The Astrophysical Journal, 604:700–706, 2004 April 1 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
ON THE POSSIBILITY OF A STRONG MAGNETIC FIELD IN THE LOCAL INTERSTELLAR MEDIUM V. Florinski, N. V. Pogorelov,1 and G. P. Zank Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521;
[email protected],
[email protected],
[email protected]
B. E. Wood JILA, University of Colorado, Boulder, CO 80309;
[email protected]
and D. P. Cox Department of Physics, University of Wisconsin, Madison, WI 53706;
[email protected] Received 2003 August 26; accepted 2003 December 12
ABSTRACT We analyze the consequences of the local interstellar magnetic field being almost 3 times larger than the Galactic average (ordered) field on the structure of the heliospheric interface in the axisymmetric case when the field is parallel to the relative direction of motion between the local interstellar medium (LISM) and the Sun. A field of such strength is expected to exist in the Local Interstellar Cloud, if the latter condensed from material inside a magnetic flux tube rebounding from the wall of the Local Bubble cavity. The analysis is performed using a newly developed multifluid neutral MHD model. We show that the bow shock ahead of the heliopause still exists for supersonic and sub-Alfve´nic LISM parameters. Our results agree well with the observations of the Ly absorption spectra and yield positions of the termination shock and the heliopause similar to those obtained from the standard super-Alfve´nic model. Subject headings: ISM: clouds — ISM: magnetic fields — MHD — shock waves — solar wind — ultraviolet: solar system
1. INTRODUCTION
Voyager 1 (the magnetic pressure would push the heliopause closer to the Sun). However, the above values are too small for the general equilibrium between the LB and the LIC (Ferlet 1999), and stronger magnetic fields should not be excluded (Slavin & Frisch 2002). It is worth noting that additional pressure will not be exerted on the forward part of the heliopause if B1 k v1 . A new model for the formation of the LIC was recently developed by Cox & Helenius (2003). It represents an attempt to explain how the LIC can be in a pressure equilibrium with the surrounding hot fully ionized gas of the LB. According to this model, the LB cavity of hot low-density gas is initially formed as a result of successive supernova explosions over a period of a few million years (Smith & Cox 2001). As the magnetic field is compressed in the shell of denser material surrounding the cavity, a coherent flux tube may detach from the cavity wall. The flux tube then springs into the hot-gas– filled cavity, owing to the magnetic tension that attempts to straighten it, and flings the material trapped inside the tube toward the center of the cavity, lengthwise along the tube. After about 3 million yr, the flux tube has reached the cavity center and slowed considerably owing to drag by the surrounding hot gas. By then, the material within the tube has collected near the center, in one or several small clouds similar to our LIC. The most intriguing prediction of the model of Cox & Helenius (2003) is the existence of a magnetic field with a strength of 5–6 G inside the LIC. From the apparent distribution of local material, and therefore of the flux tube in this model, the authors predict that the field would be directed approximately parallel to the LISM velocity vector with respect to the solar system. Thus, an axisymmetric model with the field-aligned LISM wind is a good first approximation for
The Sun is located in a rather special region of the outer part of the Galactic disk called the Local Bubble (LB). The LB is ~200 pc in size and is filled with a very tenuous gas of temperature 106 K. Besides the hot gas, the LB contains denser warm diffuse clouds, such as the Local Interstellar Cloud (LIC) surrounding the solar system (Ferlet 1999). The LIC is ~1 pc in size, contains partly ionized gas with a temperature of 6500–7000 K, and is moving at a speed of 25–26 km s1 with respect to the solar system (Frisch 2000; Lallement 2001). The number density nHe of helium atoms is measured at ~0.014 cm3 (Lallement 2001). Taking into account the helium ionization ratio (Dupuis et al. 1995; Wolff, Koester, 1999) and the cosmological abundance ratio & Lallement nH þ np =ðnHe þ nHeþ Þ, which is about 10, allows us to choose the neutral hydrogen density nH ’ 0:2 cm3. Different sources give values for model-derived electron and pro ton densities, ne ’ np , between 0.04 and 0.1 cm3 (Gloeckler, Fisk, & Geiss 1997; Frisch 2000; Lallement 2001). The strength and direction of the interstellar magnetic field (ISMF) in the LIC has not been directly measured. A value of B1 ¼ 1:6 G is consistent with observations of pulsar dispersion and rotation measures (Rand & Kulkarni 1989). Estimates of the heliospheric confinement pressure yield the upper limit of B1 3–4 G (Gloeckler et al. 1997) under the assumption that the ISMF is directed perpendicular to the local interstellar medium (LISM) velocity vector. Higher magnetic fields are ruled out in this case because the heliospheric termination shock (HTS) has not been detected by 1
Also at Institute for Problems in Mechanics, Russian Academy of Science, 101 Vernadsky Avenue, Building 1, Moscow 119526 Russia.
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STRONG MAGNETIC FIELD IN LOCAL ISM exploring the heliospheric interaction in this scenario. Such a model can also be applied to a wider class of quasi-parallel interactions, where the angle between B1 and v1 is small. Radio emission events may provide additional constraints on the direction of the magnetic field in the LISM. These emissions are believed to be produced from the passage of solar-wind–merged interaction regions into the outer heliosheath (Gurnett et al. 1993) by Langmuir waves generated as a result of instabilities in the electron distribution (Cairns & Zank 2002). Recently, Kurth & Gurnett (2003) observed that the sources of radio emissions are distributed along a line that is directed nearly parallel to the Galactic plane. The authors suggested that this might be an effect of the ISMF lying in the plane of the Galaxy and being perpendicular to the apex direction. However, this conclusion remains empirical and does not answer the question of whether a quasi-perpendicular LISM field is required to produce the observed source distribution, or if a small angle between B1 and v1 would suffice. Note that in the latter case the magnetic field would still become nearly perpendicular at the heliopause. Clearly, a quantitative investigation of the radio emission source problem is required, conducted with a fully three-dimensional MHD model. However, such analysis goes beyond the scope of the present work. The structure of the heliospheric interface has traditionally been studied using two-shock gasdynamic (Baranov & Malama 1993; Pauls, Zank, & Williams 1995; Pogorelov 1995; Zank et al. 1996; Mu¨ller, Zank, & Lipatov 2000; Myasnikov et al. 2000) or MHD (Baranov & Zaitsev 1995; Washimi, & Tanaka 1996; Pogorelov & Semenov 1997; Pogorelov & Matsuda 1998, 2000; Ratkiewicz et al. 1998) models. A comprehensive review may be found in Zank (1999). The presence of interstellar neutral atoms reduces the heliocentric distances of the HTS and heliopause, primarily by the removal of momentum from the supersonic solar wind (SW) through charge exchange (e.g., Baranov & Malama 1993; Zank et al. 1996; Mu¨ller et al. 2000). A region of enhanced hydrogen density in the outer heliosheath is also produced. A relatively weak (B 1:5 G) magnetic field directed parallel to the interstellar flow does not modify the solution appreciably. Nevertheless, the distance to the heliopause increases because of a pull exerted by magnetic tension in the outer heliosheath (Baranov & Krasnobaev 1971). The models of Linde et al. (1998), Aleksashov et al. (2000), and Florinski, Zank, & Pogorelov (2003) adequately reproduce all these effects. In this paper we use a newly developed axisymmetric multifluid neutral MHD model of the heliosphere to examine the structure of the heliospheric interface when the LISM magnetic field is 3 times greater than the Galactic average. We show that if the incoming flow is sub-Alfve´nic and the flow is field aligned (B1 k v1 ), a bow shock still exists but possesses properties that are substantially different from those of a super-Alfve´nic bow shock. Ly absorption profiles in the spectra of nearby stars are also calculated in order to show that our model is in good agreement with the observations. 2. A MULTIFLUID NEUTRAL MHD MODEL To self-consistently calculate the structure of the SW-LISM interaction region, we developed a four-fluid neutral MHD model, which is an extension of the two-fluid axisymmetric MHD model introduced by Florinski et al. (2003). In addition to the interstellar neutrals (population 1), we now include hydrogen atoms produced in the inner heliosheath (population 2 neutrals or energetic neutral atoms [ENAs]) and those born
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in the supersonic solar SW (population 3). This ensures that additional effects, such as heating of the LISM flow upstream of the bow shock by ENAs and an increase of the distance between the bow shock and the heliopause (Zank et al. 1996), are properly captured. We assume that protons and electrons have equal temperatures and describe their motion by a modified set of MHD equations in the weakly conservative form, @p þ r G ðp vp Þ ¼ Dp ; @t @ðp vp Þ BB þ r G p v p v p þ pp I 4 @t Bðr r BÞ ¼ þ Mp; 4 @ep Bðvp G BÞ þ r G ðep þ pp Þvp 4 @t ðvp G BÞðr r BÞ þ Ep ; ¼ 4 @B þ r G vp B Bvp ¼ vp ðr r G BÞ: @t
ð1Þ
ð2Þ
ð3Þ ð4Þ
Here p , vp , pg , and pp ¼ pg þ B2 =ð8Þ are the density, the velocity and the thermal and total pressures, respectively. The total energy density is ep ¼ p v 2p =2 þ B2 =8 þ pg =ð 1Þ. The MHD system is written in the symmetrizable form first suggested by Godunov (1972). It is a Galilean invariant and is proved useful for enforcing the divergence-free condition for magnetic fields (Powell et al. 1999; Kulikovskii, Pogorelov, & Semenov 2001). All three populations of neutrals are treated hydrodynamically in a way similar to that of Zank et al. (1996). Coupling between the neutral atoms and the plasma is provided by the charge exchange terms Dp , M p , and Ep , which are calculated according to Pauls et al. (1995). Our choice of a multifluid, rather than kinetic, approach was motivated by numerical efficiency concerns. Because perturbations can propagate upstream in the LISM flow (see below), a large simulation domain and, consequently, long integration times are required, making the kinetic approach impractical. In addition, kinetic models have some difficulty predicting Ly absorption along multiple lines of sight, which may indicate that some essential physics is missing (Wood, Mu¨ller, & Zank 2000). We use the following values of the LISM flow parameters at infinity: proton number density np1 ¼ 0:1 cm3, neutral hydrogen number density nH1 ¼ 0:2 cm3, velocities vp1 ¼ v H1 ¼ 25 km s1 , and temperatures Tp1 ¼ TH1 ¼ 7000 K (Frisch 2000; Lallement 2001). Thus, the LISM streams of protons and neutrals have Mach numbers Mp1 and MH1 of 1.8 and 2.55, respectively. We take B1 ¼ 4:3 G, which makes the LISM flow sub-Alfve´nic, with the Alfve´n number A1 ¼ 2ðp1 Þ1=2 v1 =B1 ¼ 0:84. The SW has density np0 ¼ 5 cm3 , velocity vp0 ¼ 500 km s1, and temperature Tp0 ¼ 1:5 105 K at 1 AU. The MHD system is solved on a variable polar (r, )-grid inside a semispherical domain 4000 AU in size, with 600 grid intervals in the r-direction and 200 intervals in the -direction. 3. THE STRUCTURE OF THE HELIOSPHERE Parker (1961) developed a simple analytic model of the heliosphere being produced by the interaction of the SW with
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pand in both directions along the symmetry axis, although upstream expansion is considerably slower because the heliosheath flow must overcome the interstellar wind dynamic pressure, while transverse expansion is suppressed by the magnetic pressure. Clearly, this scenario does not describe the real heliosphere, in which evidence from UV and radio emission strongly suggests a heliopause that is located within a few hundred AU in the nose direction. However, when plasma-neutral interactions are included, the interface structure is substantially different, and a quasisteady solution is possible. The overall structure of the heliosphere is shown in Figure 1 and resembles that of a superAlfve´nic two-shock model (e.g., Pogorelov & Semenov 1997; Pogorelov & Matsuda 2000; Florinski et al. 2003). The picture shown in Figure 1 represents a typical snapshot of a quasisteady solution. The heliopause oscillates between 128 and 137 AU on the symmetry axis, but the positions of the termination and bow shocks remain essentially unchanged at 97 and 260 AU in the upwind direction, respectively. The oscillatory nature of the heliopause reveals an instability of the interface caused by charge exchange. This Rayleigh-Taylor–like instability was first analyzed by Liewer, Karmesin, & Brackbill (1996) and Zank et al. (1996). The period of oscillations in our case is about 100 yr. The growth rate of the instabilities depends on the wavelength, which we estimate using the typical value, corresponding to a change from purely evanescent structures to propagating waves, given by V. Florinski et al. (2004, in preparation) as H
Lc ¼
2v H ; 3:44oh
ð5Þ
where oh is the charge exchange rate in the outer heliosheath. For typical heliosheath conditions, Lc 100 AU and the growth rate is 30 yr, which is consistent with our results. Note that the HTS is nearly spherical. A strong hydrogen wall is clearly visible in the bottom panel of Figure 1. Figure 2 shows radial profiles of the proton number density in the upwind, crosswind, and downwind directions. The density distribution evidently reveals a bow shock. Since parallel MHD shocks are always evolutionary if the upstream
p
Fig. 1.—Logarithm of the plasma number density and plasma flow streamlines (top), population 1 neutral hydrogen density (middle), and logarithm of the plasma temperature (bottom).
a strong magnetic field in vacuum. In the absence of MHD effects, the shape of the heliopause is completely determined by the pressure balance at two points on the interface and consists of a spherical shell with two side channels extending to infinity along the direction of the ISMF. To verify whether the structure persists when dynamic effects are included, we performed a test simulation with a fully ionized interstellar wind (i.e., no H atoms) and a magnetic field as above. This calculation showed that no steady (or quasisteady) solution is possible. The heliopause continues to ex-
Fig. 2.—Proton number density in the upwind (solid line), crosswind (dashed line), and downwind (dotted line) directions.
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4. Ly ABSORPTION PROFILES
Fig. 3.—Magnetic field perpendicular to the local bow shock normal, intersecting the shock 15 from the axis of symmetry. Note the decrease of the B? component, indicating that the shock is slow, and the rarefaction in front of the shock.
flow is supersonic and sub-Alfve´nic (see, e.g., Kulikovskii et al. 2001), the bow shock we obtained must be slow. This is confirmed by the behavior of the tangential component of the LISM magnetic field crossing the shock, shown in Figure 3. Note that if the upstream flow is subsonic and super-Alfve´nic, parallel slow shocks are inadmissible, which is the reason why purely gasdynamic subsonic models (Zank et al. 1996; Gayley et al. 1997) do not involve bow shocks, and the density smoothly increases from infinity to the surface of the heliopause. While fast waves can still propagate upwind in our case, the integral curves of fast Riemann waves (Jeffrey & Taniuti 1964; Kulikovskii & Lyubimov 1965) reveal that they can produce only small variations in density when propagating at small angles to the ISMF. It is interesting to note that, since magnetic field lines must turn toward the symmetry axis at a slow bow shock, the latter cannot be convex unless there exists a prior rotation of these lines in the opposite direction. In our model, this rotation occurs in the rarefaction region preceding the bow shock. This appears to be a purely twodimensional phenomenon, since no rotation is necessary on the symmetry axis. Ratkiewicz & Webb (2002) have recently studied the problem of the SW-LISM interaction in the supersonic and sub-Alfve´nic case on the basis of a simplified chargeexchange model. The authors did not recognize the possibility of their solution containing a bow shock and stated that a bow shock cannot be adjacent to an elliptic region of the stationary MHD system. In fact, the absence of characteristics upstream excludes only weak slow shocks. Admissible shocks in this case must be strong (Kogan 1959). It is worth noting that Aleksashov et al. (2000), using a kinetic model for the neutral hydrogen, obtained a solution that involved a bow shock in a supersonic and sub-Alfve´nic LISM, although no attempts were made to elaborate its properties. It is doubtful, however, that the bow shock can extend as far as shown in the abovementioned paper. Since the shock weakens as we move along its profile away from the symmetry axis, it must sooner or later disappear. This is clearly seen in Figure 1. Moreover, such a shock cannot degenerate into a characteristic line at infinity because the necessary characteristics do not exist. We perform some analysis of the slow shock properties in the Appendix.
The population of hot neutral hydrogen gas that permeates the heliosphere because of charge exchange processes has been detected by the Hubble Space Telescope (HST ) in observations of H i Ly lines from nearby stars. The heliospheric absorption is highly blended with absorption from interstellar gas, but the presence of the heliospheric component can be discerned as an excess of absorption on the red side of the Ly line. This absorption has been detected clearly in the lines of sight toward Cen, Sirius, and 36 Oph (Linsky & Wood 1996; Izmodenov, Lallement, & Malama 1999; Wood, Linsky, & Zank 2000). There have been several attempts to use these detections to test various models of the heliosphere (Gayley et al. 1997; Wood, Mu¨ller, & Zank 2000; Izmodenov, Wood, & Lallement 2002). Not only has heliospheric Ly absorption been measured, but analogous ‘‘astrospheric’’ absorption has also been detected surrounding the observed stars as an excess of H i absorption on the blue side of the Ly line, rather than the red side (Linsky & Wood 1996; Wood et al. 2000). With the help of the developed global interaction models, the astrospheric absorption has proved to be a very valuable diagnostic for the winds of Sunlike stars, which were not previously detectable. Nevertheless, since all Ly absorption studies to date have ignored the ISMF, a whole parameter dimension has been left unexplored. As mentioned earlier, the applicability of oneshock models is questionable, not only because they artificially assume a high-temperature environment, presumably due to contributions from magnetic and cosmic-ray pressures, but also because supersonic, sub-Alfve´nic LISM regimes are qualitatively different from subsonic, super-Alfve´nic regimes, purely gasdynamic cases being a subset of the latter. As shown by Florinski et al. (2003), Galactic cosmic rays are likely to have only a weak coupling to the thermal plasma in the LISM. Here we show that the inclusion of MHD effects results in a two-shock interface in the presence of supersonic LISM flows in both sub- and super-Alfve´nic regimes. Figure 4 compares model-calculated Ly absorption profiles with the HST measurements. The picture shows the right (red) side of the saturated Voigt profiles along six different lines of sight. Results with a strong magnetic field (B1 ¼ 4:3 G), shown by dashed lines, are compared with a calculation with no ISMF, shown by dot-dashed lines in Figure 4. When the strong field is included, a very good agreement between the data and calculations is found along all lines of sight, especially where absorption is dominated by the hydrogen wall (36 Oph and Cen), except directly toward the heliotail ( Eri), but the discrepancy is equal to or less than that obtained with conventional models (Wood et al. 2000; Wood, Mu¨ller, & Zank 2000). This result indicates that a strongly magnetized ISM must be considered a serious possibility. The no-ISMF calculation tends to overpredict absorption toward Cen and underpredict absorption toward Sirius. The H wall extends to a higher latitude in the unmagnetized case, which explains excessive absorption in the forward direction when there is a large angle between the symmetry axis and the line of sight. Note that the difference in the amount of absorption calculated for B1 ¼ 4:3 G and B1 ¼ 0 is generally small. This result is not unexpected in view of the similarity in the heliospheric structure and discontinuity positions with and without the magnetic field. It appears that Ly absorption is not very sensitive to changes in B within the parameter range
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Fig. 4.—Red side of the H i Ly line observed from six stars, sampling different angles () relative to the upwind direction of the interstellar flow into the heliosphere. Dotted lines show the interstellar contributions to the broad, saturated H i absorption, based on previous analyses of these data (see Wood, Mu¨ller, & Zank 2000). For 31 Com, Cas, and Eri, the interstellar absorption fits the data well, but for the other three lines of sight there is excess absorption from the outer heliosphere. The dashed lines show the absorption predicted by the heliospheric model discussed in the text after addition to the interstellar absorption. The dotdashed lines correspond to a calculation with the same heliospheric and interstellar parameters, but with B ¼ 0.
explored. Including an even stronger magnetic field (5–6 G) may result in a heliosheath extending farther upstream with dominant population 2 absorption. However, it may be difficult to reconcile this case with UV absorption and radio emission detections. 5. CONCLUSION We have studied the structure of the heliospheric interface when the interstellar magnetic field is 3 times the Galactic average and is parallel to the vector of the Sun’s velocity relative to the LIC. A field of this strength is possible according to a newly developed theory of the LIC’s formation (Cox & Helenius 2003). Under these conditions, the heliospheric structure contains all three known discontinuities, with the termination shock and the heliopause located at distances similar to those predicted by the previous models. The difference is that the bow shock is of a slow kind and is
preceded by a rarefaction. Unlike a fast bow shock, which extends to infinity, degenerating into a characteristic line, this shock extends only a limited distance away from the symmetry axis. Our analysis of the absorption features in the Ly spectral line for six prominent stars shows that the model can accurately predict the observed amount of absorption in each direction explored. This important result implies that a strong magnetic field in the LIC cannot be ruled out on observational grounds and must be considered an alternative to the usual super-Alfve´nic model of the heliosphere. More accurate neutral particle models, based on a kinetic formalism (Wood, Mu¨ller, & Zank 2000; Izmodenov et al. 2002), may be required to obtain better constraints on the model parameters from the Ly absorption perspective, although the amount of absorption is not very sensitive to the ISMF magnitude within the parameter range explored.
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V. F., N. V. P., and G. P. Z. acknowledge support by NSF grant ATM 02-96114 and by NASA grants NAG5-11621 and NAG5-12903. N. V. P. was also partially supported by the Russian Foundation for Basic Research grant 02-01-00948 and grant 1899.2003.1 for support of leading scientific schools
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in Russia. B. E. W. acknowledges support through NASA grant NAG5-9041. D. P. C. was supported by NASA grant NAG5-12128 to the University of Wisconsin, Madison. Numerical computations were performed on the IGPP/UCR ‘‘Lupin’’ cluster.
APPENDIX PROPERTIES OF THE BOW SHOCK IN THE ELLIPTIC REGIME Let us investigate possible topologies of MHD bow shocks with the elliptic region of the stationary system ahead of them. For the case with the plasma parameter ¼ 4pp =B2 ¼ A2 =M 2 < 1, flow regimes can be subdivided into four groups: Hf 2 [M > 1= 1=2 ], Efs2 [1 < M < 1= 1=2 ], Hs2 [1=ð 1 þ Þ1=2 < M < 1], and Ec2 [M < 1=ð 1 þ Þ1=2 ], which correspond to the fast hyperbolic, first elliptic, slow hyperbolic, and second elliptic regions of the Grad-Sears group velocity diagram, respectively (Kogan 1959; Jeffrey & Taniuti 1964; Kulikovskii & Lyubimov 1965). As mentioned above, slow parallel shocks are always evolutionary. Slow oblique shocks are admissible if the velocity projection onto the shock normal is superslow ahead and subslow behind them, the flow remaining sub-Alfve´nic on both sides. In the case of < 1, shocks are possible if the upstream flow belongs to the Hf 2, Hs2, or Efs2 region. We would like to emphasize again that in the Efs2 regime, only strong slow shocks can exist (Kogan 1959). Using standard Rankine-Hugoniot shock conditions, one can readily obtain an expression for the shock compression ratio q as n o ða2 qÞ2 Mk2 ½q þ 1 ðq 1Þ 2q qM?2 A2 ½2q ðq 1Þ q½q þ 1 ðq 1Þ ¼ 0;
ðA1Þ
where Mk and M? are the sonic Mach numbers in the directions parallel and perpendicular to the shock normal, respectively. This cubic equation possesses a large variety of solutions, featuring fast (including switch on), slow, and intermediate shocks. Because our focus is on solutions that are admitted by the evolutionary conditions, we exclude all cases with intermediate shocks. We plot the compression ratio as a function of the angle between the flow and the shock normal in Figure 5. The fast Hf 2 bow shock is strongest when parallel and has the well-known convex shape. The hypothetical slow Hs2 shock is strongest when perpendicular and might form a cusp on the symmetry axis (Kabin 2001). Both of these shocks may extend to infinity, where they tend to a characteristic line. On the diagram, this happens at the intersection of the shock curve with the line q ¼ 1 and at the Mach angle given by sin2 M ¼
A2 þ M 2 1 : A2 M 2
ðA2Þ
The Efs2 shock, however, is different because it can exist for any angle (q is always greater than 1 and B? does not change sign). This implies that this type of shock may not extend smoothly to infinity if it is adjacent to a constant state. However, as we have shown earlier, the upstream state is not constant, owing to perturbations propagating upstream at an angle to the mean field, as well as experiencing additional modification from charge exchange. Finally, the shock does not extend to infinity.
Fig. 5.—Shock compression ratios as functions of the incident angle for < 1. The solid line has M ¼ 1:7, A ¼ 1:6 (fast shock), and the dashed line has M ¼ 0:8, A ¼ 0:9 (slow shock); both lines reach q ¼ 1 at ¼ M . The dash-dotted lines show the slow Efs2 shock compression ratio: M ¼ 1:3, A ¼ 0:8 for the upper line, and M ¼ 1:8, A ¼ 0:95 for the lower line.
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Another interesting property of a slow shock in the elliptic region is that its compression ratio can either increase or decrease with shock incidence angle. The critical value, where q is independent of , is given by A2crit ¼
M 2 ð þ 2 M 2 Þ : ð 1ÞM 2 þ 2
ðA3Þ
Only slow shocks with an Efs2 upstream state have Acrit ; for all other types of flows this value of A yields q < 1. REFERENCES Aleksashov, D. B., Baranov, V. B., Barsky, E. V., & Myasnikov, A. V. 2000, Lallement, R. 2001, Ap&SS, 277, 205 Astron. Lett., 26, 743 Liewer, P. C., Karmesin, S. R., & Brackbill, J. U. 1996, J. Geophys. Res., Baranov, V. B., & Krasnobaev, K. V. 1971, Cosmic Res., 9, 568 101, 17119 Baranov, V. B., & Malama, Y. G. 1993, J. Geophys. Res., 98, 15157 Linde, T. J., Gombosi, T. I., Roe, P. L., Powell, K. G., & DeZeeuw, D. L. 1998, Baranov, V. B., & Zaitsev, N. A. 1995, A&A, 304, 631 J. Geophys. Res., 103, 1889 Cairns, I. H., & Zank, G. P. 2002, Geophys. Res. Lett., 29, 47 Linsky, J. L., & Wood, B. E. 1996, ApJ, 463, 254 Cox, D. P., & Helenius, L. 2003, ApJ, 583, 205 Mu¨ller, H.-R., Zank, G. P., & Lipatov, A. S. 2000, J. Geophys. Res., 105, 27419 Dupuis, J., Vennes, S., Bowyer, S., Pradhan, A. K., & Thejll, P. 1995, ApJ, Myasnikov, A. V., Alexashov, D. B., Izmodenov, V. V., & Chalov, S. V. 2000, 455, 574 J. Geophys. Res., 105, 5167 Ferlet, R. 1999, A&A Rev., 9, 153 Parker, E. N. 1961, ApJ, 134, 20 Florinski, V., Zank, G. P., & Pogorelov, N. V. 2003, J. Geophys. Res., 108, 1228 Pauls, H. L., Zank, G. P., & Williams, L. L. 1995, J. Geophys. Res., 100, 21595 Frisch, P. C. 2000, J. Geophys. Res., 105, 10279 Pogorelov, N. V. 1995, A&A, 297, 835 Gayley, K. G., Zank, G. P., Pauls, H. L., Frisch, P. C., & Welty, D. E. 1997, Pogorelov, N. V., & Matsuda, T. 1998, J. Geophys. Res., 103, 237 ApJ, 487, 259 ———. 2000, A&A, 354, 697 Gloeckler, G., Fisk, L. A., & Geiss, J. 1997, Nature, 386, 374 Pogorelov, N. V., & Semenov, A. Y. 1997, A&A, 321, 330 Godunov, S. K. 1972, Num. Meth. Mech. Continuous Media, 3 (1), 26 Powell, K. G., Roe, P. L., Linde, T. J., Gombosi, T. I., & De Zeeuw, D. L. 1999, Gurnett, D. A., Kurth, W. S., Allendorf, S. C., & Poynter, R. L. 1993, Science, J. Comput. Phys., 154, 284 262, 199 Rand, R. J., & Kulkarni, J. R. 1989, ApJ, 343, 760 Izmodenov, V., Lallement, R., & Malama, Y. G. 1999, A&A, 342, L13 Ratkiewicz, R., Barnes, A., Molvik, G. A., Spreiter, J. R., Stahara, S. S., Izmodenov, V., Wood, B. E., & Lallement, R. 2002, J. Geophys. Res., 107, 1308 Vinokur, M., & Venkateswaran, S. 1998, A&A, 335, 363 JeArey, A., & Taniuti, T. 1964, Nonlinear Wave Propagation (New York: Ratkiewicz, R., & Webb, G. M. 2002, J. Geophys. Res., 107, 1484 Academic Press) Slavin, J. D., & Frisch, P. C. 2002, ApJ, 565, 364 Kabin, K. 2001, J. Plasma Phys., 66, 259 Smith, R. K., & Cox, D. P. 2001, ApJS, 134, 283 Kogan, M. N. 1959, J. Appl. Math. Mech., 23, 92 Washimi, H., & Tanaka, T. 1996, Space Sci. Rev., 78, 85 Kulikovskii, A. G., & Lyubimov, G. A. 1965, Magnetohydrodynamics WolA, B., Koester, D., & Lallement, R. 1999, A&A, 346, 969 (Reading: Addison-Wesley) Wood, B. E., Linsky, J. L., & Zank, G. P. 2000, ApJ, 537, 304 Kulikovskii, A. G., Pogorelov, N. V., & Semenov, A. Y. 2001, Mathematical Wood, B. E., Mu¨ller, H.-R., & Zank, G. P. 2000, ApJ, 542, 493 Aspects of Numerical Solution of Hyperbolic Systems (Boca Raton: Zank, G. P. 1999, Space Sci. Rev., 89, 413 Chapman & Hall) Zank, G. P., Pauls, H. L., Williams, L. L., & Hall, D. T. 1996, J. Geophys. Res., Kurth, W. S., & Gurnett, D. A. 2003, J. Geophys. Res., 108, 8027 101, 21639
