What is convection? Convection and • A highly efficient way of transporting energy in the Other Instabilities stellar interior • Heat carried by advection, Prialnik 6 not diffusion, Depends linearly Glatzmair and Krumholz 11 on fluid speed, not on random Pols 3, 10 Highly idealized schematic. walk Real convection is not so ordered, but mass is conserved. Mass going up = mass going • Occurs when the heat gradient down. exceeds some critical value required for buoyancy Solar surface Kerr (1996) Filamentary flows in Rayleigh-Benard convection (high Rayleigh number 2 x 107) Volume rendering of temperature structure Consider the radial Adiabatic expansion displacement of a blob of Recall from Lecture 5 (equation of state) that the first gas. As it rises its internal law of thermodynamics can be used to define an density will decrease in "adiabatic expansion" and an adiabatic exponent. accordance with the decrease ⎛ 1⎞ in pressure in the surrounding du + Pd = 0 medium. In the absence of ⎝⎜ ρ ⎠⎟ heat exchange with its which given that surroundings it will expand P adiabatically. u =φ ρ leads to The density in d logP φ +1 = ( ) = γ the surrounding medium will d log ρ φ ad Which is denser after a decrease too in accordance and a displacement? The blob with hydrostatic equilibrium. φ+1 φ+1 or its surroundings? φ φ γ P∝ ρ or P=Kρ =Kρ a Consider a rising “blob” expanding adiabatically Such a relation between pressure and density is called "adiabatic". If the gas expands due to decreased pressure on its extremities or is compressed by increased pressure Because the hydrodynamic without heat flow across its boundary, this expression gives time is much shorter than the change in internal pressure relative to the change the float time (v << csound) in density the pressure inside the floating blob at a given radius is always equal to From our previous discussion of φ, for ideal gas γ =5 / 3; a r the external pressure at the same radius. P = P , but for radiation, γ a = 4/3. e 2 e may not equal 2 Note that the gamma here is a local gamma γ a(r), unlike the gamma that characterizes a polytrope “e” stands here for “expanded” ρe = ρ1 +δρe ρ = ρ +δρ d log P = γ d log ρ (adiabatic) δρ > δρ for stability 2 1 a e P δ e δρe dρ ⇒ = γ a r or P δρe > Δ e ρe dr where δPe is determined by the ρe 1 dPe dρ dPe dP existing pressure gradient in the Δr > Δr = Pe γ a dr dr dr dr ρ 1 dP star. So δρ = e Δr. e P dr 1 dρ 1 dP 1 e γ a < dρ and dP Background gradients Background gradients dr P dr This will be stable against ρ γ a dP dρ dP dρ , convection if , are both dr dr dr dr δρ < δρ = ρ − ρ r e 1 2 r negative so but δρe and δρ are both negative numbers so stability ⇒ δρ > δρ e d log ρ 1 stable if > That is, if adiabatic expansion leads d logP γ P P and a to a greater density (less density e = ρe ≈ ρ if Δr is small ((ρ − ρ) / ρ d log ρ 1 decrease) than the surroundings, there e unstable if < −8 d logP is no convection might be 10 ) γ a d log ρ 1 log ρ S so things stable if > , but in general we don't know the ρ < ρ S: stable gradient d logP γ e 2 ad a U: unstable gradient dT dP density gradient. Instead we have information about and adiabatic U dr dr expansion Expand dP in terms of partial derivatives. P is a function of ρ P ρe Pe = P2 2 2 T, ρ, and µ (for any EOS!) ρ1 ∂P ⎞ ∂P ⎞ ∂P ⎞ dP = dT + dρ + dµ ∂T ⎟ ∂ρ ⎟ ∂µ ⎟ ρ ⎠ ρ,µ ⎠ T,µ ⎠ ρ,T ρ > ρ 2,u e 2 ρe dP ⎛ ∂logP ⎞ dT ⎛ ∂logP ⎞ dρ ⎛ ∂logP ⎞ dµ = + + ρ2,s P ⎜ logT ⎟ T ⎜ log ⎟ ⎜ log ⎟ Pols 5.3 ⎝ ∂ ⎠ ρ,µ ⎝ ∂ ρ ⎠ µ,T ρ ⎝ ∂ µ ⎠ ρ,T µ dT dρ dµ upward displacement = χ + χ + χ T T ρ ρ µ µ These values "χ " are dimensionless numbers which ρ1 P1 ρ1 P1 basically give the powers of the given quantities to which PP21 log P P is sensitive. ρT d log ρ 1 e.g., for ideal gas P ∝ so χ = 1; χ = 1;χ = −1 Figure 5.3. Schematic illustration of the Schwarzschild criterion for ststabilityable if against convection.> A gas µ T ρ µ d logP γ a element is perturbed and displaced upwards from position 1 toposition2,whereitexpandsadiabaticallyto d logP = d logT + d log ρ + − d logµ maintain pressure equilibrium with its surroundings. If itsdensityislargerthantheambientdensity,itwill sink back to its original position. If its density is smaller,however,buoyancyforceswillaccelerateitupwards: convection occurs. On the right the situation is shown in a density-pressure diagram. A layer is stable against convection if the density varies more steeply with pressure than for an adiabatic change. The expansion of the gas element as it rises over ∆r occurs on the local dynamical timescale (i.e. with the speed of sound), which is typically much shorter than the local timescale for heat exchange, at least in the deep interior of the star. The displacement and expansion of the gas element will therefore be very close to adiabatic. We have seen in Sec. 3.4 that the adiabatic exponent γad defined by eq. (3.56) describes the logarithmic response of the pressure to an adiabatic change in the density. Writing as δρe and δPe the changes in the density and pressure of the element when it is displaced over a small distance ∆r, we therefore have δPe δρe = γad . (5.40) Pe ρe Here δPe is determined by the pressure gradient dP/dr inside the star because Pe = P2, i.e. δPe = P P = (dP/dr) ∆r. Therefore the change in density δρ follows from eq. (5.40) 2 − 1 e ρe 1 dP δρe = ∆r. (5.41) Pe γad dr We can write ρe = ρ1 + δρe and ρ2 = ρ1 + (dρ/dr) ∆r, where dρ/dr is the density gradient inside the star. We can then express the criterion for stability against convection, ρe > ρ2, as dρ δρ > ∆r, (5.42) e dr which combined with eq. (5.41) yields an upper limit to the density gradient for which a layer inside the star is stable against convection, 1 dρ 1 dP 1 < , (5.43) ρ dr P dr γad where we have replaced Pe and ρe by P and ρ, since the perturbations are assumed to be very small. Remember, however, that both dρ/dr and dP/dr are negative. Therefore, in absolute value the sign 65 continuing: ∇ is usually a positive quantity. It is related to the radial d logP d logT d log d log temperature derivative by: = χT + χ ρ ρ + χ µ µ 1 − H d log ρ d logT d logµ d logT P dT P dT ⎛ dP ⎞ P dT χ = 1 − χ − χ ∇ = = = ⎜ ⎟ = − =∇ ρ d logP T d logP µ d logP d logP T dP T dr ⎝ dr ⎠ T dr −1 χ ⎛ 1 dP ⎞ d log ρ 1 χT µ where H is the pressure "scale helight", = − ∇ − ∇ P =⎜ − ⎟ d logP µ ⎝ P dr ⎠ χ ρ χ ρ χ ρ the distance over which pressure declines by 1 e-fold d logT d logµ where ∇≡ ; ∇ ≡ d logP µ d logP For ideal gas (only) in hydrostatic equilibrum ∇ here (not to be confused with the "grad" operator) is dP ρN kT 1 N kT H N k H = −P / = A = A ; P = A a dimensionless temperature gradient describing how T P dr g g T g µ ρ µ µ behaves with pressure. d log ρ 1 χ χ ⎛ d log ρ ⎞ 1 Previously, = − T ∇ − µ ∇ and = d logP µ ⎜ d logP ⎟ continuing: χ ρ χ ρ χ ρ ⎝ ⎠ ad γ ad 1 χT χ µ 1 1 χT For an adaibatic expansion at consant composition(∇ =0; µ − ∇ − ∇µ > = − ∇ad in the displaced gas the composition does not change) χ ρ χ ρ χ ρ γ ad χ ρ χ ρ χ ρ 1− χT χ µ χT ⎛ d log ρ ⎞ 1 1 χ γ γ ad − χ − ∇ − ∇ > − ∇ = = − T ∇ ⇒ ∇ = ad = ρ = ∇ µ ad ⎜ d logP ⎟ ad ad ad χ χ χ ⎝ ⎠ ad γ ad χ ρ χ ρ χT γ ad χT ρ ρ ρ This is the "dimensionless adiabatic temperature gradient" d logT 5 / 3 −1 ⎛ d log ρ ⎞ 1 −χT ∇ − χ ∇ >− χT ∇ad ∇ ≡ For ideal gas 0.4. For stability , so µ µ ∇ad = = ⎜ ⎟ > d logP 5 / 3 ⎝ d logP ⎠ star γ ad χ ∇ + χ ∇ < + χ ∇ 1 χT χ µ 1 T µ µ T ad but − ∇ − ∇µ > χ ρ χ ρ χ ρ γ ad χ µ γ − χ 1 1 ⎛ χ ⎞ ⎛ χ ⎞ 1 1 ⎛ χ ⎞ for stabiity ∇ < ∇ − ∇ Ledoux ad ρ = ∇ ⇒ − ρ = ∇ and T − = T ∇ ad µ ad ⎜ ⎟ ad ⎜ ⎟ ⎜ ⎟ ad χT γ ad χT χT γ ad ⎝ χT ⎠ ⎝ χ ρ ⎠ χT γ ad ⎝ χ ρ ⎠ and if 0(or is ignored) Schwarzschild 1 1 χT ∇µ = ∇ < ∇ad = − ∇ad γ ad χ ρ χ ρ The dimensionless radiative temperature gradient Maximum luminosity for convective stability: is given by the chain rule and hydrostatic equilibrium and radiative diffusion ⎛ ⎞ For stability: ∇ < ∇ dT dP dT Gm T d logT 3 L(r) rad ad − κ ⎜ ⎟ = i = 4 = − 3 2 dm dm dP 4πr P d logP 4ac T ⎜ 2 ⎟ 3 P κL(r) ⎝ (4πr ) ⎠ < ∇ ~ 0.4 (ideal gas) 16πacG T 4 m ad d logT ⎞ 3 κL(r)P ∇ = = rad ⎟ 4 3 N kρ κL(r) d logP ⎠ rad 16πacG mT A 3 < 0.4 The Schwarzschild condition for stability becomes, e.g.