A&A 423, 281-299 (2004)
DOI: 10.1051/0004-6361:20035678
A. Büning - H. Ritter
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85740 Garching, Germany
Received 13 November 2003 / Accepted 2 March 2004
Abstract
We resume the discussion about irradiation-driven mass transfer
cycles in semi-detached compact binary systems. The analytical model
that describes the onset of these cycles, which occur on a thermal
timescale of the donor star, is reexamined. We take into account a
contribution of the thermal relaxation which is not related to the
irradiation of the donor star and which was neglected in previous
studies. Cataclysmic variables (CVs) containing extended giant donors
are more stable than previously thought. CVs close to the upper edge
of the period gap can undergo cycles at low angular momentum loss
rates, as has been suggested by recent magnetic braking prescriptions,
while they are stable for high braking rates.
A model for the irradiation geometry that takes into account surface elements near the terminator of the donor star indicates that possibly also low-mass X-ray binaries (LMXBs) can undergo mass transfer cycles. Regarding the braking rate, which is necessary to drive cycles, the same restrictions apply for short period LMXBs as for short period CVs. We confirm that LMXBs containing giants can undergo cyles. In terms of the irradiation efficiency parameter , CVs are susceptible to irradiation instability for while LMXBs are susceptible for .
The predictions of the analytical model are checked by the first long-term evolutionary computations of systems undergoing mass transfer cycles with full 1D stellar models. For unevolved main sequence (MS) and giant donors the analytic model provides reasonable values for the boundaries of the stable and unstable regions while CVs containing highly evolved MS donors are more stable than expected at high braking rates.
Taking into account irradiation, the minimum period of CVs is increased by up to 1-2 min, depending on .
Key words: stars: binaries: close - stars: novae, cataclysmic variables - stars: evolution - stars: mass-loss - X-rays: binaries
Observations of close binary systems have shown that a star that is illuminated by its companion can exhibit a hotter, illuminated, and a cooler, unilluminated side (see, e.g., Ritter et al. 2000; Eddington 1926, and references therein). This is the well-known reflection effect (for a review see, Vaz 1985). If the illuminated star has a convective envelope, then irradiation inhibits energy transport through the illuminated outer layers (Vaz & Nordlund 1985). In the case of a semi-detached compact binary system the reflection effect can be caused by accretion luminosity that is released near the compact star. In this case the energy transport, i.e., the intrinsic flux through the illuminated outer layers of the donor star, is coupled to the mass transfer rate. This leads to a feedback, the consequences of which are the subject of this paper.
The influence of irradiation on mass transfer was first discussed by Podsiadlowski (1991) for the simplified case of symmetrical irradiation of the donor. Later, Hameury et al. (1993) tried to simulate asymmetrical irradiation by time-periodic symmetric irradiation and concluded that a more realistic model is required.
In a series of papers, Ritter et al. (1995,1996) and King et al. (1996,1995) developed a feedback model for asymmetrical irradiation that predicts under which conditions the secular mass transfer rate becomes unstable and the system undergoes mass transfer cycles that occur on the thermal timescale of the convective envelope of the donor star.
We reexamine the above-mentioned irradiation feedback model and take into account a contribution of thermal relaxation which is not related to the irradiation of the donor star. It turns out that this term, which was neglected in previous studies of the above-mentioned analytical model, can become important, especially for giant donors.
Unfortunately, there have been almost no numerical computations to test the validity of this model and especially the boundaries of the stable and unstable regions. Only few computations have been carried out by using bipolytrope stellar models and in most cases also by using simplified models for the treatment of irradiation (Ritter et al. 1995; McCormick & Frank 1998; Ritter et al. 2000). There has been only one short-term evolution study using full 1D stellar models and a realistic treatment of irradiation published by Hameury & Ritter (1997). It served more as a "proof of concept'' for their model of the reduction of the intrinsic flux by illumination than as a numerical confirmation of the irradiation feedback model.
The main purpose of this paper is to confirm the predictions of the irradiation feedback model by numerical long-term evolutionary computations that are based on full 1D stellar models for the donor star, a realistic model for the irradiation geometry, and the tabulated results of Hameury & Ritter (1997) for the reduction of the intrinsic flux by illumination.
In Sect. 2 we describe the input physics that is necessary for the feedback model and for the numerical computations. This includes the mass loss rate prescription, the reflection effect, the irradiation geometry, and the thermal relaxation of the donor star. In Sect. 3 we develop two ordinary differential equations that describe the system adequately for our purpose, and in Sect. 4 we derive the conditions that describe the onset of instability by means of linear stability analysis.
We give a short overview in Sect. 5 of the input physics, which we use in our code, and subsequently, we present our numerical results for CVs and LMXBs containing evolved and unevolved donor stars in Sect. 6.
We consider a semi-detached compact binary system. The compact primary
star, a white dwarf, a neutron star, or a black hole, is denoted by
subscript 1, the secondary star, a main sequence star with
a convective envelope or a giant, by subscript 2. Both stars
move on circular orbits with an orbital distance A. We assume
that the secondary corotates with the orbital motion. This is justified
by the short timescales of orbital circulatization and synchronization
for such systems (e.g., Zahn 1977).
The mass ratio of both stars is defined as
Since the donor star does not have a sharp rim, the mass loss rate
must be a continuous function of the difference
We assume that the matter streaming through the L_{1}-point
is collected in an accretion disk surrounding the primary and that
a fraction
of the transferred matter is finally
accreted, i.e.:
(6) |
The accretion process releases gravitational energy and produces accretion
luminosity, mostly on or nearby the compact star so that the source
of the accretion luminosity can be treated as a point source. Hence,
the irradiating flux at a distance d is given by
On the long term the system can lose additional matter and angular momentum, e.g., by nova explosions so that the long-term average value can be less than , even is possible (e.g., Prialnik & Kovetz 1995; Hameury et al. 1989). In this case determines the (long-term) mass and angular momentum balance while determines the momentary accretion rate and therefore the accretion luminosity. As has been shown by Schenker et al. (1998), describing the long-term evolution by an average is reasonable.
A star that is illuminated by its companion shows a reflection effect
(for a review see, Vaz 1985).
Accordingly, the star has an illuminated, brighter and hotter side
with an irradiation-dependend effective temperature
and an unilluminated, darker and cooler side with an effective temperature
T_{0}. The intrinsic flux
through a
surface element is defined as the net flux, i.e., the difference between
the incoming flux
and the outgoing flux
,
where
denotes the Stephan-Boltzmann constant. It is
convenient to write the irradiating flux in units of the intrinsic
flux
on the unilluminated side:
(8) |
Convection in the interior of stars is efficient, i.e., adiabatic. It was first stated by Rucinski (1969) that the entropy, which is constant in the adiabatic convection zone, must determine the intrinsic flux through the outer layers. This means that, as was first recognized by Vaz & Nordlund (1985), the temperature gradient must be flatter on the illuminated, hotter side than on the unilluminated, cooler side: since the temperature gradient in the adiabatic convection zone is essentially tied to the adiabatic temperature gradient, the temperature gradient in the subphotospheric, superadiabatic layers must be flatter on the illuminated side. Therefore, energy transport through the illuminated outer layers is inhibited. In fact, the superadiabatic zone acts like a valve for the energy flow. This can be unterstood in terms of a simple one-zone model (Ritter et al. 1995,2000).
Ritter et al. (2000, Appendix A) have pointed out that lateral energy transport within the superadiadiabatic convection zone is negligible since the radial temperature gradient in the superadiabatic layers is much larger than the lateral temperature gradient. Hence, irradiation can be treated as a local effect and for every surface element separately. While this is most likely a good approximation for weak irradiation, i.e., typical CVs, it is possible that for sufficiently strong irradiation, i.e., typical LMXBs, non-local effects like circulation, which transports heat from the illuminated to the unilluminated side, become non-negligible. We will discuss this in more detail in Sect. 6.4.
By solving the equations of stellar structure for an unirradiated and an irradiated surface element simultaneously for a fixed entropy at the bottom of the adiabatic zone it is possible to compute numerically. This has been done by Hameury & Ritter (1997)^{} for grey atmospheres and perpendicular irradiation for a large grid of effective temperature T_{0} and surface gravity . The results are available in tabular form and we use them for our numerical calculations.
The assumption of a grey atmosphere and an irradiated Planck spectrum is not necessarily very accurate, especially for CVs and LMXBs. Various properties like a different penetration depth of optical light, UV and X-rays are neglected. A smaller penetration depth means that the external radiation field is mainly absorbed and reprocessed in the optically thin^{} layers of the atmosphere. This reduces the effect of the irradiation on the photospheric boundary conditions and therefore the intrinsic flux is less inhibited. This "efficiency'' of the irradiation for a non-grey atmosphere compared to a grey atmosphere will be taken into account formally by a free parameter . If the external radiation field has the same spectrum as the irradiated star, then is unity by definition. In the case of LMXBs, hard X-rays, which can penetrate into the subphotospheric layers, can be the dominant component of the irradiating spectrum. But since even in the case of a grey atmosphere (i.e., ) about of the irradiating flux penetrates directly down to the photosphere at an optical depth of (see e.g., Tout et al. 1989, Eq. (29)) and since even hard X-rays, except for the most energetic photons, do not penetrate sufficiently deep into the outer layers (Hameury 1996), we do not expect that can become significantly greater than unity in typical LMXBs. Therefore, in general .
Models of non-grey atmospheres irradiated by Planck spectra (Brett & Smith 1993; Nordlund & Vaz 1990) and real stellar spectra (Barman 2002; Barman & Hauschildt 2001,2002) have been computed. However, those computations are much too time-consuming for generating tables that would be applicable for our purpose, i.e., similar to those by Hameury & Ritter (1997). Nevertheless, it is possible to compare models of irradiated non-grey atmospheres with models of irradiated grey atmospheres to get an estimate of the irradiation efficiency .
Since depends on the penetration depth, irradiation is more efficient for a smaller opacity and thus for a smaller metallicity. Figure 1 shows as a function of metallicity for a specific binary system. Since the source radiates mainly in the optical, the results for grey atmospheres (crosses) do not differ much from those for non-grey atmospheres of the irradiated star, for an irradiated Planck spectrum (triangles) as well as for a real stellar spectrum (diamonds). The data shown are taken from Nordlund & Vaz (1990, Table 1) and are compared with results from Hameury & Ritter (1997) (asterisk).
Figure 1: The intrinsic flux as function of metallicity Z for a star with and an effective gravity that is irradiated by with an angle of incidence , i.e., effectively by . The star is irradiated by its companion with and a relative radius of . The data shown are taken from Hameury & Ritter (1997) (asterisk) and Nordlund & Vaz (1990, Table 1): grey atmospheres (crosses), and non-grey atmospheres irradiated by a Planck spectrum (triangles) and a real stellar spectrum (diamonds). | |
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The reduction of by irradiation also depends on the model of convection, especially the mixing length las shown in Fig. 2. The data are taken from Nordlund & Vaz (1990, Table 2) and Hameury & Ritter (1997).
Figure 2: The intrinsic flux as function of the mixing length l in units of the pressure scale height . Otherwise see Fig. 1. | |
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For a plane-parallel atmosphere the external radiation field as a
function of optical depth
and angle of incidence is given by
(11) |
Figure 3: The intrinsic flux as function of the angle of incidence . Otherwise see Fig. 1. | |
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Since
can fluctuate on short timescales and
the relation between
and
is non-linear, the effect of irradiation does not only depend on the
average
,
but also on the amplitude and period
of those fluctuations. This is taken into account formally by another
free parameter
which addresses the
duty cycle of the irradiation. For constant irradiation
is unity. For time-varying irradiation the efficiency depends on the
(time-averaged) blocked part of the intrinsic flux
which determines the heating of the convective envelope. Since
is a convex function for typical stellar envelopes as shown by Fig. 4,
i.e.,
(12) |
(13) |
Figure 4: for models Nos. 1-4 and 6 of Table 1 (solid, long-dashed, short-dashed, dash-dotted and dot-dash-dotted line, respectively). | |
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In the following we will use the symbol
,
respectively
for the effective irradiating flux
Table 1: Values of several stellar models. Nos. 1-3 are unevolved MS stars, 4 is a remnant of thermal timescale mass transfer of an evolved MS star with an initial mass of , an initial central hydrogen abundance of , an initially white dwarf primary, , and subsequent mass loss driven by strong braking according to Verbunt & Zwaan (1981) using . No. 5 is similar but with an initial mass of , an initial central hydrogen abundance of , an initially neutron star primary, and is determined by an Eddington accretion rate of , otherwise . No. 6 is a giant.
Using the definition of
in Eq. (10)
the net luminosity of the star is given by
Due to the axial symmetry of the external radiation field Eq. (18)
can be simplified:
(21) |
Figure 5: Irradiation geometry. A surface element E is irradiated by the source with an angle of incidence to the surface normal. | |
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The simplest conceivable irradiation model is the constant flux model
(Ritter et al. 1995,2000) which assumes that
has the constant average value
A more realistic model is the point source model (King et al. 1996; Ritter et al. 2000)
where the irradiation source is treated as a point source as shown
in Fig. 5. In this case the effective flux for
a surface element E is given by
(25) |
At turn-on of mass transfer and thus of irradiation a certain effective fraction s of the surface of the donor star is immediately blocked by irradiation. As a consequence the star is out of thermal equilibrium because the (unchanged) nuclear energy production is higher than the (reduced) intrinsic luminosity . The reaction of a star to spontaneous blocking of its intrinsic luminosity has been discussed by Spruit (1982) in connection with star spots: on short timescales the changes of the photospheric values on the unilluminated surface are negligible; on a thermal timescale of the convective envelope the star relaxes thermally according to the modified outer boundary conditions by adjusting R_{2}, T_{0}, and . Ritter (1994) and Ritter et al. (2000) give an analytical estimate for the effect of irradiation on the thermal equilibrium values of low-mass MS stars. The effective temperature remains basically unchanged and the radius increases by about , i.e., for typical values of s.
For our numerical computations we compute by solving the equations of stellar structure for the donor star. This will be described in more detail in Sect. 5.
Figure 6 shows schematically how the feedback between mass transfer rate, irradiation, and thermal relaxation works: as soon as mass transfer starts, the accretion rate and also the accretion luminosity increase. This enhances the irradiating flux which reduces the intrinsic luminosity . Thus, the thermal expansion rate of the star is increased which in turn leads to an increasing mass transfer rate .
Figure 6: Schematic diagram of the irradiation feedback mechanism at turn on of mass transfer. | |
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Yet, this feedback can not work forever because thermal relaxation saturates at last after about a Kelvin-Helmholtz timescale . But before the star reaches a new equilibrium state, the expansion rate starts dropping and the mass transfer rate is reduced. Then the feedback works into the opposite direction: and decrease and so does . Therefore, increases and the star starts to shrink. This reduces the mass transfer even more. In extreme cases the mass transfer even stops and the system appears as a detached system until the next cycle starts.
We will now derive an analytical model that describes the onset of mass transfer cycles. In principle, this has already been done by King et al. (1996) using a slightly different formalism. All expressions derived in Sects. 3 and also 4 will be applied to the analytical model only and will not be used for the numerical computations.
The time evolution of the Roche radius
can
be written as
The time evolution of the radius R_{2} can be splitted into
three terms: the adiabatic reaction at constant entropy and chemical
composition, the thermal relaxation at constant chemical composition,
and the nuclear evolution:
The time evolution of the equilibrium radius
of the unilluminated star is given by
With the timescale of nuclear evolution
One can easily see from Eq. (3) that the mass
transfer rate is stationary if and only if
is constant.
Setting the left-hand side of Eq. (36) to zero yields
the stationary mass transfer rate
To determine the conditions for stability of the stationary solution
we perform a linear stability analysis (see, e.g., Guckenheimer & Holmes 1983)
on the vector field
(43) |
According to the theorem of Hartmann-Grobmann a fixed point of a vector field is stable, if all eigenvalues of the Jacobi matrix of at have a negative real part, and is unstable if at least one eigenvalue has a positive real part.
Using Eq. (3) and neglecting the derivatives
of
and R_{2}, which is reasonable
for
,
,
and neglecting also the derivatives of
we
get
has the eigenvalues
The first term in Eq. (46) reflects the adiabatic
reaction of the star and its Roche radius on mass loss. For dynamically
stable systems this term is negative and therefore it contributes
to the stabilization of the fixed point. The third term in Eq. (46)
tells how thermal relaxation changes with the radius of the star compared
to its equilibrium radius. A negative value means that the larger
(smaller) R_{2}, the stronger the thermal relaxation that decreases
(increases) the radius to its (irradiation-dependend) equilibrium
value. Therefore,
(48) |
(49) |
The sufficient criterion for
is that the
absolute value of the real part of the square root in Eq. (45)
is less than the absolute value of
,
otherwise at least one eigenvalue has a positive real part. Using
Eq. (44) it can be shown that this is equivalent to
which is equivalent to
There are different possibilities for the dynamics of the system nearby the fixed point depending on and if Eqs. (47) and (50) are fulfilled:
Since K depends on
via the intrinsic luminosity
,
the derivative
can be written as
(52) |
(53) |
(54) |
(59) |
Figure 7: (Eq. (58)) for stellar models Nos. 1-4 and 6 of Table 1 (solid, long-dashed, short-dashed, dash-dotted and dot-dash-dotted line, respectively) and also s'_{k} for k=1 (Eq. (60), dotted line). | |
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Figure 8: (Eq. (57)) for the same stellar models as in Fig. 7. For comparison of the point source with the constant flux model also s'_{k}for k=1 (Eq. (60)) is shown. | |
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In terms of a bipolytrope model the thermal relaxation K can
be expressed explicitely (Kolb & Ritter 1992, Eq. (32)). If the star is either
fully convective or the nuclear luminosity of the radiative core is
identical to the total luminosity of the core, which is the case in
the stationary (i.e., quasi-equilibrium) state, then the thermal relaxation
can be expressed as
The only quantity in Eq. (61) which depends on
is
itself:
(63) |
The energy generation rate from hydrogen burning can be approximated
by a simple power law of density
and temperature T:
(71) |
Inserting Eqs. (17) and (72)
into Eq. (61) gives:
(73) |
(74) |
Alternatively, Eq. (75) can also be expressed in
quantities of the illuminated star. Using Eqs. (69), (70), (72) and
in the (irradiation dependend) equilibrium state we get the more compact
expression:
(80) |
Taking Eq. (22), eliminating A by Eq. (28),
by Eq. (39) and using
the driving timescale
including the thermal
relaxation from Eq. (35) we obtain
(81) |
(83) |
For the moment we neglect the last term of order of several in criterion Eq. (79) as King et al. (1996) did. To get mass transfer cycles the blocking of the intrinsic flux (s') as a reaction upon an enhanced mass transfer rate must increase faster than the convective envelope can relax thermally ( ) compared to the driving time scale of the mass loss ( ). Therefore, systems are the more susceptible to the onset of mass transfer cycles, the smaller the timescale of the convective envelope and the larger the driving timescale is.
Accordingly, main sequence stars with thin outer convection zones have been supposed to be most susceptible (King et al. 1996; Ritter et al. 2000). Also systems driven by the longest possible timescale, i.e., the timescale of angular momentum loss by gravitational radiation, have been supposed to be more susceptible than other systems. But there are basically no systems except, e.g., low-mass CVs in or below the period gap that are driven mainly by gravitational braking. Higher mass systems are believed to be driven mainly by magnetic breaking. For a high braking rate, e.g., according to Verbunt & Zwaan (1981), CVs with unevolved main sequence stars become unsusceptible to the irradiation instability except for the most massive systems because becomes too small in comparison to (Ritter et al. 1995,2000,1996). For giants magnetic braking is believed to be ineffective and is small compared to . Therefore, such systems and also CVs with highly evolved MS stars, which have a short , have been supposed to be a priori more susceptible to the irradiation instability than CVs with unevolved MS stars (King et al. 1997).
But the last term of order of several
in Eq. (79) can be neglected only if
(85) |
(86) |
In this section we have derived more concise and, in some sense, more general, expressions (78), respectively (79) for the onset of instability than the corresponding Eqs. (45), respectively (60) of King et al. (1996). We have also provided an explicit formula Eq. (84) for which we will use in Sect. 6 to discuss our numerical results. An important result is that the additional term of several in Eq. (79) provides an upper limit for , i.e., a lower limit for . Hence, for a sufficiently low secular mass transfer rate every system becomes stable. We will show in Sect. 6.1.3 that this limit can become important for CVs with giant donors.
We use the stellar evolutionary code of Schlattl et al. (1997) and Schlattl (1999) which goes back to the code of Kippenhahn et al. (1967). The code uses a special grid point algorithm (Wagenhuber & Weiss 1994), nuclear reaction rates of Caughlan et al. (1985) and Adelberger et al. (1998), the mixing length theory of Böhm-Vitense (1958) and Cox & Giuli (1968), the OPAL opacities (Iglesias & Rogers 1996), and for the outer layers opacities by Alexander & Ferguson (1994) and additional data from P. H. Hauschildt and J. W. Ferguson (private communication). In order to avoid numerical instabilities in calculating mass transfer it was necessary to implement an equation of state which is smooth in the independent variables P, T, and chemical composition over the whole range of application. In practice, we use the equation of state of Saumon et al. (1995) with data provided by I. Baraffe (private communication), and particularly the equation of state of Pols et al. (1995). Furthermore, we developed an implicit algorithm for the treatment of mass transfer similar to the method described by Benvenuto & de Vito (2003). For more details about our code see Büning (2003). For determining we use results given by Hameury & Ritter (1997) and additional data from J.-M. Hameury (private communication).
To destabilize the stationary mass transfer the ratio of and has to be sufficiently small, at least less than the maximum of s' which is . As can be seen from Eq. (84), especially for realistic values of the ratio has to be sufficiently small for typical system parameters which is basically the case for compact binaries only. Since the system has to be dynamically and thermally stable and since the donor star has to maintain a deep convective envelope, essentially only CVs and LMXBs are eligible.
Figure 9: s' for a CV with an unevolved MS star (No. 2 of Table 1) and an white dwarf for the point source and the constant flux model (solid and dotted line). s' is compared to f for models 1-4listed in Table 2 (dot-dash-dotted, long-dashed, dash-dotted and short-dashed line, respectively). The mass transfer is unstable and undergoes cycles if s'>f which is only the case in a certain interval whose limits depend on . Otherwise, the system is stable. | |
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Table 2: System parameters of CVs containing an donor as shown in Fig. 9.
Exemplarily for a wide range of CVs with unevolved donor stars Fig. 9
shows the left-hand and also the right-hand side of Eq. (79),
namely s' and
(87) |
Figure 10: Mass transfer rate from a slightly evolved MS star with an age of and onto an white dwarf for the point source model as function of orbital period . System parameters: , , , and (solid line). The dotted line shows the mass transfer rate for the same system without irradiation feedback. Both computations have been stopped beyond the minimum period at . | |
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As can be seen from Fig. 9, CVs with unevolved
donors are the more stable, the smaller the irradiation efficiency
is. Such systems can undergo mass transfer cycles if
.
For, e.g.,
the "island
of instability'', which is defined by s'>f, ranges from
to
.
Using
Eq. (84) is it possible to compute the
corresponding ratio for
and
:
(88) |
Figure 10 shows a CV evolution for initial parameters of model No. 3 of Table 2 using the point source model and an ad hoc braking rate of . The donor, a slightly evolved MS star, fills its Roche lobe at an orbital period of about and starts to transfer mass undergoing cycles. At about the system leaves the "island of instability'' and becomes stable. The small spike in the mass transfer rate at about is caused by the star becoming fully convective. For gravitational braking only as shown in Fig. 11 the phase of mass transfer cycles does not end until the system is in the middle of the period gap. The transition, when the star becomes fully convective, coincides with a drop in the peak mass transfer rate in this case.
Figure 11: Same as Fig. 10 but for gravitational braking only. | |
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The secular mass transfer rate is lower when taking irradiation into account (dotted versus solid line in Figs. 10 and 11) since the irradiated star has a systematically larger radius. As a consequence, the minimum period increases, in the case of pure gravitational braking by up to 2 min, depending on . This is of the same order as the correction which is caused by geometrical distortion of the mass losing star by the Roche potential (Renvoizé et al. 2002), and this is still not sufficient to explain the observed orbital minimum period of (Ritter & Kolb 2003) if only gravitational braking is taken into account.
Figure 12: Mass transfer rate as function of time for the same system as in Fig. 10. The computation stops at . | |
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(92) |
Figure 13: Mass transfer rate from a slightly evolved MS star ( ) onto a white dwarf for the point source model as function of orbital period . System parameters: , , , and strong braking according to Verbunt & Zwaan (1981). The dotted line shows the evolution without irradiationfeedback. | |
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The dependence of the temporal evolution of mass transfer cycles on the efficiency parameter is shown in Fig. 14. The duration of the high state decreases with decreasing until the mass transfer rate becomes sinusoidal with a frequency as given in Eq. (91). For even weaker feedback the oscillations are damped or even vanish completely.
Figure 14: Mass transfer rate from an unevolved MS star onto an white dwarf for conservative mass transfer , , and the constant flux model for , 0.04, 0.1, 0.2, 0.4, and 1.0 (dotted, dot-dash-dotted, dash-dotted, short-dashed, long-dashed, and solid line, respectively) as function of time . | |
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The timescale of central hydrogen burning for low-mass MS stars is much longer than a Hubble time. One way of accounting for evolved low-mass donors in CVs is prior thermal timescale mass transfer from an evolved, initially more massive star. In this case the mass losing star is more massive to such an extent that the system is initially thermally unstable ( ) and the mass transfer occurs on a thermal timescale (Schenker et al. 2002; Kolb et al. 2000).
Although the bipolytrope model is formally valid only for chemically homogeneous stars we tentatively apply it also to chemically evolved donor stars. As an example Fig. 15 shows s'and f for an remnant of thermal timescale mass transfer with a central hydrogen abundance of . It is the core of an initially more massive MS star of . According to Eq. (66) is less than of of an unevolved MS star of the same mass, and its radius is larger by . Due to its deeper superadiabatic convection zone the maximum of s'is shifted to higher values of and due to its larger radius f is also shifted to larger values of , but in total the difference to the case with an unevolved donor discussed before is comparatively small. As can be seen from Fig. 15, this particular system should be stable for but can undergo cycles for . When going to smaller donor masses or lower braking rates during the initial evolution also remnants of thermal timescale mass transfer can become unstable for as CVs with unevolved donors do. It seems that CVs with an evolved donor are not more susceptible to mass transfer cycles than CVs with unevolved donors (if at all, it is the other way around). Rather they are susceptible at a shorter driving timescale .
Figure 15: s' for a CV with an highly evolved remnant of thermal timescale mass transfer (No. 4 of Table 1) and an white dwarf for the point source and constant flux model (solid and dotted line). s' is compared to f for and , 0.3, and 1.0 (short-dashed, dash-dotted, and dot-dash-dotted line, respectively). | |
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As for unevolved CVs it is also possible to determine the range of
where mass transfer cycles should occur,
e.g., if
:
(93) |
Figure 16: Mass transfer rate from an evolved initially MS star (No. 4 of Table 1) onto an initially white dwarf for the point source model as function of orbital period . System parameters: , , , and strong braking according to Verbunt & Zwaan (1981). The dotted line shows the evolution without irradiation feedback. The onset of mass transfer cycles occurs at and . | |
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Another group of CVs are systems containing a giant or subgiant donor.
Figure 17 shows the stability diagram for a CV
with a giant donor of
(model No. 6 of Table 1).
Since
according to Eq. (66)
gives an unreliable estimate for the thermal timescale of the convective
envelope of giants, we have used the timescale
which is defined as the thermal energy U of the convective
envelope divided by the stellar luminosity. The timescale on which
the giant radius changes, when irradiation is changed, is even larger
than
by .
Giants typically
have
and
.
In our case for
the "island of instability''
ranges from 0.25 to 20 in
.
This means the ratio of
and
must be in the range
(94) |
(95) |
(97) |
Figure 17: s' for a CV with an giant (No. 6 of Table 1) and an white dwarf for the point source and the constant flux model (solid and dotted line). s' is compared to f for and , 0.3, and 1.0 (short-dashed, dash-dotted, and dot-dash-dotted line,respectively). | |
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For the system shown in Fig. 17 Eq. (96)
yields:
(98) |
There are three reasons why CVs with giant donors are more stable, the more extended the giants are: first, King et al. (1997) have neglected the last term in Eq. (79) that restricts the region of instability not only to high but also to low braking rates. Thus, systems do not get automatically unstable for sufficiently small braking rates. Second, the depth of the superadiabatic convection zone grows during giant evolution which shifts the "island of instability'' to higher . Hence, more evolved giants are less affected by irradiation. Third, more evolved giants have a larger scale height so that their "island of instability'' is smaller than for less evolved giants.
As we have discussed in the last section, CVs might undergo mass transfer cycles for given a suitable driving timscale. Another class of compact binary systems with thermally stable mass transfer, i.e., small q, are LMXBs. In the following we consider only LMXBs with neutron star primaries. In principle, the discussion for these systems is analogous to the discussion of CVs. While the radius of a white dwarf is about , the radius of a neutron star is only about . Since for a given secondary star and driving mechanism (cf. Eq. (84)) the smaller radius of a neutron star has to be compensated by a correspondingly small value of for achieving a comparable value of . Because an efficiency that small is unlikely, King et al. (1996,1997) and Ritter et al. (2000) have concluded that irradiation in LMXBs is too strong to destabilize such systems.
However, this conclusion applies only to the constant flux model but not to the point source model. For the point source model s'decreases much more slowly for large than for the constant flux model because the surface elements near the terminator are only partially blocked by irradiation, even for very high fluxes due to the high angle of incidence. Therefore, the point source model does not require to get mass transfer cycles. Instead, s' becomes larger than f for fluxes below as can be seen in Fig. 18 for a LMXB with an unevolved donor, in Fig. 19 for a LMXB with a remnant of thermal timescale mass transfer, and in Fig. 20 for a LMXB with a giant. Thus, LMXBs can become unstable for .
Figure 18: s' for an LMXB with an unevolved MS star (No. 3 of Table 1) and an initially NS for the point source and the constant flux model (solid and dotted line). s' is compared to ffor and , 0.1, and 1.0 (short-dashed, dash-dotted, and dot-dash-dotted line, respectively). | |
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Figure 19: s' for an LMXB with an remnant of thermal timescale mass transfer (No. 5 of Table 1) and an initially NS for the point source and the constant flux model (solid and dotted line). s' is compared to f for and , 0.1, and 1.0 (short-dashed, dash-dotted, and dot-dash-dotted line, respectively). | |
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Figure 20: s' for an LMXB with an giant (No. 6 of Table 1) and an initially NS for the point source and the constant flux model (solid and dotted line). s' is compared to f for and , 0.1, and 1.0(short-dashed, dash-dotted, and dot-dash-dotted line,respectively). | |
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The "island of instability'' for the LMXB with an unevolved
donor, which is shown in Fig. 18, ranges from
0.02 to 150 in
for
.
This corresponds to
(99) |
(100) |
Figure 21: Mass transfer rate from an evolved ( ) initially MS star onto an initially neutron star for the point source model as function of orbital period . System parameters: , , and strong braking according to Verbunt & Zwaan (1981). The dotted line shows the evolution without irradiation feedback. | |
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There is only a small region in the parameter space where LMXBs can
undergo significant cycles for an irradiation efficiency as large
as
and for high braking rates according to Verbunt & Zwaan (1981).
As an example we show in Fig. 22 the evolution
of a LMXB through thermal timescale mass transfer. The "island
of instability'' for the
remnant (model
No. 5 of Table 1), corresponding to an orbital
period of
,
is shown in Fig. 19
and ranges from 0.02 to 100 in
,
corresponding to
(101) |
Figure 22: Mass transfer rate from an evolved, initially MS star (No. 5 of Table 1) onto an initially neutron star for the point source model and as function of orbital period . For this computation has been used as long as the Eddington accretion rate of the neutron star is not exceeded, otherwise . The dotted line shows the evolution of the same system without irradiation feedback. | |
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Unlike CVs with giant donors LMXBs with giant donors can undergo mass transfer cycles even for very extended giants. This can be seen from the example shown in Fig. 20 which illustrates the conditions at the beginning of the evolution shown in Figs. 23 and 24. Unlike all other evolutions shown before this system evolves to longer orbital periods because it is driven by nuclear evolution, not loss of angular momentum.
Figure 23: Mass transfer rate from an giant (No. 6 of Table 1) onto an initially neutron star for the point source model, , and as function of orbital period . The dotted line shows the evolution of the same system without irradiation feedback. | |
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Figure 24: Same as Fig. 23 but as function of time . | |
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However, taking into account, that a neutron star has a quite small Eddington accretion rate of , this particular giant system would be stable, since the secular mass transfer rate is above the Eddington rate, so that there would be no irradiation feedback.
The appearance of mass transfer cycles differs between different types of systems. "Ideal'' outbursts as outlined by King et al. (1997) require a very short ratio of and which is typical for giants. They are characterised by a very short high state at a rather high mass transfer rate followed by a long lasting low state as shown by Fig. 24.
In the limit of weak irradiation feedback, as it is typically the case for CVs with and weak braking, a system leaves the high state long before saturation is reached. Only a few 10^{-3} of the stellar mass is transferred during a cycle and the recurrence timescale is rather short. In the limit of strong irradiation feedback, as it is often the case for LMXBs with , the system stays in the high state until saturation has been almost exactly reached. As a consequence of this, the mass transfer rate decreases to almost the stationary value before the system enters the low state. The duration of the high and low state are of the same order, and in the high state the system transfers mass just slightly above the stationary mass transfer rate for most of the time.
Since the timescale of mass transfer cycles, i.e., the thermal timescale of the convective envelope of the donor, is by far longer than typical observational timescales (at most the order of centuries), it is effectively not possible to observe the evolution of a single system from the high into the low state, or vice versa. Nevertheless, the occurrence of these cycles would affect the observable properties of a binary population. First, systems in the low state are more difficult to observe than in the bright high state or may even appear as detached systems if mass transfer ceases completely in the low state. If the duration of the low state is longer than that of the high state, then most of the population can be unobservable. Second, even in the high state the mass transfer does not proceed at a constant rate but declines continuously from the peak rate until the system "switches'' into the low state. Hence, even a completely homogeneous sample can show significant variations in the mass transfer rate. Third, the mean mass transfer rate of the observable systems in the high state can be significantly larger than the secular mass transfer rate which can lead to apparent discrepancies between observations and theoretical predictions if the occurrence of mass transfer cycles is not taken into account.
Therefore, mass transfer cycles could provide an explanation for, e.g., the scatter of mass transfer rates of novalike CVs above the period gap, the disappearance of bright systems at the upper edge of the period gap, or discrepancies between predicted and observed luminosities and birthrates of LMXBs as recently suggested by Pfahl et al. (2003).
We have assumed that irradiation is a local effect and can be treated for every surface element of the donor separately. If the contribution of non-local effects (e.g., circulations) to the lateral heat transport becomes non-negligible for sufficiently strong irradiation, or if there is another mechanism which heats the unilluminated side (e.g., scattering of X-rays by an extended X-ray corona), then also the effective temperature and the intrinsic flux on the unilluminated surface depends on and differs from T_{0} and F_{0}, respectively.
Observations of the eclipsing binary AA Dor show that even in the case of very strong irradiation an illuminated star can preserve a cool hemisphere with while the illuminated side is heated up to (Hilditch et al. 2003). Therefore, it seems plausible that also in the case of LMXBs at most a small fraction of the irradiating flux is transported to the unilluminated side. Nevertheless, for sufficiently strong irradiation, e.g., , even a small fraction can have a significant effect onto the unilluminated side. We note that with the general definition of s in Eq. (18) our analytical ansatz remains valid, even if the unilluminated side is heated up, and we formally get the same stability criterion. In this case the shape of s' depends on the exact prescription of those non-local effects. Here we can not make any quantitative predictions. Nevertheless, it is plausible that s' becomes greater at higher fluxes if non-local effects are taken into account. This would mean that our local treatment of irradiation provides a lower limit on the occurrence of mass transfer cycles in LMXBs. An upper limit is given by the fact that the area below the s' curve has to be less than the maximum fraction of the surface which is affected by irradiation, i.e., less than unity.
We have described the physics of irradiation-driven mass transfer cycles by a two-dimensional system of ordinary differential equations similar to King et al. (1996). An additional term of order of several times which reflects the usual thermal relaxation of the mass losing star, cannot be neglected, because it provides an upper limit for the driving timescale , and a lower limit for the timescale ratio , where mass transfer cycles can occur. Taking into account this term and also the rather large pressure scale height and the deep superadiabatic convection zone of giants we have concluded that CVs with giants above a certain core mass cannot undergo mass transfer cycles unless the efficiency parameter is unexpectedly large. On the other hand LMXBs with giants might be susceptible to irradiation instability as has been predicted by King et al. (1997).
We agree with results of King et al. (1996) and Ritter et al. (2000) regarding CVs with unevolved MS stars: for high braking rates (e.g., Verbunt & Zwaan 1981), as are also required by the period gap model (e.g., Spruit & Ritter 1983; Kolb 1993), such systems are stable except for the most massive donor stars. On the other hand, for low braking rates, as proposed by Sills et al. (2000) and Andronov et al. (2003), CVs above the period gap might undergo cycles. In this case only novalikes are affected since dwarf novae have rather large amplitude outbursts with a rather small duty cycle and therefore a small . Even for gravitational braking the susceptability to irradiation ceases within the period gap so that CVs with unevolved MS stars below the period gap are stable.
Numerical evolutionary computations have shown that the uncritical application of the predictions of the analytical model to highly evolved remnants of thermal timescale mass transfer does not yield good quantitative estimates for the boundaries of the unstable region. The timescale on which the stellar radius changes, if irradiation is changed, seems to be significantly underestimated by the bipolytrope model so that such systems are more stable at high braking rates than previously expected. The most evolved systems are susceptible to irradiation for and orbital periods roughly between 2 and 5 h, or donor masses between 0.1and , respectively. For giants the ratio of thermal energy of the envelope divided by the stellar luminosity seems to give a good estimate for .
Since in the point source model surface elements near the terminator are irradiated at a high angle of incidence, they are only partially blocked by irradiation even for high fluxes. Therefore, not only LMXBs with giant donors, as expected by King et al. (1996,1997) and Ritter et al. (2000), but also LMXBs with unevolved or evolved MS stars might be susceptible to this instability. Regarding the driving timescale for such systems, very similar restrictions apply as for CVs. LMXBs can undergo mass transfer cycles only for . The contribution of non-local effects of irradiation, e.g., circulation, which was not taken into account by our model, might be non-negligible for LMXBs. It seems plausible that this effect tends to destabilize the mass transfer. However, it is not clear in all cases whether mass transfer in LMXBs is driven by Roche lobe overflow or by irradiation-induced winds (Iben Jr. et al. 1997; Basko et al. 1977; Basko & Sunyaev 1973). In the latter case our model obviously could not be applied to these systems.
We conclude that the analytical model gives a qualitatively and also quantitatively suitable description of the onset of mass transfer cycles except for highly evolved remnants of thermal timescale mass transfer. We think that the boundaries of the unstable regions can be determined with an accuracy of a factor of .
To answer the final question of whether the observable CV or LMXB population or parts of it are undergoing mass transfer cycles we need to know two basic parameters: the driving timescale, consisting of the magnetic braking law, and the efficiency . Nevertheless, we have shown that irradiation-driven mass transfer cycles in compact binaries are possible for not unreasonable values of and .
Acknowledgements
We thank A. Weiss and H. Schlattl for providing us with their stellar evolutionary code, H. Schlattl for valuable support and helpful discussions about numerics, I. Baraffe for providing high resolution EOS tables, P. P. Eggleton for providing us with his EOS code, P. H. Hauschildt and J. W. Ferguson for providing opacity tables, and J.-M. Hameury for providing high resolution irradiation tables. We also thank our referee P. Podsiadlowski for his helpful comments and suggestions.