Stellar Wind Mechanisms & Instabilities

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Stellar Wind Mechanisms & Instabilities Stan Owocki Bartol Research Institute University of Delaware Collaborators: – Rich Townsend, Bartol/UDel – Asif Ud-Doula, N.C.S.U. – Ken Gayley, U. Iowa – Luc Dessart, Utrecht – David Cohen, Swarthmore – Joachim Puls, U. Munich – Mark Runacres, U. Brussels – Achim Feldmeier, U. Potsdam What is a “Stellar Wind” • Sustained outflow in the outer layers of a star • Net result: mass loss from star’s gravitational field • Source of mass, mom, energy to ISM (=> new stars…)

• Key issues: – What force overcomes gravity to accelerate matter outward? – What is energy source for gravitational potential & kinetic energy?

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 2 Types of Stellar Winds • Solar Coronal Wind

– V• ~ 400-800 km/s ~= Vesc -14 – Low Mdot (~10 Msun/yr << Msun/tlife) – Still important for Sun-earth connection & as prototype • Cool (super) giant (super)winds

-4 -8 – Low V• (< Vesc); high Mdot (10 - 10 Msun/yr)

– all stars with Mzams < 8 Msun become white dwarf with Mfin~<1 Msun => 5 lose 1-7Msun in ~10 yrs as AGB ! • Hot-star winds

– High V• (~3 Vesc = 2000-3000 km/s) -4 -8 – High Mdot (10 - 10 Msun/yr) -1 -4 – LBVs may have superwind phases (10 - 10 Msun/yr), e.g. h Car

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 3 Wind Driving Mechanisms • Gas pressure gradient 6 – Requires “corona” heated to ca. 10 K ; asnd ~< vesc – Key for Solar Wind & other low Mdot “coronal winds” • Wave pressure gradient – MHD waves (sound waves < gas pressure) • May apply to: high-speed solar wind; low-speed cool giant winds • Radiation pressure gradient – Acts via line-opacity in hot, massive stars – May act via continuum opacity in super-Eddington LBVs – Acts via dust opacity in very cool, giant winds • Hybrid – Periodic shocks from stellar pulsation • In cool stars levitate gas to “dustosphere” => – Radiation force on dust drives gas+dust away

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 4 Basic Wind Types & Properties

Mdot v v a g type inf go x E Mo/yr km/s km/s km/s x=? x 400- 100- gas coronal 10-14 600 Q 700 200 pressure wave 10-10- waves, W + cool-star 10-50 40-60 4-5 puls -6 10 puls.,dust Wrad 10-8- 1000- 600- 10- rad. line DL~ hot-star 10-4 3000 1000 30 abs. L/100

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 5 Forces/mass

1. gravity 2. gas pressure gradient GM g 1 dP g = 2 g = - r p r dr

3. inertia 4. driving force dv ∂v dv † g = = + v gx = ? i dt ∂t dr† = grad = gwave † † Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 6 † † Energy/mass (& assoc. speeds)

1. gravitational escape 2. gas internal GM V2 1 P E = - ≡ - g E = = 3 a2 g R 2 p g -1 r 2

3. kinetic 4. energy source v2 v2 • E s = ? † E k = Æ 2 †2 = QX +Wgx heat work

† Oleron school Oct. 04 Stellar† Wind Mechanisms - Lec. #1 7 † † Hydrostatic, planar, isothermal atmosphere

2 GM hydrostatic a dP gg ª 2 ª const. 0 = -g - R equilibrium: g P dz z ≡ r - R

exponential P(z) =†e -z / H stratification:† Po † a2 a2R2 2a2 Scale Height: H ≡ = = 2 R £ 0.001R gg GM Vg † a ª10 km /s << V ª 620 km /s e.g., Solar photosphere: † g † † H†ª 300 km << R ª 700,000 km

Oleron school Oct. 04 † Stellar Wind Mechanisms† - Lec.† #1 8 † † † Hydrodynamical Equations

∂r mass: + — ⋅ rv = 0 ∂t

dv ∂v GM —P momentum: = + v⋅ —v = - rˆ - + g dt ∂t r2 r X †

∂e internal energy: + — ⋅ ev = -P— ⋅ v - — ⋅ Fc + QX † ∂t ideal gas e.o.s.: P = ra2 = (g -1)e †

Oleron school Oct. 04 † Stellar Wind Mechanisms - Lec. #1 9 Steady, spherical expansion

2 d rvr 2 mass loss rate: ( ) = 0 M˙ ≡ 4pr rv = const. dr

dv GM 1 dP momentum: v = - - + gX dr † r2 r dr † kinetic enthalpy grav. r Ê v2 g P GM ˆ total ene†r gy: DE˙ = M˙ Á + - ˜ 2 g -1 r r Ë ¯ ro r ˙ 2 2 r = Ú dr ¢( Mg X + 4pQX r ¢ ) - 4p (r Fc ) ro ro † work heat thermal cond.

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 10 † Energy requirement

2 P 2GM sound speed a ≡ escape speed v2 ≡ r g r

˙ Ê 2 2 ˆ Ê 2 2 2 ˆ DE v• g a• vo g ao vgo ˙ = Á + ˜ -Á + - ˜ M Ë† 2 g -1¯ Ë 2 g -1 2 ¯ †

2 2 v • 2 2 v• go 2 Q 4 p (ro Fco -r• Fc• ) ª + = dr ¢ g + 4p r ¢ X + 2 2 Ú ( X M˙ ) M˙ † ro kinetic grav. work heat thermal energy energy cond. † †

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 11 Energy flux requirement

3 DE˙ Ê M˙ ˆ Ê M ˆ Ê R ˆ Ê v2 ˆ erg F O 1 • 2.2 104 E ≡ 2 ª Á -14 ˜ Á ˜ Á ˜ Á + 2 ˜ ¥ 2 4pR Ë 10 ¯ Ë MO ¯ Ë R ¯ Ë vgo ¯ cm s

˙ 5 erg solar wind: M -14 ª 2 v ª vgo FE ª1- 5 ¥10 • cm2 s † LO 10 erg FO ≡ 2 ª 6 ¥10 2 4pRO cm s cool-star winds: v << v † • go † † ˙ 4 erg K5 Giant: M ª1 M ª16M R †ª 400R FE ª 6 ¥10 -7 O O cm2 s

˙ † 4 erg Mira: M ª 2 M ª M R ª 400R FE ª 5 ¥10 2 -6 O O cm s † † † † Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 12

† † † † Solar Coronal Expansion

X-ray Corona from SOHO 1991 Solar Eclipse

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 13 Heating of upper solar atmosphere ∂e energy eqn: + — ⋅ ev = -P— ⋅ v - — ⋅ F + Q ∂t c X ∂e steady, static case: v = 0 = 0 = Q - — ⋅ F ∂t X c † For now, ignore conduction (Fc=0), consider net heating vs. radiative cooling: † 0 = Q = Q - n n L(T ) † X h e i e for energy flux F damped over length : FE 2 E ld = r L˜ (T) ld † F Solve for temperature via: L˜ (T) = E +z / H 2 ~ e r ld † Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 14

† † Cooling function L(T) Cox, & Tucker 1969

thermal instability

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 15 Coronal Heating with a Conductive Thermostat

Assume upward energy flux FE defined at some base radius R, yielding energy deposition E at some deposition radius r . o d ˙ E C

˙ E o Now consider this with downward conductive flux F leading to a conductive energy † c loss Ec in the region below rd. dT The energy balance is: E˙ = 4pR2F = 4pr 2K T 5 / 2 = E˙ o † E o dr c 2 / 7 Ê 7 F r - Rˆ Integration yields: T = Á E d ˜ 2 K r /R † Ë o d ¯ † † 2 / 7 5 2 e.g. at rd=2R: T ª 26 F5 F5 ≡ FE /10 erg/cm /s

2 / 7 or, for protons with K†o =Kop : T ª 56 F5 † † Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 16

† Outward extension of high-Temperature Corona by conduction

Now suppose Qheat=0 for r>rd => T(rd)=Tmax

Then for r> rd , energy balance is set by: 2 1 d(r Fc ) 2 — ⋅ Fc = = 0 r Fc ª const. r2 dr

Using Spitzer form for Heat Flux by Electron Conduction: dT † F = K T 5 / 2 † † c e dr

Upon integration we find: T ~ r-2/ 7 Nearly isothermal!

† Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 17 † Extended, spherical hydrostatic corona

2 -q hydrostatic GM a dP 2 2Ê r ˆ 0 = - - a = a Á ˜ equilibrium: r2 P dr oË R¯

extended P(r) È R ÊÊ Rˆ1 -q ˆ˘ = expÍ ÁÁ ˜ -1˜˙ stratification:† Á ˜ Po ÎÍ Ho†(1 - q)ËË r ¯ ¯˚˙ # decades of Ê ˆ Po R 6 pressure decline: logÁ ˜ = loge ª Ë P• ¯ Ho (1- q) T6(1- q) † observations for Ê P ˆ logÁ TR ˜ =12 TR vs. ISM: Ë PISM ¯ † hydrostatic only 0.5 for conduction T6 < < 0.7 possible for: 1- q corona with q=2/7 † Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 18 † † Spherical Expansion of Isothermal Solar Wind Momentum and Mass Conservation: dv GM a2 dr 2 v = - - d(rvr ) 2 = 0 dr r r dr dr Ê a2 ˆ dv 2a2 GM Combine to eliminate density: Á1 - 2 ˜ v = - 2 Ë v ¯ dr r r † GM † RHS=0 at “critical” radius: r = c 2a2 † v2 v2 r 4r Integrate for - ln = 4ln + c + C transcendental soln: 2 2 a a rc r † => Transonic soln: sonic radius C = -3 v(rc ) ≡ a rc = rs

NOTE: since density† scales out, isothermal model can’t predict Mass Loss Rate!

† Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 19 † † Solution topology for isothermal wind

=-3 C

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 20 Temperature sensitivity of relative mass loss

Ê a2 ˆ dv 2a2 GM Á1 - 2 ˜ v = - 2 Ë v ¯ dr r r

Subsonic region r < rs still nearly hydrostatic: † R Ê R ˆ r Á -1˜ R r( s) H Ë rs ¯ 2- ª e H ª 7.4 e-14 /T6 ro = e

For fixed ro, relative mass loss is very sensitive to temperature:

˙ 2 -5/ 2 -14 /T6 e.g., ˙ ˙ † M ~ rsars ~ PoT6 e M (26 ) =194 M (16 )! † †

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 21 † † Mass loss sensitive to temperature

M˙ ˙ M (T6 =1)

†

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 22

† Polytropic Solar Wind Momentum and Polytropic Equations:

dv GM 1 dP -5/ 3 v = - - P ~ r dr r2 r dr Energy integral: 2 2 v GM ga 2 P E = - + a ≡ 2 r g†- 1 r † Scaled at the critical (sonic) radius rc: w 2 1 ga2 e = - + † g -1 † 2 x 2(g -1)(wx 2 ) where: GM r v E 5 - 3g v2 = ga2 = x ≡ w ≡ e ≡ = 2rc rc vc GM/rc 4(g -1) † Note: for g=3/2, w=1 is a solution! † † † †

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 23 Polytropic wind solution topology for g=1.1

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 24 Critical solution for various polytropic indices g=1.01-1.6

v/vc

g=1.01

g=1.6

1.5

=1.6 1.4 g

1.3 1.2 1.1 g=1.01

r/rc

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 25 Coronal Heating with a Conductive Thermostat

Recall for conductive energy balance: ˙ E C dT E˙ 4 R2F 4 r 2T 5 / 2 E˙ o = p E = p = c ˙ dr E o † 2 / 7 † † Ê 7 F r - Rˆ † T = Á E d ˜ Ë 2 Ko†r d /R ¯

2 / 7 5 2 e.g. at rd=2R: T ª 26 F5 F5 ≡ FE /10 erg/cm /s † 2 / 7 or, for protons with Ko=Kop : T ª 56 F5 † †

Oleron school Oct. 04 † Stellar Wind Mechanisms - Lec. #1 26 Coronal Heating with a Solar Wind Thermostat

Now consider the case with an added energy loss Esw due to the solar wind.

E˙ 2 The energy balance is: SW Ê ˆ ˙ ˙ V• GM E sw = M Á + ˜ ˙ ˙ ˙ ˙ Ë 2 R ¯ E o = E c + E sw E C 2GM ª M˙ † ˙ R E o † where, from T.R. models, f c -5/ 2 -14 /T P ª † Recall: M˙ ~ P T e 6† one finds (with U =14 km/s) : o o T UT † So: ˙ E sw 3 † E˙ a È 2GM˘ 9-14 /T6 ˙ sw -14 /T6 e E c † ª 2 e ª E˙ 16U ÎÍ Ra ˚˙ 5 / 2 † c T T6

˙ ˙ Note E sw dominates E c for 2 <†T6 < 30 T † † 6 Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 27

† † Wind vs. Conductive Energy Loss

9-14 /T E˙ a È 2GM˘3 e 6 sw ª e-14 /T6 ª ˙ Í 2 ˙ 5 / 2 E c 16UT Î Ra ˚ T6 ˙ E sw E˙ † c †

†

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 28

† Heating in subsonic vs. supersonic region

˙ E SW

E˙ ↑ﬁ V ↑ ; DM˙ = 0 †ext •

† ˙ ˙ E o ↑ﬁ T ↑ﬁ M ↑↑

E˙ E˙ RS o C RS †

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 29 † † † † Magnetic Effects on Solar Coronal Expansion

X-ray Corona from SOHO 1991 Solar Eclipse

Coronal streamers

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 30 Pneuman and Kopp (1971) Iterative scheme Fully dynamic, time dependent MHD model for base dipole MHD Simulation

with Bo=1 G

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 31 Latitudinal variation of solar wind speed

Coronal streamers

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 32 Wave Heating of Corona

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 33 SOHO observations

Tion >> Tp > Te

100 MK 4 MK 2 MK

† T m ion > ion Tp mp

T^ > T|| † vion > vp †

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 34 † Cyclotron Resonance Heating

Illustration of charged particles spiraling around lines of magnetic force, surfing on the wave oscillations, and spinning up while damping out the waves. (Image credit: NASA/Dana Berry, Allied Signal Max Q digital animation group)

http://cfa-www.harvard.edu/~scranmer/APS2000/aps2000.html

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 35 Summary for Solar Wind • Mechanical heating – thermal instability makes corona hot, T> 106 K – also keeps it hot against expansion cooling • Wind driven by – gas pressure gradient – also perhaps wave pressure in high-speed polar wind • Mass loss very sensitive to temperature – solar wind acts like a coronal thermostat (in open field regions) • Subsonic vs. supersonic heating/mom. addition – subsonic increases Mdot – supersonic increases vinf

• Tp>Te – conduction not as important as classically thought – ion-specific heating

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 36 Mechanism for Solar Wind

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 37 Stability of Solar Wind

• Parker 1965 linear stability analysis – showed transonic point stable • But globally stability depends on BC – convection in solar atmosphere => seeds lot of structure • Magnetic fields very complex – reconnection induces lots of variability – e.g. Coronal Mass Ejections • Both Corona and solar wind highly variable

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 38 Magnetic Loops on Solar Limb

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 39 Solar Flare

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 40 Coronal Mass Ejections

Oleron school Oct. 04 Stellar Wind Mechanisms - Lec. #1 41