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Chapter 4 1 A degenerate star has a mass-radius relation of the form -1/3 R₂ = Km₂ where K is a constant and m2 is its mass M₂ measured in Mo. Show that if the star fills the Roche lobe in a close binary with q < 1 we have Px M₂¹. If K = 2 × 10⁹ cm show that this relation and the main-sequence relation (4.11) intersect at about P = 0.6 h, m2 = 0.07. This shows that there is a minimum orbital period for CVs, since the secondary cannot be smaller than its radius when fully degenerate. The actual minimum period is somewhat greater because mass transfer makes the secondary deviate from its thermal-equilibrium radius. For the degenerate secondary of problem 1, show that M₂ J/J 2/3-q M₂ 3 The evolution of a close binary is driven by angular momentum loss to gravitational radiation, so that j 32 G³ M₁ M₂ (M₁ + M₂) a4 J 5 c5 Use problems 1, 2 above and the results of Section 4.4 to show that for main-sequence and degenerate secondaries we have M~10-¹0 (Phr/2)-2/3 Moyr 1 and M~ 1.6 × 10-¹2(Phr/2) -14/3 ³Moyr-1 352 Problems N
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) =