AST242 LECTURE NOTES PART 5 Contents 1. Waves and ...

AST242 LECTURE NOTES PART 5 15with δ > 0. With our form for the perturbations ∝ exp(i(k · x − ωt) we find that ourperturbations exponentially damp due to thermal conductivity.Note that δ is not small if k is sufficiently large. Hence our assumption that δis small is only satisfied for moderate values of k. For k sufficiently large (or oversufficiently small distances) perturbations are rapidly damped due to conduction.The parameter δ has units of inverse time. We can estimate how much the amplitudeis reduced per wavelength traveled with the factor e −δP where P is the period,P = 2π = 2π .ω c sk4.2. Field Stability Criterion. We consider slow changes or small ω. In this limitω ≪ kc s and our dispersion relation (equation 89) reduces to(95)We can rewrite this asiωγ − 1 c2 s + ( Q T − λk 2) T 0p 0c 2 sγ = Q ρ(96) iω = γ − 1 ( )Qρ ρ 0 − Q T T 0 + λk 2 T 0 .γp 0 The ISM is nearly in pressure equilibrium. Since p ∝ ρT we can use ∂ρ ∣∂T p= − ρ Tand∂Q(97)∂ ln T ∣ = Q T T + Q ρ T ∂ρp∂T ∣ = Q T T − Q ρ ρpThus our dispersion relation (equation 95) becomes[(98) −iω = γ − 1]∂Qγp 0 ∂ ln T ∣ − λk 2 T 0pIn the case of no conductivity (λ = 0) there is instability when(99) instability ⇐⇒ ∂Q∂ ln T ∣ > 0pThis is called the Field stability criterion.Let us consider the Field stability criterion physically. Pressure equilibrium isexpected otherwise there will be large velocities generated and mixing. If increasingthe temperature increases the heating rate then the temperature will continue torise and we will have instability. If increasing the temperature decreases the heatingrate then we will have convergence to a stable situation. For the second case, givingstability, we have negative feedback. For the first case, giving instability, we havepositive feedback. I note that there can be instability even in the negative feedbackcase if the feedback (heating or cooling) is delayed, a situation that we have not