THE GLOBAL SOLAR WIND BETWEEN 1 AU AND THE TERMINATION SHOCK

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1 The Astrophysical Journal, 713:71 73, 1 April C 1. The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi:1.188/4-637x/713//71 TE GLOBAL SOLAR WIND BETWEEN 1 AU AND TE TERMINATION SOCK Y. C. Whang Catholic University of America, Washington, DC 64, USA; whang@cua.edu Received 9 April 3; accepted 1 February ; published 1 March ABSTRACT This paper studies the global solar wind between 1 AU and the termination shock, taking into account the highly latitude dependent solar wind at the 1 AU boundary and the ionization of interstellar neutral hydrogen. A system of two-fluid magnetohydrodynamics equations governs the flow of solar wind plasma; the conditions of the solar wind at 1 AU are generated from Ulysses data for the relatively stable state of the wind during the declining phase and solar minimum of the solar cycle. A one-fluid model is used to describe the flow of neutral hydrogen and its condition in the far-field boundary outside the termination shock. Simultaneous solutions are obtained using a high-resolution computational code for the two equation systems coupled by the ionization of neutral hydrogen. The ionization process leads to removal of neutral hydrogen in the heliosphere: a hydrogen cavity forms inside 4 AU; the cavity extends on the downwind side to form a long cavity wake. Solutions show how the ionization process and the wind condition at 1 AU boundary affect the spatial variation of the wind speed, temperature, pickup proton, fast Mach number, and plasma β-ratio. The wind properties inside 4 AU are axisymmetrical about the solar rotation axis; this axi-symmetry totally disappears in the outer heliosphere. All wind properties are substantially modified at an increasing heliocentric distance on the upwind side to generate an upwind downwind asymmetry dictated by the direction of the relative motion between the Sun and the interstellar medium. Key words: magnetic fields magnetohydrodynamics (MD) plasmas solar wind 1. INTRODUCTION This paper studies the global solar wind between 1 AU and the termination shock taking into account the highly latitude dependent solar wind at the 1 AU boundary and the ionization of interstellar neutral hydrogen. We use magnetohydrodynamics (MD) equations to study the flow of solar wind plasma and use fluid equations to study the flow of interstellar neutral hydrogen. The two equation systems are coupled by the ionization of neutral hydrogen to produce interstellar pickup protons. The boundary conditions at 1 AU are generated from Ulysses solar wind data when the spacecraft cruised through the relatively stable recurrent structure of the solar wind during the declining phase and solar minimum of the solar cycle. As the interstellar neutral hydrogen penetrates into the heliosphere, the neutral hydrogen may become ionized by photoionization or by a charge exchange process to produce pickup protons. We include the charge exchange process between neutral hydrogen with solar wind protons (proton of solar origin) and the process between neutral hydrogen with interstellar pickup protons (proton of interstellar origin). Solar wind research deals with problems varying over a wide range of scale sizes. In this paper, we study the global solar wind between 1 AU and the termination shock. Closely related research is about the interaction of the interstellar medium with the heliosphere. The scale sizes of the two problems differ by an order of magnitude. Research on the global heliosphere studies the distortion of the interstellar medium outside the heliopause, the flow inside the heliosphere, and associated interstellar bow shock, heliopause, and termination shock (Baranov & Malama 1993; Pauls et al. 1995; Pauls & Zank 1997; Linde et al. 1998; Izmodenov et al. 1999; Mueller et al. ; Fahr et al. ; Malama et al. 6; Pogorelov et al. 6; Labun & Muller 7). Each model study uses a simple far-field outer boundary condition to carry out numerical solution. If the flow speed of the interstellar plasma is greater than the magnetosonic speed, the interstellar heliosphere interaction is a strong one; in this case an interstellar bow shock and a hydrogen wall (a layer of enhanced hydrogen density) would form upstream of the heliosphere. Linsky & Wood (1996) have reported the detection of a hydrogen wall. Associated with the formation of a hydrogen wall, the neutral hydrogen population may contain an important component of warm, decelerated hydrogen in addition to the one with the characteristics of the pristine interstellar hydrogen. owever, since the strength and direction of the interstellar magnetic field are poorly known, we do not have adequate knowledge about the magnetosonic speed of the interstellar plasma. If the flow speed is less than the magnetosonic speed, no bow shock could form upstream of the heliopause. In the absence of bow shock, the hydrogen wall is a weak one (Zank & Muller 3). In this paper, we use a one-fluid model to describe the flow of neutral hydrogen and its condition in the far-field boundary outside the termination shock. Under this assumption, the possible effects due to the warm component of the interstellar hydrogen could be underestimated for the case of strong interstellar heliosphere interaction. The computational codes used for the two problems of different scale sizes have very different resolutions. In this paper, we study the spatial variations of the global solar wind, including latitudinal variations, with the inner boundary at R min = 1 AU. The grid spacing is logarithmic in r, weuse Δr/r =., and we use angle grid spacing Δϕ =1. The highresolution computational code can catch a rapid change of the solar wind properties. The radial grid spacing used in studying the interaction of the interstellar medium with the heliosphere is typically greater than ours by a factor of 1, and they chose the angular grid spacing at 3 or 5. Some of their solutions chose the inner boundary at R min = 1 AU or R min = 3 AU. We calculate the spatial variation of the global solar wind. The ionization process leads to removal of neutral hydrogen in the heliosphere: a hydrogen cavity forms inside 4 AU; the cavity extends on the downwind side to form a long cavity wake. Pickup protons are produced outside the hydrogen 71

2 7 WANG Vol. 713 cavity and cavity wake; they have two principal effects on the solar wind: deceleration and heating. These effects are absent in the cavity and the long cavity wake. As a result, all solar wind properties are substantially modified to generate an upwind downwind asymmetry dictated by the direction of the relative motion between the Sun and the interstellar medium. The wind properties inside 4 AU are axisymmetrical about the solar rotation axis; this axisymmetry totally disappears in the outer heliosphere. This study uses a heliocentric coordinate system. In heliocentric Cartesian coordinates (x, y, z), the x-axis is antiparallel to the upwind direction, and the xz-plane contains the axis of solar rotation. The upwind direction is parallel to the Sun s motion relative to the interstellar medium. This model assumes that all solutions are symmetrical about the xy- and xz-planes.. GOVERNING EQUATIONS.1. Ionization of Interstellar Neutral ydrogen The interstellar medium is a partially ionized gas between the stars. Interstellar neutral hydrogen can penetrate into the heliosphere and may become ionized by photoionization or by the charge exchange process. The newly created protons are immediately picked up by the solar wind electromagnetic fields and move outward as a component of the solar wind. In the solar wind frame of reference, interstellar neutral hydrogen initially moves toward the Sun with a speed of the order of the bulk speed of the solar wind V. When a newly ionized particle is picked up by the solar wind electromagnetic fields, the Lorentz force changes its motion into a gyration about the interplanetary magnetic field with a large pitch angle. The pickup protons have a thermal speed of the order of the bulk speed V that is much greater than the thermal speed of the solar wind protons. The interstellar pickup protons do not thermally assimilate into the population of the solar wind protons quickly; the two kinds of protons behave like two distinguished proton species (Isenberg, 1986). The two species have same bulk velocity V, but they have different temperatures. We use the subscripts, I, S, and e to denote the properties of interstellar neutral hydrogen, interstellar pickup protons, solar wind protons, and electrons, respectively, and use the subscript to denote conditions at 1 AU. The ionization rate of interstellar neutral hydrogen by photoionization is ( r ) q ph = ν N r. (1) ere r denotes the heliocentric distance, and ν = s 1 (Möbius 1993). The charge exchange process may take place between interstellar neutral hydrogen with solar wind protons and with interstellar pickup protons (Isenberg 1986; Fahr et al. ; Malama et al. 6). The two ionization rates are q ex1 = σ (N S V ) N, () and q ex = σ (N I V ) N. (3) ere σ is the mean cross section for charge exchange between proton and neutral hydrogen. The experimentally measured cross section for charge exchange is a decreasing function of the proton energy σ = ( log E) cm (Fite et al. 196). ere the proton energy E(in ev) = [V (in cm s 1 )/1.384] 1 1. The continuity equation that tracks the removal of interstellar neutral hydrogen by the photoionization and charge exchange processes is (N V ) = (q ph + q ex1 + q ex ). (4) Photoionization accounts for 1% of the total ionization rate. The charge exchange of neutral hydrogen with solar wind protons contributes to a loss of number flux for the solar wind proton; we can write the continuity equation for the solar wind proton as (N S V) = q ex1. (5) Pickup protons are produced by the ionization process; the continuity equation for pickup protons is (N I V) = q ph + q ex1. (6) The term q ex does not appear in the continuity and the energy equations of pickup protons because during the charge exchange between neutral hydrogen and pickup protons a new pickup proton is produced replacing the old pickup proton. Neutral hydrogen created through the charge exchange processes continues to move outward at the solar wind velocity V; its number density is very small compared with the number density of low-speed interstellar neutral hydrogen. In Equation (4), the number fluxes N S V and N I V, the charge exchange cross section, and the hydrogen velocity V are variables. In this study, we calculate these variables from the simultaneous solution of the governing equations. This equation has a well-known solution under restrictive assumptions: (1) the number flux of solar wind proton is proportional to r,() q ex is neglected, and (3) V is a constant vector along the x-direction. From Equation (4), one can obtain 1 N N x = λ x + r sin ϕ ere ϕ is the heliocentric polar angle measured from +x-axis, and λ = r (ν + σn S V ) / V is a length scale, known as the ionization radius; it is of the order of 4 5 AU. Because r sin ϕ is a constant along each streamline of the neutral hydrogen flow, this equation can be integrated to recover the well-known solution (Axford 197) N N { } λ (π ϕ) = exp. r sin ϕ This result is only illustrative; it is not going to be used in the remainder of the paper. This model covers three ionization processes for the interstellar neutral hydrogen: charge exchange between hydrogen and solar wind protons, charge exchange between hydrogen and pickup protons, and photoionization. Various source terms are formulated under a simplifying approximation that we only keep the zeroth-order terms and drop terms in the order of V /V S or smaller. For the charge exchange process, the source terms for the momentum equation and the energy equation, including high-order precisions, have been extensively investigated by olzer (197), Pauls et al. (1995), and McNutt et al. (1998)... ydrogen Flow in the eliosphere The flow of neutral hydrogen can be studied using the microscopic kinetic theory model or the macroscopic fluid model. The typical kinetic theory approach is to find the solution

3 No., 1 TE GLOBAL SOLAR WIND BETWEEN 1 AU AND TE TERMINATION SOCK 73 of the distribution function f from the Boltzmann equation in the first step and then calculate the moments of f over the velocity space to obtain the solution for fluid properties: the number density, flow velocity, and temperature (Fahr 1979; Fahr & Ripken1984; Lallement et al. 1985; Osterbart & Fahr 199; Ripken & Fahr 1983; Whang 1998; Wu & Judge 1979). The fluid equations are moments of the Boltzmann equation over the velocity space; thus, the alternative fluid approach solves the moment equations to obtain a solution for fluid properties directly. A comparison has been made between the iso-density contours of the fluid solution (Whang 1996) and the kinetic theory solution of Lallement et al. (1985); the solutions obtained from these two approaches show good agreement. We study the variation of the number density N, the bulk velocity V, and the temperature T of interstellar hydrogen in the heliosphere. The system of fluid equations for the hydrogen flow consists of the continuity Equation (4), the equation of motion GM ρ V V = ρ (1 μ ) P (7) r and the energy equation N V ln P = N 5/3 3 (q ph + q ex1 + q ex ). (8) ere the pressure P = N kt, G is the gravitational constant, M is the solar mass, and μ is the ratio of the repulsion force by solar radiation to the gravitational attraction force. μ varies with solar activity. Equation (8) means that because of the ionization process in the heliosphere, the flow of interstellar hydrogen is not isentropic. Equations (4), (7), and (8) govern the variation of N, V, and P in the heliosphere. We study the solution on the xyand xz-planes for the supersonic flow of interstellar hydrogen under the assumption that μ = 1, which means that the gravitational attraction exactly balances the repulsion by solar radiation pressure on a hydrogen atom. Equations (4), (7), and (8) are the system of equations that governs the neutral hydrogen flow; let us call it System A. The system of governing equations can be written in another form; let us call it System B. System A can be reduced to System B and vise versa; see Appendix A for details. System B consists of Equations (4), (A1), and (A5). Equation (A1) is the momentum equation and (A5) is the equation of energy conservation; they include source terms. Depending on the numerical method employed to carry out the calculation, one can choose to use System A or B. On the solution plane the angle between the streamline and the x-axis is the directional angle θ ; the bulk velocity of hydrogen V may be represented by its magnitude V and the directional angle. Let e s denote the unit vector along the streamline direction and e n the unit vector normal to the streamline direction. The unit normal vector makes an angle of θ +9 with the x-axis. The component of the equation of motion in the normal direction is e n P + ρ V e s θ =. (9) Making use of Equation (8) and the component of the equation of motion along the streamline direction, we can obtain e s ( V + 5P ) ρ = kt ρ V (q ph + q ex1 + q ex ). (1) Making use of the equation of motion, we can write the continuity equation as cot α ρ V e s P + e n θ = 3m (q ph + q ex1 + q ex ). 5ρ V (11) ere α = sin ( / ) 1 5kT 3mV is the Mach angle, and m is the proton mass. The governing equations are a system of partial differential equations of hyperbolic type; it can be studied using the method of characteristics (Whang 1996). If we multiply Equation (9)by cos α, multiply Equation (1)by± sin α, and add the results, we obtain { c ± θ ± cot α } ρ V P = h ±, (1) where c ± = cos α e s ± sin α e n, and h ± = 3m (q ph + q ex1 +q ex )sinα. 5ρ V The two unit vectors c ±, respectively, make an angle of θ ± α with the x-direction. Equation (1) is known as the characteristics equation. Equation (1) can be numerically integrated first along the two characteristic curves to calculate the variation of θ and P ; Equations (8) and (1) can then be integrated along the streamline to calculate the variation of ρ and V..3. MD Equations This model treats the interstellar pickup protons and the solar wind protons as two distinguished proton species; they have different temperatures, but the same bulk velocity V (Whang et al. 1995). The equation of motion is GM ρv V + P ρ 1 r 4 π ( B) B = (q ph + q ex1 + q ex )mv. (13) ere P = P I + P S + P e is the pressure for the mixture of three plasma species (pickup proton, solar wind proton, and electron), ρ = m(n I + N S ) is the plasma density; Gaussian units are used throughout. The variation of the magnetic field is governed by the divergence-free condition and Faraday s law of induction: B =, (14) (V B) =. (15) In order to account for the important effects of pickup protons on the global solar wind, this model includes separate continuity and energy equations for pickup protons in the system of governing equations. The two equations, respectively, track the production of the pickup proton and its contribution to the thermal state of the solar wind. During the photoionization and charge exchange of neutral hydrogen with the solar wind proton, the solar wind plasma gains energy of approximately mv / brought in by the newborn pickup proton. The energy

4 Wind Speed 74 WANG Vol. 713 equation for the pickup proton is obtained from the second moment equation of the Boltzmann equation (see Appendix B) 3 P IV + P I V = (q ph + q ex1 ) mv. (16) This equation is chiefly responsible for the thermal state of the solar wind at large heliocentric distances. We assume that P e = P S and use the polytropic law to represent the energy equation for the solar wind proton PS V =, (17) N κ S where κ = 1.8 is the polytropic index. We have derived the polytropic index and developed the method of solution for the system of MD equations (Whang 1998). This MD model that treats the solar wind proton and the pickup proton as two distinguished proton species has also been used to study the solar wind problems by Wang & Richardson (1) and Usmanov & Goldstein (6). Previously, we have studied separately the solution for the system of governing equations of the interstellar neutral hydrogen (Whang 1996), and the solution for the system of the MD equations of the solar wind (Whang 1998). These studies have not included the source term due to charge exchange between hydrogen and interstellar pickup. In this paper, we take into account the additional source term and study the simultaneous solution of the two equation systems. 3. BOUNDARY CONDITIONS In a recent paper, we reported that the out-of-ecliptic solar wind has a recurrent stable structure spanning more than 5 years in each solar cycle during the declining phase and solar minimum (Whang et al. ). The relatively stable solar wind is highly latitude dependent at 1 AU. The average wind speed and temperature are much higher at high latitudes than that near the ecliptic; the average proton number density, the number flux of solar wind protons, and the transverse magnetic fields are much lower at high latitude than that near the ecliptic; solar wind fluctuations are much larger at low latitudes. In this paper, we study the global solar wind using a1au input function of the relatively stable state solar wind generated from Ulysses data. From 199. through 1999., the spacecraft cruised through the recurrent stable structure of the solar wind during the declining phase and minimum of the solar cycle (Goldstein et al. 1996; McComas et al., 3; Phillips et al. 1996; Smith et al. 3). From 3. to.9, Ulysses again observed the relatively stable structure of the solar wind in the declining phase of Cycle 3. Figure 1 shows plots of the number flux and the wind speed of the solar wind proton at 1 AU as functions of heliolatitude ω. The open circles are rotational averages generated from Ulysses data in and The solid lines that best fit the data points may be represented by N S V = a + a 4 (cos 4 ω 1), (18) V = b + b 8 (1 cos 8 ω). (19) ere a = cm s 1, a 4 = cm s 1, b = 415 km s 1, b 8 = 35 km s 1 ; a and b are, respectively, the long-term average proton number flux and wind speed of Proton Number Flux x1 8 5x1 8 4x1 8 3x1 8 x1 8 1x Latitude Figure 1. Solar wind speed (in km s 1 ) and proton number flux (in cm s 1 ) at 1 AU boundary. the relatively stable solar wind at r = 1 AU and ω =. From the Ulysses solar wind data we also obtain the best-fit representations for the temperature and the transverse magnetic fields at 1 AU as T = c + c 8 (1 cos 8 ω), () B t = d + d 8 (cos 8 ω 1), (1) where c = K, c 8 = K, d = 3.84 nt, and d 8 =.7 nt. In data analysis, the solar wind parameters have been normalized to 1 AU values by assuming that N S r, T S r.56, and B t r 1. Equations (18) (1) are used as the boundary condition at 1 AU to study the relatively stable state of the global solar wind. These boundary conditions have been used by Labun & Muller (7) to study three-dimensional simulation of the dynamical heliosphere. Pauls & Zank (1997) have studied the global heliosphere using a similar boundary condition that consists of two components: a low-density, high-speed wind emanates from polar coronal holes extending down to 35 heliolatitude, and bounding a cool, high-density solar wind. The far-field boundary condition for the interstellar hydrogen flow on the upwind side is not as straightforward as the inner boundary condition. We made a simple assumption that at infinity the hydrogen flow is along the downwind direction, N =.1 cm 3, V = km s 1, and T = 1 4 K. We carry out the solution far beyond the termination shock on the upwind side. 4. RESULTS We use an iteration procedure to obtain a simultaneous numerical solution of the governing equations for the relatively stable state of the hydrogen flow and the global solar wind between 1 AU and the termination shock. The solution takes care of the coupling between the flow of interstellar hydrogen and the solar wind through the ionization process.

5 No., 1 TE GLOBAL SOLAR WIND BETWEEN 1 AU AND TE TERMINATION SOCK Figure. Constant contours of the hydrogen density ratio N /N on the xyplane (heavy lines) and xz-plane (dotted lines). A hydrogen cavity forms inside 4 AU; the cavity extends along the +x-axis to form a long hydrogen cavity wake. The calculation assumes that the solar wind velocity V is along the radial direction. Solutions are carried out using the radial grid spacing Δr/r =. and the polar angle grid spacing Δϕ = 1. Solutions show how the ionization process and the wind condition at the 1 AU boundary affect the hydrogen flow and all parameters of the solar wind (wind speed, temperature, pickup proton, fast Mach number, and plasma β-ratio) ydrogen Cavity and Cavity Wake Figure shows the constant contour plots for N /N.The heavy lines are the iso-density contours on the xy-plane, and the thin lines are the iso-density contours on the xz-plane. The distribution of neutral hydrogen density is not axisymmetric about the upwind downwind line (the x-axis); this is caused by the latitudinal variation of the wind speed and the proton number flux at the 1 AU boundary. On the upwind side, N changes slowly outside a hemisphere at a radius of about 6 AU and N asymptotically approaches N at large r. Closer to the Sun, N changes very rapidly inside 1 AU. The hydrogen density N diminishes sharply inside 4 AU to form a hydrogen cavity. The cavity extends along the +x-axis to form a long hydrogen cavity wake on the downwind side. 4.. Velocity and Temperature of Neutral ydrogen Figure 3 shows the iso-flow speed contours (top), the isotemperature contours (center), and the iso-flow direction angle contours (bottom) on the xy-plane. The loss of interstellar hydrogen due to the ionization process creates a gradient in the number density, and hence a gradient in the pressure of interstellar hydrogen. This pressure gradient produces a driving force to modify the flow of interstellar hydrogen in the heliosphere. The bulk velocity V and the temperature T are slightly perturbed from the upstream conditions. The flow speed increases continuously by 1% and the temperature decreases continuously by 3% in the heliosphere. The travel time of neutral hydrogen in the heliosphere is greater than the solar cycle. A neutral traveling from the termination shock to 1 AU does not only experience the solar wind conditions of the declining phase of the solar cycle, but the solar wind conditions of an entire solar cycle and hence the neutral results presented in this paper are idealized in this sense. Along the ±x-axis, θ =. Outside the x-axis, θ changes up to a few degrees; it is negative throughout the heliosphere T V -.1 θ (degree) Figure 3. Neutral hydrogen flow in the heliosphere. Flow speed (top), temperature (middle), and direction angle (bottom) are slightly perturbed from the upstream condition of V = km s 1, T = 1 4 K, and θ = Solar Wind Proton Number Flux In Figure 4, the solutions for the number flux of solar wind protons are presented as the product of a dimensionless number flux and a geometrical factor F = N SV r. () a ere a is the long-term average proton number flux of the relatively stable solar wind at r = 1 AU and ω =, and r /r is a geometrical factor. The constant contour plots for F show that: (1) outside the cavity and cavity wake, the charge exchange process causes N S V to decrease more rapidly than r. () The ionization process r

6 Z (AU) Z (AU) 76 WANG Vol NUMBER FLUX PROTON OF SOLAR ORIGIN N I /N S Figure 4. Constant contours of solar wind proton flux function, defined in Equation (),on thexy-plane (top panel) and xz-plane (bottom panel). Outside of the hydrogen cavity and cavity wake, the number flux decreases at a rate faster than r. The number flux is significantly latitude dependent, the distribution shows an upwind downwind asymmetry. sharply disappears as the hydrogen density diminishes in the cavity and the cavity wake; the number flux of solar wind proton N S V remains proportional to r. (3) At a given r, the proton number flux is much lower at high heliolatitude than that near the ecliptic caused by the latitudinal decrease of the proton number flux at 1 AU boundary Pickup Proton Figure 5 shows the constant contour plots for the ratio of two proton populations N I /N S. Due to the absence of the neutral hydrogen and ionization process, the density ratio N I /N S approaches zero in the hydrogen cavity and cavity wake. Outside of the cavity and cavity wake, the density ratio N I /N S is an increasing function of the heliocentric distance r; this ratio becomes quite significant at large r. Pickup protons can significantly modify the solar wind. All solar wind properties are substantially modified by the pickup proton effect except in the cavity and cavity wake; as a result, the solar wind generates an upwind downwind asymmetry in the outer heliosphere. The ratio of two charge exchange rates q ex /q ex1 is directly proportional to N I /N S ; thus, the charge exchange of neutral hydrogen with the interstellar pickup proton must not be ignored in calculations of the global solar wind between 1 AU and the termination shock Solar Wind Speed Figure 6 shows that (1) outside the hydrogen cavity and cavity wake the wind speed decreases at the increasing heliocentric distance. () A substantial latitudinal variation in V; the wind Figure 5. Constant contours for the ratio N I /N S on xy-plane (top panel) and xz-plane (bottom panel). speed at 1 AU boundary V is directly responsible for this variation. Pickup protons have a significant effect in influencing the solar wind outside the hydrogen cavity and cavity wake. In the absence of the pickup proton in the cavity, the wind speed is axisymmetrical about the solar rotation axis inside 4 AU. This axisymmetry totally disappears in the outer heliosphere. In the cavity and along the cavity wake on the downwind direction, the wind speed increases from 415 km s 1 at 1 AU to 4 km s 1 at 5 AU, and to 48 km s 1 at 15 AU. On the opposite direction, the solar wind decelerates significantly caused by the pickup proton effects. On a global scale, the wind speed shows the characteristics of an upwind downwind asymmetry. Solutions obtained in this study show that all solar wind parameters (pickup proton number density, the wind speed, the wind temperature, the plasma β-ratio, and the flow Mach number) are substantially modified at large heliocentric distances to generate an upwind downwind asymmetry dictated by the direction of the relative motion between the Sun and the interstellar medium. The pickup proton effect is chiefly responsible for the upwind downwind asymmetry Solar Wind Temperature Figure 7 shows the temperature outside the hydrogen cavity and cavity wake. Note that the temperature of the solar wind plasma is the mean temperature of three plasma species: solar

7 Z (AU) Z (AU) No., 1 TE GLOBAL SOLAR WIND BETWEEN 1 AU AND TE TERMINATION SOCK WIND SPEED E5 4E5 TEMPERATURE 3E E Figure 6. Constant contours for the solar wind speed (in km s 1 )onthexy-plane (top panel) and xz-plane (bottom panel). The ionization process causes radial deceleration of the solar wind in all directions outside the hydrogen cavity. The wind speed increases with heliolatitude dictated by the boundary condition wind proton, pickup proton, and electron: T = N IT I + N S T S + N e T e. N I + N S + N e The pickup proton has a thermal speed of the order of the bulk speed V that is much greater than the thermal speed of the solar wind proton. Thus, the temperature of the pickup proton T I is approximately proportional to V. T I is higher than the temperature of the solar wind proton T S by orders of magnitude, while T e and T S are of the same order of magnitude. As shown in Figure 5, the density ratio N I /N S increases following the fluid motion; consequently, the mean temperature for the mixture of three plasma components increases with the heliocentric distance outside the cavity and cavity wake. In the absence of pickup protons in the hydrogen cavity and the cavity wake, the wind temperature decreases at the increasing heliocentric distance. In the near wake, the calculated solar wind temperature decreases from Kat1AU to K at 5 AU, and to K at 15 AU. Unlike the wind on the upwind side, the solar wind is very cold in the cavity wake. Consequently, on a global scale, the solar wind temperature shows the characteristics of upwind downwind asymmetry. Figures 7 also show a substantial latitudinal variation in T. The pickup proton has two principal effects on the solar wind: the wind speed V decreases and the wind temperature T increases following the fluid motion. A quantitative relationship for V and T as functions of the wind speed at 1 AU boundary V has been established for the in-ecliptic solar wind in the upwind direction (Whang et al., 3). In a similar manner, the E5 1E5 4E5 1E5 8E5 6E5 E5 1E Figure 7. Constant contours for the temperature of solar wind plasma (in K) on the xy-plane (top panel) and xz-plane (bottom panel). The ionization process causes a radial increase in solar wind temperature in all directions outside the cavity and cavity wake. The distribution shows latitude variation and upwind downwind asymmetry. latitudinal variation of the 1 AU wind speed V is primarily responsible for the latitudinal variations of V and T of the global solar wind outside the cavity and cavity wake Plasma β-ratio and Fast Mach Number The plasma β-ratio is the ratio of the total plasma pressure P I + P S + P e to the magnetic pressure. The constant contours for β-ratio, in Figure 8, show that (1) radial increase of the plasma β-ratio in all directions outside the hydrogen cavity and cavity wake, and () a higher β-ratio at high latitudes. The higher β- ratio at high latitudes results from a latitudinal increase in wind speed at 1 AU boundary; high wind speed at 1 AU leads to high temperature in the outer heliosphere, and high temperature leads toahighβ-ratio. Outside 1 AU, the ratio of the radial magnetic fields to the transverse magnetic fields decreases rapidly at increasing r (Equations (14) and (15)). At large heliocentric distances, the magnetic pressure and the Alfvén speed are mainly contributed by the transverse fields. The fast Mach number is the ratio of the wind speed to the magnetosonic speed C f = (c + a ) 1/, where c is the gasdynamic sound speed and a is the Alfven speed. In Figure 9, we plot constant contours for the fast Mach number of the solar wind. The fast Mach number is the dimensionless parameter that dominantly controls the dynamics of supersonic solar wind and shock process. Note that although V and T are

8 Z (AU) Z (AU) X (AU) 78 WANG Vol β-ratio FAST MAC NUMBER β-ratio Figure 8. Constant contours for the plasma β-ratio on the xy-plane (top panel) and xz-plane (bottom panel) Figure 9. Constant contours for the fast Mach number of the solar wind on the xy-plane (top panel) and xz-plane (bottom panel). The distribution shows a near symmetry about the x-axis, and an upwind downwind asymmetry. not symmetrical about the upwind downwind line as shown in Figures 6 and 7, the variation of the fast Mach number is not far from axisymmetrical about the upwind downwind line. The plot of the fast Mach number and plots of other solar wind parameters all demonstrate the characteristics of the upwind downwind asymmetry of the global solar wind in the outer heliosphere. 5. SUMMARY AND DISCUSSION This paper presents the calculated spatial variation of the global solar wind caused by (1) the latitude variation of the solar wind at the 1 AU boundary and () the ionization of interstellar neutral hydrogen. The ionization process causes the removal of neutral hydrogen in the heliosphere. On the upwind side, 9% of hydrogen depletion occurs inside 6 AU; the hydrogen density changes very rapidly inside 1 AU, and the hydrogen density diminishes sharply inside 4 AU to form a hydrogen cavity. The cavity extends on the downwind side to form a long cavity wake. Pickup protons are produced outside the hydrogen cavity and cavity wake; Figures 9 describe how pickup protons affect the spatial variations of solar wind properties on the xyand xz-planes. Ionization of neutral hydrogen inside 6 AU is chiefly responsible for the formation of the cavity and cavity wake and the condition of the global solar wind. Figures 9 describe the spatial variation of the solar wind parameters on the xy- and xz-planes. In addition to showing deceleration and heating of the solar wind, we also show the density ratio of the pickup proton to the solar wind proton, fast Mach number, and plasma β-ratio. On a global scale, all solar wind parameters are substantially modified at large heliocentric distances to generate an upwind downwind asymmetry caused by pickup proton effects. Neutral hydrogen also makes a charge exchange with the interstellar pickup proton to produce a new pickup proton. This process becomes quite significant as the density ratio of the pickup proton to the solar wind proton increases at the increasing heliocentric distance. It contributes to the removal of neutral hydrogen in the heliosphere. This charge exchange process must not be ignored in calculating the ionization of interstellar neutral hydrogen in the outer heliosphere. This solar wind model includes MD effects, and both charge exchange ionization and photoionization. Using a highresolution computational code, we study the spatial variations of the global solar wind, including latitudinal variations. We obtain quantitative results for all important parameters of the solar wind: the wind speed, the density, and temperature of the solar wind proton and pickup proton, the fast Mach number, and the plasma β-ratio. We identify that the long cavity wake of neutral hydrogen causes the upwind downwind asymmetry of the global solar wind. The author thanks the referee for many helpful suggestions, D. J. McComas for the Ulysses solar wind plasma data, and A. Balogh and E. J. Smith for the Ulysses magnetic field data.

9 No., 1 TE GLOBAL SOLAR WIND BETWEEN 1 AU AND TE TERMINATION SOCK 79 APPENDIX A EQUATION SYSTEM FOR TE YDROGEN FLOW The system of fluid equations for the hydrogen flow used in this paper consists of the continuity Equation (4), the equation of motion (7), and the entropy Equation (8). Equation (8) is integrable along each streamline. Multiply Equation (4) bymv and add the result to (7); we obtain the momentum equation GM (ρ V V ) = ρ (1 μ ) P r m(q ph + q ex1 + q ex )V. (A1) The source term on the right-hand side represents the loss of momentum for the flow of interstellar hydrogen due to the ionization process. We can rearrange Equation (8) as V (N kt ) 5 3 kt [V N +(q ph + q ex1 + q ex )] = (q ph + q ex1 + q ex )kt. Now using Equation (4), we can write this equation as 3N kt V + N kt m m V = (q ph + q ex1 + q ex ) 3kT m. (A) Take a dot product of V with (7) ( V ) N V GM = (1 μ )N V r N kt V (A3) m and multiply Equation (4) byv/ : V (N V ) = (q ph + q ex1 + q ex ) V. (A4) Adding Equations (A3) and (A4) to Equation (A), we obtain [ N V ( V + 5 (q ph + q ex1 + q ex ) )] kt GM = (1 μ ) N V m r ( V + 3 ) kt. (A5) m The source term on the right-hand side describes the loss of kinetic plus thermal energy for the neutral hydrogen due to the ionization process. Depending on the numerical method employed to carry out the calculation, sometimes it is more convenient to use the momentum Equation (A1) and the energy Equation (A5) to replace Equations (7) and (8). APPENDIX B ENERGY EQUATION FOR INTERSTELLAR PICKUP PROTON The Boltzmann equation (E + 1c ) U B U f I = f I t + U x f I + e m + t ex1 + t ex t ph (B1) states that the rate at which the distribution function of the interstellar pickup proton f I (U, x,t)is altered due to photoionization and charge exchange. The particle velocity U may be expressed as the sum of the solar wind velocity V and the intrinsic velocity C: U = V + C, (B) where V is a function of x and t.ifc, x, and t are used to replace U, x, and t, as the independent variables (Chapman & Cowling 196, Section 3.13), we can write Equation (B1) as [ df I e + C x f I + dt m C f I = + t ph {E + 1c } (V + C) B dv dt + t ex1 ] C x V, (B3) t ex where d = + V. dt t If we multiply Equation (B3) by a function φ = φ(c) and integrate over the velocity space, we obtain the moment equation. The moment equation for φ = m C is the energy equation for interstellar pickup protons. Two nonzero terms on the left-hand side of the moment equation are and φ df I dt dc = d dt On the right-hand side we have { ( fi ) φ + t ( 3 P I ), φ C x V C f I dc = 5 3 P I V. ph + t ex1 } fi dc t ex = (q ph + q ex1 ) mv. The resulting second-moment equation of the Boltzmann equation d dt ( 3 P I ) + 5 P I V = (q ph + q ex1 ) mv is the energy equation of interstellar pickup protons. REFERENCES (B4) Axford, W. I. 197, in NASA Spec. Publ., ed. C. P. Sonett, P. J. Coleman, Jr., & J. M. Wilcox (SP-38; Washington, DC: NASA), 69 Baranov, V. B., & Malama, Y. G. 1993, J. Geophys. Res., 98, Chapman, S., & Cowling, T. G. 196, The Mathematical Theory of Non-uniform Gases (London: Cambridge Univ. Press) Fahr,. J. 1979, A&A, 77, 11 Fahr,. J., Kausch, T., & Scherer,., A&A, 357, 68 Fahr,. J., & Ripken,. W. 1984, A&A, 139, 551 Fite, W. A., Smith, C.., & Stebbings, R. F. 196, Proc. R. Soc. London A, 68, 57 Goldstein, B. E., et al. 1996, A&A, 316, 96 olzer, T. E. 197, J. Geophys. Res., 77, 547 Isenberg, P. A. 1986, J. Geophys. Res., 91, 9965 Izmodenov, V. V., Geiss, J., Lallement, R., Gloeckler, G., Baranov, V. B., & Malama, Y. G. 1999, J. Geophys. Res., 14, 4731 Labun, L., & Muller,. R. 7, J. Geophys. Res., 11, A915 Lallement, R., Bertaux, J. L., & Dalaudier, F. 1985, A&A, 15, 1 Linde, T. J., Gombosi, T. I., Roe, P. L., Powell, K. G., & DeZeeuw, D. L. 1998, J. Geophys. Res., 13, 1889 Linsky, J. L., & Wood, B. E. 1996, ApJ, 463, 4 Malama, Y. G., Izmodenov, V. V., & Chalov, S. V. 6, A&A, 445, 693

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