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Keywords:

  • LISM magnetic field;
  • solar wind;
  • magnetohydrodynamics

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model and Method
  5. 3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[1] Both Voyagers crossed already the termination shock (TS); Voyager 2 (V2) about 10 AU closer to the Sun than Voyager 1 (V1). The 10 AU difference in the distance reveals the scale of the shock asymmetry. Between several reasons of this asymmetry there is an asymmetric pressure from an interstellar magnetic field. Our goal is to determine what magnetic field strength and orientation produces a 10 AU asymmetry in the TS locations at V1 and V2, when only this factor would be taken into account. Here we show that such a shape of the TS, which would reflect the real positions of V1 and V2 when they crossed it is obtained for the magnetic field vector placed not far from the hydrogen deflection plane with inclination from interstellar flow ∼30°, and for the strength about 3.8 μG. Our predictions are compared with other results.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model and Method
  5. 3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[2] After a series of papers published in Science in 2005 by Burlaga et al. [2005], Decker et al. [2005], Gurnett and Kurth [2005], and Stone et al. [2005], describing Voyager 1 crossing the termination shock, another series of papers describing Voyager 2 crossing the termination shock were published in Nature recently by Burlaga et al. [2008], Decker et al. [2008], Gurnett and Kurth [2008], Richardson et al. [2008b], Stone et al. [2008], and Wang et al. [2008]. The measurements made by both Voyagers provide priceless evidences on the size of the heliosphere and the scale of the heliospheric asymmetry [Richardson et al., 2008b]. The heliospheric asymmetry can be caused by the asymmetric pressure [Stone et al., 2008] from the motion of the TS [Washimi et al., 2007; Richardson et al., 2008b], from the solar wind (SW) dynamic pressure [Richardson et al., 2008a], or from the local interstellar magnetic field (LIMF) [Pogorelov and Matsuda, 1998; Ratkiewicz et al., 1998, 2000; Ben-Jaffel et al., 2000; Opher et al., 2006; Pogorelov et al., 2007a, 2007b; Ratkiewicz et al., 2008]. According to observations [Burlaga et al., 2008; Decker et al., 2008; Gurnett and Kurth, 2008; Richardson et al., 2008b; Stone et al., 2008; Wang et al., 2008], both termination shock crossings were foreshadowed by the foreshock. Voyager 1, at north heliolatitude, entered the termination foreshock region at the heliocentric distance 85 AU and crossed the termination shock at the heliocentric distance 94 AU. Voyager 2, at south heliolatitude, entered the termination foreshock region at 75AU and crossed the termination shock at 84 AU [Richardson et al., 2008b]. Note that Voyager 2 entered both: the foreshock and the termination shock at the heliocentric distance about 10 AU closer to the Sun than Voyager 1. It would indicate that the heliosphere is a distorted bubble and confirm previous theoretical predictions [see, e.g., Pogorelov and Matsuda, 1998; Ratkiewicz et al., 1998]. The most likely reason of this distortion is the action of the interstellar magnetic field. If the interstellar magnetic field were the only factor responsible for the measured asymmetries, what should it be? In order to find the direction and magnitude of the local interstellar magnetic field we have to find such a shape of the termination shock, which could reflect real positions of V1 and V2 when they crossed the shock.

2. Model and Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model and Method
  5. 3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[3] We apply our 3D MHD model with a constant flux of H [Ratkiewicz et al., 2008]. In our calculations at the inner boundary we use the solar wind pressures observed by V2 before the termination shock crossing [Richardson et al., 2008a]. At the outer boundary we use the local interstellar medium (LISM) velocity Vis = 26.4 kms−1, flowing from He direction (λ, β) = (254.7°, 5.2°) [Lallement et al., 2005] and temperature Tis ∼ 6400 K. Other LISM physical quantities we treat as parameters of the problem. They are: the angle between the helium velocity and magnetic field vectors (called the inclination angle α) which varies from 0° to 90°, the strength of the magnetic field equal to 2.8, 3.5, 3.8 or 4.1 μG, and the plasma number density which falls in the range of 0.04–0.11 cm−3 [Izmodenov et al., 2003]. If we use the measurements of the atomic H density at the termination shock (= 0.100 ± 0.008 cm−3) [Izmodenov et al., 2003], the H number density in the LISM is nH = 0.1–0.2 cm−3.

[4] The orientation of the LIMF in space is defined by two angles: an inclination and another one associated with the hydrogen deflection plane (HDP) [Lallement et al., 2005].

[5] Figure 1 illustrates the geometry showing in the solar ecliptic coordinates the position of helium (He) inflow from a direction (λ, β) = (254.7°, 5.2°), the position of hydrogen (H) flowing from a direction (λ, β) = (252.5°, 8.8°), and defined by above H and He flow vectors the hydrogen deflection plane, called the nominal HDP. There are also shown the V1 and V2 positions, when they crossed the TS, and the angle α which corresponds to the angle between equation image and equation image. Note that for each α there is a set of magnetic field directions marked by crosses every 10°, that create a cone around He inflow vector. A position on the cone is measured from the HDP by the β angle. We made simulations for many combinations of SW and LISM physical parameters to see which of them are candidates to achieve our goal. The selection has been made using a method illustrated in Figure 2, which consists of nine panels. In these panels the TS crossing distance is indicated by horizontal solid line for V1 (at 94 AU) and dashed line for V2 (at 83.7 AU) [Stone et al., 2008]. Curves indicate the heliocentric distance to the TS in the V1 (solid) or V2 (dashed) directions as a function of angle β. The vertical segments indicate β angles for which the TS is closer to the Sun in the V2 than in the V1 direction by about 10 ± 1 AU. To get both Voyagers at the proper positions we have to have vertical segments between the two horizontal lines, with solid curve at solid, and dashed curve at dashed horizontal line, respectively. For each set of data we were looking for a range of α and β angles, that would allow us to find the best fit. In Figure 2 we present results obtained for three sets of physical data. In the 1st row we use for the SW at 1 AU velocity VSW = 400 kms−1, number density nSW = 6.36 cm−3 [Richardson et al., 2008a], and temperature TSW = 51109 K, for the LISM Vis = 26.4 kms−1, Tis = 6400 K, nis = 0.11 cm−3, nH = 0.11 cm−3, Bis = 3.8 μG. To illustrate the influence of changes in conditions of the SW we increased the SW speed at 1 AU up to 508 kms−1, and decreased the number density to 5.0 cm−3 (the 2nd row). In the 3rd row we demonstrate results with the same parameters as in the 2nd row but the strength of the LIMF increased up to Bis = 4.1 μG. The best fit was achieved for conditions presented in the 1st row for α = 30°, and β = 20°. For comparison see results for α = 25°, and for α = 35°. As shown in the 2nd row, the 10 AU difference is obtained for 30°, but both Voyagers are too far from the Sun. Their positions are improving for 40° and 45°, but the distance between them drops below 10 AU. In the 3rd row a step in α in subsequent panels is 1°. 10 AU distance is achieved, but not at desired Voyagers' positions. For α = 42° the distance between Voyagers decreases. To show asymmetry of the heliospheric configuration caused by the LIMF we use the V1V2 plane, which is determined by three points: the position of the Sun and both positions at which Voyagers crossed the TS. The V1V2 plane crosses the heliosphere showing a deformation of the TS for any particular direction of the LIMF. In Figure 3 the shape of the TS reflects the steady-state asymmetry ensuring the V2 crossing the TS at a distance to the Sun closer by 10 AU than V1 obtained for α = 30°, and β = 20°. In Figure 4 the plasma parameters such as a number density, temperature, and velocity along V1 and V2 directions are shown. Jumps of the physical parameters at the TS correspond well to positions of V1 and V2. The compression ratio in V2 direction is close to jumps of velocity and number density observed by V2 (see Figure 4 caption). Our temperature is higher according to the model (heating due to pickup ions is overestimated), however the thermal pressure is still two orders of magnitude smaller than dynamic pressure, and does not change our final results.

image

Figure 1. Geometry in SE coordinates. It shows positions of helium and hydrogen inflows, nominal HDP, V1 and V2 positions when they crossed TS; black crosses have the same α and different β.

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image

Figure 2. Heliocentric distance to TS in V1 or V2 directions as a function of angle β.

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image

Figure 3. Plasma temperature distribution showing the shape of TS. Physical parameters as in Figure 2 (best fit). Positions of V1 and V2 marked by crosses. Arrow shows projection of He inflow vector.

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image

Figure 4. Plasma physical parameters distribution along the V1 and V2 direction. Vertical lines show V1 and V2 distance to TS. Compression ratio in V2 direction is close to jumps of velocity and number density observed by V2 shown at Voyager data overview at http://web.mit.edu/space/www/voyager/voyager_data/overview_plot_mission.gif 50-days averages.

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3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model and Method
  5. 3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[6] The above analysis leads us to the conclusion that for given SW and LISM parameters, the direction of the interstellar magnetic field is restricted to α about 30° and β about 20°. This finding is correlated with measurements of 2–3 kHz radio emissions [Kurth and Gurnett, 2003; Gurnett et al., 2006]. Note that since the distortion of the heliosphere depends on the orientation of the interstellar magnetic field, the place of the largest plasma density beyond the heliopause depends also on the direction of the LIMF [Ratkiewicz et al., 2008]. (Also in this region the hydrogen wall is situated). The 2–3 kHz radio emissions are generated when a strong interplanetary shock originating from the Sun interacts with the heliopause [Gurnett et al., 2006]. It means that the signal should come from high-density regions. In Figure 5 blue triangles show the source locations of heliospheric 2–3 kHz radio emissions obtained from direction-finding measurements [Kurth and Gurnett, 2003; Gurnett et al., 2006]. The coloured squares (red colour corresponds to the largest value) show the maximum density gradients along each direction in space obtained from our MHD simulations for LIMF direction, marked by a black cross (α ∼ 30°, β ∼ 20°). Note that blue triangles and yellow to red squares are correlated as they occur approximately in the same region. Very interesting is also the location of two maxima of the intensity of the ENA fluxes (cm−2sr−1s−1eV−1) at ∼6.8 and 14 keV found by Wang et al. [2008]. The ENA fluxes by Wang et al. [2008, Figure 3] show the double-peak source structure as a function of ecliptic longitude from ∼230° to 290° corresponding to the region in our Figure 5 filled in with yellow to red squares. Both regions (in Wang et al. [2008] and in the current paper) are ∼60° wide in longitude. Our finding is also in a good agreement with results inferred from deflection of the interstellar neutral hydrogen flow across the heliospheric interface [Lallement et al., 2005]. Note that the HDP is not, in fact, uniquely defined. The interstellar neutral H flows from (λ, β) = (252.5° ± 0.5°, 8.8° ± 0.5°) and He flows from (λ, β) = (254.7° ± 0.4°, 5.2° ± 0.2°). In Figure 5 this uncertainty is reflected by three blue solid lines (with the nominal HDP in the middle). Deduced by Lallement et al. [2005] direction of the local interstellar magnetic field in galactic coordinates (l, b) = (205° to 240°, −38° to −60°) has been calculated under the assumption that α is between 30° and 60°. (The range of αs marked between HDP lines in Figure 5).

image

Figure 5. Black cross indicates our best fit solution for the magnetic field. Coloured squares show the maximum density gradients along each direction in space. Blue solid lines show three HDPs. Region of α from 30° to 60° between them has a blue background. Blue and red triangles show the source locations of heliospheric 2–3 kHz radio emissions for primary and ambiguous solutions, respectively. Blue dashed line corresponds to black dashed line in Figure 1 Gurnett et al. [2006]. Black bold line marks the plane parallel to the galactic plane.

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4. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model and Method
  5. 3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[7] We have found, using only our magnetohydrodynamic model of the heliosphere and the two experimentally obtained TS crossing positions of Voyager 1 and 2, that if the interstellar magnetic field were the only factor responsible for the measured asymmetries, with the physical parameters of the solar wind and the interstellar medium used in our calculations, the most probable strength of LIMF would be about 3.8 μG, the angle α would be restricted to ∼30° and β to ∼20°. It corresponds to the local interstellar magnetic field pointing to (λ, β) = (68° ± 6°, −35° ±5°) in the solar ecliptic coordinates. Although for the modeling we use a simplified treatment of H, the LIMF longitude is surprisingly well correlated with the longitude of the first maximum at ∼68° of the ENA flow [see Wang et al., 2008, Figure 3]. In galactic coordinates, the LIMF points to (l, b) = (210° ± 5°, −33° ± 4.3°), showing the range close to the Lallement et al. [2005] result. According to Gurnett et al. [2006], their determination of the direction of the local interstellar magnetic field is similar to the Lallement et al. [2005] result. We think that our finding, independent of the assumption made by Gurnett et al. [2006] that an inclination angle is of 90° to the sun-nose line [see Gurnett et al., 2006, Figure 3], provides a new interpretation of the direction-finding measurements [Kurth and Gurnett, 2003].

[8] The present results are more precise than findings by Lallement et al. [2005] and Ratkiewicz et al. [2008]. They are not far from the first estimation of α by Ben-Jaffel et al. [2000]. The work presented here illustrates the method, which seems to be very promising for the future diagnosis of the LISM. Using this approach we plan to build a time-dependent model including the latitudinal dependence of the solar wind.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model and Method
  5. 3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[9] Authors acknowledge a support from the Polish Ministry of Science and Higher Education (PMSHE), grant N N203 4159 33. JG acknowledges support from the PMSHE, grant 4T12E 002 30. RR and JG wish to thank S. Grzedzielski and A. Czechowski for fruitful discussions.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model and Method
  5. 3. Distribution of the 2–3 kHz Radio Emissions, ENA Fluxes, HDP, and Current Results
  6. 4. Conclusions
  7. Acknowledgments
  8. References