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4.2 Mass transfer

Although there are still many unanswered theoretical questions about the nature of the mass transfer phase, the basic properties of the evolution of a binary due to mass transfer can easily be described. The rate at which a star can adjust to changes in its mass is governed by three time scales. The dynamical time scale results from the adiabatic response of the star to restore hydrostatic equilibrium, and can be approximated by the free fall time across the radius of the star,
( ) ⌊( ) ⌋1∕2 2R3 1∕2 R 3 M ⊙ tdyn ≃ ----- ∼ 40 ⌈ ---- ----⌉min, (13 ) GM R ⊙ M
where M and R are the mass and radius of the star. The thermal equilibrium of the star is restored over a longer period given by the thermal time scale
GM 2 ( M )2 R L tth ≃ -----∼ 3 × 107 ---- --⊙--⊙ yr, (14 ) RL M ⊙ R L
where L is the luminosity of the star. Finally, the main-sequence lifetime of the star itself provides a third time scale, which is also known as the nuclear time scale:
9 M---L⊙- tnuc ∼ 7 × 10 M L yr. (15 ) ⊙

The rate of mass transfer/loss from the Roche lobe filling star is governed by how the star’s radius changes in response to changes in its mass. Hjellming and Webbink [108] describe these changes and the response of the Roche lobe to mass changes in the binary using the radius-mass exponents, ζ ≡ d lnR ∕d ln M, for each of the three processes described in Equations (13View Equation, 14View Equation, 15View Equation) and defining

dlnRL ζL = (1 + q )------ (16 ) d lnq
for the Roche lobe radius-mass exponent. If ζL > ζdyn, the star cannot adjust to the Roche lobe, then the mass transfer takes place on a dynamical time scale and is limited only by the rate at which material can stream through the inner Lagrange point. If ζdyn > ζL > ζth, then the mass transfer rate is governed by the slow expansion of the star as it relaxes toward thermal equilibrium, and it occurs on a thermal time scale. If both ζdyn and ζth are greater than ζL, then the mass loss is driven either by stellar evolution processes or by the gradual shrinkage of the orbit due to the emission of gravitational radiation. The time scale for both of these processes is comparable to the nuclear time scale. A good analysis of mass transfer in cataclysmic variables can be found in King et al. [130].

Conservative mass transfer occurs when there is no mass loss from the system. During conservative mass transfer, the orbital elements of the binary can change. Consider a system with total mass M = M1 + M2 and semi-major axis a. The total orbital angular momentum

[ 2 2 ]1∕2 J = GM--1M-2a- (17 ) M
is a constant, and we can write − 2 a ∝ (M1M2 ). Using Kepler’s third law and denoting the initial values by a subscript i, we find:
P-- [M1iM2i-]3 P = M M . (18 ) i 1 2
Differentiating Equation (18View Equation) and noting that conservative mass transfer requires M˙1 = − M˙2 gives:
P˙ 3M˙1 (M1 − M2 ) -- = ---------------. (19 ) P M1M2
Note that if the more massive star loses mass, then the orbital period decreases and the orbit shrinks. If the less massive star is the donor, then the orbit expands. Usually, the initial phase of RLOF takes place as the more massive star evolves. As a consequence, the orbit of the binary will shrink, driving the binary to a more compact orbit.

In non-conservative mass transfer, both mass and angular momentum can be removed from the system. There are two basic non-conservative processes which are important to the formation of relativistic binaries – the common-envelope process and the supernova explosion of one component of the binary. The result of the first process is often a short-period, circularized binary containing a white dwarf. Although the most common outcome of the second process is the disruption of the binary, occasionally this process will result in an eccentric binary containing a neutron star.

Common envelope scenarios result when one component of the binary expands so rapidly that the mass transfer is unstable and the companion becomes engulfed by the donor star. The companion then ejects the envelope of the donor star. The energy required to eject the envelope comes from the orbital energy of the binary and thus the orbit shrinks. The efficiency of this process determines the final orbital period after the common envelope phase. This is described by the efficiency parameter

ΔEbind αCE = -------, (20 ) ΔEorb
where ΔEbind is the binding energy of the mass stripped from the envelope and ΔEorb is the change in the orbital energy of the binary. The result of the process is the exposed degenerate core of the donor star in a tight, circular orbit with the companion. This process can result in a double degenerate binary if the process is repeated twice or if the companion has already evolved to a white dwarf through some other process. A brief description of the process is outlined by Webbink [238], and a discussion of the factors involved in determining αCE is presented in Sandquist et al. [206].

The effect on a binary of mass loss due to a supernova can be quite drastic. Following Padmanabhan [172Jump To The Next Citation Point], this process is outlined using the example of a binary in a circular orbit with radius a. Let v be the velocity of one component of the binary relative to the other component. The initial energy of the binary is given by

1 ( M M ) GM M GM M E = -- ----1--2- v2 − ----1--2 = − ----1--2 . (21 ) 2 M1 + M2 a 2a
Following the supernova explosion of M1, the expanding mass shell will quickly cross the orbit of M2, decreasing the gravitational force acting on the secondary. The new energy of the binary is then
E ′ = 1--MNSM2----v2 − GMNSM2----, (22 ) 2 MNS + M2 a
where M NS is the mass of the remnant neutron star. We have assumed here that the passage of the mass shell by the secondary has negligible effect on its velocity (a safe assumption, see Pfahl et al. [177Jump To The Next Citation Point] for a discussion), and that the primary has received no kick from the supernova (not necessarily a safe assumption, but see Davies and Hansen [46Jump To The Next Citation Point] or Pfahl et al. [178] for an application to globular cluster binaries). Since we have assumed that the instantaneous velocities of both components have not been affected, we can replace them by 2 v = G (M1 + M2 )∕a, and so
( ) E ′ = GMNSM2---- -M1-+--M2--− 2 . (23 ) 2a MNS + M2
Note that the final energy will be positive and the binary will be disrupted if MNS < (1 ∕2)(M1 + M2 ). This condition occurs when the mass ejected from the system is greater than half of the initial total mass,
1 ΔM > --(M1 + M2 ) , (24 ) 2
where ΔM = M1 − MNS. If the binary is not disrupted, the new orbit becomes eccentric and expands to a new semi-major axis given by
( ) a′ = a M1--+-M2--−-ΔM---- , (25 ) M1 + M2 − 2ΔM
and orbital period
( ′)3∕2( ′ )1∕2 P′ = P a- 2a-−--a . (26 ) a a ′

We have seen that conservative mass transfer can result in a tighter binary if the more massive star is the donor. Non-conservative mass transfer can also drive the components of a binary together during a common envelope phase when mass and angular momentum are lost from the system. Direct mass loss through a supernova explosion can also alter the properties of a binary, but this process generally drives the system toward larger orbital separation and can disrupt the binary entirely. With this exception, the important result of all of these processes is the generation of tight binaries with at least one degenerate object.

The processes discussed so far apply to the generation of relativistic binaries anywhere. They occur whenever the orbital separation of a progenitor binary is sufficiently small to allow for mass transfer or common envelope evolution. Population distributions for relativistic binaries are derived from an initial mass function, a distribution in mass ratios, and a distribution in binary separations. These initial distributions are then fed into models for binary evolution such as StarTrack [18] or SeBa [191168Jump To The Next Citation Point] in order to determine rates of production of relativistic binaries. The evolution of the binary is often determined by the application of some simple operational formulae such as those described by Tout et al. [229] or Hurley et al. [111]. For example, Hils, Bender, and Webbink [107Jump To The Next Citation Point] estimated a population of close white dwarf binaries in the disk of the galaxy using a Salpeter mass function, a mass ratio distribution strongly peaked at 1, and a separation distribution that was flat in ln(a). Other estimates of relativistic binaries differ mostly by using different distributions [17119168167].

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