Lecture 17: The solar wind o Topics to be covered: o Solar wind o Inteplanetary magnetic field The solar wind o Biermann (1951) noticed that many comets showed excess ionization and abrupt changes in the outflow of material in their tails - is this due to a solar wind? o Assumed comet orbit perpendicular to line-of-sight (vperp) and tail at angle ! => tan! = vperp/vr o From observations, tan ! ~ 0.074 o But vperp is a projection of vorbit -1 => vperp = vorbit sin ! ~ 33 km s -1 o From 600 comets, vr ~ 450 km s . o See Uni. New Hampshire course (Physics 954) for further details: http://www-ssg.sr.unh.edu/Physics954/Syllabus.html The solar wind o STEREO satellite image sequences of comet tail buffeting and disconnection. Parker’s solar wind o Parker (1958) assumed that the outflow from the Sun is steady, spherically symmetric and isothermal. o As PSun>>PISM => must drive a flow. o Chapman (1957) considered corona to be in hydrostatic equibrium: dP = −ρg dr dP GMS ρ + 2 = 0 Eqn. 1 dr r o If first term >> than second €=> produces an outflow: € dP GM ρ dv + S + ρ = 0 Eqn. 2 dr r2 dt o This is the equation for a steadily expanding solar/stellar wind. € Parker’s solar wind (cont.) dv dv dr dv dP GM ρ dv o As, = = v => + S + ρv = 0 dt dr dt dr dr r2 dr dv 1 dP GM or v + + S = 0 Eqn. 3 dr ρ dr r2 € € o Called the momentum equation. € o Eqn. 3 describes acceleration (1st term) of the gas due to a pressure gradient (2nd term) and gravity (3rd term). Need Eqn. 3 in terms of v. o Assuming a perfect gas, P = R ! T / ! (R is gas constant; ! is mean atomic weight), the 2nd term of Eqn. 3 is: dP Rρ dT RT dρ = + dr µ dr µ dr Isothermal wind => dT/dr 0 1 dP $ RT ' 1 dρ ! ⇒ = & ) Eqn. 4 ρ dr % µ ( ρ dr € Parker’s solar wind (cont.) o Now, the mass loss rate is assumed to be constant, so the Equation of Mass Conservation is: dM = 4πr2ρv = const ⇒ r2ρv = const Eqn. 5 dt d(r2ρv) dv dr2 dρ o Differentiating, = r2ρ + ρv + r2v = 0 dr dr dr dr 1 dρ 1 dv 2 € => = − − Eqn. 6 ρ dr v dr r o Substituting Eqn. 6€ into Eqn. 4, and into the 2nd term of Eqn. 3, we get dv RT # 1 dv 2& GM € v + % − − ( + S = 0 dr µ $ v dr r' r2 # & RT dv 2RT GMS ⇒ % v − ( − + 2 = 0 $ µv ' dr µr r 2RT GM o A critical point occurs when dv/dr ! 0 i.e., when = S µr r2 € 2 o Setting vc = RT /µ => rc = GMS /2vc € € Parker’s solar wind (cont.) 1 dv v 2 o Rearranging => v 2 − v 2 = 2 c (r − r ) Eqn. 7 ( c ) v dr r2 c o Gives the momentum equation in terms of the flow velocity. € o If r = rc, dv/dr -> 0 or v = vc, and if v = vc, dv/dr -> ∞ or r = rc. o An acceptable solution is when r = rc and v = vc (critical point). o A solution to Eqn. 7 can be found by direct integration: 2 2 " v % " v % " r % r Eqn. 8 $ ' − ln$ ' = 4ln$ ' + 4 c + C Parker’s “Solar # vc & # vc & # rc & r Wind Solutions” where C is a constant of integration. Leads to five solutions depending on C. € Parker’s solutions o Solution I and II are double valued. Solution II also doesn’t connect to the solar surface. v/vc o Solution III is too large (supersonic) close to the Sun - not observed. Critical point o Solution IV is called the solar breeze solution. o Solution V is the solar wind solution (confirmed r/r in 1960 by Mariner II). It c passes through the critical point at r = rc and v = vc. Parker’s solutions (cont.) o Look at Solutions IV and V in more detail. o Solution IV: For large r, v! 0 and Eqn. 8 reduces to: 2 2 # v & # r & v # r & −ln% ( ≈ 4ln% ( ⇒ = % ( $ vc ' $ rc ' vc $ rc ' 2 2 1 o Therefore, r v ! rc vc = const or v ≈ r2 € const const o From Eqn. 5: ρ = 2 = 2 = const r v rc vc € o From Ideal Gas Law: P∞ = R ! ∞ T / ! => P∞ = const € o The solar breeze solution results in high density and pressure at large r =>unphysical solution. Parker’s solutions (cont.) o Solution V: From the figure, v >> vc for large r. Eqn. 8 can be written: 2 " v % " r % " r % $ ' ≈ 4ln$ ' ⇒ v ≈ vc 2 ln$ ' # vc & # rc & # rc & const const o The density is then: ρ = ≈ r2v r2 ln(r /r ) € c => ! ! 0 as r ! ∞. o As plasma is isothermal€ (i.e., T = const.), Ideal Gas Law => P ! 0 as r !∞. o This solution eventually matches interstellar gas properties => physically realistic model. o Solution V is called the solar wind solution. Observed solar wind o Fast solar wind (>500 km s-1) comes from coronal holes. o Slow solar wind (<500 km s-1) comes from closed magnetic field areas. o Figure from McComas et al., Geophysical Research Letters, (2008). Interplanetary magnetic field B! or v! ! Br or vr o Solar rotation drags magnetic field into an Archimedian spiral (r = a!). o Predicted by Eugene Parker => Parker B Spiral: r r - r0 = -(v/!)(! - !0) o Winding angle: Bφ vφ (r , ! ) tanψ = = 0 0 Br vr Ω(r − r ) ! = 0 vr o Inclined at ~45º at 1 AU ~90º by 10 AU. € Alfven radius o Close to the Sun, the solar wind is too weak to modify structure of magnetic field: B2 1/2ρv 2 << 8π o Solar magnetic field therefore forces the solar wind to co-rotate with the Sun. o When the solar wind becomes super-Alfvenic B2 € 1/2ρv 2 >> 8π o This typically occurs at ~50 Rsun (0.25 AU). B2 o Transition between regimes occurs at the Alfven radius (r ), where 1/2ρv 2 = € A 8π M o Assuming the Sun’s field to be a dipole, B = 3 r € 1/ 6 $ M 2 ' => rA = & ) 4πρv 2 € % ( € The Parker spiral Heliosphere .