution of a mag-
netic rotator.
We describe the classification by considering an
idealized scenario of evolution of magnetic rota-
tors. Suppose the parameters ρ∞, v∞ and Ṁc of
the surrounding medium remain unchanged. We
shall also assume for a while a constancy of the ro-
tator’s magnetic moment µ. Let the potential ac-
cretion rate Ṁc at the beginning be not too high,
so that the reverse effect of radiation pressure can
be neglected, Ṁc ≤ Ṁcr. We also assume that
the star initially rotates at a high enough speed to
provide a powerful relativistic wind.
Ejectors (E). We shall call a magnetic rotator
an ejecting star (or simply an ejector E) if the pres-
sure of the electromagnetic radiation and ejected
relativistic particles is so high that the surround-
ing matter is swept away beyond the capture ra-
dius or radius of the light cylinder (if Rl > RG).
Ejector: RSh > max(Rl, RG), (92)
It follows from here that Pm ∝ R−2 while the
accretion pressure within the capture radius is
Pa ∝ R5/2 i.e. increases more rapidly as we ap-
proach an accreting star. Consequently, the radius
of a stable cavern must exceed the capture radius
(Shvartsman 1970).
It is worth noting that the reverse transition
from the propeller (P) stage to the ejector (E)
stage is non-symmetrical and occurs at a lower
period (see below). This means that to switch a
pulsar on is more difficult than to turn it off. This
is due to the fact that in the case of turning-on of
the pulsar the pressures of plasma and relativis-
tic wind must be matched at the surface of the
light cylinder, not at the gravitational capture ra-
dius. In fact, the reverse transition occurs under
the condition of equality of the Alfven radius to
the radius of the light cylinder (RA = Rl).
It should be emphasized that, as mentioned by
(Shvartsman 1970), relativistic particles can be
formed also at the propeller stage by a rapidly
rotating magnetic field (see also Kundt (1990)).
Propellers (P). After the ejector stage, the
propeller stage sets in under quite general condi-
tions, when accreting matter at the Alfven sur-
face is hampered by a rapidly rotating magnetic
field of the magnetic rotator. In this regime the
Alfven radius is greater than the corotation ra-
dius, RA > Rc. A finite magnetic viscosity causes
the angular momentum to be transferred to the
accreting matter so that the rotator spins down.
Until now, the propeller stage is one of the poorly
investigated phenomena. However, it is clear that
sooner or later the magnetic rotator is spin down
enough for the rotational effects to be of no im-
portance any longer, and the accretion stage sets
Accretors (A). In the accretion stage, the
stopping radius (Alfven radius) must be smaller
than the corotation radius RA < Rc. This is the
most thoroughly investigated regime of interac-
tion of magnetic rotators with accreting plasma.
Examples of such systems span a wide range of
bright observational phenomena from X-ray pul-
sars, X-ray bursters, low-mass X-ray binaries to
most of the cataclysmic variables and X-ray tran-
sient sources.
Georotators (G). Imagine that the star be-
gins rotating so slowly that it cannot impede
the accretion of plasma, i.e. all the conditions
mentioned in the previous paragraph are sat-
isfied. However, matter still can not fall on
to the rotator’s surface if the Alfven radius
is larger than the gravitational capture radius
(Illarionov & Sunyaev 1975; Lipunov 1982c).
This means that the attractive gravitational force
of the star at the Alfven surface is not signifi-
cant. A similar situation occurs in the interaction
of solar wind with Earth’s magnetosphere. The
plasma mainly flows around the Earth’s magne-
tosphere and recedes to infinity. This analogy ex-
plains the term “georotator” used for this stage.
Clearly, a georotator must either have a strong
Table 1: Classification of neutron stars and white dwarfs.
Abbrevi- Type Characteristic Accretion Well known
ation radii relation rate observational appearances
E Ejector Rst > RG Ṁ ≤ Ṁcr Radio pulsars
Rst > Rl
P Propeller Rc > Rst Ṁc ≤ Ṁcr ?
Rst ≤ RG
Rst ≤ Rl
A Accretor Rst ≤ RG Ṁc ≤ Ṁcr X-ray pulsars
Rst ≤ Rl
G Georotator RG ≤ Rst Ṁc ≤ Ṁcr ?
Rst ≤ Rc
M Magnetor Rst > a Ṁc ≤ Ṁcr AM Her, polars,
Rc > a ? soft gamma repeaters,
anomalous X-ray pulsars
SE Super- Rst > Rl Ṁc > Ṁcr ?
ejector
SP Super- Rc < Rst Ṁc > Ṁcr ?
propeller Rst ≤ Rl
SA Super- Rst ≤ Rc Ṁc > Ṁcr ?
accretor Rst ≤ RG
magnetic field or be embedded in a strongly rar-
efied medium.
Magnetors (M). When a rotator enters a bi-
nary system, it may happen that its magneto-
sphere engulfs the secondary star. Such a regime
was considered by Mitrofanov et al. (1977) for
WD in close binary systems called polars due to
their strongly polarized emission. In the case of
NS, magnetors M may be realized only under the
extreme condition of very close binaries with no
matter within the binary separation.
Supercritical interaction regimes. So far,
we have assumed that the luminosity at the stop-
ping surface is lower than the Eddington limit.
This is fully justified for G and M regimes since
gravitation is not important for them. For types
E, P, and especially A, however, this is not always
true. The critical accretion rate for which the Ed-
dington limit is achieved is
Ṁcr = 1.5× 10−6R8M⊙ yr−1, (93)
where R8 ≡ Rst/108 cm is the stopping radius
(Schwartzman radius or Alfven radius, see above).
We stress here that the widely used condition
of supercritical accretion rate
Ṁ & 10−8(M/M⊙)M⊙ yr
is valid only for the case of non-magnetic NS,
where Rst ≈ 10 km coincides with the stellar ra-
dius. In reality, for a NS with a typical magnetic
field of 1011 − 1012 G, the Alfven radius reaches
107 − 108 cm, so much higher accretion rates are
required for the supercritical accretion to set in.
The electromagnetic luminosity released at the NS
surface, however, will be restricted by Ledd, and
most of the liberated energy may be carried away
by neutrinos (Basko & Sunyaev 1975) (see also
section about hyper accretion).
Most of the matter in the dynamic model of
supercritical accretion forms an outflowing flux
covering the magnetic rotator by an opaque shell
(Shakura & Sunyaev 1973). The following three
additional types are distinguished, depending on
the relationship between the characteristic radii:
superejector (SE), superpropeller (SP) and super-
accretor (SA).
9.8. A Universal Diagram for Gravimag-
netic Rotators
The classification given above was based on re-
lations between the characteristic radii, i.e. quan-
tities which cannot be observed directly. This
drawback can be removed if we note that the light
cylinder radius Rl, Shvartzman radius RSh and
corotation radius Rc are functions of the well-
observed quantity, rotational period of the mag-
netor p. Hence, the above classification can be
reformulated in the form of inequalities for the ro-
tational period of a magnetic rotator.
One can introduce two critical periods pE and
pA such that their relationship with period p of a
magnetic rotator specifies the rotator’s type:
p < pE , → E or SE,
pE ≤ p < pA, → P or SP,
p > pA, → A, SA, G or M,
The values of pE and pA can be determined
from Table 2 which defines the basic nomencla-
ture, and are functions of the parameters v∞, Ṁc,
µ and Mx. The parameters p and µ character-
ize the electromagnetic interaction, while Ṁc de-
scribes the gravitational interaction. Instead of
Ṁc we introduce the potential accretion luminos-
ity L
L ≡ Ṁc
, (95)
The physical sense of the potential luminosity
is quite clear: the accreting star would be ob-
served to have this luminosity if the matter for-
mally falling on the gravitational capture cross-
section were to reach its surface.
Approximate expressions for critical periods
(Lipunov 1992) are:
0.42v
38 s, (α)
38 s, (β)
1.4 · 10−2m−1/9µ4/930 s, (γ)
We assume the next indication in this formula:
(α), Ṁc ≤ Ṁcr, p ≤ pGL; (β), Ṁc ≤ Ṁcr, p >
pGL; (γ), Ṁc > Ṁcr.
38 , s, (α)
1.2m−5/7µ
38 , s, (β)
0.17m−2/3µ
30 , s, (γ)
We assume the next indication in this formula:
(α), Ṁc ≤ Ṁcr and RA > RG; (β), Ṁc ≤
Ṁcr and RA ≤ RG; (γ), Ṁc > Ṁcr.
Here a new critical period pGL was introduced
from the condition RG = Rl:
pGL =
4πGMx
≈ 500mxv−27 s, (98)
Treating the rotator’s magnetic dipole moment
µ and Mx as parameters, we find that an over-
whelming majority of the magnetor’s stages can
be shown on a “p-L” diagram (Lipunov 1982a).
The quantity L also proves to be convenient be-
cause it can be observed directly at the accretion
stage.
9.9. The Gravimagnetic Parameter
By expecting the expression for the stopping
radius in the subcritical regime (Ṁc ≤ Ṁcr) one
can note that the magnetic dipole moment µ and
the accretion rate Ṁc always appear in the same
combination,
, (99)
as was noticed by Davies & Pringle (1981). The
parameter y characterizes the ratio between the
gravitational and magnetic “properties” of a star
and will, therefore, be called the gravimagnetic pa-
rameter. Two magnetic rotators having quite dif-
ferent magnetic fields, subjected to different ex-
ternal conditions but with identical gravimagnetic
parameters, have similar magnetospheres, as long
as the accretion rate is quite low (Ṁc ≤ Ṁcr).
Otherwise, the flux of matter near the stopping
radius no longer depends on the accretion rate at
a large distance.
In fact, the number of independent parameters
can be further reduced (see e.g. Lipunov (1992))
by introducing the parameter
Ṁcv∞
, (100)
Table 2: Parameter of the evolution equation of a magnetic rotators.
Parameter Regime
E, SE P, SP A SA G M
Ṁ 0 0 Ṁc Ṁc(RA/Rs) 0 Ṁc
κt ∼ 2/3 . 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3
Rt Rl Rm Rc Rc RA a
Plotting the rotator’s period p versus Y -
parameter we can draw a somewhat less obvious
but more general classification diagram than the
“p-L” diagram discussed above. This permits us
to show on a single plot the rotators with key
parameters Ṁc, µ and v∞ spanning a very wide
range.
In the case of supercritical accretion, another
characteristic combination is found in all the ex-
pressions:
, (101)
In analog to the subcritical “p-Y” diagram, a
supercritical “p-Ys” diagram can be drawn.
10. Evolution of Magnetic Rotators
The evolution of a magnetic rotator, which de-
termines its observational manifestations, involves
the slow changing of the regimes of its interaction
with the surrounding medium. Such an approach
to the evolution was developed in the 1970s by
Shvartsman (1970); Bisnovatyi-Kogan & Komberg
(1976); Illarionov & Sunyaev (1975); Shakura
(1975); Wickramasinghe & Whelan (1975),
Savonije & van den Heuvel (1977) and others.
Three stages were mostly considered in these pa-
pers: ejector, propeller and accretor. All these
stages can be described by a unified evolutionary
equation.
10.1. The evolution equation
Analysis of the nature of interaction of a mag-
netized star with the surrounding plasma allows
us to write an approximate evolution equation for
the angular momentum of a magnetic rotator in
the general form (Lipunov 1982a):
= Ṁksu − κt
, (102)
where ksu is a specific angular momentum applied
by the accretion matter to the rotator. This quan-
tity is given by
ksu =
(GMxRd)
1/2, Keplerian disk accretion,
G, wind accretion in a binary,
∼ 0, a single magnetic rotator.
(103)
where Rd is the radius of the inner disk edge,
Ω is the rotational frequency of the binary sys-
tem, and ηt ≈ 1/4 (Illarionov & Sunyaev 1975).
The values of dimensionless factor κt, characteris-
tic radius Rt and the accretion rate Ṁ in different
regimes are presented in Table 2.
The evolution equation (102) is approximate.
In practice, the situation with propellers and su-
perpropellers is not yet clear. In Table 2 Rm is
the size of a magnetosphere whose value at the
propeller stage is not known accurately and which
may differ significantly from the standard expres-
sions for the Alfven radius.
10.2. The equilibrium period
The evolution equation presented above indi-
cates that an accreting compact star must en-
deavor to attain an equilibrium state in which the
resultant torque vanishes (Davidson & Ostriker
1973; Lipunov & Shakura 1976). This hypothe-
sis is confirmed by observations of X-ray pulsars.
By equating the right-hand side of equation
(103) to zero, we obtain the equilibrium period:
peq ≈ 7.8π
κt/ǫ2(GMx)
−5/7y−3/7 s, (α)
peq =
A/BwL
10 s, (β)
(104)
where A ≈ 5× 10−4(3κt)µ230I−145 m−1x s yr−1, and
Bw ≈ 5.2×10−6R26m
0 /(10
2/3m2x)I
45 Ṁ−6η s yr
L37 = L/10
37, T10 = T/10 days; (α), disk accre-
tion; (β), quasi-spherical accretion.
Alternatively:
peq ≈ 1.0L−3/737 µ
30 s, disk,
peq = 10η
×(m2/30 /(102/3m2x))−1/2×
×L−137 T
10 µ30 s, stellar wind,
(105)
Let us turn to the case of disk accretion. The
above model of the spin-up and spin-down torques
possesses an unexpected property. The equilib-
rium period obtained by setting the torque to zero
is connected with the critical period pA through a
dimensionless factor:
peq(A) = 2
pA, (106)
The parameters κt and ε must be such that
peq > pA. Since κt ≈ ε ≈ 1, the equilibrium period
in the case of disk accretion is close to the critical
period, pA, separating accretion stage A and the
propeller stage P . In the case of the supercritical
accretion the equilibrium period is determined by
formula
(Lipunov 1982b):
peq(SA) ≃ 0.17µ2/330 m−1/9x s, (107)
10.3. Evolutionary Tracks
The evolution of NS in binaries must be stud-
ied in conjunction with the evolution of nor-
mal stars. This problem was discussed qualita-
tively by Bisnovatyi-Kogan & Komberg (1976);
Savonije & van den Heuvel (1977),
Lipunov (1982a) and other. We begin with the
qualitative analysis presented in the latter of these
paper.
The most convenient method of analysis of NS
evolution is using the ”p-L” diagram. It should
be recalled that L is just the potential accretion
luminosity of the NS. This quantity is equal to the
real luminosity only at the accretion stage.
In Figure 1 we show the evolutionary tracks of
a NS. As a rule, a NS in a binary is born when
the companion star belongs to the main sequence
(loop-like track). During the first 105 − 107 years,
the NS is at the ejector stage, and usually it is
not seen as a radiopulsar since its pulse radia-
tion is absorbed in the stellar wind of the normal
star. The period of the NS increases in accordance
with the magnetic dipole losses. After this, the
matter penetrates into the light cylinder and the
NS passes first into the propeller stage and then
into the accretor stage. By this time, the normal
star leaves the main sequence and the stellar wind
strongly increases. This results in the emergence
of a bright X-ray pulsar. The period of the NS
stabilizes around its equilibrium value. Finally,
the normal star fills the Roche lobe and the accre-
tion rate suddenly increases; the NS moves first
to the right and then vertically downward in the
”p-L” diagram. In other words, the NS enters the
supercritical stage SA (superaccretor) and its spin
period tends to a new equilibrium value (see equa-
tion (107)).
After the mass exchange, only the helium core
of the normal star is left (a WR star in the case
of massive stars), the system becomes detached
and the NS returns back to the propeller or ejec-
tor state. Accretion is still hampered by rapid NS
rotation. This is probably the reason underlying
the absence of X-ray pulsars in pairs with Wolf-
Rayet stars (Lipunov 1982c). Since the helium
star evolves on a rather short time-scale (≈ 105
yr), the NS does not have time to spindown con-
siderably: after explosion of the normal star, the
system can be disrupted leaving the old NS as an
ejector, i.e. as a high-velocity radio pulsar.
The “loop-shaped” track discussed above can
be written in the form:
• I+E → I+P → II+P → II+A → III+SA →
IV+P → E+E (recycled pulsar) → . . .
• I+E → I+P → II+A → III+SA → IV+E
→ (recycled ejector) → IV+P → E+E (re-
cycled pulsar) → . . .
• Another version of the evolutionary track of
a NS formed in the process of mass exchange
within a binary system is:
III+SE → III+SP → IV+P → E+E → . . .
The overall lifetime of a NS in a binary system
depends on the lifetime of the normal star and on
the parameters of the binary system. However,
Fig. 1.— Tracks of NS on the period (p) - gravimagnetic parameter (Y) diagram: track of a single NS
(vertical line) and of a NS in a binary system (looped line). For the second track, possible observational
appearances of the NS are indicated.
the number of transitions from one stage to an-
other during the time the NS is in the binary is
proportional to the magnetic field strength of the
Figure 2 demonstrates the effect of NS mag-
netic field decay (track (a) with and (b) without
magnetic field decay). The first track illustrates
the common path which results in the production
of a typical millisecond pulsar.
10.4. Evolution of Magnetic Rotators in
Non-circular Orbits
So far we have considered evolution of a mag-
netic rotator related to single rotators or those
entering binary system with circular orbits. This
approximation was appropriate for the gross anal-
ysis of binary evolutionary scenario performed
by Kornilov & Lipunov (1983a,b). This approx-
imation is further justified by the fact that the
tidal interaction in close binaries leads to or-
bital circularization in a short time (Hut 1981).
However, the more general case of a binary
with eccentric orbit must be considered for fur-
ther analysis. It is especially important because
many of the currently observed X-ray pulsars, as
well as radio pulsars with massive companions
PSR B1259-63 (Johnston et al. 1992) and PSR
B0042-73 (Kaspi et al. 1994), are in highly ec-
centric orbits around massive companions. Pre-
viously, such studies have been performed by
Gnusareva & Lipunov (1985); Prokhorov (1987).
Orbital eccentricity necessarily emerges after
the first supernova explosion and mass expulsion
from the binary system. In massive binaries with
long orbital periods & 10 days, the eccentricity
may be well conserved until the second episode of
mass exchange (Hut 1981). Here, we concentrate
on the evolutionary consequences of eccentricity.
10.5. Mixing types of E-P-A binary sys-
tems with non-zero orbital eccentric-
The impact of eccentricity on the observed
properties of X-ray pulsars has been consid-
ered in many papers (Amnuel’ & Guseinov 1971;
Shakura & Sunyaev 1973; Pacini & Shapiro 1975;
Lipunov & Shakura 1976). The most important
consequence of orbital eccentricity for the evolu-
tion of rotators can be understood without de-
tailed calculations, and suggests the existence of
two different types of binary systems separated by
a critical eccentricity, ecr (Gnusareva & Lipunov
1985).
Consider an ideal situation when a rotator en-
ters a binary system with some eccentricity. The
normal star (no matter how) supplies matter to
the compact magnetized rotator. We assume that
all the parameters of the binary system (binary
separation, eccentricity, masses, accretion rate,
etc.) are stationary and unchanged. Then a crit-
ical eccentricity ecr appears such that at e > ecr
the rotator is not able to reach the accretion state
in principle. Let the rotator be rapid enough ini-
tially to be at the ejector (E) state. With other
parameters constant, the evolution of such a star
is determined only by its spindown. The star will
gradually spindown to such a state that when pass-
ing close to the periastron where the density of
the surrounding matter is higher, the pulsar will
“choke” with plasma and pass into the propeller
regime. Therefore, for a small part of its life the ro-
tator will be in a mixed EP-state, being in the pro-
peller state at periastron and at the ejector state
close to apastron. The subsequent spindown of
the rotator leads most probably to the propeller
state along the entire orbit. This is due to the
fact that the pressure of matter penetrating the
light cylinder Rl increases faster than that caused
by relativistic wind and radiation, as first noted
by Shvartsman (1971a). So it proves to be much
harder for the rotator to pass from the P state to
the E state than from E to P state (see the follow-
ing section).
The rotator will spindown ultimately to some
period, pA, at which accretion will be possible dur-
ing the periastron passage. Accretion, in contrast,
will lead to a spin-up of the rotator, so that it
reaches some average equilibrium state character-
ized by an equilibrium period peq defined by the
balance of accelerating and decelerating torques
averaged over the orbital period. If the eccentric-
ity was zero, the rotator would be in the accretion
state all the time. By increasing the eccentricity
and keeping the periastron separation between the
stars unchanged, we increase the contribution of
the decelerating torque over the orbital period and
thus decrease peq. At some ultimate large enough
eccentricity ecr the equilibrium period will be less
than the critical period pA permitting the transi-
Fig. 2.— The period-gravimagnetic parameter diagram for NS in binary systems. (a) with NS magnetic
field decay (the oblique part of the track corresponds to “movement” of the accreting NS along the so-called
“spin-up” line), (b) a typical track of a NS without field decay in a massive binary system.
tion from the propeller state to the accretion state
at apastron to occur. The rotational torque ap-
plied to the rotator, averaged over orbital period,
vanishes, and in this sense the equilibrium state is
achieved, but the rotator periodically passes from
the propeller state to the accretion state.
Thus, X-ray pulsars with unreachable full-orbit
accretion state must exist. This means that from
the observational point of view such binaries will
be observed as transient X-ray sources with sta-
tionary parameters for the normal component.
Typically, the evolutionary track of a rotator in
an eccentric binary is
• E → PE → P → AP, e > ecr
• E → PE → P → AP → A, e < ecr
this may be the principal formation channel of
transient X-ray sources.
10.6. Ejector-propeller hysteresis
As mentioned earlier, the transition of the ro-
tator from the ejector state to the propeller state
is not symmetrical. Here we consider this effect in
more detail. In terms of our approach, we must
study the dependence of Rst on Ṁc. To find Rst,
we must match the ram pressure of the accreting
plasma with that caused by the relativistic wind or
by the magnetosphere of the rotator. This depen-
dence Rst(Ṁc) will be substantially different for
rapidly (Rl < RG) and slowly (Rl > RG) rotat-
ing stars (see Figure 3). One can see that in the
case of a fast rotator, an interval of Ṁc appears
where three different values of Rst are possible,
the upper value R1 corresponding to the ejector
state and the bottom value R3 to the propeller
state; the intermediate value R2 is unstable. This
means that the rotator’s state is not determined
solely by the value of Ṁc , but also depends on
previous behavior of this value.
Now consider a periodic changing of Ṁc caused,
for example, by the rotator’s motion along an ec-
centric orbit, and large enough for the rotator to
transit from the ejector state to the propeller state
and vice versa. Initially, the rotator is in the ejec-
tor state. By approaching the normal star, the
accretion rate Ṁc increases and reaches a criti-
cal value ṀEP , where the equilibrium points R1
(stable point corresponding to the ejector state)
and R2 (unstable) approach RG (upper kink),
where they merge (see Figure 2). After that only
one equilibrium point remains in the system, the
stopping radius Rst jumps from ≈ RG down to
Fig. 3.— Dependence of the stopping radius Rst on the modified gravimagnetic radius Ry for two possible
relations between the light cylinder radius Rl and the gravitation capture radius RG: Rl > RG (left-hand
panel) and Rl < RG (right-hand panel; here the ejector-propeller hysteresis becomes possible).
R3 < Rl, and the rotator changes to the propeller
state.
As Ṁc decreases further along the orbit and
reaches the critical value ṀEP once again, the
reverse transition from propeller to ejector does
not occur. The transition only occurs when Ṁc
reaches another critical value, ṀPE < ṀEP ,
where the unstable point R2 meets the stable pro-
peller point R3, and the stopping radius Rst jumps
from ∼ Rl up to R1 > RG. It should be noted
that for fast enough rotators, a situation is pos-
sible when the step down from the ejector state
occurs in such a manner that the stopping ra-
dius Rst < Rc and the rotator passes directly to
the accretion state. The reverse transition always
passes through the propeller stage: A → P →
E. In principle, transitions from the ejector state
to supercritical states SP or SA are also possible
(Prokhorov 1987). In the case of slow rotators
(Rl > RB) ), the “E-P” hysteresis is not possible,
and transitions between these states are symmet-
rical.
10.7. E-P transitions for different orbits
Orbital motion of the rotator around the nor-
mal companion in an eccentric binary draws a hor-
izontal line on the ”p-Y” diagram, with the begin-
ning at a point corresponding to Ṁc(ap), and the
end at a point corresponding to Ṁc(aa) (here ap
and aa are the periasrton and apastron distances,
respectively). The length of this segment is de-
termined by the eccentricity. Since Y ∝ Ṁc , the
rotator moves along this segment from left to right
and back as it revolves from the apastron to the
periastron. At each successive orbital period, this
line slowly drifts up to larger periods. The evolu-
tion of this system is thus determined by the or-
der the critical lines on the diagram are crossed by
this “line”. It is seen from the ”p-Y” diagram that
the regions with and without hysteresis are sepa-
rated by a certain value of the parameter Y = Yk.
Since Y ∝ Ṁc ∝ 1/r2 , four different situations are
possible depending on the relationship of the bi-
nary orbital separation a with critical value acr,
corresponding to Yk (see e.g. Lipunov (1992);
Lipunov et al. (1996b)).
1. ap > acr. In this case no hysteresis oc-
curs and transitions E–P and the reverse
take place in symmetrical points of the orbit.
The rotator passes the following sequence of
stages: E → EP → P → . . . Here EP means
a mixed state of the rotator at which it is
in the ejector state during one part of the
orbit, and in the propeller state during the
rest of the orbital cycle.
2. aa > acr > ap. In this case, the hysteresis
occurs at the beginning of the mixed EP-
state (state EPh), but as the rotator slows
down the hysteresis gradually decays and
disappears. E-P transition for this system
is: E → EPh → EP → P → . . .
3. aa < acr. The hysteresis is possible in prin-
ciple, but the shape of the transition depends
on the eccentricity. Suppose a pulsar has
spun down so much that the first transition
from the ejector to the propeller occurred at
the periastron. If the eccentricity were smal
e < ecr, (do not confuse this ecr with the
critical eccentricity introduced in the pre-
vious section) the reverse transition to the
ejector state would not occur even at the
apastron, and the evolutionary path would
be E → P → . . .
4. If e > ecr , the track is E → EPh → P →
. . . It should be noted that just after the first
EP transition (as well as before the last), the
system spends a finite time in the E and P
states at every revolution.
The value of ecr can be expressed through the
orbital parameters as
ecr ≃
cr − a1/7a
cr + a
, (108)
To conclude, we note that the hysteresis during
the ejector-propeller transition may be possible for
single radio pulsars also. For example, when the
pulsar moves through a dense cloud of interstellar
plasma, the pulses can be absorbed. The radio
pulsar turns on again when it comes out from the
cloud. The hysteresis amplitude for single pulsars
can be high enough because of small relative ve-
locities of the interstellar gas and the pulsar, so
that RG ≫ Rl.
11. Summary
The “Scenario Machine” is the numerical code
for theoretical investigations of statistical proper-
ties of binary stars, i.e. this is population synthesis
code (Lipunov et al. 1996b). It includes the evo-
lution of normal stars and the evolution of their
compact remnants. This is especially important
for studies of the neutron stars (Lipunov 1992).
We always include the most important observa-
tional discoveries and theoretical estimations into
our code.
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This 2-column preprint was prepared with the AAS LATEX
macros v5.2.
	Basic equations and initial distributions
	Evolutionary scenario for binary stars
	Basic evolutionary states of normal stars
	Main sequence stars
	Post main-sequence stars
	Roche lobe overflow
	Wolf-Rayet and helium stars
	Stellar winds from normal stars
	Change of binary parameters: mass, semi-major axis and eccentricity
	Mass change
	Semi-major axis change
	The change of eccentricity
	Special cases: supernova explosion and common envelope
	Three regimes of mass accretion by neutron stars
	Ordinary accretion
	Super-accretion
	Hyper-accretion
	Mass accretion by black holes
	Accretion induced collapse and compact objects merging
	Additional scenarios of stellar wind from massive stars
	Evolutionary scenario B
	Evolutionary scenario C
	Evolutionary Scenario W
	The ``Ecology'' of Magnetic Rotators
	A Gravimagnetic Rotator
	The Environment of the Rotator
	The Stopping Radius
	The Stopping Radius in the Supercritical Case
	The Effect of the Magnetic Field
	The Corotation Radius
	Nomenclature
	A Universal Diagram for Gravimagnetic Rotators
	The Gravimagnetic Parameter
	Evolution of Magnetic Rotators
	The evolution equation
	The equilibrium period
	Evolutionary Tracks
	Evolution of Magnetic Rotators in Non-circular Orbits
	Mixing types of E-P-A binary systems with non-zero orbital eccentricity
	Ejector-propeller hysteresis
	E-P transitions for different orbits
	Summary
