Convection and Other Instabilities

Transcription

1 What is convection? Convection and Other Instabilities rialnik 6 Glatzmair and Krumholz 11 ols 3, 1 Highly idealized schematic. Real convection is not so ordered, but mass is conserved. Mass going up = mass going down. A highly efficient way of transporting energy in the stellar interior Heat carried by vection, not diffusion, Depends linearly on fluid speed, not on random walk Occurs when the heat grient exceeds some critical value required for buoyancy Solar surface Kerr (1996) Filamentary flows in Rayleigh-Benard convection (high Rayleigh number x 1 7 ) Volume rendering of temperature structure

2 Consider the rial displacement of a blob of gas. As it rises its internal density will decrease in accordance with the decrease in pressure in the surrounding medium. In the absence of heat exchange with its surroundings it will expand iabatically. Adiabatic expansion Recall from Lecture 5 (equation of state) that the first law of thermodynamics can be used to define an "iabatic expansion" and an iabatic exponent. du + d 1 = which given that u =φ Which is denser after a a displacement? he blob or its surroundings? he density in the surrounding medium will decrease too in accordance with hyostatic equilibrium. les to and ( ) d log = φ + 1 = γ φ φ+1 φ φ+1 φ or =K =K γ a Such a relation between pressure and density is called "iabatic". If the gas expands due to decreased pressure on its extremities or is compressed by increased pressure without heat flow across its boundary, this expression gives the change in internal pressure relative to the change in density From our previous discussion of φ, for ideal gas γ a =5 / 3; for riation, γ a = 4/3. Note that the gamma here is a local gamma γ a (r), unlike the gamma that characterizes a polytrope Consider a rising blob expanding iabatically r e stands here for expanded Because the hyodynamic time is much shorter than the float time (v << c sound ) the pressure inside the floating blob at a given rius is always equal to the external pressure at the same rius. e =, but e may not equal

3 Background grients, d e = 1 +δ e = 1 +δ r d log = γ a d log (iabatic) δ δ e = γ e a e e where δ e is determined by the existing pressure grient in the star. So δ e = e e 1 γ a Δr. his will be stable against convection if δ e < δ = 1 but δ e and δ are both negative Background grients, d r δ e > δ for stability δ e > d Δr or e 1 e d Δr > e γ a Δr 1 d < 1 1 γ a e d and are both = negative so numbers so stability δ e > δ hat is, if iabatic expansion les to a greater density (less density decrease) than the surroundings, there is no convection e = and e if Δr is small (( e ) / might be 1 8 ) stable if d log > 1 γ a unstable if d log < 1 γ a log 1,u e,s S: stable grient U: unstable grient 1 S U ols 5.3 log so things stable if d log > 1 γ a, but in general we don't know the density grient. Inste we have information about and Expand in terms of partial derivatives. is a function of,, and µ (for any EOS!) =,µ = log log = χ +,µ,µ d + µ + log log + χ d + χ dµ µ µ µ,, dµ d + log logµ hese values "χ " are dimensionless numbers which, dµ µ basically give the powers of the given quantities to which is sensitive. stable if d log > 1 γ a e.g., for ideal gas µ so χ = 1; = 1;χ µ = 1 = d log + d log + d logµ

4 continuing: = χ d log + d log + χ µ d logµ d log = 1 χ d log χ d logµ µ d log = 1 χ χ µ µ where d log ; µ d logµ here (not to be confused with the "gr" operator) is a dimensionless temperature grient describing how behaves with pressure. is usually a positive quantity. It is related to the rial temperature derivative by: = d log = where H = 1 = 1 1 = H = is the pressure "scale helight", the distance over which pressure declines by 1 e-fold For ideal gas (only) in hyostatic equilibrum H = / = N k A µ 1 g = N k A µg ; H = N k A µg reviously, d log = 1 χ χ µ µ and d log For an aibatic expansion at consant composition( µ =; in the displaced gas the composition does not change) d log = 1 = 1 χ 1 γ χ = χ his is the "dimensionless iabatic temperature grient" For ideal gas = 5 / / 3 1 χ χ µ µ > 1 χ = 1 χ 1 1 = 1 χ d log =.4. For stability but χ = and χ = 1 = γ χ = χ star > 1, so 1 1 = χ χ χ continuing: 1 χ χ µ χ µ > 1 = 1 χ χ χ χ µ µ > χ χ χ µ µ > χ χ + χ µ µ < + χ d log for stabiity < χ µ χ µ Ledoux and if µ = (or is ignored) < Schwarzschild

5 he dimensionless riative temperature grient is given by the chain rule and hyostatic equilibrium and riative diffusion dm = dm i = Gm d log 4πr 4 = 3 κ L(r) 4ac 3 4πr r = d log r 3 κl(r) = 16πacG m 4 ( ) he Schwarzschild condition for stability becomes, e.g. r < For ideal gas for example γ a =5 / 3, =1, χ = 1 = χ = / 3 5 / 3 =.4 For riation γ a = 4 / 3, =, χ = 4 = χ = 4 / 3 16 / 3 =.5 Maximum luminosity for convective stability: For stability: r < 3 κl(r) 16πacG 4 m < ~.4 (ideal gas) 3 N A k κl(r) 16πacG µ 3 m <.4 L(r)<.4 16πacGµ 3 m(r) 3N A kκ < µ 3 e.g. µ =.61,κ =1, =1 6, M =M κ m(r) erg s 1 L < erg s 1 Where is convection important? L(r) > µ 3 κ m(r) erg s 1 Convection will be important in regions where the opacity is high or the energy generation is concentrated in a small mass giving a large ratio of L(r)/m(r). Examples: he cores of massive stars powered by the CNO cycle which is very temperature sensitive. Regions of high opacity especially in ionization zones and near stellar surfaces. Kramers opacity increases at low. he sun Manfred Schussler Max lank Insitut fur Aeronomie he interiors of massive stars where L/M is large

6 Convection in the sun and massive stars ols, Fig 5.4 Main sequence stars Heat transport by convection and mixing length theory he sun In a region where the temperature grient is even slightly superiabatic, there will be efficient transport of energy by vection. Carrying entrained energy and dispersing it after some distance is much more efficient than riative diffusion or conduction. It turns out that the efficiency of the convection can be simply and fairly accurately (based upon 3D simulations) characterized by a representative length scale called the mixing length. his is how far a relatively intact plume or bubble of hot gas will rise iabatically before dispersing and depositing its energy by diffusive and turbulent processes.

7 o conserve mass, there will also be matter falling back down from large rii to smaller ones, but the downflowing matter now carries less energy than when it went up, the difference having now been dissipated. he net heat being transported up (per gram) is approximately the heat capacity, c = u, times the temperature difference, Δ, between the gas inside the convective plume and outside of it evaluated at r + m where r is the rius where the transport starts and m is the mixing length. hat is, Δ is given by the difference between the temperature grient in the external medium times m, and the temperature change that happens to a rising iabatic bubble after floating the same distance. Note: Heat transport = C Δ Δ= = d log actual = d log = Δ m > m = = H as previously discussed (slide 14) 1) he "mixing length", m, is a characteristic scale over which a given convection element dissipates. he total extent of a convective region can itself be greater (or even less) than the mixing length. ) connects directy to the equation of state, pressure is a better variable here than rius. is dimensionless and the log's are base "e". Δ= = d log actual m = d log = 3) We defined the reciprocal of 1 = as the "pressure scale height" (the rial distance over which the pressure declines by a factor of e). Almost universally the mixing length m is taken to be a (parametric) multiplier, α, times the pressure scale height. m = α 1 = αh p.3 < α <1.5 is < H = is >

8 hen Where m = α Δ= = m H d log d log 1 = αh p m H actual m ( )=α ( ) actual in the star = α iabatic from the EOS he heat being transported per unit area is then F=v c Δu = v c c Δ hat is the flux is the mass flux in a given convective element, v c, times the excess heat content (erg/gm) in the element compared with its surroundings. c, the heat capacity,can be obtained from the equation of state but is 5N A k for an ideal gas. It remains to determine v c,the typical convective speed. he acceleration, a, is given by a = Δ g where Δ is the density contrast between the floating plume and its surroundings, Δ = inside outside < As the plume floats (v c <<sound speed), it remains in hyostatic equilibrium like its surroundings and in pressure equilibrium with its surroundings. So Δ/ =. For an ideal gas d == p + dµ µ so, if Δµ=, -Δ/ Δ/. a= Δ g = Δ g Recall also Δ= α Gm(r) where g = r ( ) he typical velocity while moving a distance m = 1/ at v c = m t v c m αg m m / a = a / = g m m ( ) = m =αh Δ α H g ( )

9 ( ) = v c l αg m α H g ( ) ( ) and F= c Δ v c Δ= α = c α ( )v c = c α H g ( ) 3/ H L(r) = 4πr F = 4πr c α g ( ) 3/ Given the degree to which the actual temperature grient is superiabatic, ( ), this gives the convective heat flux. ypically α =1/3 to 1.5. How superiabatic is the temperature in the convection region? H L(r) = 4πr c α g ( ) 3/ Evaluate approximately for the solar convective zone, r R, = 1.4 g cm 3, from Virial theorem about K For ideal gas and HE, H = 1 1/ 1 = N A k µg = N A k µg L 4πR c N k A µ ( ) 3/ ( ( ) 3/ ()(.6)) 1/ ( ) 33 = (4)(π )C ( ) (1.4)(3 1 6 )[( )( )(3 1 6 )] 1/ =.11 5 N A k = g cancels in ( ) ( ) Δ for α=1; Δ= α 1 8 How superiabatic is a convection region? While one might quibble about factors of, the deviation from the iabatic grient is clearly very small. his is generally true in other places where convection happens as well. Exceptions: near stellar surfaces, in the outer layers of red supergiants, and in the final stages of massive star evolution. Examining the various terms,,, R, L, we see that the excess scales as Δ R L/3 M 5/3 (ols p 7) At very high L, say 1 6 L and large R the temperature grient could start to become substantially superiabatic. his rarely happens (SN progenitors), but the theory can also break down near stellar photospheres. he small degree of super-iabicity also implies that the required convective speeds are quite modest, very subsonic. v c = α H g ( ) N A k N = 1 4 A k µ ( ) µ 1/ ~ 15 meters/s Note that this is also ~1 4 1/ 1 4 c sound

10 Stars may become unstable for a variety of reasons Stellar Stability Glatzmaier and Krumholz 11 (p 17) Insensitivity of the pressure to temperature as in a degenerate gas with a temperature sensitive energy generation rate oo great a component of riation or relativistic degeneracy pressure Burning in too thin a shell Opacity and recombination iven pulsations First, why are they usually stable in the first place? Non-degenerate gas with some small amount of riation = N A k µ a 4 u gas = 3 u r = 3 he Virial theorem says that the pressure and binding energy are related by M Ω= 3 dm We can note immediately the neutral stability of a star supported entirely by ration or relativistic particles. hen E = U + Ω M = 3 dm 3 dm = he star has no net energy and its rius is undefined. Such is the case with the Chanasekhar mass white dwarf but also stars that are too massive and β too great. M But so long as ideal gas remains a significant contributor Ω = 3 M 3 u + 1 gas 3 u r U gas = 1 (Ω + U r ) dm = ( U gas + U r ) E = Ω + U r + U gas = 1 (Ω + U r ) = U gas What does this say about the mass-averaged temperature? U gas = 3 M N A k µ dm = 3 N A km µ where the in the denominator is because u is the internal energy per gram and = 1 M M dm

11 Conservation of energy then requires that L nuc L = de dt = 3 M N A k µ dt In thermal equilibrium the left hand side is zero, but assume a small positive imbalance L=L nuc - L dt = 3 µ δl N A k M he negative sign is important here and relates again to the negative heat capacity of stars as imposed by the Virial theorem. (L nuc L) goes down and vice versa he star is stable. Instability of a degenerate gas Obviously if the pressure is insensitive to the temperature and the energy generation is very insensitive to the temperature, a very unstable situation exists that is prone to runaway. emperature rises but there is no expansion and cooling. Energy generation rises and increases the temperature still more, etc. o examine this in a bit more detail, consider the relation between central pressure and density we derived for polytropes. c = C n GM /3 c 4/3 where C n is a slowly varying function of polytropic index, but we might have n = 3 in mind. aking the time derivative and dividing by the original equation one has dt dividing gives 4 = C n GM /3 3 c 4/3 c = C n GM /3 = 4 d c 3 c 1/3 d c Now assume the pressure is of the form = a b where dt A small perturbation in pressure thus perturbs the temperature and density according to ( ) c = a a 1 c b c d c + b a c b 1 c which when divided by c = c a c b gives c c = a d c c +b c c Substituting the results for the polytrope, c = 4 d c, c 3 c 4 d c = a d c +b c 3 c c c for ideal gas a = b = 1, for fully degenerate gas a = 4 3 to 5 3, b =, and so on. and c c = 1 4 b 3 a d c c

12 Cases c c 1) Ideal gas - a = b = 1 = 1 4 b 3 a d c c c = 1 d c c 3 c expansion (d c < )causes a decrease in, stable ) Degenerate gas - b << 1, a >4/3 c = 1 c b a 4 d c 3 c Expansion actully causes increased heating. a is always 4/3 for completely degenerate electrons. he temperature rises causing a runaway in energy generation that only stops when the gas has become sufficienty non-degenerate (a < 4/3). Examples helium flash, ype Ia supernovae. hin shell instability Let = + << δm <<M he pressure in the shell is give by = - M Gm(r) dm = - 4πr 4 m( ) R Gm(r) r and δm <<m( ) Change the pressure in the thin zone slightly, e.g., by burning. his les to a new outer rius = + + δr = old + δr (δr still << ) + δ = 1+ δ M Gm(r) dm 1+ δr 4π(r + δr) 4 m( ) 1 4 δr 1 4 δr δ = 4 δr (the integral is -) 4 M m( ) Gm(r) dm 4πr 4 δ = 4 δr but δm = 4π = 4π ( + δ)( + δr) So + δ = δ + δr δr = 1+ δr = δr δ = 4 δ 1 he mass does not change when the zone expands 1 δr = 1 δ 4 δ = 4 δ Using the same representation of the EOS as before, = a b So that δ δ = a δ + b δ a 4 = b δ δ = 1 b 4 a r shell = 4 δ << so δ is small even if δ is large Since a and b are positive numbers and << thin shells are always unstable. he instability is removed when becomes large or the fuel runs out.

13 Examples: ( ) Asymptotic Giant Branch Stars CS envelope stellar wind convective envelope H burning rialnik p He burning H, He deg. core He C, O.1 R. 1 5 R. ~1 pc?.5 1. M..1 few M. ~.5 R..1. M. Classical novae X-ray bursts on neutron stars Dynamical instability he instabilities discussed so far manifest over a long time typically the nuclear time, but there are other more violent instabilities that affect hyostatic equilibrium. Quite basically, when a star, or a portion of a star contracts, its pressure goes up but so too does the force of gravity. If gravity goes up sufficiently faster than pressure, then an unbalanced state remains unbalanced and may collapse even faster. If on the other hand pressure goes up faster, the star when compressed will expand again and perhaps oscillate, but it is stable against collapse. Very roughly: dm = Gm(r) 4πr 4 m r 4 Now consider an iabatic compression: =K γ Also m / r 3 so for hyostatic equilibrium m /3 4/3. If the density changes due to expansion or contraction, leing to new pressure and density, ' and ', hyostatic equilibrum demands ' = ' If the pressure increases more than this, i.e., ' > 4/3 ' 4/3 there will be a restoring force that will le to expansion. If it is less, the contraction will continue and perhaps accelerate. So ' = If, for a given mass, ' ' γ γ > ' > ' 4/3 one has stability, hyostatic equilibrium is satisfied. If not, things are unstable. hus if γ > 4 3 the star is stable and if γ < 4 it is not. 3 his is a global analysis and doesn't necessarily apply to small regions of the star but illustrates the importance of γ =4/3. We will return to this later.

14 Glatzmaier and Krumholz treat this more correctly using perturbation theory. he force equation is r = - Gm r Now multiply by dm 1 dm r = - Gm dm 4πr r Initially forces are balanced and =- Gm dm 4πr r But perturb and include r. A change in r by δr causes changes in and so that +δr, +δ, +δ, so dm d r dt 1+ δr r = - Gm 1+ δr dm 4π r 1+ δr d 1+ δ Keep only first order terms dm d r dt 1+ δr r = - Gm dm 4π r δr 1+ δr 1+ r d 1+ δ r dm δr = 1 δr r Gm dm 1+ δr + δ r 4πr but =- Gm r dm 4πr so dm δr = Gm δr dm δr 3 r + δ 4πr o go further we must assume a behavior for (). Assume, because the time scale is too short for heat transport, that the compression is iabatic, i.e., = K γ. hen since log = log K+γ log = = γ d log = γ d = γ d (1+ ε) n 1 + nε Also linearizing dm= 4πr dm= 4π 1+ δr r 1+ δ δr 1+ r 4π r 1+ δr r δ 1+ δr 1+ r 4πr 1+ δr + δr + δ +... = 4πr 1+ 3δr + δ But dm is precisely 4π so δ = 3δr δ = = 1 γ 3γ δr δ δ dm δr = Gm = 3γ δr 3 δr dm δr = Gm δr dm 4πr 3 r + δ δr 4πr 3γ δr = Gm dm 4πr r ( 3γ ) But for the unperturbed configuration we h So Gm dm = 4πr r dm δr = 4πr ( 4 3γ ) = Gm dm( 4 3γ ) δr δr δr

15 dm δ r = Gm ( ) Gm δr = 3γ 4 dm( 4 3γ ) 3 his is an equation for a harmonic oscillator. It has solution δr δr δr = Ae iωt ω = ± ( 3γ 4) Gm 3 Kippenhahn and Weigert 34.1 If ω is real, get oscillatory motion. If ω is imaginary (γ < 4 3 ) get exponential growth or decay on a time scale τ 1 iω 1 G i.e., a hyodynamic time scale Dynamic Instabilities Especially for stars with near 4/3, the nuclear energy generation may oscillate periodically. Compression les to heating which les to expansion and cooling but the net energy yield in a cycle is positive so the instability persists. his is called the epsilon instability Heat transport depends on the temperature grient and opacity. If this combination behaves in such a way as to trap heat when the gas in the outer layers of a star is compressed and heated and release it when the gas expands and cools, then the star may be subject to the kappa instability ulsational instability by the κ mechanism requires that (ols Chap 1, esp p 158) or d logκ d logκ > = logκ log =κ + κ > + logκ log d log For Kramer's opacity κ 3.5 and if κ = 1 and κ = 4.5,.4, so the condition is not generally satisfied.

16 wo notable exceptions: H opacity where the opacity depends on a positive power of the temperature (Mira variables) Kramer's opacity but <.3 as in ionization zones (Cepheid variables and RR-Lyrae stars) here are two important ionization zones: H I H II and He I HeII both near 15, K He II HeIII (i.e., He ++ ) near 4, K ols 3.5 suppression of the iabatic index Δ = d log by partial ionization of a pure hyogen gas. D returns to its standard ideal gas value when the hyogen is either neutral or fully ionized Ionization in the outer layers of the sun. he second shallow depression is He + turning into He ++ If the ionization zone lies very close to the surface and generally in a region that is alrey convective, it has little effect. Not enough mass (or heat capacity) participates. his is the case for eff > 75 K If the ionization zone is located too deep, at high density, the effect of partial ionization is diminished. he iabatic grient is not greatly supressed and the heat trapped is not enough to ive an oscillation of the heavy overlying material. his happens if eff < 55 K Between 55 and 75 K, instability can ccur.

17 eriod-luminosity relation for Cepheids As we showed previously when γ <4/3, the instability is dynamical τ = 3γ 4 ( ) Gm 3 1/ 1 ~ G For more massive Cepheids the rius is larger and the average density less, hence the period is longer Also Cepeids obey a mass luminosity relation somewhat like main sequence stars. So brighter Cepheids have longer periods. Obtaining a good match with the observations remains a challenge. 1 A&A Equivalence to Glatzmaier Chap 11 F= c α 1/ gh p ( ) 3/ Substituting, for ideal gas and HE, H = 1 = H ( )= N k A µg δ 1 = N A k µg F = c α 1/ gh p ( ) 3/ N = c α A k µ N = c A k µ g 1/ N A k µg δ 3/ α δ 3/ 3/

18 α H g ( ) = H δ v c v c = =αh α H g H δ g δ For an ideal gas and hyostatic equilibrium H = / = N k A µg v c = = N A k µg α H g H δ = α N k A µ which is Glatzmaier if β=1/ g δ