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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 isused 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 ideasthat 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 withthe 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 jointscenario has allowed to construct a two-dimensionalclassification of possible states of binary systemscontaining NS (Kornilov & Lipunov (1983a,b);Lipunov (1992)). According to this classification,we will distinguish four basic evolutionary stagesfor a normal star in a binary system:

I — A main sequence (MS) star inside its Rochelobe (RL);

II — A post-MS star inside its RL;

III — AMS or post-MS star filling its RL; the massis transferred onto the companion.

1E-mail: lipunov@xray.sai.msu.ru2E-mail: pk@.sai.msu.ru3E-mail: mystery@xray.sai.msu.ru4E-mail: a78b@yandex.ru

IV — A helium star left behind the mass-transferin case II and III of binary evolution; maybe in the form of a hot white dwarf (for M ≤2.5M⊙), or a non-degenerate helium star (aWolf-Rayet-star in case of initial MS mass> 10M⊙).

The evolution of single stars can be representedas a chain of consecutive stages: I→ II→ compactremnant; the evolution of the most massive singlestars probably looks like I → II → IV → compactremnant. The component of a binary system canevolve like I→ II→ III→ IV→ compact remnant.

In our calculations we choose the distributionsof initial binary parameters: mass of the primaryzero age main sequence component (ZAMS), M1,the binary mass ratio, q = M2/M1 < 1, the orbitalseparation 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⊙;

(1)

Especially important from the evolutionary pointof view is how different are initial masses of thecomponents (see e.g. Trimble (1983)). We haveparametrized it by a power-law shape, assumingthe primary mass to obey Salpeter’s power law:

f(M) = M−2.351 , 0.1M⊙ < M1 < 120M⊙, (2)

1

f(q) ∼ qαq , q = M2/M1 < 1; (3)

We should note that some apparently reason-able distributions – such as both the primary andsecondary mass obeying Salpeter’s law, or “hier-archical” distributions involving the assumptionthat the total binary mass and primary’s mass aredistributed according to the Salpeter mass func-tion – all yield essentially flat-like distributions bythe mass ratio (i.e. with our parameter αq ≃ 0).

We assume that the neutron star is formedin the core collapse of the pre-supernova star.Masses of the young neutron stars are randomlydistributed in the range Mmin

NS – MmaxNS . Initial

NS masses are taken to be in the range MNS =1.25 − 1.4M⊙. The range of initial masses of theyoung NSs was based on the masses of the neu-tron star in the B1913+16 binary system and ofthe radio pulsar in the J0737-3039 binary. In theB1913+16 system (the Hulse-Taylor pulsar, radiopulsar + neutron star) the mass of the neutronstar, which is definitely not accreting matter fromthe 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 + radiopulsar), which likewise does not accrete from anoptical 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 isassumed to be Mn – Mb; stars with initial massesM > 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 thataccording to some stellar models, a very massivestar (≈ 50 − 100M⊙) can leave behind a neutronstar as a remnant due to very strong mass lossvia powerful stellar wind, so we account for thispossibility 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 anadditional, presumably randomly oriented in spacekick velocity w (see Section 4 below for more de-tails).

The magnetic field of rotating compact ob-jects (neutron stars and white dwarfs) largelydefine the evolutionary stage of the compactobject 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 magneticdipole moment of the newborn neutron star istaken 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 acomprehensive review Chanmugam (1992)). Weassume that the magnetic fields of neutron starsdecays exponentially on a timescale of td (usuallywe take this parameter to be equal to 108, 5 · 107and 107 years). A radio pulsar is assumed to be“switched on” until its period P (in seconds) hasreached the “death-line” defined by the relationµ30/P

2d = 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 valueBmin = 8 · 107 G and do not decay further:

B =

B0 exp(−t/td), t < td ln(B0/Bmin),Bmin, t ≥ td ln(B0/Bmin).

(5)

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 neutronstars (the Oppenheimer-Volkoff limit) is MOV =2.0M⊙ (in general, it is a free parameter in thecode; 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 thatthe progenitor mass before the collapse has a massMcr. The masses of the black holes are calcu-lated as Mbh = kbhMPreSN , where the parameterkbh = 0.0 − 1.0, MPreSN is the mass of the pre-supernova star.

We consider binaries with M1 ≥ 0.8M⊙ with aconstant chemical (solar) composition. The pro-cess of mass transfer between the components istreated as conservative when appropriate, that isthe total angular momentum of the binary system

2

is assumed to be constant. If the accretion ratefrom one component to another is sufficiently high(say, the mass transfer occurs on a timescale fewtimes shorter than the thermal Kelvin-Helmholtztime 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 (forexample, stages with strong stellar wind or thosewhere the loss of binary angular momentum oc-curs due to gravitational radiation or magneticstellar wind) are treated using the well knownprescriptions (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 astronomymade during the last decades stimulated the as-tronomers to search for particular evolutionaryways of obtaining each type of observational ap-pearance of white dwarfs, neutron stars and blackholes, the vast majority of which harbours in bi-naries. Taken as a whole, these ways costitute ageneral evolutionary scheme, or the “evolutionaryscenario”. We follow the basic ideas about stel-lar evolution to describe evolution of binaries bothwith normal and compact companions.

To avoid extensive numerical calculations inthe statistical simulations, we treat the continu-ous evolution of each binary component as a se-quence of a finite number of basic evolutionarystates (for example, main sequence, red super-giant, Wolf-Rayet star, hot white dwarf, etc.), atwhich stellar parameters significantly differ fromeach other. The evolutionary state of the binarycan thus be determined as a combination of thestates of each component, and is changed once themore rapidly evolving component goes to the nextevolutionary stage.

At each such stage, we assume that the stardoes not change its physical parameters (mass, ra-dius, luminosity, the rate and velocity of stellarwind, etc.) that have effect on the evolution ofthe companion (especially in the case of compactmagnetized stars). Every time the faster evolvingcomponent passes into the next stage, we recalcu-late its parameters. Depending on the evolution-

ary stage, the state of the slower evolving star ischanged or can remain unchanged. With some ex-ceptions (such as the common envelope stage andsupernova explosion), states of both componentscannot change simultaneously. Whenever possiblewe use analytical approximations for stellar pa-rameters.

Prior to describing the basic evolutionary statesof the normal component, we note that unlikesingle stars, the evolution of a binary compo-nent is not fully determined by the initial massand chemical composition only. The primary starcan fill its Roche lobe either when it is on themain sequence, or when it has a (degenerate) he-lium or carbon-oxygen core. This determines therate of mass transfer to the secondary compan-ion and the type of the remnant left behind. Wewill follow Webbink (1979) in treating the firstmass exchange modes for normal binary compo-nents, whose scheme accounts for the physicalstate of the star in more detail than the simpletypes of mass exchange (A, B, C) introduced byKippenhahn & Weigert (1967). We will use bothnotations A, B, C and D (for very wide systemswith independently evolving companions) for evo-lutionary types of binary as a whole, and Web-bink’s notations for mass exchange modes for eachcomponent separately [Ia], [Ib], [IIa], [IIb], [IIIa],etc.

3. Basic evolutionary states of normalstars

The evolution of a binary system consisting ini-tially of two zero-age main sequence stars can beconsidered separately for each components until amore massive (primary) component fills its Rochelobe. Then the matter exchange between the starsbegins.

The evolutionary states of normal stars will bedenoted by Roman figures (I-IV), whereas thoseof 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 clearphysical meaning. We will implicitly express themass 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

3

in units of 108 cm s−1 and the accretion rates Monto compact objects in units of 10−8M⊙ yr−1,unless other units are explicitely used.

3.1. Main sequence stars

At this stage, the star is on the zero-age mainsequence (ZAMS) and its size is much smaller thanthe Roche lobe radius. The time the star spendson the main sequence is the core hydrogen burningtime, tH , which depends on the stellar mass only(Iben & Tutukov 1987):

1.0 + 0.95 79m , m ≥ 79.0,

103.9−3.8 logm+log2 m, 79 > m ≥ 10,2400m−2.16, 10 > m ≥ 2.3,104m−3.5, m < 2.3,

(6)

The radius of the ZAMS star is assumed to be

r =

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)

(8)

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 modeof the first mass exchange (which is determined bythe initial orbital period and masses of the com-ponents; see Webbink (1979)) determine the massand the type of the core that will be formed duringstage I. For example, for single stars and primariesin “type C” binaries that fill its Roche lobe havinga degenerate core, we use the expressions

mc =

0.1mmax, (α)0.446 + 0.106mmax, (β)0.24m0.85

max, (γ)min(0.36m0.55

max, 0.44m0.42max), (δ)

0.44m0.42max, (ǫ)

≈ MCh, (ζ)0.1m1.4

max, (η)

(9)

where mmax is the maximum mass the star hadduring the preceding evolution. We assume thenext 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 duringthe first mass transfer will be treated as a rapidlyrotating “Be-star” with the stellar wind rate dif-ferent from what is expected from a single star ofthe 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 stilldoes not fill its Roche lobe. The duration of thisstage for a binary component is not any more afunction of the stellar mass only (as in the caseof single stars), but also depends on the initialbinary type (A, B, or C) (see Iben & Tutukov(1985, 1987)):

tII =

0, (α),2tKH , (β),6300m−3.2, (γ),tHe, (δ),

(10)

In type A systems, the primaries fill their Rochelobes when they belong to the main sequence. Weassume 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), typesC, 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 mosttime of helium burning with such large radius. Inthe framework of our approximate description, wetake the radius of the giant star to be equal tothe maximum value, which depends strongly onthe mass of its core and is calculated accordingto Webbink’s mass transfer modes as follows (seealso Iben & Tutukov (1985, 1987)):

rII =

3000m4c, (α)

1050(mc − 0.5)0.68, (β)10m0.44

c , (γ)(11)

We assume the next indication in this formula:(α), mode [IIIA] or [IIIB] with He core; (β),

4

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 theRoche lobe size; in such cases we put it equalto 0.9RL during the stage II. The most sensi-tive to this crude approximation are binaries withcompact 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 ouranalysis than the stages at which the optical starfills its Roche lobe. A more detailed treatmentof normal star evolution (given, for example, byPols & Marinus (1994)) can reduce such uncer-tainties.

Luminosities of giants are taken from de Jager(1980):

log lII =

15.92m6

c

(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)

(12)

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 fromAllen (1973):

logTeff =

4.50, (α)3.60, (β)3.70, (γ)

(13)

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 withmass higher than 10M⊙ calculate using formula

rII = exp 2.3(0.5 log lii − 2 logTeff + 9.7).(14)

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 beformed if the Roche lobe overflow occurs in thetype C system (where the primary has a well de-veloped core) even for q < 1; otherwise (for typeB systems) we use the condition q ≤ qcr = 0.3 forthe CE stage to occur. Radius of the star at theRoche lobe filling stage is taken to be that of theequivalent Roche lobe radius (Eggleton 1983):

RL

a(1− e)=

0.49q2/3

0.6q2/3 + ln(1 + q1/3), (16)

Here a is the binary orbital separation and e is theorbital eccentricity, q is the arbitrary mass ratio.

For q < 0.6 a more precise approximation canbe used:

RL

a= 0.4622

(

q

1 + q

)1/3

. (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 limitedby the Roche lobe. If the stellar size exceeds theRoche lobe, the star can lose matter on a timescale close to the dynamical one until its radiusbecomes smaller than the new Roche lobe size.

Now we consider how the RL-filling star losesmatter. Let the star be in equilibrium andReq(M) = RL(M) at the initial moment of time.When a fraction of mass δm is transported to thecompanion, the mass ratio q and semi-major axisa of the binary changes depending on the masstransfer 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 canbe bigger or smaller than the equilibrium radiusReq(M − δm) of the star. The thermal equilib-rium establishes on the thermal time scale TKH ≈

5

GM2/RL, so after that the stellar radius relaxesto the equilibrium value Req(M − δm).

Relations RL(M − δm), Rad(M − δm) andReq(M−δm) determine the mode of mass transferduring the RL overflow stage. Following Webbink(1985), one usually introduces the logarithmicderivative ζ = d lnR/d lnM (R ∝ M ζ). It lo-cally fits the real dependence R(M). Three valuesof ζ are relevant:

ζL =d lnRL

d lnM,

ζad =d lnRad

d lnM, (18)

ζeq =d lnReq

d lnM,

where “L”, “ad” and “eq” correspond to the valuesof radii discussed above.

Three possible cases are considered dependingon ζi:

1. If ζad < ζL, the star cannot be inside its RLregardless of the mass loss rate (dM/dt < 0).Such stars lose their matter in hydrodynami-cal time scale. Mass loss rate is limited onlyby the speed of sound near the inner La-grangian point L1. ζeq is unimportant be-cause the size of the star becomes bigger andbigger than RL. The equilibrium is impossi-ble.

2. ζeq < ζL < ζad. The star losing mass cannotbe in thermal equilibrium, because otherwiseits size would exceed RL. Nevertheless, inthis case Rad < RL. So the hydrodynamicalequilibrium is established. As a result, thestar loses mass on thermal time scale.

3. ζL < ζad, ζeq . In this case the size of the starlosing mass becomes smaller than its RL.The evolutionary expansion of the star orthe binary semi-major axis decrease due toorbital angular momentum loss via magneticstellar wind (MSW) or gravitational radia-tion (GW) support the permanent contact ofthe star with RL. The star then loses masson a time scale dictated by ots own evolu-tionary expansion or on a time scale corre-sponding to the orbital angular momentumloss.

For non-degenerate stars, Req increases mono-tonically with M . On the other hand, the expo-nent ζad is determined by the entropy distributionover the stellar radius which is different for starswith radiative and convective envelopes. It can beshown that stars with radiative envelopes shouldshrink in response to mass loss, while those withconvective envelopes should expand 5.

Therefore, stars with convective envelopes in bi-naries should generally have a higher mass lossrate than those with radiative envelopes underother equal conditions. The next important fac-tor is the dependence RL(M). It can found bysubstituting 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 whenthe total mass and the orbital angular momen-tum of the system do not change, one readily getsthat the binary semi-major axis decreases whenthe mass transfer occurs from the more massive tothe less massive component; RL decreases of theprimary correspondingly. When the binary massratio reaches unity, the semi-major axis takes on aminimal value. In contrast, if the less massive starloses 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 onthermal time scale until the masses of the compo-nents become equal 6. The next stage of the firstmass exchange poceeds in more slower (nuclear)time scale. If the primary has convective enve-lope, the mass transfer can proceed much fasteron a time scale intermediate between the thermaland hydrodynamical one, and probably on the hy-drodynamical time scale. In that case the faststage of the mass exchange ends when the massof 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 suchequation 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 envelopeis lost and the stellar core is stripped (the core has othervalues ζad and ζeq).

6

The process of mass exchange strongly dependson stellar structure at the moment of the RL over-flow. The structure of the star in turn dependson its age and the initial mass. The moment ofthe RL overflow is determined by the mass of thecomponents and by the initial semi-major axis ofthe system. to calculate a diagram in the M − a(or M − Porb) plane which allows us to concludewhen the primary in a binary with given initial pa-rameters fills its RL, what is its structure at thatmoment and what type of the first mass exchangeis expected. We use the diagram calculated byWebbink (1979) (see also the description of mod-ern stellar wind scenarios below).

We distinguish different sub-stages of the RLoverflow according to the characteristic timescalesof the mass transfer:

stage III:

This is the most frequent case for the first masstransfer phase. The primary fills its RL and themass transfer proceeds faster than evolutionarytime scale (if outer layers of the star are radia-tive, then it is thermal time scale, if outer layersare convective, then time scale is shorter, up tohydrodynamical time scale). This stage comes tothe end when mass ratio in the system changes(“role-to-role transition”), i.e. when the mass ofthe donor (mass losing) star is equal to mass ofsecond companion (for radiative envelopes) or 0.6of the mass of the second companion (for convec-tive envelopes). This stage also stops if the donorstar totally lost its envelope.

stage IIIe:

This is the slow (evolutionary driven) phase ofmass 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 reversalduring the first stage of mass exchange for binariesof type B (see van den Heuvel (1994)), e.g. as inwide low-mass X-ray binaries.

stage IIIs

This is the specific to super-accreting com-pact companions substage of fast mass transfer atwhich matter escapes from the secondary compan-ion carrying away its orbital angular momentum.Its duration is equal to

tIIIs = tKHq(1 + q)

2− q − 2q2, (19)

q = Ma/Md < 0.5,

here and below subscripts “a” and “d” refer tothe accreting and donating mass star, respectively.For systems with small mass ratios, q < 0.5, thistimescale corresponds to an effective q-time short-ening of the thermal time for the RL-overflowingstar.

stages IIIm,g

At these stages, the mass transfer is controlledby additional losses of orbital angular momentumJorb caused by magnetic stellar wind or gravita-tional wave emission. The characteristic time ofthe evolution is defined as τJ = −(Jorb/Jorb),and in the case of MSW is (see Verbunt & Zwaan(1981); Iben & Tutukov (1987))

τMSW = 4.42a5mxλ

2MSW

(m1 +m2)2m4op

, (20)

Here mop denotes mass of the low-mass opticalstar (0.3 < m < 1.5) that is capable of producingan effective magnetic stellar wind (because onlysuch stars have outer convective envelopes whichare prerequisit for effecftive MSW), λ is a numer-ical parameter of order of unity. We have usedthe mass-radius relation r ≈ m for main sequencestars in deriving this formula. The upper limitof the mass interval and empirical braking law formain-sequence G-stars are taken from Skumanich(1972), the lower limit is determined by absence ofcataclysmic variables with orbital periods Porb ≈3h (Verbunt 1984; Mestel 1952; Kawaler 1988;Tout & Pringle 1992; Zangrilli et al. 1997). Weuse λ = 1 (see for details Kalogera & Webbink(1998)).

The time scale of the gravitational wave emis-sion is

τGW = 124.2a4

m1m2(m1 +m2)× (21)

×(

1 +73

24e2 +

37

96e4)

(1− e2)−7/2,

Wether the evolution is governed by MSW orGW is decided by which time scale (τMSW orτGW ) turns out to be the shortest among all ap-propriate evolutionary time scales.

stage IIIwd

This is a special case where the white dwarfoverflows its RL. This stage is encountered for very

7

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 masstransfer is unstable, since the less massive WDfills its RL first. It can be shown that the massexchange is always stable in such systems if themass ratio q < 0.8. This condition always holdsin WD+NS and WD+BH systems. WD loses itsmatter until its mass decreases to that of a hugeJupiter-like planet (∼ a few 10−3M⊙), where theCOulomb 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-livingdisk around it. If the second star is a neutron starand its rotation had been spun up by accretionsuch that a millisecond radio pulsar appeared, theplanet can be evaporated by relativistic particlesemmited by the pulsar (Paczynski & Sienkiewicz1983; Joss & Rappaport 1983; Kolb et al. 1998;Kalogera & Webbink 1998).

The mass loss rate at each of the III-stages iscalculated according to the relation

M = ∆M/τi, (22)

where ∆M is the a priori known mass to be lostduring the mass exchange phase (e.g. ∆M = M1−(M1 +M2)/2 in the case of the conservative stageIII, 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 assumedto 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 hydrogenenvelope 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 leftas a remnant. The life-time of the helium star isdetermined by the helium burning in the stellarcore (Iben & Tutukov 1985)

tHe =

1658m−2, (α)1233m−3.8, (β)0.1tH , (γ)6913m−3.47, (δ)≃ 10, (ǫ)0.1tH , (ζ)

(24)

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 invan den Heuvel (1994)), the envelope is lost anda CO stellar core is left with mass

mc =

1.3 + 0.65(m− 2.4),m ≥ 2.5,0.83m0.36,m < 2.5,

(25)

The mass-radius dependence in this case is(Tutukov & Yungelson 1973)

rWR = 0.2m0.6. (26)

3.5. Stellar winds from normal stars

The effect of the normal star on the compactmagnetized component is largely determined bythe rate M 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 takento be equal to the Keplerian velocity at the stellarsurface:

v∞ =√

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

8

(de Jager 1980)

m = 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 assumev∞ = 3vp and for massive star we take the max-imum between the stellar wind rate given by deJager’s formula and that given by Lamers (1981)

m = max(52.3αwl

v∞, 102.33

l1.42r0.61

m0.99), (30)

M ≥ 10M⊙;

For red super-giants we use Reimers’s formula(Kudritzki & Reimers 1978):

m = max(52.3αwl

v∞, 1.0

lr

m), (31)

M ≥ 10M⊙;

For a Wolf-Rayet star the stellar wind loss ratecan significantly increase (up to 10−5M⊙ year−1).We parametrize it as

MWR = kWRMWR/tHe, (32)

where the numerical coefficient is taken to bekWR = 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 themass 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 areable to calculate how the mass of the faster evolv-ing star changes (e.g. due to the stellar wind orRL overflow), that is we can estimate the quantity∆M = Mi −Mf . Then we set the characteristicmass loss rate at this stage as

Mo = ∆M/∆t, (33)

Next, we should calculate the change of mass forthe 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 − Mo ×∆t, (34)

Accordingly, the mass of the accreting star is

Mf = Mi + Mc ×∆t, (35)

where Mc is the accretion rate of the capturedmatter.

For stages without RL overflow the accretionrate of the captured stellar wind matter is

mc = 3.8×10−2

(

m

a(v2w + 0.19(m1 +m2)/a)

)2

mo .

(36)

At stages where RL overflow ocurs and bothcomponents are normal (non-degenerate), we willassume that the accretor can accomodate mass atthe rate determined by its thermal time, i.e.

mc = mo

(

tKH(donor)

tKH(accretor)

)

. (37)

This means that the evolution can not befully conservative, especially during the first masstransfer where the primary component usually hasa shorter thermal time scale.

The mass increase rate by compact accretorsis assumed to be limited by the critical Edding-ton luminosity (see, however, the possible hyperaccretion stage discussed below):

LEdd =4πGMmp

σT≈ 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

Mcr = RstopLEdd

GM. (39)

Thus, the mass of the accreting compact star atthe end of the stage is determined by the relation

Mf = Mi +min(Mc, Mcr)×∆t . (40)

9

3.8. Semi-major axis change

The binary separation a changes differently forvarious mass exchange modes. First, we introducea measure of non-conservativeness of the mass ex-change as the ratio between the mass change ofthe accretor and the donor:

β ≡ −(M iaccr −Mf

accr)/(Midonor −Mf

donor). (41)

If the mass exchange is conservative (β = 1,i.e. Ma + Md = const) and one can neglect theangular momenta of the components, the orbitalmomentum conservation implies

afai

=

(

M iaM

id

MfaM

fd

)2

. (42)

In a more general case of quasi-conservativemass transfer 0 ≤ β < 1, the orbital separationchanges differently depending on the specific an-gular momentum carried away from the system bythe escaping matter (see van den Heuvel (1994)for more detail). We treat the quasi-conservativemass transfer by assuming the isotropic mass lossmode in which the matter carries away the spe-cific orbital angular momentum of the accretingcomponent (ja)

Jout = (1− β)Mcja . (43)

From here we straightforwardly find

afai

=

(

qfqi

)3(1 + qi1 + qf

)

(

1 + βqf

1 + βqi

)3+2/β

. (44)

In this formula q = Maccr/Mdonor and thenon-conservative parameter β is set to be theminimal value between β = 1 and the ra-tio TKH(donor)/TKH(accr) (TKH(donor) andTKH(accr) is the thermal time of donor and ac-cretor, respectively).

When no matter is captured by the secondarycompanion 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, weuse another well-known formula

afai

=M i

1 +M i2

Mf1 +Mf

2

. (45)

In this case the orbital separation always increases.

When the orbital angular momentum is carriedaway by GW or MSW with no RL overflow, thefollowing approximate formulas are used:

afai

=

(1−∆t/τMSW )1/4, for MSW,(1−∆t/τGW )1/5, for GW,

(46)

In a special case of a white dwarf filling itsRL (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

afai

=

(

mf

mi

)−2/3

, (47)

where mi is the initial mass of the WD donor andmf is its mass at the end of the mass transfer.

3.9. The change of eccentricity

Tidal interaction between components, as wellas the orbital angular momentum loss due to MSWor GR decrease the eccentricity of the binary sys-tem. The tidal interaction is essential in veryclose binaries or even during the common envelopestage. MSW is effective only in systems with low-mass late-type main sequence stars (see above),GW losses become significant only in short-periodbinaries.

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 fasterthan major semi-axis of the orbit decreases duringcommon envelope stage. We suppose that tcyr =1/3tCE. We accept that tcyr for RL-filling starsis equal to its Kelvin-Helmholtz time. Detachedsystems change their eccentricity during the nextcharacter time (Zahn 1975; Press & Teukolsky1977; Zahn 1989b; Zahn & Bouchet 1989b)

tcyr = tKH

(

1 + e

1− e

)3/2(R

RL

)−5

. (49)

10

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 chooseminimal value of tcyr. Resonances at very high ec-centricities are not taken into account (Mardling1995a,b).

Orbits of the systems with MSW become circu-lar during tcyr = τMSW (see 46).

In case of GW analytical exact solutions wereobtained for a(t) and e(t) (Peters & Mathews1963; Peters 1964).

4. Special cases: supernova explosion andcommon envelope

Supernova explosion in a binary is treated asan instantaneous mass loss of the exploding star.The additional kick velocity can be imparted tothe newborn neutron star due the collapse asym-metry (see below for discussion). In this case theeccentricity and semi-major axis of the binary af-ter the explosion can be straightforwardly calcu-lated (Boersma 1961) (see necessary formulas alsoin 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 arecalculated;

2. then the mass of the exploding star Mpr -Mremnant is changed and the arbitrarily di-rected kick velocity w is added to its orbitalvelocity;

3. after that the transition to the new system’sbarycenter is performed (at this point thespatial velocity of the new center of mass ofthe binary is calculated);

4. in this new reference frame the new total en-ergyE′

tot and the orbital angular momentumJ ′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 newJ ′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 theMaxwellian form:

f(w) ∼ w2

w20

e−w2

w20 . (50)

We suppose that the absolute value of the ve-locity that can be added during the formation ofa black hole depends on the mass loss by the col-lapsing star, the value of the parameter w0 duringthe 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 byPaczynski (1976)) is not fully understood as yet,so we use the conventional energy considerationto find the binary system parameters after the CEby introducing a parameter αCE = ∆Eb/∆Eorb,where ∆Eb = Egrav − Ethermal is the binding en-ergy of the ejected envelope matter and ∆Eorb isthe drop in the orbital energy of the system duringthe spiral-in phase (van den Heuvel 1994). Thisparameter measures the fraction of the system’sorbital energy that comes during the spiral-in pro-cess to the binding energy (gravitational minusthermal) of the ejected common envelope. Thus

αCE

(

GMaMc

2af− GMaMd

2ai

)

=GMd(Md −Mc)

Rd,

(52)

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 separationai and the initial mass ratio Ma/Md, where Ma isthe mass of the accreting star).

On the CE stage the luminosity of the accret-ing star can reach the Eddington limit so that thefurther increase of the accretion rate can be pre-vented by radiation pressure. This usually hap-pens at accretion rates M ≃ 10−4−10−5M⊙ yr−1.However, Chevalier (1993) suggested that whenthe accretion rate is higher (M ≃ 10−2− 10−3M⊙yr−1), the energy is radiated away not by high-energy photons only, but also by neutrinos (seealso Zeldovich et al. (1972) and the next section).On the typical time scale for the hyper accretionstage of 102 yr, up to ∼ 1M⊙ of matter may beincident onto the surface of the neutron star.

11

5. Three regimes of mass accretion byneutron stars

A considerable fraction of observed neutronstars have increased their masses in the course oftheir evolution, or are still increasing their masses(e.g., in X-ray sources). But how large can thismass increase be? It is clear that the only originof a mass increase is accretion. It is evident thatthe overall change in the mass of a neutron staris determined not only by the accretion rate, butalso by the duration of the accretion stage:

∆M =

∫ Ta

0

Mdt = MTa, (53)

where M is the mean accretion rate and Ta is thelifetime of the accretion stage. We emphasize that,in the case under consideration, the accretion rateis the amount of matter falling onto the surfaceof the neutron star per unit time, and can differsignifcantly from the values indicated by the clas-sical Bondi-Hoyle formulas. Three regimes of ac-cretion are possible in a close binary containing aneutron star: ordinary accretion, super-accretion,and hyper-accretion.

5.1. Ordinary accretion

The ordinary accretion regime is realized whenall matter captured by the gravitational field of theneutron star falls onto its surface. This is possibleonly if the radiation pressure and electromagneticforces associated with the magnetic field of the starand its rotation are small compared to the gravita-tional force. In this case, the increase in the masswill be precisely determined by the gas dynamicsof the accretion at the gravitational-capture ra-dius or, if the donor fills its Roche lobe, by thebinary mass ratio and the evolutionary status ofthe optical component. In this case, the accretoris observed as an X-ray source with luminosity

Lx = MGMx

R∗, (54)

where Mx and R∗ are the mass and radius of theneutron star. The accretion rate M is determinedby the Bondi-Hoyle formula

M = πR2Gρv , (55)

where RG is the gravitational-capture radius ofthe neutron star, v is the velocity of the gas flowrelative to the neutron star, and ρ is the densityof the gas.

The X-ray luminosity of the accretor Lx andits other main parameters can be used to estimatethe mass ∆M accumulated during the accretionphase:

∆M =LxR∗Ta

GMx, (56)

5.2. Super-accretion

Regime of super accretion was considered, forinstance, in paper Lipunov (1982d). Despite theabsence of detailed models for supercritical diskaccretion (supercritical accretion is realized pre-cisely via an accretion disk), it is possible to esti-mate the main characteristics of the process – theaccretion rate, magnetosphere radius, and evolu-tion equations. Accretion is considered to be su-percritical when the energy released at the radiuswhere the accretion exceeds the Eddington limit:

MGMx

Rstop> LEdd = 1.38×1038(Mx/M⊙) erg s−1,

(57)

where Rstop is either the radius of the neutron staror the magnetosphere radius RA.

For strongly magnetized neutron stars withmagnetic fields B ≫ 108 G, all matter arrivingat the magnetosphere is accreted onto the mag-netic poles, where the corresponding gravitationalenergy is released. If the black body temperatureT , roughly estimated as

SσT 4 = MGMx

R∗, (58)

is higher than 5 × 109 K (S is the area of thebase of the accretion column), most of the energywill escape from the neutron star in the form ofneutrinos, and, hence, will not hinder accretion(Zeldovich et al. 1972; Basko & Sunyaev 1975).In this case, the rate at which the neutron staraccumulates mass will be

M ≃ Mcrit

(

RA

R∗

)2

≫ Mcrit, (59)

12

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-envelopestage 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 usingthe Bondi-Hoyle formulas is four to six orders ofmagnitude higher than the critical rate and, aswas suggested by Chevalier (1993), this may re-sult in hyper-accretion, when all the energy is car-ried away by neutrinos for the reasons describedabove. There are currently no detailed theoriesfor hyper-accretion or the common-envelope stage.The amount of matter accreted by the neutronstar can be estimated as

∆M =

∫ Thyper

0

1

4

(

RG

a

)2

Mdt ≃ (60)

≃ 1

4(Mopt −Mcore)

(

Mx

Mopt

)2

;

where Thyper is the duration of the hyper-accretionstage, RG is the gravitational-capture radius ofthe neutron star, a is the initial semi-major axisof the close binary orbit, Mcore is the mass of thecore of the optical star, and Mopt and Mx are thetotal 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 duringthe common envelope stage as much as ∼ 1M⊙(Bogomazov et al. 2005), up to MOV . Such NSscollapse 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 = µMc2, (61)

where µ = 0.06 for Schwarzschild black hole andµ = 0.42 (maximum) for extremal Kerr BH.

We use the Bondi-Hoyle formulas to estimatethe accretion rate onto BH.

Powerful X-ray radiation is able to originateonly if an accretion disc has formed around theblack hole (Karpov & Lipunov 2001). For spher-ically symmetric accretion onto a black hole theX-ray luminosity is insignificant. A very low stel-lar wind velocity is necessary to form an accretiondisc (Lipunov 1992)

V < Vcr ≈ (62)

≈ 320(4η)1/4m3/8T−1/410 R

−1/88 (1 + tan2 β)−1/2,

where η is averaged over the z-coordinate dynamicviscous coefficient, m = Mx/M⊙, Mx is the rela-tivistic star mass, T10 = T/10, T is the orbitalperiod in days, R8 = Rmin/10

8cm, Rmin is theminimal distance from the compact object up towhich free Keplerian motion is still possible andβ is the accretion disk axis inclination angle withrespect to the radial direction. For black holesRmin = 3Rg, where Rg = 2GMbh/c

2.

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 dependson chemical composition of the white dwarf

MCh =

1.44M⊙, He WD,1.40M⊙, CO WD,1.38M⊙, ONeMg WD,

(63)

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 thecollapse of a ONeMg WD is not clear. It can leadto the formation of a neutron star (the accretioninduced collapse, AIC). Some papers come to thedifferent conclusion about the result of AICs (seee.g. Garcia-Berro & Iben (1994); Ritossa et al.(1996)). The question about the NS formationduring the WD collapse remains undecided. Nev-ertheless, some NSs could have been formed fromAIC WD (van Paradijs 1997). In the “ScenarioMachine” code the possibility of NS formationduring ONeMg WD is optional.

13

The merging of two compact objects in a binaryWD system (e.g., due to the GW losses) shouldlikely to be similar to AIC. During the mergingof two helium WDs, one object with a mass ofless than MCh (for example, 0.5M⊙ + 0.6M⊙ →1.1M⊙) can form. At the same time, if the totalmass of the components exceeds 1.0 − 1.2M⊙, athermonuclear burning can happen. It is likelythat the merging of a ONeMg WD with anotherWD can form a NS.

For the typical NS mass ≈ 1.4M⊙, the binaryNS+NS merging event can produce a black hole.Massive (≈ 2.8M⊙) neutron star can be formedonly if the NS equation of state is very hard andMOV ≃ 2.8− 3.0M⊙.

BH+BH merging should produce a rapidly ro-tating black hole with the mass equal to the totalmass of the coalescing binary.

8. Additional scenarios of stellar windfrom massive stars

8.1. Evolutionary scenario B

In the end of 1980s and in the beginning of1990s the series of new evolutionary tracks werecalculated. The authors used new tables of opaci-ties (Rogers & Iglesias 1991; Kurucz 1991), newcross-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 coincidentwith previous calculations. More massive starshad much stronger stellar winds.

A massive star loses up to 90% of its ini-tial mass in the main-sequence, supergiant, andWolf-Rayet stages via stellar wind. There-fore, the presupernova mass in this case canbe ≈ 8 − 10M⊙, essentially independent of theinitial 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 andin 1998 a new version of the evolutionary scenariowas 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 initialmass M0 > 15M⊙).

log M =

1.67 logL− 1.55 logTeff − 8.29, (α)logL+ logR− logM − 7.5, (β)0.8 logL− 8.7, (γ)logL− 10, (δ)

(64)

We assume the next indication in this formula:(α), H burning in the core; (β), giant, M0 ≥40M⊙; (γ), giant, M0 < 40M⊙; (δ), Wolf-Rayetstar.

Note that in this subsection we used the massM is in M⊙, the luminosity L is in L⊙ and the ra-dius R is in R⊙. With these new calculations, theWebbink diagram described above changed signif-icantly Vanbevern et al. (1998).

In this scenario, the total mass loss by a star iscalculated using the formula

∆M = (M −Mcore), (65)

where Mcore is the stellar core mass (66). If themaximum mass of the star (usually it is the initialmass of a star, but the mass transfer in binarysystems is able to increase its mass above its initialvalue) Mmax > 15M⊙, the mass of the core in themain sequence stage is determined using (66α),and in giant and supergiant stages using (66β). Inthe 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)2

(β)0.83m0.36

WR (γ)1.3 + 0.65(mWR − 2.4) (δ)mcore = 3.03m0.342

opt (ǫ)(66)

These evolutionary scenarios have some pecu-liar properties with respect to the classical sce-nario. One of them is that the strong stellar windfrom massive stars leads to a rapid and significantincrease of the system’s orbit and such stars can-not fill its RL at all.

There are three observational facts that conflictwith the strong stellar wind scenarios:

14

1. A very high M is a problem by itself. Theobservers calculate this quantity for mostof OB and WR stars using the emissionmeasure EM ∝

∫

n2edl. The estimate of

M using EM is maximal for homogenouswind. However, there are evidences (seee.g. Cherepashchuk et al. (1984)) thatstellar winds of massive stars are strongly“clumpy”. In this case the real M mustbe 3-5 times less. This note is especiallyimportant for the scenario with high stellarwind.

2. Very massive Wolf-Rayet stars do exist.There are at least three double WR+OB sys-tems including very massive WR-stars: CQCep 40M⊙, HD 311884 48M⊙ and HD9274077M⊙ (Cherepashchuk et al. 1996). Suchheavy 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 thestar, according to the high mass loss sce-nario, cannot fill its RL.

8.3. Evolutionary Scenario W

The evolutionary scenario W is based on thestellar evolution calculations by Woosley et al.(Woosley et al. (2002), Fig. 16), which repre-sents the relationship between the mass of therelativistic remnant and the initial mass of thestar. We included into population-synthesis codetwo models with W-type stellar winds, which welabel Wb and Wc. In models Wb and Wc, themass-loss rates were computed as in scenario Band scenario C, respectively. The use of thesemodels to calculate the wind rate in a scenariobased on Woosley’s diagram (Woosley et al.(2002), Fig. 16) is justified by the fact thatscenarios B and C are based on the same nu-merical expressions for the mass-loss rates fromSchaller et al. (1992); Vanbevern et al. (1998);Nieuwenhuijzen & de Jager (1990) that wereused 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, whichdraw energy from nuclear reactions, there are ob-jects in the Universe whose radiation is caused by astrong gravitational and magnetic field. The well-known examples include neutron stars and whitedwarfs. The property that these objects have incommon is that their astrophysical manifestationsare primarily determined by interaction with thesurrounding 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 thefirst Monte Carlo simulation of the NS evolution(Lipunov 1984). In addition to NSs, this schemehas been shown to be applicable to other types ofmagnetized rotating stars.

By virtue of the relationship between the grav-itational and electromagnetic forces, the NS invarious states can manifest itself quite differentlyfrom the astronomical point of view. Accordingly,this leads to the corresponding classification of NStypes and to the idea of NS evolution as a gradualchanging of regimes of interaction with the envi-ronment. The nature of the NS itself turns out tobe important also when constructing the classifica-tion scheme. This indicates that there should be awhole class of quite different objects which have anidentical physical nature. To develop the theorydescribing properties of such objects (in a sense, itshould establish “ecological” links between differ-ent objects), it proved to be convenient to use sym-bolic notations elaborated for the particular caseof NS. We start this subsection with recollectingthe magnetic rotator formalism (mainly accordingto the paper Lipunov (1987)).

9.1. A Gravimagnetic Rotator

We call any gravitationally-bounded objecthaving 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 aresufficient – the mass M , the total angular mo-mentum J = Iω (I is the moment of inertia and ωis the angular velocity), and the magnetic dipole

15

moment µ. Given the rotator radius R0, one canexpress the magnetic field strength at the polesB0 by using the dipole moment B0 = 2µ/R3

0. Theangle β between the angular moment J and themagnetic 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 byan 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. Arotator without a magnetic field would capture astationary flow of matter, Mc, which can be es-timated using the Bondy-Hole-Lyttleton formu-lae (Bondi & Hole 1944; Bondi 1952; McCrea1953):

Mc = δ(2GM)2

(a2∞ + v2∞)3/2ρ∞, (67)

where δ is a dimensionless factor of the order ofunity. When one of the velocities, a∞ or v∞, farexceeds the other, the accretion rate is determinedby the dominating velocity, and can be written ina 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 infinity7.

As already noted, the matter surrounding aNS or a WD is almost always in the form ofa 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 andtherefore the accretion is not a purely gas dynamicprocess. In general, such interaction should bedescribed by the magneto hydrodynamical equa-tions. This makes the already complicated pic-ture of interaction of the compact star with thesurrounding medium even more complex.

The following classification of magnetic ro-tators is based on the essential characteristics

7RG = 2GMv2

.

of the interaction of the plasma surroundingthem with their electromagnetic field. Thisapproach was proposed by Shvartsman (1970)who distinguished three stages of interactionof magnetic rotators: the ejection stage, thepropeller stage, which was later rediscoveredby Illarionov & Sunyaev (1975) and named assuch, and the accretion stage. Using this ap-proach, Shvartsman (1971a) was able to pre-dict the phenomenon of accreting X-ray pulsarsin 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 themagnetic rotator with the surrounding plasma isnot yet understood in detail. However, even thefirst approximation reveals a multitude of interac-tion models. To simplify the analysis, we assumethe electromagnetic part of the interaction to beindependent of the accreting flux parameters, andvice versa.

Henceforth, we shall assume in almost all casesthat the intrinsic magnetic field of a rotator is adipole field (Landau & Lifshiz 1971):

Bd =µ

R3(1 + 3 sin2 θ)1/2, (68)

This is not just a convenient mathematical sim-plification. We will show that the magnetoplasmainteraction takes place at large distances from thesurface of the magnetic rotator, where the dipolemoment 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 theratio of the quadrupole magnetic moment q to thedipole moment µ in direct proportion to the radiusof the collapsing star, q/µ ∝ R.

It should be emphasized, however, that the con-tribution of the quadrupole component to the fieldstrength at the surface remains unchanged.

The light cylinder radius is the first importantcharacteristric of the rotating magnetic field:

Rl =c

ω, (69)

where c is the speed of light.

A specific property of the field of the rotatingmagnetic dipole in vacuum is the stationarity of

16

the field inside the light cylinder and formation ofmagneto dipole radiation beyond the light cylin-der. The luminosity of the magnetic dipole radia-tion is equal to (Landau & Lifshiz 1971):

Lm =2

3

µ2ω4

c3sin2 β = kt

µ2

R3l

ω, (70)

where kt =23 sin

2 β.

This emission exerts a corresponding brakingtorque

Km = −(

2

3

)

µ2 sin2 β

c3ω3cω, (71)

leading to a spin down of the rotator. Althoughmagnetic dipole radiation from pulsars do not ex-ists, almost all models predict energy loss quantitynear this value. At the same time we do not takeinto account possibly complicated angular depen-dence of such loss.

9.3. The Stopping Radius

Now we consider qualitatively the effect of theelectromagnetic field of a magnetic rotator on theaccreting plasma. Consider a magnetic rotatorwith a dipole magnetic moment µ, , rotational fre-quency ω, and mass M . At distances R ≫ RG

the surrounding plasma is characterized by thefollowing 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 electromagneticfield, however, will obstruct this process, and theaccreting matter will come to a stop at a certaindistance.

Basically, two different cases can be considered:

• When the interaction takes place beyond thelight cylinder, Rstop > Rl. This case firstconsidered by Shvartsman (1970, 1971a). Inthis case the magnetic rotator generates arelativistic wind consisting of a flux of differ-ent kinds of electromagnetic waves and rel-ativistic particles. The form in which themajor part of the rotational energy of thestar is ejected is not important at this stage.What is important is that both relativisticparticles and magnetic dipole radiation willtransfer their momentum and hence exert

pressure on the accreting plasma. Indeed,random magnetic fields are always presentin the accreting plasma. The Larmor radiusof 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 therelativistic wind will be trapped by the mag-netic field of the accreting plasma and thuswill transfer its momentum to it.

Thus, a relativistic wind can effectively im-pede the accretion of matter. A cavern isformed around the magnetic rotator, and thepressure 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 sizeof the stopping radius, which we call theShvartzman radius RSh.

• The accreting plasma penetrates the lightcylinder Rstop < Rl. The pressure of the ac-creting plasma is high enough to permit theplasma to enter the light cylinder. Since themagnetic field inside the light cylinder de-creases as a dipole field, the magnetic pres-sure is given by

Pm =B2

8π≈ µ2

8πR6, (73)

Matching this pressure to the ram pressureof the accreting plasma yields the Alfven ra-dius RA.

The magnetic pressure and the pressure ofthe relativistic wind can be written in thefollowing convenient form:

Pm =

µ2

8πR6 , R ≤ Rl,Lm

4πR2c , R > Rl,(74)

We introduce a dimensionless factor kt suchthat the power of the ejected wind is

Lm = ktµ2

R3l

ω, (75)

17

Assuming kt = 1/2 we get for R = Rl acontinuous function Pm = Pm(R).

The accreting pressure of plasma outside thecapture radius is nearly constant, and hencegravitation 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 andexerts pressure on the “wall” equal to the dy-namical pressure. For spherically symmetricaccretion we obtain

Pa =

Mv∞4πR2

G, R > RG,

Mv∞4πR2

G

(

RG

R

)1/2, R ≤ RG,

(76)

Here we used the continuity equation Mc =4πR2

Gρ∞v∞. When presented in this form,the pressure Pa is a continuous function ofdistance.

Summarizing, for the stopping radius we get

Rstop =

Ra, Rstop ≤ Rl,RSh, Rstop > Rl,

(77)

The expressions for the Alfven radius are:

RA =

(

2µ2G2M2

Mcv5∞

)1/6

, RA > RG,(

µ2

2Mc

√2GM

)2/7

, RA ≤ RG,

(78)

and for the Shvartzman radius:

RSh =

(

8ktµ2G2m2ω4

Mcv5∞c4

)1/2

, RSh > RG,

(79)

9.4. The Stopping Radius in the Super-critical Case

The estimates presented above for the stoppingradius were obtained under the assumption thatthe 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 thiseffect after Lipunov (1982b). Suppose that theaccretion rate of matter captured by the magneticrotator is such that the luminosity at the stoppingradius exceeds the Eddington limit (see equation38).

We shall assume, following Shakura & Sunyaev(1973), that the radiation sweeps away exactlythat amount of matter which is needed for theaccretion luminosity of the remaining flux to beof the order of the Eddington luminosity at anyradius:

M(R)GM

R= Ledd, (80)

This yields

M(R) = McR

Rs, Rs =

κ

4πcMc, (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 ofthe accreting plasma is now obtained as anotherfunction of the radial distance

Pa ≈ ρv2 ≈ M(R)

4πR2v =

Mc

√2GM

4πRsR−3/2, R ≤ Rs,

(82)

in contrast to the subcritical regime when Pa ∝R−5/2. Matching Pa and Pm (see the previoussection) for the supercritical case gives

RA =

(

µ2κ

8πc√2GMx

)2/9

, (83)

RSh =

(

κktµ2ω4

4πc5√2GM

)2

, (84)

The critical accretion rate Mcr is defined by theboundary of the inequality

Mc ≥ Mcr, (85)

and, correspondingly, is

Mcr =4πc

κRst. (86)

18

The dependence of the Alfven radius on the ac-cretion rate is such that the Alfven radius (be-yond the capture radius) slightly decreases withincreasing accretion rate as M−1/6, while it de-cerases below the capture radius as M−2/7 andattains its lowest value for the critical accretionrate Mc > Mcr, beyond which it is independent ofthe external conditions.

We also note that in the supercritical regime,the pressure of the accreting plasma increasesmore slowly (as R−3/2) when approaching themagnetic rotator than the pressure of the relativis-tic wind (as R−2) ejected by it. This means thatin the supercritical case a cavern may exist evenbelow the capture radius.

The estimates presented here, of course, aremost suitable for the case of disk accretion. Infact, the supercritical regime seems to emergemost frequently under these conditions. This canbe simply understood. Indeed, the accretion rateis proportional to the square of the capture radiusMc ∝ R2

G. At the same time, the angular momen-tum of the captured matter is also proportional toR2

G. Hence, at high accretion rates the formationof 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 radiusexceeds the radius of the star, Rst > Rx. We takethe Alfven radius RA for Rst, since it is the small-est of the two quantities RA and RSh. Hence, wecan estimate the lowest value of magnetic field ofa star which will influence the flow of matter

µmin =

(

Mcv5

∞R6

x

4G2M2x

)1/2

, RA > RG,(

Mc

√GMxR

7/2x

)1/2

, RA ≤ RG

Mc ≤ Mcr,

(

4πc√2GMxR

9/2x

κ

)

, Mc > Mcr,

(87)

The case RA ≥ RG, Mc ≥ Mcr is consideredmost frequently, and for this case we get the fol-lowing numerical estimations

µmin = 1026R7/46 M

1/217 m1/4

x G cm3, (88)

or, equivalently,

Bmin = 107R5/46 M

1/217 m1/4

x G, (89)

Most presently observed NS have magneticfields ∼ 1012 G and dipole moments ∼ 1030 Gcm3, so the magnetic field must necessarily betaken into account when considering interactionof matter with these stars.

9.6. The Corotation Radius

The corotation radius is another importantcharacteristics of a magnetic rotator. Supposethat an accreting plasma penetrates the lightcylinder and is stopped by the magnetic field at acertain distance Rst given by the balance betweenthe static magnetic field pressure and the plasmapressure. Suppose that the plasma is “frozen” inthe rotator’s magnetic field. This field will dragthe plasma and force it to rotate rigidly with theangular velocity of the star. The matter will fallon to the stellar surface only if its rotational ve-locity is smaller than the Keplerian velocity at thegiven distance Rst:

ωRst <√

GMx/Rst, (90)

Otherwise, a centrifugal barrier emerges andthe rapidly rotating magnetic field impedes the ac-cretion of matter (Shvartsman 1970; Pringle & Rees1972; Davidson & Ostriker 1973; Lamb 1973;Illarionov & Sunyaev 1975). The latter authorsassumed that if ωRst ≫

√

GMx/Rst, the mag-netic field throws the plasma back beyond thecapture radius. They called this effect the “pro-peller” regime. In fact, matter may not be shed(Lipunov 1982a), but it is important to note thata stationary accretion is also not possible.

The corotation radius is thus defined as

Rc = (GMx/ω2)1/3 ∼ 2.8×108m1/3

x (P/1s)2/3 cm,(91)

where P is the rotational period of the star.

If Rst < Rc , rotation influences the accretioninsignificantly. Otherwise, a stationary accretionis not possible for Rst > Rc.

9.7. Nomenclature

The interaction of a magnetic rotator with thesurrounding plasma to a large extent depends onthe relation between the four characteristic radii:the stopping radius, Rst, the gravitational captureradius, RG, the light cylinder radius, Rl, and thecorotation radius, Rc. The difference between the

19

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 ofmagnetic rotators. The classification notation andterminology is described below and summarized inTable 1, based on paper by Lipunov (1987).

Naturally, not all possible combinations of thecharacteristic 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 magneticrotators. Under the same intrinsic and externalconditions, the same rotator may gradually passthrough 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 anidealized scenario of evolution of magnetic rota-tors. Suppose the parameters ρ∞, v∞ and Mc ofthe surrounding medium remain unchanged. Weshall also assume for a while a constancy of the ro-tator’s magnetic moment µ. Let the potential ac-cretion rate Mc at the beginning be not too high,so that the reverse effect of radiation pressure canbe neglected, Mc ≤ Mcr. We also assume thatthe star initially rotates at a high enough speed toprovide a powerful relativistic wind.

Ejectors (E). We shall call a magnetic rotatoran ejecting star (or simply an ejector E) if the pres-sure of the electromagnetic radiation and ejectedrelativistic 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 theaccretion pressure within the capture radius isPa ∝ R5/2 i.e. increases more rapidly as we ap-proach an accreting star. Consequently, the radiusof a stable cavern must exceed the capture radius(Shvartsman 1970).

It is worth noting that the reverse transitionfrom the propeller (P) stage to the ejector (E)stage is non-symmetrical and occurs at a lowerperiod (see below). This means that to switch apulsar on is more difficult than to turn it off. Thisis due to the fact that in the case of turning-on ofthe pulsar the pressures of plasma and relativis-

tic wind must be matched at the surface of thelight cylinder, not at the gravitational capture ra-dius. In fact, the reverse transition occurs underthe condition of equality of the Alfven radius tothe radius of the light cylinder (RA = Rl).

It should be emphasized that, as mentioned by(Shvartsman 1970), relativistic particles can beformed also at the propeller stage by a rapidlyrotating magnetic field (see also Kundt (1990)).

Propellers (P). After the ejector stage, thepropeller stage sets in under quite general condi-tions, when accreting matter at the Alfven sur-face is hampered by a rapidly rotating magneticfield of the magnetic rotator. In this regime theAlfven radius is greater than the corotation ra-dius, RA > Rc. A finite magnetic viscosity causesthe angular momentum to be transferred to theaccreting matter so that the rotator spins down.Until now, the propeller stage is one of the poorlyinvestigated phenomena. However, it is clear thatsooner or later the magnetic rotator is spin downenough for the rotational effects to be of no im-portance any longer, and the accretion stage setsin.

Accretors (A). In the accretion stage, thestopping radius (Alfven radius) must be smallerthan the corotation radius RA < Rc. This is themost thoroughly investigated regime of interac-tion of magnetic rotators with accreting plasma.Examples of such systems span a wide range ofbright observational phenomena from X-ray pul-sars, X-ray bursters, low-mass X-ray binaries tomost of the cataclysmic variables and X-ray tran-sient sources.

Georotators (G). Imagine that the star be-gins rotating so slowly that it cannot impedethe accretion of plasma, i.e. all the conditionsmentioned in the previous paragraph are sat-isfied. However, matter still can not fall onto the rotator’s surface if the Alfven radiusis larger than the gravitational capture radius(Illarionov & Sunyaev 1975; Lipunov 1982c).This means that the attractive gravitational forceof the star at the Alfven surface is not signifi-cant. A similar situation occurs in the interactionof solar wind with Earth’s magnetosphere. Theplasma 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

20

Table 1: Classification of neutron stars and white dwarfs.Abbrevi- Type Characteristic Accretion Well knownation radii relation rate observational appearances

E Ejector Rst > RG M ≤ Mcr Radio pulsarsRst > Rl

P Propeller Rc > Rst Mc ≤ Mcr ?Rst ≤ RG

Rst ≤ Rl

A Accretor Rst ≤ RG Mc ≤ Mcr X-ray pulsarsRst ≤ Rl

G Georotator RG ≤ Rst Mc ≤ Mcr ?Rst ≤ Rc

M Magnetor Rst > a Mc ≤ Mcr AM Her, polars,Rc > a ? soft gamma repeaters,

anomalous X-ray pulsars

SE Super- Rst > Rl Mc > Mcr ?ejector

SP Super- Rc < Rst Mc > Mcr ?propeller Rst ≤ Rl

SA Super- Rst ≤ Rc Mc > Mcr ?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 regimewas considered by Mitrofanov et al. (1977) forWD in close binary systems called polars due totheir strongly polarized emission. In the case ofNS, magnetors M may be realized only under theextreme condition of very close binaries with nomatter 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 sincegravitation is not important for them. For typesE, P, and especially A, however, this is not alwaystrue. The critical accretion rate for which the Ed-dington limit is achieved is

Mcr = 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

M & 10−8(M/M⊙)M⊙ yr−1

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 magneticfield of 1011 − 1012 G, the Alfven radius reaches107 − 108 cm, so much higher accretion rates arerequired for the supercritical accretion to set in.The electromagnetic luminosity released at the NSsurface, however, will be restricted by Ledd, andmost of the liberated energy may be carried awayby neutrinos (Basko & Sunyaev 1975) (see alsosection about hyper accretion).

Most of the matter in the dynamic model ofsupercritical accretion forms an outflowing fluxcovering the magnetic rotator by an opaque shell(Shakura & Sunyaev 1973). The following threeadditional types are distinguished, depending onthe relationship between the characteristic radii:superejector (SE), superpropeller (SP) and super-accretor (SA).

21

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. Thisdrawback can be removed if we note that the lightcylinder radius Rl, Shvartzman radius RSh andcorotation radius Rc are functions of the well-observed quantity, rotational period of the mag-netor p. Hence, the above classification can bereformulated in the form of inequalities for the ro-tational period of a magnetic rotator.

One can introduce two critical periods pE andpA such that their relationship with period p of amagnetic rotator specifies the rotator’s type:

p < pE , → E or SE,pE ≤ p < pA, → P or SP,p > pA, → A, SA, G or M,

(94)

The values of pE and pA can be determinedfrom Table 2 which defines the basic nomencla-ture, and are functions of the parameters v∞, Mc,µ and Mx. The parameters p and µ character-ize the electromagnetic interaction, while Mc de-scribes the gravitational interaction. Instead ofMc we introduce the potential accretion luminos-ity L

L ≡ McGMx

Rx, (95)

The physical sense of the potential luminosityis 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:

pE =

0.42v−1/47 µ

1/230 L

−1/438 s, (α)

1.8v−5/67 m1/3µ

1/330 L

−1/638 s, (β)

1.4 · 10−2m−1/9µ4/930 s, (γ)

(96)

We assume the next indication in this formula:(α), Mc ≤ Mcr, p ≤ pGL; (β), Mc ≤ Mcr, p >pGL; (γ), Mc > Mcr.

pA =

400v−5/47 µ

1/230 L

−1/438 , s, (α)

1.2m−5/7µ6/730 L

−3/738 , s, (β)

0.17m−2/3µ2/330 , s, (γ)

(97)

We assume the next indication in this formula:(α), Mc ≤ Mcr and RA > RG; (β), Mc ≤Mcr and RA ≤ RG; (γ), Mc > Mcr.

Here a new critical period pGL was introducedfrom the condition RG = Rl:

pGL =4πGMx

v2∞c≈ 500mxv

−27 s, (98)

Treating the rotator’s magnetic dipole momentµ and Mx as parameters, we find that an over-whelming majority of the magnetor’s stages canbe 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 accretionstage.

9.9. The Gravimagnetic Parameter

By expecting the expression for the stoppingradius in the subcritical regime (Mc ≤ Mcr) onecan note that the magnetic dipole moment µ andthe accretion rate Mc always appear in the samecombination,

y =Mc

µ2, (99)

as was noticed by Davies & Pringle (1981). Theparameter y characterizes the ratio between thegravitational and magnetic “properties” of a starand 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 gravimagneticparameters, have similar magnetospheres, as longas the accretion rate is quite low (Mc ≤ Mcr).Otherwise, the flux of matter near the stoppingradius no longer depends on the accretion rate ata large distance.

In fact, the number of independent parameterscan be further reduced (see e.g. Lipunov (1992))by introducing the parameter

Y =Mcv∞µ2

, (100)

22

Table 2: Parameter of the evolution equation of a magnetic rotators.Parameter Regime

E, SE P, SP A SA G M

M 0 0 Mc Mc(RA/Rs) 0 Mc

κt ∼ 2/3 . 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3Rt Rl Rm Rc Rc RA a

Plotting the rotator’s period p versus Y -parameter we can draw a somewhat less obviousbut more general classification diagram than the“p-L” diagram discussed above. This permits usto show on a single plot the rotators with keyparameters Mc, µ and v∞ spanning a very widerange.

In the case of supercritical accretion, anothercharacteristic combination is found in all the ex-pressions:

Ys =κµ2

v∞, (101)

In analog to the subcritical “p-Y” diagram, asupercritical “p-Ys” diagram can be drawn.

10. Evolution of Magnetic Rotators

The evolution of a magnetic rotator, which de-termines its observational manifestations, involvesthe slow changing of the regimes of its interactionwith the surrounding medium. Such an approachto the evolution was developed in the 1970s byShvartsman (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 thesestages can be described by a unified evolutionaryequation.

10.1. The evolution equation

Analysis of the nature of interaction of a mag-netized star with the surrounding plasma allowsus to write an approximate evolution equation forthe angular momentum of a magnetic rotator inthe general form (Lipunov 1982a):

dIω

dt= Mksu − κt

µ2

R3t

, (102)

where ksu is a specific angular momentum appliedby the accretion matter to the rotator. This quan-tity is given by

ksu =

(GMxRd)1/2, Keplerian disk accretion,

ηtΩR2G, 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 M in differentregimes 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 isthe size of a magnetosphere whose value at thepropeller stage is not known accurately and whichmay 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 theresultant torque vanishes (Davidson & Ostriker1973; 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−1/237 T

−1/610 s, (β)

(104)

23

where A ≈ 5× 10−4(3κt)µ230I

−145 m−1

x s yr−1, and

Bw ≈ 5.2×10−6R26m

2/30 /(102/3m2

x)I−145 M−6η s yr−1,

L37 = L/1037, T10 = T/10 days; (α), disk accre-tion; (β), quasi-spherical accretion.

Alternatively:

peq ≈ 1.0L−3/737 µ

6/730 s, disk,

peq = 10η−1/2k M

−1/2−6 ×

×(m2/30 /(102/3m2

x))−1/2×

×L−137 T

−1/610 µ30 s, stellar wind,

(105)

Let us turn to the case of disk accretion. Theabove model of the spin-up and spin-down torquespossesses an unexpected property. The equilib-rium period obtained by setting the torque to zerois connected with the critical period pA through adimensionless factor:

peq(A) = 23/4κ1/2t

ε7/4pA, (106)

The parameters κt and ε must be such thatpeq > pA. Since κt ≈ ε ≈ 1, the equilibrium periodin the case of disk accretion is close to the criticalperiod, pA, separating accretion stage A and thepropeller stage P . In the case of the supercriticalaccretion the equilibrium period is determined byformula

(Lipunov 1982b):

peq(SA) ≃ 0.17µ2/330 m−1/9

x 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 thequalitative analysis presented in the latter of thesepaper.

The most convenient method of analysis of NSevolution is using the ”p-L” diagram. It shouldbe recalled that L is just the potential accretionluminosity of the NS. This quantity is equal to thereal luminosity only at the accretion stage.

In Figure 1 we show the evolutionary tracks ofa NS. As a rule, a NS in a binary is born whenthe 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 isnot seen as a radiopulsar since its pulse radia-tion is absorbed in the stellar wind of the normalstar. The period of the NS increases in accordancewith the magnetic dipole losses. After this, thematter penetrates into the light cylinder and theNS passes first into the propeller stage and theninto the accretor stage. By this time, the normalstar leaves the main sequence and the stellar windstrongly increases. This results in the emergenceof a bright X-ray pulsar. The period of the NSstabilizes around its equilibrium value. Finally,the normal star fills the Roche lobe and the accre-tion rate suddenly increases; the NS moves firstto the right and then vertically downward in the”p-L” diagram. In other words, the NS enters thesupercritical stage SA (superaccretor) and its spinperiod tends to a new equilibrium value (see equa-tion (107)).

After the mass exchange, only the helium coreof the normal star is left (a WR star in the caseof massive stars), the system becomes detachedand the NS returns back to the propeller or ejec-tor state. Accretion is still hampered by rapid NSrotation. This is probably the reason underlyingthe absence of X-ray pulsars in pairs with Wolf-Rayet stars (Lipunov 1982c). Since the heliumstar 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, thesystem can be disrupted leaving the old NS as anejector, i.e. as a high-velocity radio pulsar.

The “loop-shaped” track discussed above canbe 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 ofa NS formed in the process of mass exchangewithin a binary system is:III+SE → III+SP → IV+P → E+E → . . .

The overall lifetime of a NS in a binary systemdepends on the lifetime of the normal star and onthe parameters of the binary system. However,

24

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 observationalappearances of the NS are indicated.

25

the number of transitions from one stage to an-other during the time the NS is in the binary isproportional to the magnetic field strength of theNS.

Figure 2 demonstrates the effect of NS mag-netic field decay (track (a) with and (b) withoutmagnetic field decay). The first track illustratesthe common path which results in the productionof a typical millisecond pulsar.

10.4. Evolution of Magnetic Rotators inNon-circular Orbits

So far we have considered evolution of a mag-netic rotator related to single rotators or thoseentering binary system with circular orbits. Thisapproximation was appropriate for the gross anal-ysis of binary evolutionary scenario performedby Kornilov & Lipunov (1983a,b). This approx-imation is further justified by the fact that thetidal interaction in close binaries leads to or-bital circularization in a short time (Hut 1981).However, the more general case of a binarywith eccentric orbit must be considered for fur-ther analysis. It is especially important becausemany of the currently observed X-ray pulsars, aswell as radio pulsars with massive companionsPSR B1259-63 (Johnston et al. 1992) and PSRB0042-73 (Kaspi et al. 1994), are in highly ec-centric orbits around massive companions. Pre-viously, such studies have been performed byGnusareva & Lipunov (1985); Prokhorov (1987).

Orbital eccentricity necessarily emerges afterthe first supernova explosion and mass expulsionfrom the binary system. In massive binaries withlong orbital periods & 10 days, the eccentricitymay be well conserved until the second episode ofmass exchange (Hut 1981). Here, we concentrateon the evolutionary consequences of eccentricity.

10.5. Mixing types of E-P-A binary sys-tems with non-zero orbital eccentric-ity

The impact of eccentricity on the observedproperties 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 importantconsequence of orbital eccentricity for the evolu-tion of rotators can be understood without de-

tailed calculations, and suggests the existence oftwo different types of binary systems separated bya critical eccentricity, ecr (Gnusareva & Lipunov1985).

Consider an ideal situation when a rotator en-ters a binary system with some eccentricity. Thenormal star (no matter how) supplies matter tothe compact magnetized rotator. We assume thatall the parameters of the binary system (binaryseparation, eccentricity, masses, accretion rate,etc.) are stationary and unchanged. Then a crit-ical eccentricity ecr appears such that at e > ecrthe rotator is not able to reach the accretion statein principle. Let the rotator be rapid enough ini-tially to be at the ejector (E) state. With otherparameters constant, the evolution of such a staris determined only by its spindown. The star willgradually spindown to such a state that when pass-ing close to the periastron where the density ofthe surrounding matter is higher, the pulsar will“choke” with plasma and pass into the propellerregime. 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 stateclose to apastron. The subsequent spindown ofthe rotator leads most probably to the propellerstate along the entire orbit. This is due to thefact that the pressure of matter penetrating thelight cylinder Rl increases faster than that causedby relativistic wind and radiation, as first notedby Shvartsman (1971a). So it proves to be muchharder for the rotator to pass from the P state tothe E state than from E to P state (see the follow-ing section).

The rotator will spindown ultimately to someperiod, 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 itreaches some average equilibrium state character-ized by an equilibrium period peq defined by thebalance of accelerating and decelerating torquesaveraged over the orbital period. If the eccentric-ity was zero, the rotator would be in the accretionstate all the time. By increasing the eccentricityand keeping the periastron separation between thestars unchanged, we increase the contribution ofthe decelerating torque over the orbital period andthus decrease peq. At some ultimate large enougheccentricity ecr the equilibrium period will be lessthan the critical period pA permitting the transi-

26

Fig. 2.— The period-gravimagnetic parameter diagram for NS in binary systems. (a) with NS magneticfield 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 stateat apastron to occur. The rotational torque ap-plied to the rotator, averaged over orbital period,vanishes, and in this sense the equilibrium state isachieved, but the rotator periodically passes fromthe propeller state to the accretion state.

Thus, X-ray pulsars with unreachable full-orbitaccretion state must exist. This means that fromthe observational point of view such binaries willbe observed as transient X-ray sources with sta-tionary parameters for the normal component.

Typically, the evolutionary track of a rotator inan eccentric binary is

• E → PE → P → AP, e > ecr

• E → PE → P → AP → A, e < ecr

this may be the principal formation channel oftransient X-ray sources.

10.6. Ejector-propeller hysteresis

As mentioned earlier, the transition of the ro-tator from the ejector state to the propeller stateis not symmetrical. Here we consider this effect inmore detail. In terms of our approach, we muststudy the dependence of Rst on Mc. To find Rst,

we must match the ram pressure of the accretingplasma with that caused by the relativistic wind orby the magnetosphere of the rotator. This depen-dence Rst(Mc) will be substantially different forrapidly (Rl < RG) and slowly (Rl > RG) rotat-ing stars (see Figure 3). One can see that in thecase of a fast rotator, an interval of Mc appearswhere three different values of Rst are possible,the upper value R1 corresponding to the ejectorstate and the bottom value R3 to the propellerstate; the intermediate value R2 is unstable. Thismeans that the rotator’s state is not determinedsolely by the value of Mc , but also depends onprevious behavior of this value.

Now consider a periodic changing of Mc caused,for example, by the rotator’s motion along an ec-centric orbit, and large enough for the rotator totransit from the ejector state to the propeller stateand vice versa. Initially, the rotator is in the ejec-tor state. By approaching the normal star, theaccretion rate Mc increases and reaches a criti-cal value MEP , 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 onlyone equilibrium point remains in the system, thestopping radius Rst jumps from ≈ RG down to

27

Fig. 3.— Dependence of the stopping radius Rst on the modified gravimagnetic radius Ry for two possiblerelations between the light cylinder radius Rl and the gravitation capture radius RG: Rl > RG (left-handpanel) and Rl < RG (right-hand panel; here the ejector-propeller hysteresis becomes possible).

R3 < Rl, and the rotator changes to the propellerstate.

As Mc decreases further along the orbit andreaches the critical value MEP once again, thereverse transition from propeller to ejector doesnot occur. The transition only occurs when Mc

reaches another critical value, MPE < MEP ,where the unstable point R2 meets the stable pro-peller point R3, and the stopping radius Rst jumpsfrom ∼ Rl up to R1 > RG. It should be notedthat for fast enough rotators, a situation is pos-sible when the step down from the ejector stateoccurs in such a manner that the stopping ra-dius Rst < Rc and the rotator passes directly tothe accretion state. The reverse transition alwayspasses through the propeller stage: A → P →E. In principle, transitions from the ejector stateto 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 Mc(ap), and the

end at a point corresponding to Mc(aa) (here apand aa are the periasrton and apastron distances,respectively). The length of this segment is de-termined by the eccentricity. Since Y ∝ Mc , therotator moves along this segment from left to rightand back as it revolves from the apastron to theperiastron. At each successive orbital period, thisline 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 bythis “line”. It is seen from the ”p-Y” diagram thatthe regions with and without hysteresis are sepa-rated by a certain value of the parameter Y = Yk.Since Y ∝ Mc ∝ 1/r2 , four different situations arepossible 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 reversetake place in symmetrical points of the orbit.The rotator passes the following sequence ofstages: E → EP → P → . . . Here EP meansa mixed state of the rotator at which it isin the ejector state during one part of theorbit, and in the propeller state during therest of the orbital cycle.

2. aa > acr > ap. In this case, the hysteresisoccurs at the beginning of the mixed EP-state (state EPh), but as the rotator slowsdown the hysteresis gradually decays anddisappears. E-P transition for this systemis: E → EPh → EP → P → . . .

3. aa < acr. The hysteresis is possible in prin-ciple, but the shape of the transition dependson the eccentricity. Suppose a pulsar hasspun down so much that the first transitionfrom the ejector to the propeller occurred atthe periastron. If the eccentricity were smale < ecr, (do not confuse this ecr with thecritical eccentricity introduced in the pre-vious section) the reverse transition to the

28

ejector state would not occur even at theapastron, and the evolutionary path wouldbe E → P → . . .

4. If e > ecr , the track is E → EPh → P →. . . It should be noted that just after the firstEP transition (as well as before the last), thesystem spends a finite time in the E and Pstates at every revolution.

The value of ecr can be expressed through theorbital parameters as

ecr ≃a1/7cr − a

1/7a

a1/7cr + a

1/7a

, (108)

To conclude, we note that the hysteresis duringthe ejector-propeller transition may be possible forsingle radio pulsars also. For example, when thepulsar moves through a dense cloud of interstellarplasma, the pulses can be absorbed. The radiopulsar turns on again when it comes out from thecloud. The hysteresis amplitude for single pulsarscan be high enough because of small relative ve-locities of the interstellar gas and the pulsar, sothat RG ≫ Rl.

11. Summary

The “Scenario Machine” is the numerical codefor theoretical investigations of statistical proper-ties of binary stars, i.e. this is population synthesiscode (Lipunov et al. 1996b). It includes the evo-lution of normal stars and the evolution of theircompact remnants. This is especially importantfor studies of the neutron stars (Lipunov 1992).We always include the most important observa-tional discoveries and theoretical estimations intoour code.

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