THE INTERACTION OF THE SOLAR WIND WITH THE INTERSTELLAR MEDIUM W. I. Axford An invited review
INTRODUCTION The interaction between the solar wind and the inter- stellar medium has been a topic of interest for many POSSIBLE years [Davis, 1955, 1962; Parker, 1961, 1963; Axford TERMINAL SHOCK et al., 19631 , but progress has been rather slow largely 100 AU I because of the paucity of relevant observations. Recently, however, the situation has changed as a result of the clear evidence for the penetration of interstellar gas into the inner solar system obtained from OGO 5 [momas and Krassa, 1971; Bertaux and Blamont, ,’. \ 19711. Furthermore, the prospect that there will be space probes traveling toward the outer regions of the solar system within a few years (see fig. 1) has created a need for a better understanding of the subject so that the most effective use can be made of the missions. This review is a survey of the work, both published and unpublished, that has been carried out up to the Figure 1 Spacecraft trajectories for the proposed time of writing. To the best of our knowledge, the “Grand Tour ’’ missions to Jupiter-Saturn-Pluto bibliography is complete up to mid-1971. First, the (1976-1 9 77), and Jupiter- Uranus-Neptune (19 79). me expected characteristics of the solar wind, extrapolated trajectories carry the spacecraft in the general direction from the vicinity of the earth are described, and several of the projection of the solar apex into the ecliptic simple models are examined for the interaction of the plane, the direction in which the distance to the solar solar wind with the interstellar plasma and magnetic wind termination is a minimum. mefinal portion of the field. The possibility that there is a substantial neutral trajectories (dashed) can be vaned, since it depends on component of the interstellar gas is ignored. In later sec- the last planetary encounter. tions, we consider various aspects of the penetration of neutral interstellar gas into the solar wind, and describe Allen, 19631. The sun is located on the inner edge of the the dynamic effects of the neutral gas on the solar wind local (Orion) spiral arm at a distance of -8 parsec north (to the extent that they are understood). Finally, we of the plane of the galaxy. The solar motion relative to discuss problems associated with the interaction of neighboring stars is 20 kmsec-’ in the direction cosmic rays with the solar wind. a=271”, 6 =+30°, or lrr=57”, b11=+22” (the “solar We briefly summarize here the present status of apex”). The nearest star is more than 1 parsec away, and knowledge concerning the properties of the interstellar the mean density of stars in the solar neighborhood is medium in the vicinity of the solar system [see, e.g., -0.057 solar masses per cubic parsec. Since the region containing the solar wind (the “heliosphere”) has dimen- sions that are at most of the order of 102-103 AU The author is with the Departments of Physics and Applied ( 609 sure that the solar wind interacts directly with the inter- Staelin and Reifenstein, 1969; Goldstein and Meisel, stellar medium and that there is no overlapping of stellar 1969; Hewish, 19701, and of other polarized galactic winds in our vicinity. It should be noted that the sun sources [Morris and Berge, 1964;Milne, 19681 . Of these appears to be on the fringes of the Gum nebula; how- various types of observation, only the 21-cm Zeeman ever, according to Brandt etal. [1971] it lies some effect and the pulsar Faraday rotation measurements 60 parsecs outside and the local interstellar medium may provide a value for the magnetic field strength, and these not be directly affected by the presence of the nebula. refer only to some weighted mean of the line-of-sight The mean interstellar electron density ne in the disc component. Magnetic fields of the order of 10 pG have of the galaxy can be determined in two ways: (1) by been found in some absorbing clouds; however, it seems measurements of low frequency radio wave absorption likely that the field strength in such clouds is not typical [Ellis and Hamilton, 1966; Smith, 1965; Bridle, 1969; of the interstellar magnetic field as a whole, and is in Bridle and Venugopal, 1969; Alexander et al., 19691 ; fact amplified as a result of the contraction of the clouds and (2) by measurements of the frequency dispersion of [Verschuur, 1969~1.The pulsar observations provide the pulsar emissions [Rlkington et al., 1968; Lyne and most satisfactory results for our purposes since they Rickett, 1968;Habingand Pottasch, 1968;Davies, 1969; refer to a region relatively close to the sun. If it is pos- Prentice and Ter Haar, 1969a,b; Davidson and Terzian, sible to obtain reasonably good values for the rotation 1969; Mills, 1969; Gould, 1971; Grewingand Walmsley, measure (RM=0.85 JneB dZ, with B in pG and 1 in 19711 . The first method yields the quantity (n:Te-3'2)l, parsecs) and the dispersion measure (DM = Jnedl) from where 1 is the distance in the galaxy along the line of the ratio RM/DM, one can obtain a value of the mean sight and Te is the electron temperature; this tends to line-of-sight component of B, weighted with respect to overemphasize the contributions from denser regions the electron density. The measured values of BIIrange [Gould, 19711 . The second method yields be)l and thus from 0.7 pG to 3 pG, and it seems reasonable therefore permit, us to estimate 02,) directly if we know the dis- to assume that the largest value is representative of the tance to the pulsar by independent means (as in the case interstellar magnetic field strength. This value is consis- of the Vela and Crab pulsars), or if an estimate can be tent with that found using the mean value of ne made of the neutral hydrogen content in the line of sight obtained from pulsar measurements together with the (e.g., from 21-cm absorption of the pulsar emission, as in rotation measures of extragalactic sources [Davies, the case of CP0329) so that a comparison can be made 19693. Furthermore, a magnetic field strength of this with 21-cm emission in the same direction. In fact, the order is required to account for the nonthermal galactic interpretation of the observations is not easy because of radio emission [Felten, 1966; Webber, 1968; Goldstein the confusion introduced by dense WI regions, and the et al., 1970al , although there is some uncertainty con- structure associated with the spiral arms of the galaxy. cerning the unmodulated spectrum of cosmic ray elec- According to Davidson and Terzian [1969] the mean trons. These and other arguments concerning the electron density in the disc of the galaxy is interstellar magnetic field have been reviewed by 0.03-0.10 ~m-~.However, the result is model dependent, Burbidge [ 19691 . and according to Gould [ 19711 ,who notes that the path Interstellar atomic hydrogen can be detected by to the Crab pulsar crosses the interarm region where the observations of (1) the 21-cm line in emission and density could be low, the local mean electron density in absorption [Kerr and Westerhout, 1965; Kerr, 1968, the galactic plane is -0.12 cm-3 . 19691 , including absorption of pulsar emissions [deJager Evidence for the presence of a large-scale interstellar etal., 1968; Guelin etal., 1969; Gordon etal., 1969; magnetic field arises from observations of (1) the polari- Gordon and Gordon, 1970;Hewish, 19701 ;(2) Lyman a zation of starlight [Davis and Greenstein, 1951; absorption in the ultraviolet spectra of nearby early type Mathewson, 1968) ; (2) the polarization of galactic non- stars [Morton, 1967; Jenkins and Morton, 1967; thermal radio emission [Mathewson and Milne, 19651 ; Gzrruthers, 1968, 1969, 197Oa; Morton etal., 1969; (3) the Zeeman effect at 21-cm wavelength [Verschuur, Jenkins et al., 1969; Smith, 1969; Wilson and 1968, 1969a,b,c, 1970; Davies etal., 19681 ; (4) the Boksenberg, 19691 ; (3) the absorption of soft X rays Faraday rotation of polarized extragalactic radio sources from discrete sources [Gorenstein et al., 1967; Fritz [Gardner and Whiteoak, 1966; Gardner and Davies, et al., 1968; Rappaport et al., 1969; Grader et al., 1966; Gardner et al., 1967; van de Hulst, 1967; Berge 19701 ; and (4) the backscattering of solar Lyman a and Seielstad, 1967; Mathewson and Nicholls, 19681 ; [Chambers etal., 1970; Barth, 197Oa; Thomas and and (5) the Faraday rotation of pulsar emissions [Smith, Krassa, 1971; Bertaux and Blamont, 19711 . Observa- 1968a,b; Radhakrishnan et al., 1969; Ekers etal., 1969; tions of 21-cm radio emission suggest that the mean 610 density of neutral hydrogen in the vicinity of the sun is observations (all of which refer to path lengths of the h~)x 0.7 ~m'~.The 21-cm absorption measurements order of 100 parsecs or more), and indeed is quite con- yield an estimate for h~)l(where I is the distance to the sistent with most of the ultraviolet absorption source), provided one assumes a value for the spin measurements. temperature T,;it is generally assumed that Ts - 100" K. It has been conjectured that the interstellar gas might However, discrete sources such as pulsars do not show contain a significant fraction of molecular hydrogen the absorption one would expect on the basis of [Gould and Salpeter, 1963; Gould et al., 19631, and in measurements made using extended sources. This cool, dense, dusty regions where the presence of a suggests that the interstellar medium may have a "raisin number of radicals has been detected from their radio pudding" structure, with most of the 21-cm absorption emissions, this is to be expected [Hollenbach etal., occurring in small, dense clouds with low spin tempera- 19711 . Ultraviolet absorption measurements failed to ture [Clark, 1965; Kerr, 1969; Gordon and Gordon, show the presence of molecular hydrogen initially 19701 . The ultraviolet absorption measurements show [Gzrruthers, 1967, 197&] ; however, recent measure- ments indicate that in some regions the concentrations much the same effect; that is, the column density of molecular and atomic hydrogen are comparable obtained from Lyman a! absorption lines in the spectra [Werner and Harwit, 1968; Ozrmthers, 1970bl. In the of early-type stars is usually, but not always, about vicinity of the sun, however, where the atomic hydrogen one-tenth the value obtained from 21 -cm emission density is quite low, and there is no evidence for dust, it measurements made in the same region of the sky is to be expected that molecular hydrogen is not an [Ozrmthers, 197Oa for a review of this topic]. Where important constituent of the interstellar gas. Lyman and 21-cm absorption measurements have been a In regions, the temperature of the interstellar gas made for essentially the same path (between the sun and WI is usually 5-1OX lo3 and is determined by a balance the Orion nebula) the results suggest that Ts 20" OK, between heating due to photoelectrons, and cooling due [Ozrmthers, 19691 . to recombination, collisional excitation, and free-free There seems to be a discrepancy between the 21-cm transitions [Spitzer, 1968a,b] . In W regions, the situa- emission and Lyman a! absorption measurements that tion is more complex; taking into account the effect of needs to be resolved. Since the 21-cm measurement heating by lowenergy cosmic rays [Hayakawa et al., refers to the entire line of sight, it is possible that the 1961; Field, 1962; Spitzer and Tomasko, 1968; answer is simply that there is a low density Pikel'ner, 19681 and cooling by collisional excitation it (nH X 0.1 ~m-~)region near the sun, which affects the is found that two thermally stable states are possible, absorption measurements in at least some directions. In one with neutral hydrogen densities of the order of contrast, Jenkins et al. [ 19691 have reported values of 0.1 ~m-~,TX IO4 OK, and ne/nH ~0.1,and the other GzH) from Lyman a absorption measurements that with neutral hydrogen densities of the order of exceed those obtained from 2 I-cm emission measure- lO~m-~,TS 300" K, and ne/nH= [Field etal., ments. It is possible that the 21-cm emission process is 19691 . There are, however, some difficulties associated not completely understood [Fischel and Stecher, 1967; with cosmic ray heating [Werneret al., 19701 ,especially Storer and Sciama, 19681 ; however, more detailed com- in view of our lack of knowledge of the galactic cosmic parisons are needed before we can be sure that a real ray spectrum at low energies [Goldstein et al., 1970b; discrepancy exists. The soft X-ray absorption measure- Gleeson and Urch, 1971al. Limits placed on the low ments indicate that in the direction of the Crab nebula, energy cosmic ray flux by indirect observations [Habing hH)x 0.3 cm-3 [Rappaport et al., 1969; Grader et al., and Pottasch, 1967; Greenberg, 1969; Fowler etal., 19701 , which is consistent with the 21-cm measure- 1970; Solomon and Werner, 19701 appear to make ments. However, in the direction of ScoXR-1, where cosmic ray heating ineffective. It has been suggested that Jenkins et al. [ 19691 report OZH) x 1.4-3.0 cmW3,the heating by soft X rays may be more important [Silk and soft X-ray absorption measurements are somewhat Werner, 1969; Sunyaev, 1969; Werner et al., 19701, and contradictory [Fritz et al., 1968; Grader et al., 19701. there are observational means of distinguishing between As far as the region in the immediate vicinity of the sun the two modes of heating [Silk and Brown, 1971; is concerned, recent measurements of the intensity of Bergeron and Souffin, 1971; Habing and Goldsmith, backscattered solar Lyman a can be interpreted as 19711. In any case, there is evidence from 21 cm and implying that hH)-0.03-0.12, depending onthe model other observations in support of a two-component used [Blum and Fahr, 197Oa; Thomas, 19711. This is a model of interstellar HI regions with properties as relatively low value, but it is not inconsistent with other described above [Clark, 1965; Radhakrishnan and 61 1 Murray, 1969; Riegel and Jennings, 1969; Hjellming THE INTERPLANETARY MEDIUM BEYOND et al., 1969; Rohlfi, 19711 , and this would also be com- THE ORBIT OF EARTH patible with the Lyman a! absorption measurements To date, in situ observations of the interplanetary [Jenkins, 19701. medium have been carried out only in a very restricted It is to be expected that the interstellar gas has a region lying close to the ecliptic, between the orbits of velocity component of 20 km sec-’ from the direction Venus (0.7 AU heliocentric distance) and Mars (1.5 AU). of the solar apex. In addition, however, there is likely to Extensive reviews of these observations have been given be a component associated with a general streaming of by Hundhausen [1968a, 19701, Axford [1968], Ness the interstellar gas relative to the stellar population, per- [1967, 19681, Lust [1967], Dmtis [1970], and Brandt haps associated with the “density wave” structure [1970]. For information concerning other regions we described by Lin et al. [ 19691 [see also Humphreys, have had to rely on inferences drawn from less direct 1971; and Burton, 19711, and a “turbulent” component observations, notably those involving comets, radio tech- that is evident from observations of interstellar absorp- niques, and cosmic rays [Axford, 19681. Within the tion lines [Munch, 19681 and 2 1-cm observations [Kerr, next decade we can reasonably hope for a considerable 1968, 1969; Goldstein and MacDonald, 19691 . Since the extension of the region of in situ observations as a result “streaming” and “turbulent” components of the inter- of the planned missions to the vicinity of Mercury, to Jupiter and the outer planets, and possibly out of the stellar gas velocity can easily amount to 10 km sec-l or ecliptic following a Jupiter swingby. In this section we more, we should not be surprised if the direction of consider the extrapolation of theoretical models of the relative motion between the sun and the gas differs interplanetary medium into the unexplored regions at substantially from the direction of the solar apex. great distances from the sun, neglecting external influ- Indeed the OGO 5 measurements of scattered solar ences (notably those due to the neutral interstellar gas). Lyman a! suggest that the direction of relative motion From the theoretical point of view the solar wind is lies close to the ecliptic plane [Thomas and Krassa, probably understood rather well in a qualitative sense, 1971; Bertaux and Blamont, 1971; Thomas, 19711, although as pointed out by Blum and Fahr E19711 this although the complexity of the phenomenon is such that we are unlikely ever to be able to produce a complete, may result in part from the asymmetry of the solar Lyman a! emission. quantitative model. Recent reviews of theoretical work have been given by Parker [1965a, 19691 and Holzer A first estimate of the radius of the region in which the solar wind is supersonic can be obtained by equating and Axford [ 197Oal. As a result of the lack of observa- tions we must rely on theoretical models, even if they %e solar wind ram pressure to the pressure of the inter- stellar gas and magnetic field (see p. 616). On the basis are oversimplified, to extrapolate our knowledge of the solar wind to heliocentric distances beyond the orbit of of the observations described above, the radius is Mars, and to high heliolatitudes. The results of calcula- nominally of the order of 100AU if the effects of tions based on one such model are shown in figures2 neutral interstellar gas, the interplanetary magnetic field, and 3 [Leer and Axford, 19711. In this model, it is and cosmic rays are neglected. This is well beyond the assumed that the solar wind velocity is radial every- orbit of Pluto, the most distant of the known planets, where, and that the protons are heated by hydromag- and it is probably too far to be considered as a major netic waves emitted by the sun in a region with a objective of the proposed Outer Planet Grand Tour characteristic radius of 5 R, [Barnes, 1968, 19691. missions (fig. 1). However, according to the analyses Otherwise, it is assumed that the interplanetary magnetic presently available, all the effects that have been field does not affect the dynamics apart from its effects neglected in this estimate tend to reduce the distance by on the electron thermal conductivity and on the proton perhaps 10 to 15 percent, and hence together they could temperature anisotropy. Whatever assumptions are made give rise to a significant and favorable change in our first concerning the processes that take place near the sun (in estimate. In figure 1, therefore, we have shown the solar r 5 0.5 AU), it is to be expected that at large distances wind termination as occurring somewhere in the region from the sun (r 2 1 AU), the predictions of any steady 50 to 100 AU from the sun, with -50 AU being a not flow model are essentially as shown in figure 2 provided unrealistic possibility. It is recommended that a major the effects of the interstellar medium are not taken into effort be made to improve our understanding of this account. In particular the solar wind speed is very nearly problem, and to provide better values for the various constant and the number density varies inversely as rz parameters involved. beyondr = 1 AU. 612 Io3 The interplanetary magnetic field, which is controlled by the solar wind, by the rotation of the sun, and by conditions in the solar photosphere, should be such that lo5 at large distances the field lines form conical Archimedean spirals in the manner described by Parker [1958, 19631 ; the spirals are determined by the inter- section of the surfaces IO2 lo4 c c u n .In E lo3 x E c a where u is the solar wind speed, i2 is the angular velocity of the sun, and @,e,$) are spherical polar coordinates. 10 IO2 The angle made by the spiral relative to the radial direc- tion (the “garden hose” angle) is given by ri2 sin 8 IO 9 = tan-’ (y) The garden hose angle is shown as a function of radial = distance and heliolatitude in figure 3(a). The compo- IO I o2 nents of the magnetic field can be written rlo Figure 2 variation of the solar wind number density n Boa2i2sin9 and radial velocity u with radial distance from the sun in ) (3) (Br&,B$ = cs ,O , Tu units of solar radii a, according to the model of Leer and Axford 119711. The dashed lines refer to the case in where Bo is the field strength at the corresponding point which there is no heat input [ Hartle and Sturrock, at the surface of the sun (r =a). The magnetic field 19681 . strength is also shown as a function of radial distance 500 u 50a u 50 u 1au MAGNETIC FIELD STRENGTH ELECTRON TEMPERATURE ,T, 1°K) PROTON TEMPERATURE ANISOTROPY ,lT;-TpTp a b C d e Figure 3 Contours of various quantities in the interplanetary medium in a polar plot in which the abscissa lies in the ecliptic plane and the ordinate coincides with the axis of rotation of the sun, according to the model of Leer and Axford [I9711 . 613 and heliolatitude in figure 3(a). It should be noted that in the ecliptic plane (0 = z/2) the direction of the mag- netic field becomes essentially azimuthal at large dis- tances from the sun (i.e., 9 > 80" for r > 5 AU), and the field strength varies as l/r. In contrast, in the polar regions (0 = OJ) the magnetic field lines are radial and the field strength varies as 1/?. As a result of this behavior the ratio of solar wind dynamic pressure to the magnetic field pressure 8znEu2 Pv= 7 (4) tends to a constant value (h, = 500) at large values of r near the ecliptic plane, but increases as r2 at high helio- latitudes. Here and elsewhere, we have taken the mean mass of the solar wind ions 5 to be 2X10'-24 gm to allow for the presence of helium and heavier elements. Wherever the garden hose angle is small the electron temperature is dominated by heat conduction; hence Y IO -7 Id -2/7 IO IO2 I o3 Te Teo (k) r /a Figure 4 lirze characteristic expansion rate (vex,), the electron collision frequency (vee), and the proton colli- where Teo is the electron temperature at some reference sion frequency (vpp)in the ecliptic plane as functions of level r=ro. As the direition of the magnetic field distance from the sun in units of solar radii, according to becomes more azimuthal however, the effects of adia- the mcdel of Leer and Axford [I9 711. batic expansion become more important, and ultimately Te ~xr-'~.The electron temperature is shown as a func- once more, so that collisions suppress the anisotropy and tion of heliocentric distance and heliolatitude in fig- Tp ar4l3. These effects are all evident in figures 3(c) ure 3(b). As shown in figure 4, the electron collision fre- and 3(d), which show the proton temperature aniso- quency is everywhere comparable with, or greater than, tropy and the mean proton temperature, respectively, as the characteristic expansion rate (vexp = -(u/n)(dn/dr)), functions of heliocentric distance and heliolatitude. and hence we do not expect that the electron The model from which the above description of the temperature should be anisotropic anywhere under behavior of the solar wind is drawn involves several normal conditions. In contrast the proton collision assumptions that are not strictly valid. The most impor- frequency may be less than vexp beyond a few solar tant of these is that the flow is assumed to be steady and radii and remain so until the magnetic field direc- (near the sun) spherically symmetric. Unsteady flow tion becomes essentially azimuthal. Thus, we can expect may affect the behavior of the solar wind at great helio- the proton temperature to be anisotropic, since when it centric distances since the interaction of fast and slow approaches a constant speed the expansion of the solar streams beyond the orbit of the earth can result in a wind takes place anisotropically (the radial dimension of mean flow that is considerably hotter than suggested in an element of fluid remains unchanged, while the trans- figure 3(d), with correspondingly smaller proton collision verse dimensions expand in proportion to r). In regions frequencies and hence a larger temperature anisotropy where the magnetic field direction is essentially radial we than suggested in figure 3(c). Departures from spherical should find Tpll > Tpl, while in regions where the field symmetry that do not involve unsteady effects must is essentially azimuthal we should find that Tpll a l/r2, occur since at any given time solar active regions tend to Tpl 0: l/r so that eventually Tpl > Tpll. However, in the be concentrated in latitude; however, apart from pro- latter case the mean temperature Tp = (TP" + 2Tp1)/3 ducing a variation with heliolatitude of the asymptotic may decrease sufficiently rapidly for the proton collision solar wind speed and perhaps slightly changing the frequency to become comparable to the expansion rate temperature contours in figures 3(b), (c), and (d),it is 614 unlikely that qualitatively significant effects can arise in is an increasing function of heliocentric distance in this way. regions where the magnetic field lines are almost radial The effects on the solar wind of the interplanetary (i.e., fl a r’ ‘I7) and a decreasing function of heliocentric magnetic field and of the rotation of the sun have been distance in regions where the field lines are almost neglected in the model calculation apart from the con- azimuthal and the plasma is cooling adiabatically trol exerted by the field on the electron thermal conduc- @ a r-413 ). If the stability of neutral sheets against tivity and on the proton temperature anisotropy. Close reconnection is determined by p, it seems possible that to the sun, however, where the solar wind speed is com- the sector boundaries may begin to break up somewhere beyond the orbit of earth where fl is small [cJ: Davis, parable with or less than the Alfvkn speed, the magnetic 19701. This process may be aided by the divergence of field is capable of inducing significant nonradial motions the flow, which causes elements of plasma in the neutral of the plasma 19671 Indeed, it seems [Weber and Davis, . sheet to drift apart where the magnetic field is azi- likely that the solar wind has almost the full angular muthal; in contrast, near the sun where the field is velocity of the sun out to perhaps 20 R,. This rotational nearly radial, the separation between elements of plasma component of the solar wind velocity should decay in the neutral sheet stays almost constant. It is empha- inversely with heliocentric distance and become negli- sized that if such an instability should occur it is likely gible at the orbit of earth and beyond. A meridional to be of a macroscopic rather than a microscopic nature, component of the solar wind velocity should be induced since it has been shown that quasilinear diffusion can near the sun, causing the magnetic field to bend toward easily stabilize a neutral sheet against microscopic the solar equatorial plane as suggested by observations of tearing instabilities [Biskamp et al., 1970) . radio sources near the sun [Hewishand Wyndham, 1963; We have neglected the possible effects of plasma Slee, 19661. Accordingly, the solar wind in the polar instabilities associated with anisotropies of the velocity regions must expand more rapidly than spherical geom- distributions of solar wind particles. Perhaps the most etry would suggest, and hence it can reach a higher important of these is the instability resulting from the velocity at a given heliocentric distance than it would asymmetry of the electron distribution function associ- achieve at the same distance in the solar equatorial ated with heat conduction [Forslund, 19701 ; this can plane. It seems possible that the observations of reduce the hffective electron thermal conductivity and Dennison and Hewish [1967], which suggest that the also alter its dependence on temperature, so that the solar wind speed increases at high heliolatitudes, could distribution of electron temperature could be signifi- be explained in this way. It is unlikely, however, that cantly different from that shown in figure 3(b). Other any of these effects lead to substantial departures from instabilities may be associated with anisotropy of the the description given in figures 2 and 3 of the behavior proton temperature [Parker, 1963; Kennel and ScarA of the solar wind at large distances from the sun. 1968; Hollweg and Volk, 19701 ; however, the anisot- It has been assumed in the previous discussion that ropy is not observed to be large [Hundhausen etal., the solar wind can be considered to be a perfect electri- 1967; Eviatar and Schulz, 19701 nor is it expected to cal conductor, and hence the magnetic field behaves as if be so (see, e.g., fig. 3(c)).Thus any instabilities that may it were frozen into the plasma. Field lines cannot result from this source might be expected to .produce only a rather low level of plasma turbulence. On the reconnect in these circumstances, and hence adjacent other hand, there is at least indirect evidence for impor- regions of inward- and outward-directed fields are sepa- tant instabilities involving the protons and other ions, in rated by neutral sheets that spiral away from the sun that the heavier ions usually appear to have the same indefinitely in the same manner as the field lines them- bulk speed, but a higher temperature than the protons selves. These neutral sheets can be identified with [Snyder and Neugebauer, 1964; Robbins et al., 1970; “sector boundaries” [ WiZcox and Ness, 1965; WiZcox, Bame et al., 1970; Egidi et al., 19701. Furthermore, it is 19681, and their observed persistence can be taken to possible that the solar wind ions are heated via the elec- indicate that the configuration is stable against reconnec- trons, rather than directly as we have assumed, in which tion in the sense that this would not lead to a state of case some form of plasma turbulence must be generated lower energy [Axford, 19671. As shown in figure 3(e), to provide sufficiently high electron-proton energy the parameter 0, defined as exchange rates and to reduce the proton temperature anisotropy to the observed low values [Nishida, 1969; Toichi, 19711. As far as the solar wind at great distances is concerned, plasma turbulence may play an important 61 5 role in providing an effective interaction with ions of only a negligible neutral component. Many of the essen- interstellar origin [Holzer and Axford, 19711 . tial features of this interaction were discussed by Davis As a first approximation, the composition of the solar [1955, 19623 in his work concerning the effect of the wind might be expected to be that of material having solar wind cavity on cosmic rays. A more detailed anal- something like the cosmic abundance of elements in ysis was carried out by Parker [ 19611 ,with the assump- ionization equilibrium at the temperatures of tion that the effects of the interplanetary (solar) 1-2X106 “K that exist in the solar corona [cf: Holzer magnetic field can be neglected. This discussion is based * and Axford, 1970bI. The observed variability of the largely on Parker’s work [1963]. helium abundance in the solar wind [Robbins etal., It was pointed out by Clauser [1960] and Weymann 1970; Hundhausen, 19701 and some theoretical work [1960] that the solar wind should undergo a shock [Jokipii, 196Q; Geiss etal., 1970al suggest that the transition so that it can come into equilibrium with the approximation might not be yholly adequate, although interstellar medium [McCrea, 1956; Holzer and Axford, there is no indication that it is seriously wrong [Bame 1970aI. The situation is closely analogous to the dis- etal., 1970; Geiss et al., 197Obl. As pointed out by charge from a Lava1 nozzle into the atmosphere Hundhausen et al. [ 1968) , the ionization state of the [Liepmann and Roshko, 19571, and to the flow of solar wind plasma is essentially “frozen” at a heliocen- water over a dinner plate (see appendix). In the ebsence tric distance of at most a few solar radii, where the of a neutral component of the interstellar gas, a shock electron temperature is not substantially less than it is in must always exist if the pressure exerted by the inter- the lower corona. At great distances from the sun, radia- stellar medium is finite. The suggestion made by Faus tive recombination times are very long indeed and one [ 19661 that the solar wind might terminate as a “free cannot expect this process to play an important role in expansion” is incorrect; indeed, the solar wind itself can determining the dynamics of the solar wind. For be regarded as a free expansion within the shock transi- example, for an ion with charge Ze in the region beyond tion. r.= 1 AU where Te a r*I3, the radiative recombination If the effects of the interplanetary ma&etic field are time rrec [Allen, 19631 is such that neglected, and the solar wind is assumed to behave as a perfect fluid with specific heat ratio y = 513, then since 3X109Te3’4uldn/drI 2x107 (7) the Mach number (M) of the supersonic solar wind is rrecvexp % =- Z2n2 22 very large at the expected position of the shock transi- tion, the shock is “strong.” In this case the which is a large number even for the more highly ionized Rankine-Hugoniot relations yield constituents (2 = 15). In high latitude regions where Te a r-21 , rrecvei is larger and increases with increasing r. Neverthe!&, changes in the state of ioniza- tion may occur as a result of neutralization of solar wind ions on iinterplanetary dust grains [Bandermann and and Singer, 1965; Banks, 19711, evaporation of the dust grains themselves followed by photoionization [Banks, private communication, 197 11 , evaporation of material from comets and planets, and most importantly the penetration of interstellar gas into the inner solar sys- Here subscripts 1 and 2 denote conditions on the tem, as discussed in a later section (p. 623). The sugges- tion that the solar wind should contain a substantial upstream (supersonic) and downstream (subsonic) sides (solar) neutral component [Akasofu, 1964; Wax et al., of the shock, respectively, and the subscript n indicates 19701 encounters difficulties [Brandt and Hunten, that the component of velocity normal to the shock is 1966; Cloutier, 1966;Axford, 19691 ,and would lead to involved; p is the total pressure, and p = nG is the mass a much larger flux of He’ ions than is observed. density of the gas. In the subsonic region beyond the shock, the Bernoulli equation holds along each stream- : INTERACTION OF THE SOLAR WIND WITH line IONIZED INTERSTELLAR GAS AND MAGNETIC FIELD Here we consider the nature of the interaction between the solar wind and the local interstellar medium for the case where the latter is part of an HII region containing together with 616 Let us consider the case in which the interstellar gas is (11) at rest with respect to the sun and there is no interstellar magnetic field. The configuration is spherically symmet- where the subscript denotes the stagnation value s ric, and hence (16) must apply to all if we = On combining equations (lo) and (11) and using (u '1. interpret po as the pressure of the interstellar medium the definition the fo'Owing M2 =u2k2=pu21yp' (and therefore of the solar wind) at infinity. Since the equations can be obtained for p, p, c2,and T: Mach number in the subsonic region can only decrease with heliocentric distance (M" 2 u=u2(:) r>rs and thus at large distances the sun would appear to be a point source of incompressible fluid. Next let us consider the case in which there is an interstellar magnetic field which is uniform and has It'is evident from (9) that kf is small on the subsonic strength Bo at great distances from the sun. We gener- side of the shock transition, and if it remains small it is alize Parker's [1963] treatment of this problem by the permissible to neglect terms O(@), so that (10) reduces inclusion of static interstellar gas, which exerts a uni- to form pressure pg, and also by allowing for the effects of 12 compressibility in the region of subsonic solar wind -Psu +P'PS 2 (15) flow. It is assumed that the magnetic scalar potential in the external region (fig. 5) is given approximately by Also from (13) we see that p = ps +O(M),so that if M2 << 1 the, flow can be considered as being incompressible. In general the shock is not spherical, and hence the flow in the subsonic region contains vorticity, and the stagnation pressure is not the same along all streamlines. However, for streamlines that lead to stagnation points at the interface between the solar plasma and the inter- where (r,0,@) are spherical polar coordinates with 0 = 0 stellar medium, the conditions for equilibrium require the axis of symmetry, and 1 is the radius of the diamag- that ps = po, where po is the local pressure of the inter- netic sphere that would produce the same external field. stellar gas and magnetic field. In the simple configura- The magnetic field lines are given by tions of interest, these streamlines are normal to the shock and hence unl = u1 (the solar wind speed on the supersonic side of the shock). For these streamlines it 8 = [I (19) can be deduced from equations (8), (9), and (10) that [il-(--]sin2 -(+)!(+) where r = b is the radial distance to the point where the field line intersects the plane e = ~12.If the interface between the interstellar medium and the solar wind is where, for y = 513, K = 1.13. Since n, = nge2/rs2, determined by (19) with b =bo, then at large distances where rs is the radial distance to the shock and the from the sun (z = r cos 6 large) the distance of the inter- subscript e denotes quantities evaluated at the orbit of face from the axis of symmetry is given by earth (re = 1 AU), this equation can be used to deter- mine rs. Note that this should be expected to be the minimum distance to the shock, and that rs may be (20) substantially different on other streamlines. / ~ 617 where Q is the total solar wind mass flux. Given Bo, pg M,, and the parameters characterizing the supersonic solar wind, we can solve equations (20) through (23) for cot bo, rs, and 1. As in the analogous problem of the flow due to a point source in a rotating fluid [Burnu, 19551, it is necessary to specify the exit conditions to obtain a unique solution. In the case of the solar wind, the appro- priate condition would seem to be that the fluid should escape freely to infinity, and hence we have taken k2= 1 together with reasonable values of Bo and pg to obtain the solution shown in figure 5. The above solutions are likely to be inadequate as representations of the actual interaction between the solar wind and the interstellar medium, since any relative motion between the sun and the interstellar gas must cause the solar wind plasma in the subsonic region to be blown away. This effect is illustrated in the third prob- lem treated by Parker 119611 : the case in which there is no interstellar magnetic field but a steady interstellar Figure Calculated shape of the region of subsonic 5 wind having a small Mach number relative to the sun. solar wind flow (shaded), for the case when the solar The solution to this problem is obtained easily if it is wind plasma is entirely confined by a uniform inter- assumed that the solar wind can be replaced by a point stellar magnetic field. The solar wind is supersonic in source of incompressible fluid, and that the interstellar r The condition of pressure balance at the stagnation line r = b, 0 = n/2, yields in terms of cylindrical polar coordinates (s,@,z),where C is a constant on each streamline, and ro = r,(p2uZ2/pP)lI4. The streamline defining the inter- g face between the solar wind and the interstellar gas is given by C = -1 if the interstellar wind flows in the posi- tive z direction. The stagnation point occurs at = 0, Similarly, along the interface of the exit channels, where s z = -ro, and for z >> r , the solar wind is confined to a hP +Md as z -+ w, we require for pressure equilibrium 4 that circular cylinder of radius Bo, as shown in figure 6(u). The major defects of this model, as far as the solar wind is concerned, are (1) the assumption that the inter- stellar gas behaves as an incompressible fluid, and (2) the implicit assumption that rs2 << ro2; that is, u~/V>>(p~/p~)~/~.Although it is not easy to satisfy Finally, from mass conservation, and assuming that the these requirements with acceptable values for the various flow in the exit channels becomes uniform as z -+ 00, quantities involved, we expect that the general nature of the flow remains more or less the same, as indicated in figure 6(a), even for the extreme case of a hypersonic interstellar wind. Thus, the value of rs can still be calcu- lated from (16), provided po is taken to be the stagna- tion pressure of the interstellar wind (po=pg+pgP). 618 ar Figure 6(a) Streamlines in a model of the interaction between the solar wind and the interstellar wind in which both fluids are treated as being incompressible [Parker, 1941,19631 . Figure 6@) Gzlculated shape of the interaction region This will represent only the minimum value of rs, how- of the solar and interstellar winds assuming hypersonic ever, since the cavity within which the solar wind is flow [Baranov et al., 19701. The shape of the stream- supersonic must become progressively more asymmetric lines, and the positions of shock fronts are indicated by as the Mach number of the interstellar wind increases. the (sketched)dashed lines. Provided we restrict our attention to the flow in the vicinity of the axis of symmetry in the upstream direc- tion, it should be possible to obtain estimates for the radius of curvature of the solar wind shock and for the positions of the interface and of a possible shock in the interstellar medium, using the approximate theory for a where subscripts n,t indicate normal and tangential com- hypersonic shock standoff region due to Lighthill ponents, respectively, v is the mean speed of the com- [Hayes and Probstein, 1966; pp. 232-2541. If the radius pressed gas in the layer, s is the distance measured along of curvature of the solar wind shock exceeds rs on the the iayer from the stagnation point at 6 = 0, and axis of symmetry as indicated in figure 6@), the shock is not normal to the flow everywhere, and the vorticity (r2 + rr213’2 induced in the subsonic region can play an important R= r2 + 3” - yr~f role in turning the flow into the wake. is its radius of curvature. Note that the equation for An alternative procedure using Busemann’s method momentum conservation in the direction of the normal [Hayes and Probstein, 1966; pp. 78,1561 has been adopted by Baranou et al. [1970), who treat the sub- contains a term representing the centrifugal force acting sonic region as a thin layer separating two hypersonic on the plasma in the layer, but this does not affect the streams. In this case the details of the region of subsonic distance to the stagnation point (r = rs) since CPv = 0 at flow are ignored and one simply writes equations repre- 0 = 0. With some manipulation v and CP can be elimi- senting conservation of mass and momentum for the nated from these equations, leaving a third-order, non- layer as a whole, which can be represented as a curve of linear differential equation for t(0) that must be solved the form r = rS$(0).Thus, the mass flaw is numerically (see fig. 6(b)). The thickness of the layer, which can be obtained using the Lighthill method, would be expected to be of the order of 30 percent of @ = mZpgllZ+ 2m2(1-co~e)p2u2 (25) the radius of curvature of the layer at the stagnation at a point (r,6) in the layer. The momentum equations point tR = 5r~/3)* are The closely analogous problem of the interaction between the solar wind and the ionized head of a comet @v (26) has been treated as being equivalent to a source of pgVnz ‘~2~22,+ 2,,.,.Rsin e incompressible fluid in a supersonic flow by Zoge 619 [1966a,b; 19681. Other aspects of the case in which the interstellar wind is supersonic have been considered by IOoo 2 Dokuchaeu [ 19641 , Brengauz [ 19691 , and Sakashita ng: lnlentellar electron density (19701, who note that the shock waves produced by h-'1 randomly moving stars surrounded by extensive stellar '$ 2 Heliocentric distance to shock 200 (AM wind regions could be a significant source of energy for the interstellar gas. It should be noted, however, that IO0 provided the stellar winds are supersonic near the stars, there can be no drag whatsoever on the stars themselves. 50 The momentum imparted to the interstellar gas is taken up by the outer subsonic part of the stellar wind region as it is turned to form a comet-like wake behind the star. 20 In fact, this must be essentially correct even if the stellar wind is subsonic everywhere, provided the dimensions of 10 the star are small compared with those of the associated stellar wind region. 5 Although we have not been able to construct a com- 3 I '1111' I I ,,,I 03 .05 I 2 5 I 23 5 10 pletely satisfactory model of the interaction between the INTERSTELLAR MAGNETIC FIELD STRENGTH (10'5poussl solar wind and an ionized interstellar medium, three significant results have been established: (1) in the Figure 7 Minimum distance to the solar wind shock presence of an interstellar wind the region of subsonic transition (rs), as a function of interstellar magnetic field solar wind flow should form a comet-like tail in the strength for various values of the interstellar electron wake of the sun; (2) the cavity within which the solar density. me curves have been calculated from equa- wind is supersonic can be expected to be asymmetric; tion (28) assuming a! = 2.25, Tg=104 OK, and (3) the minimum radius of the cavity is determined by V=20 kin sec-'. the maximum pressure exerted by the interstellar gas and magnetic field, since the point of maximum external implausible values of the parameters involved in equa- pressure must be a stagnation point of the (subsonic) tion (28) are required, or else an entirely different model interior flow. Thus, we can calculate the minimum should be considered, as discussed later (p. 623). radius of the solar wind shock transition from equa- In the previous discussion we have ignored the effects tion (16) by taking po to be the maximum external of the interplanetary magnetic field on the flow of the pressure: solar wind. This is a reasonable procedure in the super- sonic region in the absence of effects associated with neutral interstellar gas, since pV 2 500, and in any case the Maxwell stresses are very nearly in self equilibrium. However, in the subsonic region it is easily seen that inconsistencies arise if the effects of the magnetic field where a! is a factor that allows for the enhancement of are neglected and the flow is treated as if it were incom- the interstellar magnetic field due to the presence of the pressible. For example, if we assume that the flow is solar wind (cf: eq. (21)). The minimum rs according to spherically symmetric and radial, and that the magnetic (28), for a range of possible values of Bo and ng, and field is azimuthal, then since fured values of the remaining parameters, can be read directly from figure 7. It should be noted that in the ruB = constant (29) absence of any ionized interstellar gas, rs = 140 AU if we assume that Bo x 3X gauss as suggested by the we see that with u er-2 (eq. (17)), B ar and hence the observations described earlier. If we assume that magnetic field strength increases indefinitely. Although ng = 0.05 cme3 then with V = 20 km sec-' our estimate the flow is likely to be nonradial and the dimensions of for rs is reduced to 100 AU. For the transition to occur the subsonic region are restricted as indicated in fig- in the vicinity of the orbit of Jupiter as suggested by ures 6(a) and (b), the tendency for the magnetic field Lanzerotti and Schulz [1969], or even closer to the sun strength to increase is unavoidable. Indeed, a very similar as suggested by Brandt and Michie [1962], quite situation is found in the earth's magnetosheath where 620 the interplanetary magnetic field strength near the solar wind stagnation point is often found to be comparable to that of the geomagnetic field just within the magneto- pause [ahilland Amazeen, 19631 . The manner in which the magnetic field can be enhanced in the subsonic region, so that it ultimately dominates the plasma behavior, is demonstrated rather well by the case of steady, radial flow with spherical divergence, in the presence of a purely azimuthal mag- netic field [Ganfill, 19711. Let us assume that the region of subsonic flow begins at a shock transition occurring at r = rs: the Rankine-Hugoniot equations can be written in normalized form I 2 5 - 10 20 50 100 r = r/rS - Figure 8 Solution of equations (34)(37) with initial B2Z2 = 1 (3 3) conditions determined by equations (30)-(33)and with &, = 500. Note that the presence of the magnetic field has caused the total pressure FT to decrease to zero as where F=p/p,, Z=u/u1,B=B/B1, P=p/pIul2, and - r+-, rather than remaining constant @!))as it does in subscripts 1,2 denote upstream and downstream condi- tions, respectively. The equations of motion in the sub- the field-free case Wom Ganfill, I974 . sonic region can be integrated to yield make a substantial change in the total pressure jjT=jj+jiii2 +B2/&,, although the changes are not insignificant for moderate values of At F=2 for example, eT is reduced by about 12 percent from the value FTo obtained in the absence of magnetic field. In a more realistic model with nonradial flow it should be expected that the effectiveness of the mag- netic field in reducing the total pressure beyond the shock transition is more marked than in the model with strictly radial flow. Lateral flow along the magnetic field lines permits the gas pressure to decrease and be com- pensated by a corresponding increase in the pressure of the magnetic field. It is easily shown that the reduction where r ='r/rs. For a given value of PV, equations (30) in total pressure Ap across a layer of thickness d and through (33) can be solved for F2 and p2, and equa- radius of curvature R is given approximately by tions (34) through (37) can in turn be solved for E, E, 5, and F as functions of 7. It should be noted that the normalization results in the elimination of rs from the equations, leaving &, as the only parameter. The solution for a value of &, appropriate to the ecliptic plane is where @I2>is the mean square magnetic field strength in shown in figure 8. it is evident that Fmust be large to the layer. This result applies equally well to the earth's 62 1 magnetosheath, and one might expect the values of formed by the region of subsonic solar wind flow could Ap/ps to be comparable in the two cases. Observations be observed from a distance; however, this does not made in the magnetosheath suggest that &Ips could appear to be feasible. The suggestion that free-free easily be 0.1 -0.2, and hence we can reduce the minimum transitions in this region might produce a detectable flux distance to the solar wind shock transition from our of soft X rays with a roughly isotropic distribution was previous estimate of 100 AU to perhaps 80 to 90 AU. originally made by Reiffel [1960]. The spectrum of the We have assumed that the interplanetary magnetic observed isotropic X-ray background is shown in fig- field can be considered as being frozen into the solar ure 9, together with calculated spectra based on the wind plasma in the above discussion. However, magnetic (favorable) assumptions that the electron and proton field directional discontinuities must exist within the temperatures are equal in the shell of hot solar wind solar wind plasma (sector boundaries), and at the inter- plasma, and that the thickness of the shell is 4rs. The face with the region of interstellar plasma and magnetic contribution from free-bound transitions associated with field. Throughout most of the region of subsonic solar the ultimate recombination of all stellar wind plasma wind flow, it might be expected that sector boundaries originating from stars within 330 parsecs of the sun is are stable against reconnection since 0 tends to be 0(1) also shown. It is evident that neither the solar wind nor or greater. However, there is no obvious reason why the stellar winds in general can be considered to be likely interface with the interstellar region should be stable sources for the observed X-ray flux [Bowyer and Field, against reconnection, and hence we can expect that the 1969; CranJill, 19711 . interstellar and interplanetary (solar) magnetic fields are The shock transition that terminates the supersonic connected in much the same manner as the inter- solar wind may be collision free, in which case its planetary magnetic field connects with the geomagnetic thickness is no more than a few proton gyroradii (about field [Dungey, 1961;Levy etal., 19641. 10" cm for 6 = v/2, rs 2 100 AU). However, it is also It would be of great interest if the shell of hot plasma possible that the transition is as wide as the system will Figure 9 A comparison of observations of the diffuse soft X-ray flux (dashed line), with estimates of the X-ray flux resulting from (a)free-fiee emission from the region of subsonic solar wind flow (straight solid lines), and (b)recombinations of stellar wind ions (bar graph). The thickness of the region of subsonic solar wind flow is taken to be 4rs, with Te = 184 eV, and n = 4nge2Jrs2 with ne = 5 ~m-~.Even though we have assumed a rather large thickness for the subsonic region, it is evident that for reasonable values of rs (i.e., rs 2 50 AU) the fiee-free emission fails to account for the observed flux by a factor of the order of lo3. The bar graph represents 0.05 keV averages of the fiee-bound emission resulting fiom the recombination of heavy ions that have been injected into the interstellar medium hy stellar winds. It is assumed that a steady state has been reached with the recombination rate equal to the injection rate, and that all stars within 330parsecs of the sun contribute. The star density is taken to be 0.08parseF3, and all stellar winds are assumed to be similar to the solar wind in composition and total flux. Ground state recombina- tions of all appropriate ion states of He, C, N, 0, Ne, Mg, Si, and Fe have been included. It is evident that the recombination radiation is also insufficient to account for the observed soft X-ray flux [Ckanfill, 1971 1 . hv (keV) 622 permit [Schindler, private communication] , in which Q = 4X 103’sec-’ ; the points of intersection of the two case the shock structure would be determined by sets of lines give the values of rII for the various com- proton-proton collisions. The thickness of such a shock binations of values of TI and TI^ The thickness of the is presumably of the order of X cos’ J/ a: A(u/r!2 sin ionization front separating the HI and HI1 regions is of where h is the mean free path. Since at large distances the order of A = l/uLnI, where uL FZ 6.3X lo-‘ ’ cm’ is vu< l/vexp = r/2u (fig. 4), it is evident that the thick- the photoionization cross section for hydrogen atoms at ness must be less than 1/(2rs sin2 0) AU. Thus, in the the Lyman limit. In cases where rIz < A x 5X lo4 AU ecliptic plane (0 = 7r/2), we expect that the shock thick- our assumption that the HZI region is well defined is ness is less than 10’ ’ cm, even if the shock is collision invalid and a more detailed analysis must be carried out dominated. [Gould et al., 1963; Lenchek, 1964; Williams, 19651. The thickness of the ionization front is somewhat NEUTRAL [NTERSTELLAR GAS IN THE greater than A since the absorption coefficient decreases VICINITY OF THE SUN approximately as f3 beyond the Lyman limit, and con- If there were no relative motion between the sun and the sequently the radiation reaching the outer part of the surrounding interstellar gas, and if the complications WI region is such that the effective absorption coeffici- introduced by the solar wind could be ignored, the sun ent is much less than uL. Nevertheless, the value of rII could produce a quite substantial HZI region if the calculated from equation (39) has some signif:lcance as a density of the ionized gas is not too large [Newkirk et al. ‘‘characteristic’’ dimension, even in circumstances where 1960; Brandt, 1964~;Lenchek, 1964; Williams, 19651. the HII region is not sharply bounded. However, if there is relative motion, the situation is In principle, the solar WI region could contribute to drastically altered, and even with quite modest speeds of the low frequency cutoff in the spectrum of galactic the order of 20 kmsec-’ it can be shown that the nonthermal radio noise, as suggested by Lenchek neutral interstellar gas can penetrate almost unattenu- [1964]. However, the density required to produce sig- ated into the inner solar system within the orbit of nificant absorption at frequencies of the order of 1 MHz Jupiter [Blum and Fahr, 1969, 197Oa; Holzer, 1970; is quite large (nII= 10 ~m-~)even if the electron Holzer and Axford, 197 1 ;Semar, 19701 . In this section, temperature is as low as lo3 OK. we assume that the interstellar medium in the vicinity of In the absence of any interstellar magnetic field, there the solar system is part of an HI region, and examine the would be little point in considering a static HII region of manner in which the neutral gas is affected by the solar the type described above, since the solar wind would wind and by ionizing radiation from the sun. push the interstellar gas to such a great distance that Let us consider first the very simple model in which ultimately the WZregion would either overlap those of the nearby interstellar gas is stationary with respect to other stars or be limited by recombination of the solar the sun, with the solar WI region assumed to be fully wind plasma. In the latter case, the solar wind would ionized and having the same pressure as the surrounding terminate at a recombination front [Newman and HI region. On equating the flux of ionizing photons Axford, 19681. The interstellar magnetic field should emitted by the sun Q to the total recombination rate not be expected to have any important direct effect on a we find that the radius of the HII region rII is given by static, approximately spherical solar HII region; how- Stromgren’s result: ever, it would effectively shield the HII region from the solar wind, which would be channeled harmlessly away in the manner indicated in figure 4. The chief defect of these models of a solar WI region is the requirement that there be no relative velocity where a = 3X lo-’ Ti/ is the coefficient for recom- between the sun and the local interstellar gas, when in bination to all levels [Allen, 1963, p. 901 . The condition fact a relative speed of the order of 20 km sei-’ might of pressure equilibrium yields be considered more appropriate. An atom of interstellar gas, approaching the sun with speed V=20 km se? , has a mean free path against loss by ionization given by Figure 10 gives these relationships for various values of Tz and TIz for the case nI = 0.2~rn-~,with 623 I o6 10 ‘ 6 nI (crnS3) U) Io5 IO0 4 2x 2x10 ‘ IO-^ io- io-* Figure 10 The values of rII and nI (solid lines) are plotted for various values of TII and TI/TII according to equations (39)and (40). The intersection of these lines defines repre- sentative allowed sets of parameters describing model solar WI regions. The ionization fiont thickness A is shown for two values of TI/TII (dashed lines). As an example, for TlI= 3x10’ OK, and TI= 30” K, we see that rII = 7X104 AU, and A = IO4 AU. where the pj are rate coefficients for all possible ioniza- The flow of neutral interstellar gas past the sun has tion processes (table 1). In general, 0 = po(re2/r2),where been studied by Fahr [1968a], Blum and Fahr [1969, Po is a constant for each atomic species. It is to be 1970a1, Holzer [1970], Holzer and Axford [1971], expected therefore that a substantial fraction of inter- lrnsley [1971], and Semar [1970] . The following dis- stellar atoms approaching the sun can penetrate to cussion is based on Holzer’s treatment of the problem within a heliocentric distance of the order of (essentially the same as that of Blum and Fahr), with a ri = (Jorez/V). From table 1 we see that interstellar slight generalization to allow for solar radiation pressure. hydrogen atoms can easily penetrate to within the orbit As a first approximation, solar radiation pressure can be of Jupiter, while interstellar helium atoms can penetrate assumed to produce a radial outward force that varies to the orbit of earth. Clearly then, we must expect to be inversely with the square of the distance from the sun. involved in quite a different situation than that of a Thus it can be taken into account by introducing an static, spherical WI region that effectively excludes “effective” gravitational constant (l+)G, where ,u is the neutral interstellar gas from the region occupied by the ratio of the force on an atom resulting from radiation solar wind. pressure to the force of gravity. For hydrogen, the large 624 Table 1 Loss coefficients at r=re(fl=flo), and penetration distances for V = 20 km sec-' (ri). The solar ultraviolet flux has been taken from Hinteregger [19701 Species Reaction u(cm)' Reference Hthv+Ht t e Banks and Kockarts [1972] 1.5 H H t Ht + Ht t H 2x1~ Fite et al. [ 19621 4 4 H t Het2 + H+ t He+ 4X10-16 Fite et al. [1962] 0.036 He + hu +Het + e Lowry et al. [ 19651; Cairns and 0.62 Samson [ 19651 He He+Ht+HeftH 1.5X lo-' ' Afrosimov et al. [ 19691 0.03 0.5 He t Hett + 2He+ 2x 1~ 7 Hertel and Koski [1964] 0.00 18 C c tlw+c+t e McGuire [ 19681 40 30 N t hv+ Nt t e Henry [1968] 1.7 N 4 N t Ht + Nt f H 1.8X lo-' Stebbings et al. [1960] 3.6 0 t hu+ 0' t e Huffman [1969] 2.4 0 3 0 t H+ + 0' t H 8X IO- Fite etal. [1962] 1.6 Ne t hu +Net + e Ederer and Tomboulian [ 19641, 1.9 Ne Samson [I9651 1.6 Ne t Ht -+ Net t H Afrosimov et al. [ 19691 0.2 Si Si + hu -+ Sit + e McGuire [1968] 60 45 Art hv + Ar+ t e Cairns and Samson [ 19651; 2.8 Ar Rustgi [1964] ;Madden et al. 3.6 [ 19691 Art Ht + Art + H 10-15 Koopman [ 19671 2 Fe t hv+ Fe't e McGuire [19681 2.4 Fe 1.8 Fe t Ht -+ Fet+ H < 10-16 Lee and Hasted [1965] solar Lyman 01 flux results in 1.1 being of order unity, and important collisions with solar wind particles are hence the effects of radiation pressure are very impor- ionizing, it is appropriate to make use of a free particle tant [Wilson, 1960; Brandr, 1961; Rnsley, 19711, which treatment allowing for losses associated with ionization, is contrary to the conclusion of Fahr [1968a] . However, with loss coefficients as given in table 1. Taking polar for other species p << 1, and hence solar gravity deter- coordinates (r$) in any such plane, it can readily be mines their dynamic behavior. In a more accurate treat- shown that the trajectories are hyperbolae with the sun ment it would be necessary to take into account the as focus, given by variation of p with position resulting from Doppler shifts and the reduction in the intensity of the solar radiation A resulting from scattering. It should also be noted that - = 1 t B sin(@ t a) (42) the solar Lyman 01 flux varies with solar activity espe- r cially at the center of the line where 100 percent varia- tions can occur [Blamontand Madjar, 19711. wjth If we neglect the thermal velocities of interstellar 1I2 atoms ( 625 where 8' = 0 is antiparallel to the incident flow velocity V, p is the angular momentum of the atom about the sun, u,, u8 are velocity components in the r, 8' direc- tions, M is the mass of the sun, and A= P2 (1-I.1L)GM (45) 1 12 *= (1 t y) (46) 1 sin a! = - - (47) B Figure 1l(b) Same as (a), but with the effects of radia- In general, any point in the plane (r,8') is a point of tion pressure exceeding gravity. A parabolic forbidden intersection of two trajectories having angular momenta (hatched) is Produced with the sun at the focus- given by (48) since the angular momenta given by equation(48) become imaginary. In the case 1.1 = 1 ,the trajectories are where C = /( 1 - p)GM, and the plus sign is associated with p, and the minus sign with p2.As indicated in straight lines parallel to 8' = 0 and p2 = 0. figure ll,pl >O,p, -3 AU hEhqhG 5AU 4 3 2 I -m ~'0,V=20kmser' If we take ?I, = p/V, then it can be shown that Figure 1l(a) Examples of intersecting particle trajec- (5 2) tories for a case Of a Cold interstekr wind approaching Also, since the displacement dE is along a trajectory the sun (at the origin)from the right of the diagram at 20 km sec", and with radiation pressure absent (p= 0). ur de' Note that all trajectories cross the axis of symmetry on (5 3) the downwind side of the sun. h@= 7 626 Hence on putting h4 = rjsin 0'1, and taking 0 = flore2/r2, intersect at (r$) strikes the sun before reaching the we obtain from (5 1) point, its contribution to n must be discounted. Note that if p -+ 1, C -+ 00, then u = V, b = Vr sin e', p2 = 0, and (54) reduces to the simple result NP~~XP @ge2ej/lpil) n(r,e)= i tnr sin e [r2 sin2e + 4r( 1- cos e)/q '2 c -Pore2 6 (54) n(r.0) = N exp (vr sin e) (55) where B is the polar angle from the axis of symmetry (0 < 8 < R and lsin 8' 1 = sin e), N is the number density Equal number density contours for hydrogen and of the incident stream at infinity, and Bj = 8 if pj > 0, helium have been calculated with 1.1 = 0,0.8 , 1 .O ,and 1.2 Bj = (271-0) if Pj < 0. If either of the trajectories that for the case of hydrogen (figs. 12(a)-(d)) and withp = 0 (b)p = 0.8 2 (c)p=l.O (d)p = 1.2 Figure 12 Contours of equal density of neutral interstellar hydrogen. The sun is at the center of each diagram and the orbit of earth is shown as a dashed line. The plane of the diagram is the plane containing the velocity vector of the interstellar wind and the sun. The three-dimensional contours are of course misymmetric. Note the high density on the axis of symmetry (e = A) for 1.1 < 1, and the paraboloidal void which appears when p exceeds 1. 627 for helium (fig. 13). Note that n(r, 0) is finite on 0 = 0 in Johnson and Axford [1969], the average rate of loss of each case, but if p < 1, n(r,0)+ = as 0 + n as shown in a atmospheric He by evaporation is approximately slightly different context by Danby and Camm [ 19571 . 7 atoms eme2 sec-' ; hence, allowing for contribu- This singularity disappears if one relaxes the assumption tions from other sources, we find that N(H) = that the incident stream of atoms has no thermal veloc- 0.05-0.1 ~m-~.At certain times of the year, the ities [Danby and Bray, 19673. For p 2 1 ,n(r,n)= 0,and relative velocity between the earth and the interstellar for p > 1 an empty region develops in the wake of the gas can be as high as 80 km sec- . This is sufficient for sun. In the upstream direction the interstellar neutral gas heavier atoms to be trapped in a target foil, thus raising penetrates to a heliocentric distance of order the possibility that the isotopic composition of the inter- ri = flore'/V, as argued previously, and hence there are stellar gas (especially neon and argon) might be mea- likely to be observable effects that permit us to deter- sured directly [Geiss, private communication] . mineN(H), N(He), and V. It might also be possible to estimate the density of Fahr [1968a,b;19691 has examined the possibility the neutral interstellar gas in the vicinity of the solar that the local fluxes of neutral interstellar gas and of system from observations of the state of ionization of neutral hydrogen produced in the solar wind by charge the solar wind. For example, charge exchange between exchange should be sufficient to cause changes in the solar wind protons and interstellar hydrogen atoms pro- structure of the upper atmosphere. However, on taking duces a neutral hydrogen component of the solar wind; reasonable values for the various parameters concerned, similarly, photoionization of interstellar helium atoms, one finds that it is unlikely that observable effects can together with other less important processes, results in be produced. Holzer and Axford [1971] have argued the appearance of He+ ions in the solar wind (fig. 14). that accretion of interstellar gas could be an important The hydrogen flux in the solar wind is estimated to be less than of the proton flux if N(H) = 0.1 ~m-~, and is probably too small to be easily measured. The expected flux of He+ ions in the solar wind is shown in figure 14(d) for the case N(H)= 1 ~m-~.There have been reports of a singly ionized helium component of the solar wind [Bame et al., 1968; Wove etal., 19661, with a flux of the order of 1O-j times that of He'' ions. On the basis of these results, Holzer and Axford [ 197 11 have estimated that N(H) = 0.1 crnb3, which is consis- tent with the estimate based on the terrestrial 3He bud- get. More recent observations by Bame etal. [1970] showed no evidence for the presence of 4He+ ions in the solar wind possibly because they were masked by other ions with the same charge-mass ratio (Si", S+' , A+9, Fe+14) or because they had a bulk flow velocity and temperature significantly different from those of other ions and could not be easily identified in an energy-per- Figure 13 Contours of equal number density of inter- unit-charge spectrum. There is clearly a need for further stellar helium in the plane containing the sun and the observations using a detector which can identify 4He+ velocity vector of the interstellar gas. The orbit of the ions unambiguously [Ogilvie et al., 1968; Ogilvie and earth is shown as a dashed line. Wilkerson, 19691 . The interstellar gas can be detected directly by obser- .item in the terrestrial budget of 3He but not of other vations of resonant scattering of solar photons, notably gases unless conditions were vastly different in the past. Lyman a (A1216 HZ) and A584 Hel [Morton and fircell, It is evident from the calculations described above that 1962; Kurt, 1965, 1967; Kurt and Dostovalov, 1968; the number density of neutral helium near the earth is Kurt and Syunyaev, 1968; Young et al., 1968;Chambers only slightly less (-25 percent) than it is in the interstellar etal., 1970; Mange and Meier, 1970; Thomas and gas. If we assume that [4He]:[3He] k3000:l Krassa, 1971; Bertaux and Bhamont, 1971; Wallace, [Cameron, 19681 ,it is estimated that the average flux of 1969; Tinsley, 1969; Reay and Ring, 1969; Barth, interstellar 3He atoms accreted by the earth is approxi- 1970a,b; Barth etal., 1967, 1968; Metzger and Clark, mately 50 N(H) atoms cm-' sec-' . According to 1970;Byram et al., 1961; Bowyer etal., 1968;Johnson 628 1.7~ 1.01 3.0 x --+ t - -+- - 40 AU 30 20 40 W 40 AU 30 40 AU ___, _.__i) V V p:O jL= 0.8 (a) ,Ot s- 40AU 30 20 (C) ,Ot (d) 1 Figure 14 Contours of equal flux of hydrogen in the solar wind resulting from charge exchange with interstellar hydrogen, relative to the solar wind proton flux (assumed to be 2x10' ern- sec-' at 1 AU). The plane of the diagram is defined by the velocity vector of the interstellar gas and the sun. There is an enhancement of solar wind hydrogen along the downwind axis of symmetry in (a) and (b) where p = 0.0 and 0.8, respectively, but not in (c)where E.C = 1. The flux of He+ ions in the solar wind resulting from ionization of the interstellar gas, relative to the solar wind proton flux, is shown in (d). me interstellar gas density has been assumed to be 1 cm-3 in these calculations, with N(H) = 0.92 cm", and N(He) = 0.08cmW3. For other values of N(H) and N(He) the figures should be rescaled appropriately. et al., 1971 ; Ogawa and Tohmatsu, 19711. From such Germogenova [ 19671 , Blum and Fahr [ 1969, 1970b,c, observations, we can in principle deduce the composi- 19711, Tohmatsu [1970], Bowers [1970], Tinsley tion and density of neutral interstellar gas flowing past [1971],Holzer [1971], 7homas [1971], and Wallis the solar system, the velocity vector, and also the tem- [1971]. perature. Accordingly, they are of great significance for Let us consider a simple model for the scattering of our understanding of the behavior of the solar wind at solar radiation by neutral atoms in the interplanetary great distances from the sun. Interpretations of these medium. The flux J(r,B;v) of solar photons in the fre- observations in terms of various models of the inter- quency range (v, v + dv) satisfies the equation action between the solar wind and the interstellar medium have been given by Patterson et al. [1963], ia - (r2J)=-ad Brandt [ 1964b, 19701 ,Hundhausen [ 196833 ,Kurt and r2 -ar 629 where a(r,O;v) is the absorption coefficient, and we have at the center of the line is only 70 percent of the maxi- neglected multiply scattered photons. If r, r' are the mum intensity in quiet solar conditions, and the maxi- position vectors of a point in space relative to the sun ma occur at k0.2518 from the center (corresponding to and earth, respectively, then the intensity of singly Doppler shifts such that lur I % 60 km sec-' ). If it is con- scattered photons in a fixed direction ?' is given by sidered that the above analysis is unnecessarily compli- cated in view of the fact that the solar emission line is quite variable, and that multiple scattering has been 1- S(v)= 4a a(r;u)n(r)J(r;v)dr' (57) neglected, a simplification can be achieved by assuming that the medium is optically thin. In this case the right hand side of equation(56) is equated to zero, so that where the integration is carried out along the line of instead of sight from the earth, v refers to the frequency before scattering takes place, and absorption of the scattered photons is neglected. The total intensity of singly scattered photons from a given solar emission line is therefore with the integration being carried out along the radius vector from the sun, we neglect the exponential term re J(r;v) = -J(re;v) It does not matter that the frequency of the scattered r2 photons need not be the same as that of the absorbed photons (in the observer's frame) since observations are in (57). In any case, these calculations will usually tend usually carried out with broadband detectors. However to overestimate the intensity of the scattered light for it might be important in some cases to allow for Doppler given N, as a result of the neglect of multiple scattering. shifts in calculating the absorption coefficient if the Examples of calculated total intensity profiles of solar emission line is not broad and featureless. In the scattered solar radiation from neutral hydrogen case of Lyman a for example (see fig. 15), the intensity (A1216 HI) are shown in figures 16(a) and (b) and from neutral helium (A584Hei) in figures 16(c) and (d). In 19 APRIL. 1960 figures 16(a) and (c) the earth is shown situated on the upstream axis of symmetry (ae = 0), and in figures 16(b) PLATE 10 and (d) it is shown situated at 90" from the axis of (QUIET REGION1 symmetry (ae = 90", 270"). We have not attempted to obtain an exact fit between the models and the observa- tions (fig. 17), but instead have simply taken N(H) = 0.1 cmW3, N(He) = 0.008 ~rn-~,and V = 20 km sec-' , with the interstellar wind vector in the plane of the ecliptic; this choice provides a reasonable match to the observations if p = 1. The direction of the interstellar wind is determined by the direction of maxi- mum Lymana intensity, although as pointed out by Blum and Fahr [1971], some distortion may be pro- duced as a result of asymmetries in the solar Lymana ' 0~4 ' 0'2 ' 00 ' 0'2 ' 0'4 ' 0'6 ' 0:s emission. The velocity of the interstellar wind is deter- WAVELENGTH (A) mined by the parallax effect shown in figure 16(b) (see also figs. 17(a)-(c)). The value of N(H) is determined Figure 15 Solar Lyman a profiles for two regions on from the maximum intensity of scatbred Lyman a, the sun obtoined by Meier and Pvinz [I9701. The central although it should be noted that there is some ambiguity core is produced by the earth's hydrogen corona, and since it is also necessary to account for the observed the &shed line indicates the expected unattenuated minimum intensity of -250 R [27zomas, 19711. Note intensity. that for small values of p the peak Lyman a emission 630 800 700 c2 600 - p = 0.8 .-0 -01 =.. 6? W 500 G (I 400 UJ 52 E ,-, 300 I cucp 200 IO0 "0 30 60 90 120 150 180 Figure 16(a) Gzlculated scattered intensity of Lyman a (AI216 HI) corresponding to the results shown in figure 12 for various values of p, with N(H)= 0.1 cm-3 Figure 16(b) The same as (a) but with the earth at and V=20 km sec-". The earth is on the axis of sym- *90" from the axis of symmetry (i.e., &e=90", 270") metry on the upstream side of the sun (ae=O). For showing the effect of parallax. With ~~0.7,and the ,u < 1.0 there is a singularity at 8 = 180 " corresponding same values assumed for the various parameters involved, to the singularities on the axis of symmetry shown in the direction of the region of maximum emission inten- $@res 12(aj and (b), sity can vary by about 60" throughout the year. occurs at 8 = 90", and there would not be a single maxi- be very prominent unless observations are made exactly mum as observed (fig. 17). If p > 1, the minimum inten- along the axis of symmetry (ae = = 180O). Preliminary sity is zero according to our calculations, which are calculations indicate that for ae # 180" the secondary based on the assumption that TI = 0. intensity maximum is not at all pronounced and could The observed minimum intensity of -250 R can be be easily missed by observations made with wide-angle accounted for in any of three ways: (1) the effect of detectors [Johnson, 19711. If the minimum intensity radiation pressure is such that ,u < 1 ; (2) there may be a represents galactic Lyman a, then this should be distin- uniform galactic background of Lymana [Kurt and guishable from the scattered solar Lyman a since the line width is expected to be considerably greater. S'nyaev, 1968; Tinsley, 1969, 197 1; Adams; 197 1] ; Furthermore, as pointed out by Blum and Fahr or (3) the interstellar hydrogen has a relatively high [197Oc], galactic and local components of the diffuse temperature lo4 as discussed earlier [Thomas, (2'1% OK) Lyman a radiation can be distinguished by the fact that 1971J . If TI is small and the effects of radiation pressure the former must be constant in time, while the latter do not exceed those of gravity (p 0.7), then a small must reflect changes in the solar Lymancw flux, espe- bright patch should occur near the center of the region cially the 27-day variations [Meier, 19691 . Observations of low intensity as a result of the high neutral hydrogen of such time variations could remove these uncertainties, density in the vicinity of the axis of symmetry [Blum as could observations made at greater distances from the and Fahr, 197Obl. However, the bright patch may not sun (beyond the orbit of Jupiter) where the galactic 63 1 I I I I I I " -180 -120 -60 0 60 120 I80 0 (degrees1 Figure 16(d) The same as (e), but with ae = 270", showing the displacement of region of maximum intensity. a,= oo - 20, I I I I I 1 I I I I I I 30 60 90 120 150 180 8 (degrees) Figure 16(c) Intensiv of scattered W84 HeI; we have used the same parameters as in (a)and (b), and assumed that the interstellar gas contains 8 percent He. Note that there is a maximum in the direction of the sun as well as in the direction of the downstream axis of symmetry. m x 00 The former maximum occurs in the case of helium but 05: 00 30 0 60 0 900 1200 1500 I800 not hydrogen because the helium is easily able to pene- 8 (degrees1 trate to within the orbit of earth. The effect of varying r.(He) fkom 0.5 to 1.0 is shown for the case see= 180". Figure 16(e) Intensity of scattered A304 Hellresulting he solar line width is assumed to be 0.023 .A [e.g., from the presence of singly ionized helium in the solar Donahue and Kumer, 19711 which is probably too wind for ae = 0", and ri(He) = 0.5. Most of the scattered small, however the scattered intensity vanes approxi- light comes from the direction of the sun. The scattered mately inversely with the line width, and for other intensity is approximately inversely proportional to the values the scattered intensity can be obtained by simply assumed solar line width, which in this case has been rescaling this diagram. taken to be 0.06 A component should be dominant. The suggestion by two ways of determining whether or not the tempera- Thomas [1971] that TI is relatively large is comistent ture of the interstellar gas is large: (1) from measure- with theoretical work on the intercloud component of a ments of the intensity of scattered X584He1, which neutral interstellar gas, This is certainly a possibility if should show a very pronounced maximum in the direc- tion of the axis of symmetry ae = 180°, and to some the background soft X-ray flux is sufficient to heat the extent for other values of ae if TI is small (figs. 16(c) interstellar gas [Silk and Werner, 1969; Werner etal., and (d)); and (2) from measurements of the width of the 19701, but probably not if it is necessary to rely on scattered solar Lyman a line, made when ae = 90", 270", heating by low-energy cosmic rays. There are at least and away from the sun (8 = 90", 270°), where scattering 632 should take place in the line center if TI is small. In this intensity minimum occurs in the direction away from case, a hydrogen absorption cell should have a the sun except when ae = 180" (figs. 16(c) and (d)). pronounced effect on the observed radiation unless there These effects could also be found in the case of Lyman a is a substantial galactic component [Blamont, private for observations made in the region r> 5 AU. In our communication, I97 I] . calculations we have adopted a solar line width of The resonantly scattered helium radiation (A584 He4 0.0238 [Donahue and Kumer, 19711 ; this line width is of particular interest since helium atoms penetrate may be too small, in which case the expected scattered easily to within the orbit of earth, and hence there is intensity should be reduced proportionately downward always a maximum in the direction of the sun. The from the values sfiown in figures 16(c) and (d). There (a) SUI on 13 September I969 363 300 240 IW 120 60 (b)SU 2 on 15 December 1969 Figure 17 Contour maps of the solar Lyman 01 background in ecliptic coordinates for the three spinups (SUS) of OGO5: (a)SUl on 13September 1969; (b)SU2 on 15 December 1969; and (c) SU3 on I April 1970. These figures have been taken from Thomas and Krassa [I9 711. Similar diagrams have been given by Bertaux and Blamont [197I]. (d)Contour map of scattered h304HeII radiation based on the observations of Johnson et al. [1971],and taken from 0gawaand.Tohmatsu 119711. 633 Figure 17 Continued. have been only a few observations of the diffuse helium center of the solar line, and have a half width of about radiation to date; these indicate an upper limit of -2 R la as a result of the random velocities of the hydrogen for the intensity of the diffuse 5848 radiation from atoms produced in this region. This emission has been extraterrestrial neutral helium in a direction away from considered by Patterson et al. [ 19631 and Hundhausen the sun [Youngetal.,1968;Johnsonetal., 1971; Ogawa [1968b] in an attempt to account for an observation of and Tohmatsu, 19711. We suggest that these observa- -540 R of Doppler-broadened Lyman a reported by tions be pursued for various combinations of values of Morton and Purcell [ 19621. However, this observation ae and 8, and also outside the influence of the neutral showed only that there is more or less isotropic emission helium envelope of the earth. outside k0.048 from the center of the line, which is There must be diffuse Lyman a and A304 HeZZ radia- easily accounted for by scattering from interstellar tion resulting from the presence of neutral hydrogen and hydrogen, since it is moving at velocities exceeding singly ionized helium in the solar wind as indicated in 20 km sec-l, and will in general produce Lyman a emis- figure 14. The Lymana produced in this way can be sion displaced about 0.18 from the line center. The cal- distinguished from that described above since it should culations of Patterson et al. [ 19631 and Hundhausen be red-shifted by about 1.58 from the center of the [ 196883 indicate that the isotropically moving hydrogen solar line. However, calculations by Gregory [1971] atoms must be produced in the region r = 10 to 20 AU indicate that for p = 0.8, N(H) =0.1 cmW3,and cye = 0, to explain the observed emission intensity. However, the intensity of the scattered Lyman a from this source since it seems likely that as noted, the transition to sub- is only -3.5 R in the direction 8 = 0, which is probably sonic flow does not occur within 50 AU, this should not too small to be distinguished from the galactic back- be an important source of diffuse Lyman cy. ground. The intensity of scattered 304a radiation is It would be of interest to extend the calculations shown in figure 16(e) for the case ae = 0; as in the case described in this section to include other emissions. of radiation scattered by neutral helium, maximum Undoubtedly other hydrogen and helium lines are emission occurs in the direction of the sun. For present in the diffuse radiation with intensities propor- 8 120' to 1SO', the intensity of the scattered radia- = tional to those of the A1216 and A584 HeZ emissions tion is -1 R, which is consistent with an observation of HZ [Reay and Ring, 1969; Tohmatsu, 19701. Furthermore, 1.O 50.2 R reported by Ogawa and Tohmatsu [ 197 11 . Another possible source of diffuse Lyman a radiation the possibility of scattering by other species is also of is scattering from fast hydrogen atoms produced by interest, and on the basis of table 1 it would appear that charge exchange in the region of subsonic solar wind neon in particular might produce a detectable emission, flow beyond the shock transition. In this case the emis- since ri(Ne) is small and neon is a relatively abundant sion should be centered at -0.38 to the red side of the element. 634 1; In 1. I I i I: i I ...* ...... :i? = i :P $1 21 0' 0: 10. I, ; 8. I- Y) a Y 63.5 INTERACTION OF THE SOLAR WIND WITH NEUTRAL INTERSTELLAR GAS In this section we consider the effect on the solar wind of the neutral component of the interstellar gas. Various (63 ) aspects of this interaction have been considered by several authors [Axford et al., 1963; Patterson et al., Here cs(w;) is the plasma sound speed, m is the 1963; Axford and Newman, 1965; Dessler, 1967; Kern proton mass, and c is the speed of light. The rates of and Semar, 1968; Hundhausen, 1968a; Fahr, 1968a,b. mass, momentum, and energy production are repre- 1969, 1970; Blum and Fahr, 1969, 1970a,b; Semur, sented by q, p,and @, per proton mass per unit 1970; Holzer, 1970, 1971; Holzer and Axford, 1971; volume, respectively, with the subscripts p and c refer- WaIIis, 1971 ; Bhatnagar and Fahr, 19711 . In this discus- ring to the processes of - photoionization and charge sion of the problem, we shall follow the treatment given exchange. Thus by Holzer [ 19711, which is similar to the work of Wallis [1971] but somewhat more detailed. It has been shown that the interstellar neutral gas penetrates to within a few astronomical units of the sun before becoming significantly affected by losses due to photoionization and charge exchange. Since the solar wind is expected to extend to a heliocentric distance of the order of 100 AU in the absence of any interaction with neutral interstellar gas, it is evident that as a first approximation we can treat the solar wind as if it were flowing through a uniform background of neutral gas with density N, temperature Tn, and velocity V relative to the sun. Even with this assumption, the problem is very difficult since there must be nonradial flow in the region of supersonic solar wind flow as well as in the @ =m(V-u) subsonic region; furthermore, the effects of the inter- planetary magnetic field should not be ignored. How- ever, a simplification can be achieved if we consider only the directions parallel and antiparallel to the velocity vector of the interstellar gas, since the flow can then be treated as being approximately radial. In this case, as noted earlier, the azimuthal component of the inter- planetary magnetic field is dominant in r 2 5 AU and 128k V=ap (aT+ Tn)+(u-V>" accordingly we may assume that Br =O and [s 1 B = BG 0: (UT)-' . If we represent the solar wind as a steady, radial, where up is a mean photoionization cross section, 8 is spherically symmetric flow of a protonelectron plasma the average energy of a photoelectron produced in the through an atomic hydrogen gas, then neglecting viscos- photoionization process, T(=Te+ Tp) is the plasma ity and heat conduction, the equations of motion can be temperature, aT is the proton temperature, and a, is the formulated as follows: resonant charge exchange cross section. The velocity of the interstellar neutral gas Q is taken to be radial. 1d The electric field the current (i), and the mag- -- (nur2) = q (E), r2 dr P netic field satisfy the equations 4n 1d Id VXB= -j -- (nu"' ) = - - - (ncs2) -G3tz n r2 dr Y dr - r2 C 636 Thus, with the assumption that the magnetic field is In the special case where E = G = V=T, = 8 = 0, azimuthal, we find that equations(74) and (75) reduce to a single first-order differential equation in terms of the Mach number M = u/cs: (72) C 1dM y+l y-1 - (J!P-l)=T +-(W-l) Mdr- r and 4p Y2-% +-J!P3y-14 --(-34 +-27 We can now write the equations of motion in the form (80) 1 du 2 Gm - - (u2-cs2-cA2) =; ~,2-~+ 9 (74) Note that v/u is a function only of M, since in this u dr special case (69) reduces to 112 v=upu I+-- r u dr ( l9?!; ;) Many features of the solutions of the more general equa- with the following definitions: tions (74) and (75) can be understood from an examina- tion of the solutions of (80). In effect, this is the prob- lem considered by Semar [1970], who carried out a numerical integration of the equations of motion (61), (62), and (63), for this case taking y = 2, a! = 1, and omitting the term enclosed in. the bracket in (81). qP Unfortunately, the existence of a singular point at M=I c;-z (7-1); d (76) was overlooked in Semar’s analysis, which consequently implies that a shock-free (“transonic”) transition to sub- sonic flow will always occur. In fact, as pointed out by Wallis [1971] and Holzer [1971], this is not necessarily a! Equation (80) has a singular point at which (M“- 1) and the right-hand side of (80) vanish simultaneously. If this point is defined to be at r = rc, then (77) 1 rc = (3-7-z~f, 4fc + [4R2fc2+ (20 + 4y)Rfc + (~-3)~]~/~1 (82) ~2 Bo2u02r02 where, using (64) and (69), and assuming that the CA2 = (79) =zm47rm33r2u neutral gas is optically thin in the region ro < r < rc, we have where the subscript o denotes quantities evaluated at a (83) point r = ro close to the sun. 637 Id I Note that under typical solar wind conditions and with g IO' 0.05 < N < 1 .O ~m-~,re 0: l/N. The form of the characteristics passing through the singular point can be determined by expanding (80) 10-1 about M= 1, r = rc. Defining X = M-1 and E = (r/rc)-l, 00 05 IO 15 20 we obtain to first order in h and RADIAL DISTANCE (rk) (a) b2 +4a >O where IO' (2y-l)(y+ 1) 'c y+ 1 "r bzy-1- --- 2 101 4 rc+R 4 'Cfc 10-2 00 05 10 15 20 RADIAL DISTANCE (r/r) (b) b2 +4a Fire 18 CRaracteristics of equation (58) [from A characteristic passing through the singular point has a Holzer, 1971] . Note that a transonic solution is possible dope at the singular point given by S = lim(A/t). Hence S E- in (a) but not in (b) where the singular point is a focus. can take @ne one of the two values On examination of the possible solutions displayed in figures 18(a) and (b) one finds that only one character- istic leads to infinity in each case, and that these are such that the pressure p + 0 as r -j m; there are no solu- tions that yield a finite pressure at infinity. For the case If b2 t 4a >a> the characteristics passing through the of moderate and low interstellar densities, the character- singular pointtcan have slopes St or S-. If b2 + 4a = 0, all istics are as shown in figure 18(a), and it is always characteristics passing through the singular point have possible to find a solution starting with suitable condi- slope St = S-= b/2. If b2 t 4a < 0, the characteristics in tions at r = 5 AU, say, which contains a smooth transi- the vicinity of the singular point are spirals and the tion to subsonic flow and eventually has vanishing corresponding solutions are multivalued in M. A family pressure at infinity. For somewhat higher interstellar of solutions of (SQ 3 for which b2 t 4a > 0 is shown in densities (N? 10 ~m-~,depending on the value chosen figure 18(a) and a family for which b2t 4a < 0 is shown for other parameters), the characteristics are likely to be in figure 18(b). as shown in figure 18(b); in this case, again it is possible 638 to find a solution starting with suitable conditions at 11111111111~1- r = 5 AU, and with vanishing pressure at infinity, but it is - necessary to insert a shock at a correctly determined - - I - location near the singular point to make the solution -U - single valued. In the numerical treatment carried out by c Semar [ 19701 these complexities have passed unnoticed, u) 0W although the results obtained are probably a fairly good I - representation of the solution except in the vicinity of the singular point. It should be noted, however, that the 00 : form for the collision frequency v used by Semar is 0 50 100 150 200 inadequate if the Mach number is not large, and the RADIAL DISTANCE TO SONIC POINT (AU) former has an important effect on the properties of the (a) Radial distance to the sonic point (see characteristics in the subsonic region. $gs. 16(u), (b))for various assumed values of the relative The solutions of equation (80) are useful in that they uelocity of the interstellar gas, and interstellar hydrogen suggest what we can expect in the more general case densities in the range 0.0 to 1.0 ~m'~. involving a pair of differential equations - namely, (74) and (75) - that must be solved simultaneously for two unknowns u and cs' Holzer [ 19711 has treated this prob- lem by a straightforward integration of (74) and (75), having assigned values for u, p, i? and B at a point relatively close to the sun (r = 5 AU) where the solar wind can be expected to be essentially unaffected by the presence of neutral interstellar gas. The integration proceeds in the direction of increasing r, until the Mach number MT= u/(cs2 t CA~)''~decreases to unity, or (if 3 a shock transition is inserted) until the solutions w I00 approach some asymptotic form that can be recognized. 5 50 100 150 200 In this procedure, there is some danger that singular RADIAL DISTANCE IAU) points might be missed and that the solutions might be (b)Solar wind velocity as a function of radial distance unstable; however, the results obtained appear to be fiom the sun for the case in which the interstellar gas has satisfactory. Examples of solutions of equations (74) a velocity of 20 seC' towards the sun, and for var- and (75) obtained in this manner are shown in fig: km ious assumed values the interstellar hydrogen density. ures 19(a) and (6) and 2O(a)-(c). The initial conditions of Note that for the preferred value (0.1 cK3)the solar have been taken to be uo = 400 km sec-', wind velocity reduced to GOOkm sei-' at a distance no = 0.2 cso = 15 km sec-', Bo = 0.7X10-5 gauss is of 100 AU where the shock transition might be expected at r = ro = 5 AU, together with a range of values for the to occur (see fig. 7). density (N) and velocity (V) of the interstellar neutral gas. Figure 19 Effect of interstellar neutral gas hydrogen The location of the sonic point (M = 1; r = IC) on the solar wind [ fiom Holzer, I9 71 ] . is shown in figure 19(a) for a range of values of N and V; this yields an outer limit for the position of any possible shock transition in the flow. Note that the location of assume that Nz0.1 cm-3 and V=-20kmsec-', as the sonic point is quite sensitive to the density of the suggested by the observations described in the preceding neutral interstellar gas, varying roughly as N- ' . Further- section, it can be seen that the solar wind speed is more, there is a substantial difference between the values reduced from 400kmsec-' at 5AU, to about of rc on the upwind (V < 0) and downwind (V> 0) 270 km sec- at 100 AU, which we have estimated %obe sides of the sun, which should produce a distinct asym- the likely position of the shock transition. Since photo- metry in the region of supersonic solar wind flow - for ionization is a relatively unimportant process the mass example, for N= 0.1 ~m'-~and V=20, -20 km sec-', flux in the solar wind is essentially unagected by the rc = 250 AU, 400 AU, respectively. The variation of deceleration and hence the momentum flux (or ram solar wind speed with distance is shown in figure 19(b) pressure) is reduced in proportion to the velocity. This for V=-20 km sec-' and various values of N. If we result can also be deduced easily by noting that the 639 10' Y (L *3 *Y a 10'' IO-' 10' 10' 103 RADIAL DISTANCE (A U) (a) Variation of the Mach number with radial distance (b) Variation of the plasma densiry n, totd pres- for various assumed positions of the shock transition. sure pT, and magnetic field pressure B21Sn, as a Note that the solutions terminate at a finite distance finction of radial distance for the case where the shock from the sun if the shock is inserted too close to the is inserted at 15 AU. critical point. 10' IO' - U UIW 10 ' > k sU >W IO 10-1 IO' IO' 10' 10' io5 RADIAL DISTANCE (A U 1 (e) Variation of the solar wind speed u, the sound speed cs, and the Alfie'n speed CA, with radial distance for the case in which the shock transition is inserted at 15 AU. Figure 20 Variation of solar wind parameters with distance for the case V=20 kmlsec-', and N= 1.0 emd3 worn Holzer, 19711. Note that the solutions are somewhat similar to those shown in figure 7 for distances less than a few hundred AU. At very large radial distances the solutions may not be at all realistic; however, it is inter- esting that the asymptotii forms create a balance between the stresses associated with the magnetic field and the frictional interaction between the plasma and the neutral inter- stellar wind. 640 mean free path of solar wind protons against charge exchange is to cool the plasma, thus permitting the mag- exchange is approximately 300 AU if N = 0.1 ~m-~,and netic field strength to grow more rapidly than it would hence on reaching a heliocentric distance of 100 AU, the otherwise (fig. 8). Charge exchange should not be impor- plasma must lose (l-e-1’3) R!. 1/3 of its momentum per tant in the upwind subsonic region as far as the unit mass. Formally, this is equivalent to making the pressure/momentum balance is concerned since the approximation MT~>> 1 in (74) and putting streaming velocities are low. However, the increase of q1 =-[(y t 1)/2]vu. Thus we see that loss of magnetic field strength enhances the inward force due to momentum due to charge exchange within the region of the tension stress associated with the field, and as supersonic solar wind is likely to have a significant effect noted earlier, this could produce a small but significant on the location of the shock transition (in the sense that radial pressure gradient in the subsonic region. This rs might be reduced by 10 to 15 percent from the value effect, together with the loss of solar wind momentum estimated previously. due to charge exchange, could easily render our previous Solutions of equations (74) and (75) for the estimate of rs = 100 AU for the location of the shock postshock flow have also been considered by Holzer transition too large by perhaps 20 to 30 percent [Kern [1971], again with the assumption that the flow is and Semar, 19681. However, a better self-consistent strictly radial; an example with V=20 km sec-’ and treatment is required before we can be sure that this is N= 1 ~m’-~is shown in figures 2O(a)-(c). There appear the case. to be no solutions that extend to infinity in the upwind On the downwind side of the sun (in the “tail” of the direction (V< 0); in the downwind direction (V> 0), heliosphere) we can expect the solar wind in the super- however, such solutions exist, provided the transition to sonic region to lose momentum due to charge exchange, subsonic flow takes place within a certain critical radius so that the shock transition (if any) is not as far from (r=rI). The Mach number is shown as a function of the sun as it would be otherwise. In the subsonic region, radial distance in figure 20(a). Note that if the shock the plasma should at first be cooled by charge exchange location is such that rs >rl ,the solution in the subsonic until p drops to such low values that the magnetic field region bends upward, intersecting the line MT= 1 with has an important influence in controlling the flow. infinite slope, and does not extend to r+-. In cases Momentum exchange due to photoionization and charge where rs 64 I recombination unless the temperature of the interstellar of the interplanetary magnetic field does not signifi- neutral gas is relatively low (TI= lo2 OK). cantly affect particles with energies of the order of 100 GeV and higher, the regular (Archimedes spiral) THE INTERACTION BETWEEN GALACTIC field can bend the particle trajectories so that any anisot- COSMIC RAYS AND THE SOLAR WIND ropy becomes obscured. This is evident in figure (21), Although galactic cosmic rays must pass through the which also shows contours on which particles have gyro- outer regions of the heliosphere to reach the earth, they radii in the local mean interplanetary magnetic field do not appear to carry much information about these equal to the scale length of the field (BlVB). Within regions. Nevertheless, it is useful to consider various each contour, the trajectories of particles with energies aspects of the behavior of galactic cosmic rays within the less than the designated energy can be completely turned heliosphere, especially as it is likely that the situation by the magnetic field. McCracken [unpublished, but see will change when observations from deep space probes become available. At very high particle energies Q200GeV), it is 50 found that the solar wind and the interplanetary mag- netic field have no effect on the mean cosmic ray inten- sity observed at the earth in the sense that there is no solar cycle modulation or evidence for Forbush decreases. It might be expected, therefore, that if the cosmic ray “gas,” including these particles, has a velocity Vcr relative to the solar system, then this should be detectable as a sidereal diurnal variation of the intensity. That is, the differential intensity in the kinetic energy range (T, T+dT) should have the form j(T,t) =jo(T)[l + t(T)cos $(t)l (90) where t is the time, $(t) is the angle between the detector acceptance direction and Vcr, and ((T) is the anisotropy given by Figure 21 Contour diagrams similar to those shown in figure 3 indicating regions of the interplanetary medium in which cosmic rays of a given energy have a gyroradius in the local magnetic field which is less than the charac- teristic scale of the field (BfVB). The diagram is some- where v is the particle speed corresponding to kinetic what misleading in the low latitude region beyond the energy T, and Cis the Compton-Getting factor: few astronomical units since it does not take into account the sector structure of the interplanetary mag- ia netic field. Since the sectors have a characteristic scale of C(T)=1-- -(aTU) only a few astronomical units, the trajectories of parti- 3U aT cles with gyroradii much larger than this will be essentially unaffected by the field. where a=(T+ 2Eo)/(T+ Eo), Eo is the particle rest energy, and U(T)= 477j(T)/v is the differential number density in (T, T+dT) [Gleeson and Axford, 1968a; Antonucci et al., 19701 has examined the nature of the Forman, 19701 . At high energies (T>> Eo), it is a good sidereal variations to be expected in the energy range approximation to take a = 1, v =e, and j(T)aT-@ 50-500 GeV (the range appropriate to underground where p 2.5. Thus C=(2+@)/3=1.5, and if muon detectors). He has shown that diurnal and semi- Vcr V = 20 km sec-’ , we should expect that diurnal intensity variations are to be expected, with the ( 0.03 percent, in principle an observable value. diurnal variation varying in amplitude throughout the Unfortunately, the situation is not as simple as this year in such a manner that sidebands in the modulation analysis suggests, since although the irregular component should occur at 364 and 368 cycles/yr. 642 There is some evidence for the existence of a sidereal Urch [1971] have been helpful; however, for many anisotropy obtained from detectors at 60 to 80 m.w.e. purposes it is more convenient to use various approxi- [EZliot et al., 1970; Antonucci et al., 19701 ;however, it mate forms of equations (93) and (94) such as those of is not wholly convincing. For example, underground Fisk and Axford [1969] and Gleeson etal., [1971] that muon detectors at depths less than about 165 m.w.e. are yield asymptotically valid solutions. Gleeson and Axford sensitive to particles having energies less than 100 GeV [1967,19683] and FiskandAxford [1969] have shown [AhZuwaZia, 19711, and the solar diurnal modulation that an adequate approximation for TZ 200 (to which such particles are clearly subject) can produce Me V/nucleon at the earth can be achieved by neglecting pseudosidereal effects that can contaminate any genuine the term S on the left of equation(94); the resulting sidereal variation [Swinson, 197 11 . It is probably neces- equation sary to make use of extensive air shower observations to obtain a good value for the cosmic ray anisotropy, since au cUu=K (95) the particles involved have energies in excess of 10” eV -ar and should not be ncjticeably affected by the inter- planetary magnetic field or solar modulation. Unfortu- is termed the force-field equation. nately, the intensity of cosmic rays with energies If K(r, T) is a separable function of r and T - that is, N> lo1 eV is low, and so far it has been possible only to K(r, T)= K 1(r)~2(T)- this equation can be integrated to place an upper limit of about 0.1 percent on the yield a result similar to Liouville’s theorem: anisotropy. At energies less than 100 GeV, galactic cosmic rays are modulated in intensity in a variety of ways by their interaction with the solar wind and interplanetary mag- netic field; Jokipii [1971] has reviewed the theory of some of these effects. In particular, the cosmic rays where E = T + Eo is the total energy of the particles, and undergo a solar cycle modulation that appears to be the CP can be regarded as a “potential” associated with the result of a reduction of intensity in the inner solar sys- (apparent) force exerted on the cosmic ray distribution. tem due to outward transport of the particles by the In particular, for the case in which K~ = am/c, solar wind with associated energy changes [Parker, 1963, which appears to be a reasonable choice in the vicinity 19653; GZeeson and Axford, 1967; Gleeson, 1969; of the earth for rigidities PN> 1 Gv [Jokipii and Jokipii and Puker, 19701. The equations describing the Coleman, 19681 , cf, = Ize @(r),where behavior of the differential number density U and current density S,assuming that the solar wind is steady and radial, and taking only radial diffusion into account, are It is found that equations (96) and (97) provide a very good description of the (1 l-yr) solar modulation of protons, a particles, and heavier nuclei for energies (94) greater than 100-200 MeV/nucleon. The “potential difference” between the earth and infinity is typically found to be in the range 150-500 MV, and the modula- where u is the solar wind speed and K(r, T)the effective tion shows the predicted Z dependence [Gleeson and radial diffusion coefficient for cosmic rays in the Axford, 1968b; Lezniak and Webber, 19711. Goldstein (irregular) interplanetary magnetic field. et al. [197Ob] and Gleeson and Urch [1971a] have It is rather difficult to obtain solutions of equa- found from numerical solutions of the full equa- tions (93) and (94) with the boundary conditions appro- tions (93) and (94) that galactic cosmic rays with initial priate to the modulation problem (U+ U,(T) asr +-, energies less than 100-200 MeV/nucleon are effectively ?S -+ 0 as r + 0), although classes of exact solutions excluded from the interplanetary medium at the earth’s exist if it is assumed that cw can be treated as a constant orbit (fig. 22), which is consistent with the idea that the [%k and Axford, 19691. hely numerical treatments modulation can be described in terms of a force field. of the problem such as those of Fisk [1971a], Gleeson Clearly it should be considered a prime task of any deep and Urch [1971b], Lezniak and Webber, 119711, and space mission to measure the spectra of various species 643 I directly by (97) as - Id‘ v) On taking a value for the diffusion coefficient consistent != z with that obtained by Jokipii and Coleman [ 19681 ,one 3 finds that the gradient is of the order of 20-30 percentlAU for 1 GeV protons, and decreases at somewhat higher energies approximately as 1/P. There is some evidence that gradients of this general magnitude exist, at least for P _> 1 GV, from spacecraft observations [O’Gal/agher and Simpson, 1967; O’Gallagher, 19671 , measurements of radio activity in meteorites [Forman et al., 19711 , and observations of the associated cosmic ray anisotropy perpendicular to the ecliptic [ Yoshida et al., 19711. If such gradients were to exist throughout the region in which the solar wind is supersonic (say, r 5 50 AU) then the total energy density of galactic cosmic rays lo4 outside the heliosphere would be enormously larger than that observed at the earth. In fact, it seems unlikely that -1 \ - the unmodulated spectrum for T_>1 GeV has a steeper slope than a spectrum with j(T) a T-P with p 2.5. -5 \ In I I 1111111 I I I11111 I I I I lllL This spectrum matches the (apparently unmodulated) I” I I o2 I I o4 observed spectrum in T_>lo2 GeV. It would require IO o3 that the energy density of galactic cosmic rays be greater PROTON KINETK ENERGY (MeV) by only a factor of 2 or 3 outside the heliosphere than near the earth for T 1 GeV, and hence that the radial Figure 22 A series of essentially monoenergetic proton _> spectra in interstellar space (solid line) and the corre- gradient cannot be maintained at the value quoted above sponding modulated spectra at earth (dashed lines) for more than a few AU from the sun. This argument is obtained from numerical solutions of equations (93)and essentially the same as that of Gleeson and Axford (94) [from Goldstein etal., 1970bl. ne ‘yorce-field” [1968b], who for convenience assumed that solution (eq. (96)) provides a very good fit to the K (r) a e”rO in (97), and found that if the “potential envelope of the modulated spectra for energies above difference” between the earth and infinity is about lo2 MeK It is important to note that particles 100-200 MV, then ro must be of the order of 1 AU. with energies 5 200MeV in interstellar space (i.e., Additional confirmation that the modulation of unmodulated peaks 1-3) make a negligible contribution galactic cosmic rays with T_>1 GeV occurs in a region to the spectrum at earth, where this energy range is that is effectively contained within a few AU of the sun dominated by particles having higher energies in inter- arises from comparisons of the solar cycle variation of stellar space (i.e., unmodulated peaks 4-6) and which the cosmic ray intensity (measured by neutron monitors, have lost energy as a result of the modulating process. In for example) and various indices of solar activity effect, galactic cosmic rays with energies less than [Simpson, 1962; Simpson and Wang, 1967, 1970; 100 MeVare unobservable at earth. Hatton et al., 1968; Guschina et al., 1970; Kolomeets etal., 1970; Wang, 19701. It is argued that the lag obtained from a cross correlation analysis involving of cosmic rays as a function of radial distance from the these quantities should provide a rough estimate of the sun as a basis for determining the unmodulated spectra time required for the solar wind to traverse the modu- and the variation of the diffusion coefficient with lating region. A cross correlation between cosmic ray distance. intensity and the geomagnetic activity index Kp yields a The radial gradient of the cosmic ray density or inten- lag of the order of 6 to 12 months, which suggests that sity at energies greater than -200 MeV/nucleon is given the radius of the modulating region might be as large as 644 50 to 80 AU [Donnan and Donnan, 19661. However, it has been found that if a cross correlation is made s=ul [ [~tfa,-l)JA between the coronal green line (A Fe XIV 5303), the lag is only of the order of 1 month, which yields a radius of -7 AU for the modulating region of particles detected by neutron monitors [Simpson and Wang, 1967; Barker and Hatton, 1970;Kolomeets et al., 19701. where B is a constant obtained in the integration of It is not possible to draw similar conclusions for the (loo), and it is assumed that U a T-p everywhere. low energy end of the galactic cosmic ray spectrum, Beyond the shock transition (inr >rs), we can treat the simply because the degree of modulation at low energies flow as if it were incompressible to a first approxima- is expected to be so large that we are completely igno- tion, so that u = (1/4)~~(r~/r)~;hence, equations (93) rant of the form of the unmodulated spectrum (fig. 2 1). and (94) become Furthermore, at the lowest energies, the spectrum observed near the earth appears to consist mostly of particles of solar origin [kfbrd, 19701. The effects of adiabatic deceleration associated with the expansion of the solar wind dominate the behavior of these particles. It seems likely that galactic cosmic ray protons and heavier nuclei with energies of the order of 1-10 MeV/nucleon are to a large extent excluded from the region of supersonic solar wind flow, although they where C(T)=CT7u arises from the integration of (93). may penetrate the region of subsonic flow more easily The appropriate solutions of these equations are such since the effects of convection in this region are less that U, S+Oasr+w: important. At the shock transition separating these regions it is possible for significant acceleration of low- energy particles to occur, thus producing, in a sense, a local “source” of cosmic rays [Jokipii, 19681. This effect can be demonstrated by means of appropriate solutions of equations (93) and (94).Let us suppose that the effective radial diffusion coefficient is small (K < (99) au Note that the enhancement of the cosmic ray density u1(U-U0) = K1 - ar resulting from the presence of the shock (C/A) depends on the value of the diffusion coefficient in the subsonic having put U=UIand(a/ar)(r2S) =r2(aS/ar), and inte- region, and may be very large if Kz/ulrs << 1. The grating (93). Thus we find that if K = K is assumed to be cosmic ray intensity near the earth is not significantly independent of r and T, and a is taken to be constant, affected by the presence of this enhancement since if then K << ur the particles are convection dominated and downstream boundary conditions have little influence. It is possible to solve equations (93) and (94) exactly with the solar wind distribution assumed in the above analy- sis, provided a is taken to be constant, K is independent 645 of T, and Uo:T+ [Fisk, 19691 . The solutions are con- where $“ STdT is the net radial flux of energy in 0 sistent with those obtained here in the limit urs/K >> 1 ; cosmic rays; since udr/dr > 0, it is evident that there is a they also show that it is possible to produce an enhance- net transfer of energy from the solar wind to the cosmic ment of the intensity of galactic cosmic rays in the rays [cf.Jokipii and Parker, 19671 . vicinity of the shock transition under suitable condi- In the treatment of the problem by Axford and tions. On this basis, therefore, it should be expected that Newman [1965] and others, the right-hand side of (1 12) a low-energy cosmic ray enhancement similar to those was omitted; since this term represents an effective heat associated with propagating shocks [Fisk, 197 lb] is sink for the solar wind, the results obtained by these associated with the shock transition terminating the authors are such that the Mach number decreases toward supersonic solar wind, and this may provide useful unity too rapidly with increasing radial distance. Wallis confirmation of the presence of the transition. [1971] has pointed out that when equa- The possibility that the solar wind itself might be tions (109)-(112) are reduced to a single equation for significantly affected by cosmic rays has been considered the Mach number in terms of radial distance, the by several authors [Axford, 1965;Axfordand Newman, characteristics of the equation have properties similar to 1965; Dorman and Dorman, 1968; Modisette and those described in the preceding section for the case of Snyder, 1968; Sousk and Lenchek, 1969; Wallis, 19711. interaction with interstellar neutral gas. In particular, In the early analyses it was assumed that there is no there is a singular point at a large radial distance, the exchange of energy between the solar wind and the nature of which depends on the values assumed for the cosmic rays, and that the cosmic ray pressure n satisfies various parameters involved. It is found, however, that if the convection-diffusion equation the interaction is sufficiently strong that a significant dn effect is produced on the solar wind, then the cosmic ray K-=un dr pressure required in the interstellar medium is unaccept- ably high (more than 10 times the value observed at the position of the earth). An example of a solution where K(r) is the “effective” diffusion coefficient and obtained by Sousk and Lenchek [1969] for the case where K is independent of r is shown in figure 23; it is evident that quite unreasonable cosmic ray pressures are necessary in this case to reduce the solar wind speed by as much as 25 percent. As a corollary to calculations of In more recent treatments [Sousk and Lenchek, 1969; this sort, it is found that the cosmic ray pressure cannot Wallis, 1971), which take the energy exchange into be prevented from becoming excessive unless the net account, equation (108) is used together with the diffusion coefficient K increases more strongly than fdlowing equations for the conservation of mass, linear with radial distance. This is consistent with the momentum and energy d the solar wind plasma: conclusion of Simpson and Wang [ 19671 and others that the dimensions of the modulating region must be rela- d tively small (-7 AU). (pur2)= 0 dr- Cosmic rays are affected by the regular component of the interplanetary magnetic field as well as by irregular. ities, and tis a consequence they tend to rotate with the magnetic field about the sun. The aesulting anisotropy is observed at the earth as a soIar diurnal variation of the cosmic ray intensity, which is of the order of 0.3 to 0.4 percent, given approximately by On combining equations (1 10)-(112), we can show that p/pT = constant. The right-hand side of equation (1 12) is easily shown from (933 to be given by where L? is the angular velocity of the sun [Ahluwalia and Dessler, 1962; Parker, 1964; Axford, 19651 . This anisotropy has yet to be detected 8t low energies (T_<2 GeV) from spacecraft; it is possible that it is absent at energies less than about 200 MeV/nucleon 646 which case the anisotropy given by (1 15) must dis- appear. This effect is probably not as pronounced at the orbit of earth as suggested by Jokipii and Parker [ 19691 -60 w [Subramanian, 19711; however, it is expected to increase in importance with increasing heliocentric -§o to EF distance and ultimately may be capable of keeping E+ within reasonable limits. A breakup of the interplanetary ma'gnetic field sector structure as described earlier would serve the same purpose. It should be noted that in the presence of a large value of E+, the medium comprising the solar wind plasma and the cosmic ray gas would be liable to streaming instabilities [Wentzel, 19691 , which 2: ul in turn would tend to reduce both K(I and KL and thus increase the overall modulation. ul The problem of how cosmic rays penetrate the region occupied by the solar wind and interplanetary niagnetic 0 20 40 60 80 100 I20 140 field is of some interest. In the absence of direct connec- DISTANCE FROM SUN tAU1 tion between the interplanetary and interstellar magnetic Figure 23 Variation of the solar wind speed V, sound fields it is difficult to understand how lowenergy elec- speed C, and cosmic ray pressure n, with distance from trons, for example, could enter the interplanetary the sun for the case in which the effective diffusion region with any reasonable efficiency. Parker [ 1968 J has coefficient K is constant [from Sousk and Lenchek, suggested that the effect of field line wandering 19691. It should be noted that in order for the solar described above might be adequate to bring any given wind speed to drop to 300 km sec-I, the cosmic ray field line sufficiently close to the interface between the pressure must rise by a factor of 10 over its value at the solar plasma and interstellar region to allow it to receive orbit of earth, which seems unacceptably large. nesolu- its full quota of cosmic rays, regardless of where the tions have not been carried through the singular point of field line intersects the surface of the sun. Alternatively, the equations where the solar wind and the sound speed we would argue that the access of cosmic rays can take are equal [cf. Wallis, 19 71] . place most easily if the interplanetary field is connected directly to the interstellar medium. There is little reason where c(7) =O [Gleeson and Axford, 1968aj. It is for believing that this is not so, and in the case of the difficult to escape the conclusion that this anisotropy interplanetary magnetic field and the geomagnetic field, must increase with increasing heliocentric distance, and for example, the evidence for interconnection is very become very large indeed at distances of the order of compelling [Akasofu and Axford, 19711 . 50 AU. However, the anisotropy can be suppressed in Schatten and Wilcox [ 1969, 19701 have suggested two ways: by (1) a gradient of the intensity perpendic- that interconnection of the interplanetary and inter- ular to the ecliptic, and (2) isotropy of the diffusion stellar magnetic fields could explain the 20-year cycle of tensor. The enormous intensity gradients necessary for the diurnal variation of the cosmic ray intensity at the any significant effect in this respect are difficult to earth reported by Forbush [1967, 19691. It is argued accept, but the second possibility may well be impor- that the effect is related to the 20-year solar magnetic tant. A more accurate form of equation (1 14) is cycle, and that the access of cosmic rays is controlled to some extent according to whether or not the solar mag- netic field is favorably directed so ihat cosmic rays have easy access to the heliosphere. In fact, the situation is quite complex, and it is not obvious that this is the correct explanation. However, it should be noted that a where KII and KI are the diffusion coefficients parallel similar situation exists in the case of access of aniso- and perpendicular to the mean magnetic field direction, tropic fluxes of solar energetic particles into the earth's respectively. It has been pointed out byJokipii [1966b, magnetosphere, where the direction of the inter- 19711 and Jokipii and Parker [ 19691 that as a result of planetary magnetic field can lead to a pronounced the wandering of magnetic field lines with respect to the asymmetry in the particle fluxes observed at low alti- direction of the mean field it is possible for KL + K 11, in tudes in the polar magnetosphere [Reid and Sauer, 647 1967; Engelman etal., 1971; Van Allen etal., 19711. where subscripts 1 and 2 refer to upstream and down- The possibility that interconnection of the interplane- stream conditions, respectively. For given values of Q tary and interstellar magnetic fields might affect the and H, these equations are sufficient to determine u(r), intensity of low-energy cosmic ray electrons has been h(r), and rj, provided we also specify conditions at r = R. discussed by Fisk and Van Hollebeke [ 19711 in an attempt to explain the “quiettime increases” of these ACKNOWLEDGEMENTS particles [Simnett et al., 19711 . Kovar and Dessler The author wishes to thank P. M. Banks, C. A. Barth, J. [ 19671 have suggested that the sidereal anisotropy of E. Blamont, W. H. Chambers, C. W. Cranfdl, C. T. galactic cosmic rays might be related to the asymmetry Gregory, T. E. Holzer, H. E. Johnson, T. N. L. Patterson, of the outer heliosphere. As noted previously, however, B. J. Rickett, F. Scherb, C. E. Thomas, B. A. Tinsley, M. the particles that can be expected to show a sidereal P. Ulmer, and M. 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Sonett For the distant solar wind the energy density decreases as l/rz at least. At great distance, say a hundred astronomical units, a simple calculation suggests that the galactic cosmic ray energy density exceeds that of the solar wind. The cosmic ray gas is, of course, an extremely hot gas and it is hard to visualize a supersonic flow in such a gas. Now, for the mixture of these two gases (solar wind and cosmic rays) is it still likely that one can talk about some kind of shock transition? W I. Axford I think it is quite reasonable to treat the two gases separately at least as a first approximation, since the coupling is not very strong. One sees plenty of shock waves in the interplanetary medium moving through the cosmic ray gas. A strong shock does affect the cosmic rays noticeably, but I do not think the shock wave itself is much affected. I described these effects briefly at the end of my talk: as the termination of the supersonic solar wind flow is approached we expect that the intensity of cosmic rays will gradually increase to the interstellar level and then remain roughly constant in the sub- sonic region. There may be some acceleration of low energy cosmic rays at the shock as discussed by Jokipii [Astrophys. J., 152, 799,19681 , but the more energetic cosmic rays do not seem to be noticeably affected. If this acceleration of low energy particles is significant and the unmodulated energy density of low energy cosmic rays is much larger than observed near the earth (which is possible) then one definitely would have to take into account the effect of the cosmic rays on the shock wave. Nevertheless there should still be a shock wave of sorts even if it is somewhat smeared out. C l? Sonett It’s true from a structural standpoint, since the gyroradii are larger than the solar system, but there are still JXB forces so I’m not sure that one can completely ignore the fact that the cosmic ray gas has an enormous temperature. Another question is in relation to Geiss’ comment regarding the ratio 4He/3He. The values that he showed for the Apollo samples were 1860-2720, while the value for 658 certain classes of meteorites was about 4000. I wonder, in view of what you said this morning, whether the ’He gradient might be such that the density is higher farther out. This class of meteorites have spent most of their lives outside 1 AU and perhaps that would account for the concentration. W. I. Axford I think that the He concentration in meteorites could be expected to differ from that found in the lunar samples, since the latter are saturated with solar wind gases, whereas meteorites have the outer layers (which should also contain solar wind material) ablated away. Presumably the He concentration in meteorites is affected by cosmic ray implantation and spallation, and any other ’He deep within the meteorites must be primordial in some sense. There is a radial gradient in the density of interstellar 3He, but the density within 10 AU is less than that of the solar wind, and of course the particle energy is very much less. I doubt that interstellar ’He could be important as far as the interior of meteorites is concerned, and it seems unlikely that it could have much effect on the concentrations in lunar dust. C l? Sonett Also, there’s the component from charge exchange. W. I. Axford With the presently accepted values for the density of interstellar gas near the sun, I doubt that this could double the 3He content anywhere in the solar wind within 100 AU from the sun. J. R Jokipii I would like to comment on Dr. Sonett’s point and then ask a question. It seems to me that if there are regions where the pressure of the cosmic rays is great enough, then the cosmic rays will indeed affect the supersonic or the subsonic nature of the gas. I’m not sure whether this will ever happen or not. W. I. Axford Yes, it can in principle (see p.645),but the answer depends upon how well the cosmic ray gas is coupled to the solar wind gas, and we have no way of deter- mining this without actually exploring the termination region. J. R. Jokipii Then one must consider the effect of sound waves actually being propagated through the cosmic ray gas. The sound speed is essentially the total pressure divided by the density, provided only that the wavelengths are long. enough that the cosmic rays and plasma are coupled. We’re talking about 0.01 to 0.1 AU, which is the cyclotron radius of a typical cosmic ray and this is small on this scale. Anyway, that’s just the comment. The question I have concerns the interstellar density of hydrogen. Presum- ably your value is not intended to be an average for the interstellar density of hydrogen. It seems to me from some work of Parker and perhaps previous work of others that there is trouble with the virial theorem if the density of matter is too low. W. I. Axford That is a problem and, of course, people have been searching for the extra mass for many years. But the mass of neutral hydrogen would not be enough even if the number density was as much as 1 ~m-~.My assumed value of 0.1 cm-3 is not neces- sarily the general density, but it appears to be very close to the local value (within -100 pc). As far as the “missing mass” isconcerned,theresimply is not enough atomic hydrogen in the galaxy to matter, and there are observations which suggest that the total amount of molecular hydrogen is also not significant in this respect. The only suggestion that seems to work is that the missing mass is in the form of black holes - which has the nice feature of being almost inipossible to reject on observational grounds. As far as the confinement of cosmic rays within the galaxy is concerned, I think we must begin to revise our models for the behavior of cosmic rays, perhaps by considering “galactic wind” type models in which there is more emphasis on “flow” rather than “leakage” of the cosmic ray gas out of the galaxy. J. C Brandt About the mass of the galaxy, that has been missing for decades, I don’t think that it’s time to get too worried about it. But there are large areas in the galaxy where the number density is apparently 0.1/cm3 and I think we just have to live with that. 659 M. Dtyer I have a question about the term “shock wave.” Whenever we hear the term we think of Rankine-Hugoniot conditions, and we happily make comparisons between these conditions, including magnetic effects, and observations. Are we still talking about the same kind of thin layer in this context, that is, need we worry about these things on this scale, or will the meaning of the word “shock wave” be less well-specified in this context? W I; Axford I don’t see why there should be any problem at all. The scale of the system is very large - 100 AU. As long as there is some mechanism which will make the shock look “thin” on this scale the Rankine-Hugoniot conditions must be satisfied. Since these represent conservation of mass, momentum and energy, there can hardly be much wrong with them. However, one might worry about the effective value of the ratio of specific heats, which appears prominently in the Rankine-Hugoniot relations. I doubt that the number of degrees of freedom can differ significantly from 3 per particle, but elec- tron heat conduction could have a noticeable effect on the transition, as it presumably does in the case of shocks observed in the vicinity of the earth. C R Sonett Would you care to comment on the time-dependent ptoblem, for example, the time-dependent problem that would arise because of the motion of the heliosphere through the local arm. This part of the galaxy is very spotty with clouds and as the solar system moves through it, the background density varies, the field direction changes, and so on. Specifically, how does the information on a change in boundary then propagate back into the solar system or heliosphere? W. I; Axford The relaxation^' time for the system to reach equilibrium is of the order of the time the solar wind takes to get to the terminating shock wave. Since the solar wind travels an astronomical unit every 4 days, the time scale for the system to reach equilibrium is about a year. This is short compared with any possible time scale for changes occurring outside the solar system. At a number density of 0.1 ~m-~,the mean free path is about 10’ ’ cm. At 20 km/sec, which is roughly the velocity of the interstellar gas past the sun, an interval of 5.10” sec is required for a “cloud” with a scale size equal to one mean free path to go past the solar system. So a cloud which is only one mean free path across takes about 2000years to go past the sun at this speed. If you take 1000 mean free paths for the size of an interstellar cloud, one sees that the heliosphere should be stationary over time scales of a million years. C l? Sonett Is that true also for the wave field? Supposing there were a wave field on the outside, as we suspect there probably is. W. I; Axford Well, the waves that I would imagine to be important are Alfvin waves. The Alfvin speed is also quite small, something like 30 km/sec, so the time scale is still going to be of the order of lo6 years if the wavelengths are comparable to cloud dimensions. 660