Geometry of a binary

In a binary configuration having a primary star with mass M₁ (primary stands usually for the more massive star in the binary), a secondary star with massM₂, and a separationa, the gravity of these stars defines equipotential surfaces, which are a function of their masses and the separation. The Roche geometry defines the gravitational potential in the corotating frame of the binary, as illustrated in Fig 1.10. The Roche lobes are the largest closed equipotential surfaces that enclose each of the stars. The Roche lobes of both stars meet at the Lagrange point L1. The equivalent radius (R_RL) of a Roche lobe is defined by setting the Roche lobe’s volume equal to a sphere occupying the same volume. In the case for the secondary star, its Roche lobe is given by Eggleton’s expression (Eggleton, 1983),

R_RL a =

0.49q2/3

0.6q2/3+ln₍1+_q1/3₎, (1.6) where q = M₂/M₁ is the mass ratio. Equation 1.6 is an updated version of the simpler analytical approximation of Paczyński (1971),

R_RL a =0.462 q 1+q 1/3 , (q <0.8) (1.7)

The binary configuration is classified according to whether one, none, or both of the stars fill their Roche lobe. If none of the stars fills their Roche lobe the binary is called

detached system, if only one of the star fills the Roche lobe, the binary issemi-detached,

and if both stars are filling their Roche lobes, the system is incontact(Figure 1.10).

In general, the evolution of the orbital separation of a close binary is governed by the following equation,

Û a a =2 Û J J −2 Û M₁ M₁ −2 Û M₂ M₂ + Û M M, (1.8)

withJthe angular momentum,M₁andM₂the primary and secondary masses,M= M₁+M₂ the total mass and dots representing time derivatives. It can be deduced from this equation that the evolution of the orbit can be driven by changes in angular momentum (e.g. losses via emission of gravitational waves or magnetic braking, which will be explained in more details later), or if a fraction of the total mass is lost from the binary system (e.g. ejection of mass due to nova eruptions, which are thermonuclear runaways on the white dwarf surface),

Figure 1.10: Three dimensional (and projected onto two dimensions) representation of the Roche potential in a binary with a mass ratioq= 2. L₁, L₂and L₃are Lagrange points where the gravitation of the two stars and the centrifugal potential are cancelled. The thick solid black line that crosses L1represents the shape of the Roche lobes. Figure taken from Postnov

& Yungelson (2014). On the right side, a sketch that represents the binary configurations (top to bottom) of none, one, or both stars filling their Roche lobes.

or mass variations of one of the stars. A key example is mass transfer through the L1point

from one star that fills its Roche lobe to the other, which is the scenario that, CVs, the systems of interest in this thesis, present.

Angular momentum losses: magnetic braking & gravitational radiation

Since this thesis focuses on CVs, I will explain the two processes that are thought to operate within the standard theory of Cataclysmic Variable (CV) evolution that lead to_JÛ_{in equation} 1.8: magnetic braking and gravitational radiation.

Magnetic braking has its origin in the weak stellar wind of the low-mass companion. Assuming a dipole, the magnetic field lines near the rotational equator are closed, therefore the wind is mainly emitted from the poles. The ionised particles in the wind are forced by the magnetic field to corotate with the star out to the Alfvén radius. When the matter decouples from the magnetic field the particles are accelerated to high speeds leaving the companion and in this process carry away substantial amounts of angular momentum, resulting in a braking of the stellar rotation. This phenomenon has been measured indirectly by its effect on the rotation rate of single stars (Kraft, 1967; Schatzman, 1962; Reiners & Basri, 2008).

However, due to tidal forces the rotation of the companion in a CV is synchronised with the binary orbit, consequently angular momentum is extracted from the binary orbit. In the literature many descriptions have been developed in order to model magnetic braking (Verbunt & Zwaan, 1981; Rappaport et al., 1983; Mestel & Spruit, 1987; Kawaler, 1988; Andronov et al., 2003; Ivanova & Taam, 2003) and there are huge differences among them, by at least three orders of magnitude (see Knigge et al., 2011, for a review). However, within many studies Rappaport et al. (1983)’s prescription has been used due to the easy manipulation of the strength and shape by changing theγexponent,

Û J_MB=−5×10−29k₂2f−2 2π P_orb 3 M₂ M R₂ R 4 R₂ R γ−4 . (1.9)

k₂ is the radius of gyration of the part of the star coupled to the magnetic wind, usually∼0.1 for low-mass stars and f is a constant that lies within∼0.7–1.8 (for details see Schatzman, 1962; Huang, 1966; Mestel, 1968; Eggleton, 1976; Whyte & Eggleton, 1980; Verbunt & Zwaan, 1981). The origin of the magnetic field in low-mass stars is unclear, but it is believed that it is generated by a shell dynamo in the transition region between the radiative core and the convective envelope (Charbonneau & MacGregor, 1997; MacGregor & Charbonneau, 1997).

Another mechanism of angular momentum loss is gravitational radiation. According to the theory of general relativity, all close compact binaries warp the spacetime with ripples. These waves are gravitational radiation that is described by Einstein’s quadrupole formula (Paczyński, 1967): Û J_GR= −32 5 G7/2 c5 M₁2M₂2(M₁+M₂)1/2 a7/2 (1.10)

withGthe gravitational constant, andcthe speed of light. It can be clearly deduced that when binaries have sufficiently short orbital periods, angular momentum loss through gravitational wave emission becomes very efficient, and it is a strong function of the masses of the binary. Thus, compact and dense objects such as neutrons are strong space time distorters, and their gravitational waves can be directly measured, as the two colliding neutron stars detected in 2017, GW170817 (Abbott et al., 2017).

Stability of mass transfer

In semi-detached systems with wide orbits in which all the material from the mass losing star (M_donor) accretes onto the accreting star (M_acc), i.e. the total mass of the system (M_acc+M_donor) remains constant (conservative scenario), the stability of the mass transfer is

dictated by how the donor Roche lobe evolves as response to the mass transfer. The change of the Roche lobe can the obtained from differentiating the Roche lobe approximation of Paczyński (1971), given in equation 1.7,

Û R_RL R_RL = Û a a + 1 3 Û M_donor M_donor, (1.11) Û

M_donoris negative, since it is the mass losing star. In this expression,aÛ/acan be obtained from equation 1.8, resulting in:

Û R_RL R_RL =2 Û J J −2 Û M_donor M_donor 5 6 −q (1.12) From this expression, if the angular momentum is conserved,_JÛ_{=0, then in situations when} q > 5/6 ∼ 1, i.e. the mass of the donor star is the largest, the Roche lobe of the donor shrinks as a consequence of its mass loss, increasing the mass transfer towards very high rates. In contrast, ifq < 5/6 ∼ 1, the Roche lobe grows, but in fact, as seen in equation 1.11, the change of the Roche lobe depends on the mass loss and the change of the orbit. Therefore, the binary will expand its orbit resulting the system to detach and therefore the accretion to stop. This fact is in contradiction with the ongoing accretion observed in CVs, which haveq <5/6∼1 and therefore it was the empirical motivation to include the angular momentum losses in the models of CV evolution, as explained above.