- Chapter 4 1 A Degenerate Star Has A Mass Radius Relation Of The Form 1 3 R Km Where K Is A Constant And M2 Is Its M 1 (131.09 KiB) Viewed 45 times
An interesting consequence of the form (4.7) for R₂/a for q 0.8 is that the mean density p of a lobe-filling star is determined solely by the binary period P: P 3M2 35π 4T R²2 8GP² 110P²g cm hr (4.10) where we have used (4.1) to eliminate a. Equation (4.10) shows that, for binary periods of a few hours, stars with mean densities typical of the lower main sequence 4.4 Roche geometry and binary evolution 55 (~ 1-100 g cm ³) can fill their Roche lobes. If one assumes a structure for the lobe-filling star, and thus a relation R₂ (M₂), (4.10) fixes its properties uniquely for a given period. For example, if we assume that the lobe-filling star is close to the lower main sequence we know that its radius and mass are approximately equal in solar units, i.e. m₂ = R₂/Ro (e.g. Kippenhahn & Weigert (1990)). Thus 3M₂ 3M 1 1.4 -3 P = = g cm 4T R₁2 4π R³ m² m₂ where we have used the solar mean density po = now give a period-mass relation 1.4 g cm-³. This relation and (4.10) m₂≈ 0.11 Phr (4.11) and a period-radius relation R₂≈ 0.11P₁ Ro = 7.9 × 10⁹ Phr cm. (4.12) =
Note that these very simple relations come about because of our assumption that the secondary was close to the lower main sequence: however, since the star is in a binary and losing mass to the companion it is not obvious that its structure will be the same as an isolated star of the same mass. Thus we cannot in general assume a main-sequence (or any other) structure without checking carefully the conditions for this to hold (see below). The vigilant reader will have realized that the process of mass transfer in a binary will change its mass ratio q. Equally true, if not quite so obvious, is that the period P and separation a will be altered by the same process, because of the redistribution of angular momentum within the system. Since the Roche geometry is determined by a and q, it is important to ask whether the effect of these changes is to shrink or swell the Roche lobe of the mass-losing star. In the former case, the lobe-overflow process will be self-sustaining at least for some time, and indeed may run away; in the latter case the mass transfer will switch off unless some effect, such as the nuclear evolution of the mass-losing star, can increase its radius at a sufficient rate. The determining quantity for these questions is the orbital angular momentum J. Writing for the binary's angular velocity (= 2π/P), this is J = (M₁a + M₂a²²2)w, where a1 = - (M/F) a, a2 = = (Mi) a are the distances of the two stars from the centre of mass, and M = M₁ + M₂. Substituting for a₁, a2 above and using (4.1) gives 56 Accretion in binary systems Ga 1/2 J = M₁ M₂ M (4.13) Usually it is a good approximation to assume that all the mass lost by the secondary
1/2 Ga J = M₁ M₂ (4.13) M Usually it is a good approximation to assume that all the mass lost by the secondary star is accreted by the primary, so that M₁ + M₂ = 0, M₂ <0 (the treatment can be extended to cover cases where this does not hold). Then logarithmically differentiating (4.13) with respect to time gives 2j 2(-M₂) M₂ + (1- -M₁). (4.14) Conservative mass transfer is characterized by constant binary mass and angular mo- mentum; setting J = 0 in (4.14) and remembering that M₂ < 0, we see that the binary expands (à > 0) if conservative transfer takes place from the less massive to the more massive star: more matter is placed near the centre of mass, so the remaining mass M₂ must move in a wider orbit to conserve angular momentum. Conversely, transfer from the more massive to the less massive star shrinks the binary separation. The Roche lobe size is affected by the change in mass ratio as well as separation. Logarithmic differentiation of (4.7) gives R₂ à M₂ + R2 a 3M2' so combining with (4.14) yields 2.j 2(-M₁) (8 - M²₂) (5 R₂-2/+ M2 (4.15) = J M₂ 6 Clearly there are two cases, depending on whether q is larger or smaller than . For q> conservative mass transfer shrinks the Roche lobe down on the mass-losing star, and any angular momentum loss (J < 0) will accentuate this. Unless the star can contract rapidly enough to keep its radius smaller than R₂, the overflow process will become very violent, proceeding on a dynamical or thermal timescale depending on whether the star's envelope is convective or radiative. The possibility that the star could shrink makes the critical value of q for this type of unstable mass transfer depend on its mass-radius relation, so it may differ slightly from. The rapid overflow will stop once enough mass has been transferred to make a smaller than this critical value. All interacting binaries must undergo unstable mass transfer when the more massive star expands as it evolves off the main sequence and other episodes can occur