In the previous Section, the evolution of isolated stars and their final products have been described. However, as discussed above, the majority of the Solar–type stars in our Galaxy, and up to'80%of massive stars, are born in binary systems. Evolution of such objects can differ from that of isolated stars because, if the system is tight enough, the two stars will interact at some point in their lives thus influencing their mutual evolution (Kopal,1978).
The two stellar components of a binary systems are in a rotating frame of reference and the presence of a centrifugal force modifies the gravitational poten- tial. The effective potential originating from the combination of the centrifugal and gravitational potentials is called Roche potential. The equipotential surfaces of the Roche potential are spherical close to each stellar surface (where the gravitational
potential of one of the stars is dominant) while, moving away from the two stellar objects (where the centrifugal potential is stronger), they become ellipsoidal (Fig- ure 1.10). The inner equipotential surfaces enclosing both stars are called Roche lobes.
The two lobes touch in one point in between the two stars, the first (or inner) Lagrangian pointL1. Here, the gravitational pulls from the two stars cancel each other thus a particle passing trough L1 will move from a region where it is gravitationally bound to one star to a region where it is gravitationally bound to the other. As described in Section1.1.1, a star, once exhausted the nuclear fuel in its core, will leave the main sequence and start expanding. If this star is part of a close binary, then it will get in touch with its Roche lobe. Alternatively, a star can fill its Roche lobe if, because of angular momentum losses (see Section 1.4.2), the orbital separation decreases thus reducing the Roche lobe size.
The shape of the equipotential surfaces of the Roche potential is completely determined by the system mass ratio, q = M2/M1, where M2 is the mass of the Roche–lobe filling star (the secondary) and M1 is the mass of its companion (the primary), while the orbital separationa determine their size. Particularly, the size of the Roche lobes is defined by the equation (Eggleton,1983):
RL=
a0.49x2/3
0.6x2/3+ ln(1 +x1/3) (1.4)
wherex =q returns the size of the secondary’s Roche lobe, whilex= 1/q returns the size of the primary’s Roche lobe. Whether the components of a binary system will interact or not depends only on these two parameters,q and a, and, regardless of the mechanism that brings the secondary in touch with its Roche lobe, it will start mass transfer onto the other stellar component.
1.3.1 Mass transfer in binaries
A star is in hydrostatic equilibrium, i.e. the thermal pressure that drives the ex- pansion of the gas is counterbalanced by the gravitational force that holds the star together. However, when a star gets in touch with its Roche lobe, its photosphere around L1 does not feel the gravitational pull anymore and it is free to expand into the surrounding vacuum, entering the Roche lobe of its companion. Here, the effective gravity pulls the matter toward the detached primary, giving rise to a mass transfer stream (Figure1.11).
Because of the transfer of mass and the redistribution of angular momentum in the system, the mass ratio, the orbital separation and thus the orbital period change. The way in which these changes occur determines whether the accretion process is stable or not and a self–sustained mass transfer will be established only
Figure 1.11: The binary system rotates with angular speedΩand the matter inside the Roche lobes feels an effective gravity (black arrows). When one of two stel- lar components gets in touch with its Roche lobe, its photosphere near L1 is not in hydrostatic equilibrium anymore because the gravitational pull vanishes in L1. Since the internal pressure of the gas is not counterbalanced by gravity anymore, it expands freely into the Roche lobe of the companion, giving rise to a mass–transfer stream. Figure from Shu(1982).
when the effect of such variations forces the mass–losing star to stay constantly in touch with its Roche lobe (Frank et al.,2002).
The starting point to derive how these quantities vary is the orbital angular momentumJ: J =M1M2 Ga M 1/2 (1.5) where M = M1 +M2 is the total mass of the system. The logarithmically dif- ferentiation of this equation provides the relationship governing the evolution of a mass–transferring system: ˙ a a = 2 ˙J J − 2 ˙M1 M1 −2 ˙M2 M2 +M˙ M (1.6)
It is usually assumed that all the mass lost by one star is accreted by the other thus no mass is lost by the system (M˙ = 0), and Equation1.6can be written as (King,1988): ˙ a a = 2 ˙J J − 2 ˙M2 M2 (1−q) (1.7)
Under the assumption of conservative mass transfer, i.e. also the total angular mo- mentum of the system is conserved (J˙= 0), sinceM˙2 <0 by definition, it becomes clear from Equation1.7that the system expands (a >˙ 0) if mass is transferred from the less massive to the more massive while the binary shrinks (a <˙ 0) in the opposite case. In fact angular momentum must be conserved so moving matter closer to the centre of mass implies a wider orbit for the secondary (first case). On the contrary, when matter is moved further away from the centre of mass,the orbital separation has to be reduced (second case).
The consequent variation of the Roche lobe radius is given by the relation- ship: ˙ RL RL = 2 ˙J J − 2 ˙M2 M2 5 6 −q (1.8) and therefore two different cases can be identified accordingly to the value ofqwith respect to 5/6 (Ritter,1976):
• q >5/6, under the assumption of conservative mass–transfer, the Roche lobe size is reduced thus keeping the secondary in touch with it. In this case, the mass transfer process proceeds in an violent and possibly catastrophic way and it stops when enough matter has been accreted on the companion so that
q.5/6.
• q . 5/6, in this case the size of the Roche lobe increases under conservative mass transfer. The secondary loses contact with the Roche lobe and accretion is halted. However, if the secondary expands owing to nuclear evolution (i.e. leaves the main sequence becoming a giant) or if the system shrinks by losing angular momentum, then the mass transfer can proceed in a stable way.
Finally, the stability of the mass transfer process is also related to the re- sponse of the secondary to the mass loss. If the time–scale at which the secondary loses mass (Knigge et al.,2011):
tM˙2 =
M2 ˙
M2
(1.9) is much longer than its thermal time scale defined in Section 1.1.1, then the sec- ondary has enough time to adjust its radius to the new mass configuration and
accretion is stable (Section 1.5.2). On the contrary, when the mass loss occurs faster than the thermal time–scale, the star has not the time to find a new equilib- rium configuration and it is driven out of thermal equilibrium and expands. In this situation a runaway mechanism starts and mass transfer becomes unstable, leading to a common envelope phase (Section1.3.2).
1.3.2 Common envelope
Unstable mass transfer in binaries leads to the formation of a common envelope (CE). In this configuration, the core of the donor and its companion orbit each other inside a common envelope that, if it is not co–rotating with the system, it can exert a drag force onto the two stellar components. In this case, the two stars transfer their orbital energy to the envelope and, as a consequence of the energy loss, they spiral in and get closer while the envelope gains energy and is expelled. The duration of the common envelope phase is thought to be very short, lasting
.103yr.
The physics behind this phase is not well modelled yet and it is usually parametrised with the so–called “αprescription”. In this approach, the efficiency at which the two stars transfer their energy to the envelope, i.e. the fraction of orbital energy that is used to expel the envelope, is defined by the parameter (Paczynski,
1976;Tutukov and Yungelson,1979;Webbink,1984):
α= ∆Ebind ∆Eorb
(1.10) where∆Ebindis the binding energy of the envelope and∆Eorb is the change in the orbital energy. These two quantity are defined as follow:
∆Ebind=−
GMDMe
λRL,A
(1.11) whereMDandMeare the initial mass of the donor and of the envelope, respectively,
RL,A is the Roche lobe radius of the accretor and λ is a parameter (' 1) that accounts for the envelope density distribution, while
∆Eorb =− G 2 McMA af − MDMA ai (1.12) whereMcis the mass of the core of the donor,MAis the mass of the accretor andai andaf are the initial and final orbital separations, respectively. Using the definition of binding and orbital energy, Equation 1.10can be rewritten as:
GMDMe λRL,A =αG 2 McMA af − MDMA ai (1.13)
This is the “standardα–formalism”, in which is commonly assumed that the donor mass does not change during the common envelope phase. From the study of a large sample of post common–envelope binaries (PCEBs), Zorotovic et al. (2010) have studied the energy balance problem described by Equation1.13and have shown that the current population of PCEB can be reproduced assumingα'0.2−0.3.