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DYNAMICAL STABILITY OF SPHERICAL STARS We consider first a spherical star in a hydrostatic equilibrium. r0 , ρ0 , and P0 are the radius, density, and pressure, assumed to be known function of the mass fraction, Mr , with 0 < Mr < M , where M is the total mass of the star. Hydrostatic equilibrium means that dP0 GMr =− , dMr 4πr04 (d.1a) 1 dr0 . = dMr 4πr02 ρ0 (d.1b) We shall make now a homologous, adiabatic perturbation, with radius of every mass shell changed by the same factor: r = r0 (1 + x), where x is a very small number that may vary with time, but not in space. Mass conservation expressed with eq. (d.1b) demands that ρ = ρ0 (1 − 3x). We shall assume that the change is adiabatic, with a constant adiabatic exponent γ : P = Kργ , P − P0 ρ − ρ0 =γ . P0 ρ0 (d.2) It follows that P = P0 (1 − 3γx). Naturally, we assume that density and pressure perturbations are very small, i.e. |(ρ − ρ0 )/ρ0 | ≪ 1, |(P − P0 )/P0 | ≪ 1. The perturbed quantities no longer satisfy the equation of hydrostatic equilibrium, but rather the equation of motion: GMr ∂P ∂2r = − 2 − 4πr2 . 2 ∂t r ∂Mr (d.3) As the only time variable quantity is x we may write the equation of motion in the form d2 x GMr dP0 ∂2r = r = − 2 (1 − 2x) − 4πr02 (1 + 2x − 3γx) . 0 ∂t2 dt2 r0 dMr (d.4) Notice, that all quantities with subscript ”0” satisfy the equation of hydrostatic equilibrium. Therefore, combining equations (d.1a) and (d.4) we obtain GMr d2 x = 3 (4 − 3γ) x. dt2 r0 (d.5) Now we make another crude approximation, Mr /r03 ≈ ρav = const, and we obtain d2 x = Gρav (4 − 3γ) x = σ 2 x; dt2 This equation has a solution σ 2 ≡ Gρav (4 − 3γ) . x = x1 eσt + x2 e−σt . (d.6) (d.7) If σ 2 < 0 then the motion is oscillatory, i.e. the star is dynamically stable. If σ 2 > 0 than there is a solution that increases exponentially, i.e. the star is dynamically unstable. Therefore, the star is dynamically unstable if γ < 4/3. We obtained this result in a crude way. The proper analysis leads to the requirement that average value of γ within the star has to be less than 4/3 for the star to be dynamically unstable. The instability or the oscillations develop on a dynamical time scale, which is defined as τd ≈ σ −1 ≈ (Gρav ) d—1 −1/2 . (d.8) The critical value for the adiabatic exponent, γcr = 4/3, is a consequence of the gravitational force varying as r−2 . If the gravitational acceleration was given as −GMr /r2+α , then the critical value would be γcr = (4 + α)/3. One of the effects of general relativity is to make gravity stronger than Newtonian near a black hole, but the effect is there even when the field is weak. For r ≫ rSch ≡ 2GM/c2 we have roughly α ≈ rSch /r. This means that dynamical instability may set in when γ is just very close to, but somewhat larger than 4/3. d—2