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1 Stellar Winds and Magnetic Fields 1 Stellar winds and magnetic fields by Viggo Hansteen The solar wind is responsible for maintaining the heliosphere, and for being the driving agent in the magnetospheres of the planets but also for being the agent by which the Sun has shed angular momentum during the eons since its formation. We can assume that other cool stars with active corona have similar winds that play the same roles for those stars (1). The decade since the launch of SOHO has seen a large change in our understanding of the themrally driven winds such as the solar wind, due to both theoretical, computational, and observational advances. Recent solar wind models are characterized by low coronal electron temperatures while proton, α-particle, and minor ion temperatures are expected to be quite high and perhaps anisotropic. This entails that the electric field is relatively unimportant and that one quite naturally is able to obtain solar wind outflow that has both a high asymptotic flow speed and maintaining a low mass flux. In this chapter we will explain why these changes have come about and outline the questions now facing thermal wind astrophysicists. The progress we have seen in the last decade is largely due observations made with instruments onboard Ulysses (2) and SOHO (3). These observations have spawned a new understanding of solar wind energetics, and the consideration of the chromosphere, corona, and solar wind as a unified system. We will begin by giving our own, highly biased, \pocket history" of solar wind theory highlighting the problems that had to be resolved in order to make the original Parker formulation of thermally driven winds conform with observational results. Central to this discussion are questions of how the wind's asymptotic flow speed and mass flux are set, but we will also touch on higher order moments such as the ion and electron temperatures and heat fluxes as 1 2 Stellar winds and magnetic fields well as the possible role of Alfv´en waves and particle effects in driving the solar wind outflow. 1.1 A pocket history There are many ways to start a history of the solar wind, but being Nor- wegian it seems natural to begin with Kristian BirkelandOn the basis of laboratory studies and observations from stations in the arctic, Birkeland concluded that the Sun continuously emits electric corpuscular radiation (4; 5). The laboratory studies were performed using his \Terella" designed to study the interaction of electrons from the Sun with the Earths magnetic field where cathode rays were impinging on a magnetized sphere. But also emission of particles from the Sun was studied, by using the magnetized sphere as a cathode. Birkeland became so convinced of the correctness of this view that he assumed that all stars were emitting electrons and ions and that interstellar space was filled with \electrons and flying ions of all kinds". Unfortunately the work carried out by Birkeland went largely forgotten and it was first some 50 years later that the concept of a solar wind became universally known and accepted. One could also begin such a history with Chapman and Ferraro's studies through the 1930's (6) where they pointed out that magnetic storms could be explained by clouds of ions ejected from the Sun or with Forbush's discovery (7) that the intensity of cosmic rays was lower when solar activity was high. The concept of ionized gas outflow from the Sun was, however, first firmly established through the work of Biermann (8, 9) where it was pointed out that light pressure from the Sun was not sufficient to account for the speed at which gases are blown away from the heads of the comets. The other vital ingredient in the thermal solar wind | a hot, extended solar corona | had by the 1950's been established by the observational interpreta- tive work of Grotrian (10, 11) and Edl´en (12) on emission lines such as Fe x and Fe xiv and the theoretical work of Alfv´en (13, 14), Biermann (15, 16), and Chapman (17, 18). Much could and should be said about the fact that the Sun is surrounded by a corona of gas of roughly 1 MK temperature as discovered by these authors. Here we will in the first instance simply note that a temperature of some millions of degrees implies that the sound speed cs | essentially the mean ion speed | is much smaller that the Sun's escape speed vg; cs ≈ kT=m ≈ 100 km=s vg = 2GMS=RS = 618 km=s (1.1) q q where k is Boltzmann's constant, T the coronal temperature, m the mean 1.1 A pocket history 3 particle mass, G the universal gravitational constant, and MS and RS the solar mass and radius, respectively. Mass and momentum balance radially away from the Sun along a flow tube with a cross section A at heliocentric distance r can be written d (ρvA) = 0 (1.2) dr dv dp GM ρv = − − ρ S (1.3) dr dr r2 with ρ the mass density, v the flow speed. Then p = 2nkT is the gas pressure in an electron { proton plasma with n representing the electron or proton number density such that ρ = mn where m is the mean particle mass which is given by m ≈ mp=2 for an electron { proton plasma. Alfv´en (13) and Chapman (17) showed that the consequence of the thermal conduction in a million degree corona is to extend the corona; ie the tempera- ture falls off slowly with distance from the Sun. Thus, in a hypothetical static atmosphere, we find a pressure at infinity given by d GM (p) = −nm S (1.4) dr r2 r GMSm dr p(r) = p0 exp − (1.5) 2k r2T (r) ZRS (1.6) Thus, if the temperature falls less rapidly than (1=r), we find that limr!1 p(r) > 0, we expect a non-vanishing pressure at infinity when the corona is extended. In particular, we find that for reasonable temperatures and densities n0, T0 at the `coronal base' this pressure is much larger than any conceivable interstellar pressure. Using this argument Parker (19) deduced that the solar wind must expand supersonically into interstellar space and solved the equations of the solar wind under the assumption of a spherically symmetric, single fluid, isothermal out- flow. In this case the equations of mass and momentum conservation (1.3,1.3) can be to give 1 dv 2 4kT GMS v − 2kT=mp = − 2 (1.7) v dr (mpr r ) n o Parker showed that the trans-sonic passes through a critical point rc mpGMS 2kT rc = where vc = ; (1.8) 4kT s mp and that such a flow could be made to match any pressure as r ! 1. Let us examine this trans-sonic wind solution in somewhat greater detail. 4 Stellar winds and magnetic fields If we integrate the force balance, Eq 1.3, from the coronal base to the critical point rc we find a density ρc at the critical point given by mpGMS 3 ρc = ρ(rc) = ρ0 exp − + : (1.9) 2kT RS 2 Note that this density is almost exactly the same as if there had been no solar wind flow: The subsonic corona in the solar wind is essentially a static atmosphere. We can also find the resultant mass flux for the wind by examining the density and the velocity at critical point. 2 rc (nv) = ncvc r r2 C / ρ T −3=2 exp − (1.10) 0 T where ρ0 is the density at the coronal base. The mass flux is proportional to the density at the coronal base and depends exponentially on the coronal temperature. For various reasons, Parker's solution had difficulties in finding general ac- ceptance, but debate quickly stilled after the predictions of a supersonic out- flow were confirmed by Mariner ii spacecraft observations analysed by Neuge- bauer & Snyder (20). 1.2 The Parker spiral The wind also carries with it the solar magnetic field. Let us briefly consider the outflow of ionized, magnetized gas from a rotating star with a monopole magnetic field. The salient aspects of such a flow are found even when only considering the equatorialplane and restricting attention to solutions where all variables are functions of r only. We make use of polar co-ordinates. We find that the equation of mass conservation then can be written 1 @ ρv r2 = 0; (1.11) r2 @r r while the φ component of the momentum equation is given by @v v 1 @B B ρ v φ + v r = B φ + B r (1.12) r @r φ r µ r @r φ r 0 or 1 d 1 1 d ρvr (rvφ) = Br (rBφ) (1.13) r dr µ0 r dr 2 Mass conservation imples that ρvrr is constant, while a divergence free field 1.3 (Some) solar wind properties 5 2 3 requires that Brr is constant. Multiplying Eq. (1.13 with r we see that 2 Brr 1 rvφ − 2 rBφ = constant = L: (1.14) ρvrr mu0 Under the assumptions above, the induction equation is 1 d (r(v B − v qB )) = 0 (1.15) r dr r φ φ r For a star rotating with angular velocity Ω, radius Rs and with a monopole field Bφs ≈ 0 we find that the induction equation implies 2 r(vrBφ − vφBr) = constant ≈ −Rs(RsΩ)Brs = −Ωr Br (1.16) 2 We can now solve Eq. (1.14) and Eq. (1.16) for vφ and Bφ and using MA = 2 2 2 2 (vr =vA) with vA = (Br /µ0ρ) we find 2 2 MA(L=r Ω) − 1 vφ = Ωr 2 (1.17) MA − 1 and 2 BrΩr 1 − (L=r Ω) 2 Bφ = − 2 MA: (1.18) vr MA − 1 ! 2 2 Both expressions show that we must have 1 − (L=r Ω) = 0 when MA − 1 = 0.
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