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Comparison of measured and calculated magnetic fields along the Ulysses orbit
Accepted Manuscript Comparison of measured and calculated magnetic fields along the Ulysses orbit N.S. Svirzhevsky, G.A. Bazilevskaya, A.K. Svirzhevskaya, Yu.I. Stozhkov PII: DOI: Reference:
S0273-1177(14)00521-3 http://dx.doi.org/10.1016/j.asr.2014.08.009 JASR 11914
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
3 February 2014 1 August 2014 9 August 2014
Please cite this article as: Svirzhevsky, N.S., Bazilevskaya, G.A., Svirzhevskaya, A.K., Stozhkov, Yu.I., Comparison of measured and calculated magnetic fields along the Ulysses orbit, Advances in Space Research (2014), doi: http:// dx.doi.org/10.1016/j.asr.2014.08.009
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Comparison of measured and calculated magnetic fields along the Ulysses orbit N.S. Svirzhevsky, G.A. Bazilevskaya, A.K. Svirzhevskaya, Yu.I. Stozhkov Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia
Abstract The existence of close relations between the temperature, density and velocity of the solar plasma and the heliospheric magnetic field (HMF) was shown along the space probe Ulysses orbit. A simple mathematical formula describing a relation between the HMF and the solar plasma temperature and density was introduced and the expected values of the HMF were calculated using daily and hourly Ulysses data. Correlation coefficients and regression equation between the values of the measured and calculated magnetic fields have been defined. An origin of the peaks in the magnetic field which are observed in the heliospheric sector zone near the corotating interaction regions is discussed as well as the specific role of plasma density and temperature in the formation of magnetic peaks. Keywords: heliospheric magnetic field; solar wind; plasma density; plasma temperature 1. Introduction In this work we have considered a possibility to express the magnetic field in the heliosphere by means of the local parameters of heliospheric plasma – temperature and density. The proposed relationship between the magnetic field and the plasma parameters is expressed in the following form (Svirzhevsky et al., 2014): Bcalc = Kn T + B0 ,
(1)
where n is the proton number density and T is the temperature of the heliospheric plasma, K and B0 are parameters, which were defined from comparison of measured and calculated magnetic fields along the Ulysses orbit. The measured magnetic fields, in dependence on the distance r to the Sun, can be described by two sets of K and B0 parameters, namely, at r = 1–2.5 AU K = 0.003 and B0 = 0.6 nT, and at r = 2.5–5.4 AU K = 0.006 and B0 = 0.2 nT. The derivation of Eq. (1) is given in the Appendix. The derivation is based on the assumption that the thermal movement of electrons with the average velocity exceeding the average thermal velocity of protons violates the neutrality of the plasma and causes the formation of the negative volume charge and the electric field in the heliospheric plasma. As a result of that the proton current occurs in the plasma which tends to restore its neutrality, as well as creates the magnetic field in it. The dependence of the magnetic field on the plasma density and temperature was obtained in a preliminary model which does not take into account some feature of the heliospheric magnetic field, in particular, its Parker spiral form. We also intend to discuss below the relationship between corotating interaction regions (CIRs) and peaks in the magnetic field which are observed in the heliospheric sector zones. The interactions between fast solar wind from coronal holes and slow solar wind from the belt of coronal streamers form regions of compressed plasma with enhanced density and temperature. ----------------------------------------------Corresponding author Email address: [email protected] Tel: +74954854100 1
These interactions, which lead to the formation of the CIRs, are considered as one of the primary processes shaping the structure of the interplanetary medium (Riley et al., 1996). Usually the CIRs are well developed at the distance of 1 AU (Hundhausen, 1973; Broiles et al., 2012). In some cases, observed at 1 AU CIRs may be identified as individual CIRs or/and as merged ones at a distance of about 5 AU (Broiles et al., 2013; Jian et al., 2011). At distances of 5–15 AU single CIRs merge into the corotating merged interaction region zones (CMIR zones), whose existence was confirmed by Voyager 2 observations (Burlaga et al., 2003). Many of the observed onboard the Ulysses spacecraft CIR characteristics were identified and examined in details in (Gosling et al., 1993; Bame et al., 1993; Gosling et al., 1995; Riley et al., 1996; Gosling and Pizzo, 1999; Broiles, et al., 2012). Various theoretical approaches to the modeling of corotating interaction regions were considered (Pizzo and Gosling, 1994; Lee, 2000; Odstrcil, 2003; Jian et al., 2011). We want to point out some features of CIRs, which can be set using the magnetic field strength, plasma density and temperature values. In particular, it will be shown that the forward and reverse shocks, bounding the CIRs, coincide with the forward and rear fronts of magnetic peaks. We can also note the difference between the magnetic peaks occurring in the interactions of fast and slow solar winds and created by shocks from coronal mass ejections. Different role of plasma density and temperature in the formation of magnetic peaks will also be shown. 2. Measured and calculated magnetic fields along the Ulysses orbit Ulysses was the first space probe which made observations of the magnetic fields and heliospheric plasma along a near-polar orbit around the Sun. The Ulysses measurements enable us to determine the relations between the HMF and the plasma parameters at different heliolatitudes and distances to the Sun. We can also compare the correlation between measured and calculated magnetic fields during the minimum and maximum of solar activity. Ulysses began to move to the south pole of the Sun after its encounter with Jupiter in February 1992 and to the middle of 1993 it was in the sector zone and regularly crossed fast solar wind streams. Comparison of the measured on board the Ulysses magnetic fields and calculated in accordance with Eq. (1) is shown in Fig. 1 for 1993. The Ulysses daily averaged data on magnetic fields and heliospheric plasma taken at website http://helio.estec.esa.nl/ulysses/archive are used in Fig. 1. The large magnetic peaks with a periodicity approximately that of the solar rotation are seen in Fig. 1. In Fig. 2 the HMF in the northern hemisphere is shown during the 1996 solar activity minimum. Moderate fluctuations are typical for the magnetic field, which is carried away by the solar wind from a coronal hole. Ulysses was located in the field of one polarity for more than a year and a half, and not a single considerable peak in the magnetic field was detected during this time. Only in the beginning of October 1996 Ulysses at 27° north heliolatitude entered a sector zone with an alternating sign of the HMF azimuth component and with large peaks of the magnetic field (exceeding the background field by a factor of 5–10). The observed magnetic field BUlys and calculated field Bcalc are quite consistent in this period, the coefficient of correlation between them is R = 0.884. Comparison of calculation with experimental data obtained during high solar activity of 2000 at high latitudes and in the area of the South Pole found that the correlation between the BUlys and Bcalc still remained high, the correlation coefficient is R = 0.832, despite the fact that the structure of the magnetic field during this period, with uncertain position of the heliospheric current sheet, has become more complex. Comparison of the measured and calculated values of magnetic fields during the crossing of the sector zone in 2007 is shown in Fig. 3. Unlike 1995, the sector zone in 2007 was located symmetrically relative to the helioequator. The coefficient of correlation between BUlys and Bcalc is R = 0.898.
2
В, nT
R=0.856
Heliolatitude, deg
4
0 5.07- 3.83 AU
3.5
B(Ulyss es ) Bcalc
-10
Heliolatitude
3
-20
2.5 2
-30
1.5
-40
1 -50
0.5 0 1993
-60 1993.2
1993.4
1993.6
1993.8
Years
1994
Fig. 1. Comparison of the measured magnetic field BUlys (grey) and calculated magnetic field
Bcalc (black) in the presence of fast solar wind streams from a coronal hole. The heliolatitude of the spacecraft is shown by a dashed line. Ulysses' distances from the Sun at the beginning and end of the examined time interval and the correlation coefficient R between two sets are also shown.
B, nT
R=0.884
Heliollatitude, deg
3
60 B(Ulysses)
3.06 - 4.71 AU.
Bcalc
2.5
50
Heliolatitude 2
40
1.5
30
1
20
0.5
10
0 1996
0 1996.2
1996.4
1996.6
1996.8
1997 Years
Fig. 2. Comparison of the measured BUlys and calculated Bcalc magnetic fields during the 1996 solar activity minimum. Notations in the Figure are the same, as in Fig. 1. Adopted from Svirzhevsky et al., 2014.
3
B, nT
1.67- 1.34 -1.55 AU
Heliolatitude, deg
10 9
60 B(Ulysses) Bcalc Heliolatitude
R=0.898
8
50 40 30
7
20
6
10
5
0
4
-10 -20
3
-30 2
-40
1
-50
0
-60
2007.4
2007.5
2007.6
2007.7
Years
2007.8
Fig. 3. Comparison of the measured and calculated values of magnetic fields during the crossing of the sector zone at the final orbit of Ulysses around the Sun. B, nT
R=0.893
2 B(Ulyss es )
4.47 - 3.69 AU
1.8
Bcalc - Bo 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 2006
2006.1
2006.2
2006.3
2006.4
2006.5 Years
Fig. 4. Comparison of the observed BUlys magnetic field with the values B − B0 = Kn T , B0 = 0.2 nT in the first half of 2006. Adopted from Svirzhevsky et al., 2014. In Fig. 4 the measured magnetic fields BUlys are compared with the values B − B0 = Kn T , where B0 = 0.2 nT. High correlation between BUlys and Kn T can be interpreted as an indication that the magnetic fields are formed by currents that flow in the same place where the measurements of plasma density and temperature are made. The origin of the parameter B0 included in the calculated magnetic field module B is not understood. The value of B0 = 0.2 nT used in calculations at distances r = 2.5–5.4 AU to the Sun is not negligibly small and it seems the B0 is decreasing with the distance more slowly then 1 / r 2 . For example, the dependence of 4
the B0 on the distance to the Sun in October 1995–December 1997 (65° N – the ecliptic) is shown in Fig. 5. Bo, nT
1 Oct 1995 - Dec 1997
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 2.5
3
3.5
4
4.5
5
r, AU
Fig. 5. The dependence of the B0 on the distance to the Sun in October 1995–December 1997. In Fig. 6 the BUlys – Bcalc scattering diagram constructed for the entire sets of the Ulysses daily data for 1990–2008 is shown. The regression equation is Bcalc = (0.16 ± 0.011) + (0.97 ± 0.0067) BUlys , with the correlation coefficient R = 0.874 ± 0.003 . Bcalc, nT
1990-2008
num ber of points 6576
16 14 12 10 8 y = 0.9711x + 0.1598 6
R=0.874
4 2 0 0
2
4
6
8
10
12
14 16 B(Ulysses), nT
Fig. 6. The scattering plot for the whole sets of the Ulysses data in the regression equation and the correlation coefficient R are also shown. Adopted from Svirzhevsky et al., 2014. In order to check the dependence of the regression and correlation coefficient R on points with the peak values of magnetic fields the regression equation was calculated under restrictions BUlys ≤ 4 nT, Bcalc ≤ 4 nT. It was obtained in that case Bcalc = (0.24 ± 0.008) + (0.87 ± 0.0063) BUlys , with
5
the correlation coefficient R = 0.867 ± 0.003 and number of points 6301. It means that high correlation between Bcalc and BUlys is not caused by the peak values of magnetic fields. The regression equations and correlation coefficients were also calculated using selected samples of hourly data. The result of calculations for the 3 year period from 2003 to 2005 without constraints on BUlys and Bcalc gives the regression equation: Bcalc = (0.14 ± 0.01) + (0.62 ± 0.01) BUlys , the correlation coefficient equals R = 0.696 ± 0.003, and the number of points is 26040. When the strengths of the magnetic fields intensities are limited as BUlys < 4 nT and Bcalc < 4 nT, the obtained regression equation is: Bcalc = (0.17 ± 0.01) + (0.56 ± 0.01) BUlys , with the correlation coefficient being R = 0.729 ±
0.003 at the number of points of 25822. After elimination of the peak values of Bcalc and BUlys the correlation coefficient increased. 5
1000
B(Ulysses)
4.5
Bcalc
a)
B, nT
4
900
K=0.007
800
Vsw solar wind
3.5
700
3
600
2.5
500
2
400
1.5
300
1
200
0.5
100
0
Vsw, km/s
1992
0 240
260
280
300
320
340
Day of year
1.4
500000 density
1.2
b)
temperature
1 300000
0.8 0.6
200000
0.4
Temperature, K
Density, 1/cm3
400000
100000 0.2 0 240
250
260
270
280
290
300
310
320
0 330 340 Day of year
Fig. 7. a) The observed (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on August 27 – December 5 1992 at distances to the Sun 5.28–5.12 AU and south heliolatitudes 16.1°–21.4°. b) Plasma density (black) and temperature (grey) observed in the same place. 6
3. Magnetic field peaks and corotating interaction regions
To discuss the relationship between magnetic fields, solar wind velocity, plasma density and temperature we consider in more details days 240–340 of 1992 (Fig. 7, a, b). In the Fig. 7a four large (2.5–3.5 nT) magnetic peaks (measured and calculated) are seen. Each peak in the magnetic field coincides with the leading front of the fast solar wind stream that is with the interaction region of fast and slow solar wind streams. As it is seen in Fig. 7 the variations of plasma density and temperature taken separately are not similar to the variations of magnetic fields. During the formation of magnetic peaks at first density increases, and temperature peaks appear by the time when the density are already decreasing. Variations of plasma density and temperature are considerably different in the case of small (nearly 1 nT) magnetic peaks formation. But the combination of plasma density and temperature in the form Kn T describes both high and low magnetic peaks with a quite good accuracy. B, nT
7-11 Oct 1992
Days 281-285
Vsw, km /s 800
3.5 a) 3
700 2.5 600
2 1.5
500
1 B(Ulys) Bcalc Vsw
0.5
400
0
300
6720
6744
n, 1/cm3 1.4 1.2
6768
7-11 Oct 1992
6792
6816
Days 281-285
6840 Tim e, hours
T, K 600000
density tem perature
b)
1
500000 400000
0.8 300000 0.6 200000
0.4
100000
0.2 0 6720
0 6744
6768
6792
6816 6840 Time, hours
Fig. 8. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 7–11 October 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shock, calculated by Riley et al. (1996), CIR 10. Time (in hours) is counted out since the beginning of 1992.
7
We have considered two magnetic peaks of Fig. 7 (days 282–285 and 305–310) using hourly Ulysses data (Figs 8 and 9). The CIRs in these days were analyzed by Riley et al. (1996) and, in particular, the time of crossing of shock forward and reverse fronts, bounded the CIRs, were identified. In Figs 8 and 9 we can see that the rear fronts of magnetic peaks (and the reverse shocks) coincide with the rapid and large decreases in temperature. The forward front of a magnetic peak arises usually after the density peak increase with the delay of several hours. There are, however, cases where the front of a magnetic peak and a density peak coincides, as in Fig. 9. In a few considered cases analyzed by Riley et al. (1996) shock fronts match up to 1 hour with the fronts of the magnetic peaks. In Figs 8 and 9 the typical behavior of solar wind velocity within the CIRs is also shown. The speed of the solar wind within the CIR increases, sometimes with small abrupt steps on the fronts of the magnetic peak. B, nT 4.5 4 3.5
1-5 Nov 1992
Days 306-310
B(Ulys) Bcalk Vs w
Vsw, km/s 900
a) 800
3
700
2.5 600 2 1.5
500
1
400
0.5 0 7320
300 7344
7368
7392
7416 7440 Time, hours
n, 1/cm 3
T, K 600000
2
b)
density tem perature
500000 400000
1.5
300000 1 200000 0.5
100000
0 7320
0 7344
7368
7392
7416
7440 Time, hours
Fig. 9. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1–5 November 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shocks, calculated by Riley et al. (1996), CIR 11. Time (in hours) is counted out since the beginning of 1992. Between the observed BUlys and calculated Bcalc according to Eq. (1) magnetic fields in almost every developed CIR there is a characteristic distinction, which is not visible on the averaged daily data. In the calculated magnetic field Bcalc there is a gap approximately in the center of the peak, whose origin is due to temporal behavior of the plasma density and temperature. The observed magnetic field BUlys at this point of the peak is usually enhanced. In 8
general, the consistency between the short-term details in the calculated and observed magnetic fields within the CIRs is not so good, although both magnetic field values match very well on the daily basis. In addition to magnetic peaks formed in the interactions of slow and fast solar wind streams, the Ulysses data contain magnetic peaks formed by coronal mass ejections (CMEs). In some cases, their identification is very simple, because the temperature of the plasma on the fronts of these peaks reaches coronal values. An example of the event from the coronal mass ejection is shown in Fig. 10. The behavior of the plasma density and temperature, as well as of the speed of the solar wind within such peaks is different from what we have seen in magnetic peaks of CIRs. 1 - 6 August 2005
B, nT
Days 213-218
Vs w, km/s
8
900
7
800 700
6
600
5
500
B(Ulyss es ) Bcalc solar wind
4 3
400 300
2
200
1
100
0
0
5089
5113
n, 1/cm3
5137
5161
5185
5209 Hours
5233
T, K
1 - 6 August 2005
3.5
1600000
3
1400000
Density
2.5
1200000
Tempetatute 1000000
2 800000 1.5 600000 1
400000
0.5 0 5089
200000
5113
5137
5161
5185
5209
0 5233
Hours
Fig. 10. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1–6 August 2005. b) The plasma density (black) and temperature (grey). Time (in hours) is counted out since the beginning of 2005. Identification of the CIR and CME events is not always easy, especially in the case of small CMEs (see for example Borovsky and Denton, 2006; Yermolaev et al., 2012). In Fig. 11, we present the event, which was analyzed by Broiles et al. (2013) and was identified as a CIR. However, according to the variations in the solar wind speed, temperature and density, it can be concluded that this event was formed by the CME in spite of the gap-structure was observed.
9
B, nT
24-29 Feb 2004
Days 55-60
Vsw, km/s
2.5
600 580
2
560 540
1.5
520 500
1
480 460
B(Ulys ) Bcalc Vsw
0.5
0 1296
1320
1344
1368
1392
440 420
1416
400 1440
Time, hours n, 1/cm3 0.8 0.7
24-29 Feb 2004
Days 55-60
T, K 250000
density tem perature
200000
0.6 0.5
150000
0.4 100000
0.3 0.2
50000
0.1 0 1296
0 1320
1344
1368
1392
1416 1440 Tim e, hours
Fig. 11. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 24–29 February 2004. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of the shock forward and reverse fronts, calculated by Broiles et al. (2013). Time (in hours) is counted out since the beginning of 2004. At the forefront of the shock from CMEs there is a sharp increase of the solar wind velocity and the temperature of plasma. Then the speed of the solar wind slowly decreases, whereas in the case of a CIR solar wind speed gradually increases. The plasma density increases significantly in the center of the event. Small increases in density there are also on the fronts of the peak. Note that in the theory of shocks there is a limit on the increase in density at the forefront of a wave, density may not be increased more than fourfold. Sometimes this rule was violated for the CIRs registered onboard the Ulysses spacecraft (Riley et al., 1996). The forward and rear fronts of the magnetic peak, associated with a CME, are produced by rapid changes in temperature. The central part of the peak is formed due to the density increase. The gap in the central part of the magnetic peak is absent. 4. Conclusions
The main purpose of this publication consists in demonstration of the possibility to describe the HMF along the orbit of Ulysses through local parameters of the heliospheric plasma – density and temperature. As it has been shown above the values of the calculated magnetic fields Bcalc = Kn T + B0 are consistent with the observed magnetic fields in the heliosphere at 10
distances of 1–5.4 AU and at various heliolatitudes with a good precision. The term Kn T in the above expression is responsible for magnetic field variations. This is clearly seen in Fig. 5, where both the observed magnetic field BUlys and Bcalc − B0 are presented. The high correlation between these two values can be considered as an indication that the observed magnetic field is generated by a current, whose density is proportional to n T and which flows at the same place where the plasma density and temperature are measured. The total magnitude of Bcalc includes the field B0 , whose origin is unclear. The value B0 = 0.2 nT, as seen in Figs 4 and 5, is not negligibly small in comparison with observed fields at distances 2.5–5.4 AU from the Sun and apparently it decreases with distance r slower then 1 / r 2 . We have also considered the relationship between the corotating interaction regions and peaks in the magnetic field. The regions of compressed plasma with enhanced density and temperature (CIRs) are formed in the case when the fast and slow wind streams in the heliosphere propagate in the same radial direction. As a magnetic field supposedly depends on plasma density and temperature, the formation of an interaction region is simultaneously the formation of a magnetic peak. Magnetic field peaks are generated in the heliosphere in the interactions between fast and slow solar wind streams. It was shown that theoretically calculated shock fronts match up to 1 hour with the fronts of the magnetic peaks, so the magnetic peaks can be considered as the magnetic images of the CIRs. It was shown (on daily and hourly data) that the plasma density and temperature in various ways are involved in the formation of magnetic peaks. A density peak on the leading part of a fast solar wind stream is connected with the forward shock in a CIR as well as with the leading edge of a magnetic peak. The trailing edge of the magnetic peak with its step-like temperature decrease may be a location of the intersection of the reverse shock by the Ulysses spacecraft. A gap between the density and temperature peaks corresponds to a rarefaction region. Spatial extents of the examined magnetic peaks are about 1 AU at solar wind velocity 700 km/s, which agrees with distance between forward and reverse shocks in a developed CIR. We have pointed out the possibility of using data on plasma density and temperature in the identification of the CIRs and CMEs. It was shown that the forward front of the CME magnetic pick is formed by the sharp increase in temperature, whereas the forward front of the CIR is formed by the sharp increase in plasma density. It was noted the different behavior of the solar wind velocity in the CIRs and CMEs and some other differences. We have also noted the difference between the magnetic peaks produced by the CIRs and by the CMEs. Nevertheless the forward and rear fronts of magnetic peaks coincide with the forward and reverse shocks of CMEs. It was proposed a possible explanation of the correlation between the measured magnetic field and plasma temperature and density. In the Appendix the derivation of Eq. (1) is given. The derivation is based on analogy between currents in long straight wires and currents in the plasma. Our basic assumption is that the thermal motion of electrons with the averaged velocities exceeding the averaged thermal velocities of protons will produce some uncompensated negative volume charge density in the plasma, which is the source of the electric field E ≠ 0 . Because the heliospheric plasma has high conductivity and protons are free to move in response to that electric field, the proton current occurs in the plasma. As a result of that the density of the positive charges increases and the neutrality of the plasma will be restored and the electric field turn to zero, E = 0. The magnetic field in the plasma appears simultaneously with the proton current. Acknowledgments
11
This work was partially supported by the Russian Foundation for Basic Research (grants 1402-00905a, 14-02-10006k) and by the "Fundamental Properties of Matter and Astrophysics" Program of the Presidium of the RAS. The authors are grateful to the scientific teams of Ulysses for the data on the magnetic fields and heliospheric plasma presented in Internet.
Appendix. Relationship between the heliospheric magnetic field and the plasma temperature and density
To derive the expression Bcalc = Kn T + B0 (A1) we used the method of determination of the magnetic field near a long straight wire carrying an electric current considered by E. Purcell in his textbook (1985). In Purcell's approach the force acting on the test charge q, moving along a wire, is calculated in the test charge rest frame. If the charge is at rest there is no magnetic force acting on it, and an electric field only needs to be taken into account. The electric force, calculated in the test charge rest frame, is then converted to a "laboratory" frame of reference and analyzed. In order to understand the origin of magnetic and electric fields around current carrying wires, consider two inertial frames − the "laboratory" S and the test charge rest frame S'. It is supposed that the coordinates of the two frames are oriented so that the velocity u of the charge q is parallel to the x axis. Suppose that the current in a wire, which is at rest in the frame S, comes from movement of both negative ρ and positive σ charges. The line charge densities are not Lorentz invariant. Denote the line density of the positive and negative charges in their rest frames as ρ 0 and σ 0 , respectively. Then the relation between ρ and ρ 0 can be expressed by the formula 2 2 ρ = ρ 0γ (v − ) where γ ( v− ) = 1 / 1 − v − / c , and v − is a relative velocity of the negative charge rest frame with respect to S (i.e. the drift velocity of the electrons in the wire). The relation ρ = ρ 0γ (v − ) is a consequence of the Lorentz transformations of a 4-vector current density. For the line charge density σ , as well as for the line charge densities ρ ' and σ ' in the primed frame S' we can find the same relations: σ = σ 0γ (v + ) , ρ ' = ρ 0γ (v' − ) , σ ' = σ 0γ (v' + ) , where v + , γ (v+ ) , v ' − , γ (v' − ) , v ' + and γ (v' + ) have the same sense as v − and γ (v− ) . Now we can write an expression for the net charge line density in the frame S' as:
γ (v+' ) γ (v −' ) ρ . σ− σ '− ρ ' = γ (v − ) γ (v+ )
(A2)
2(σ '− ρ ' ) , and the electric force acting on the test charge r' q is F ' = qE ' . In the frame S the charge q experiences the force:
Then the electric field is given by E ' =
F=
F' 2q u = (σ − ρ ) − 2 [σv+ − ρv − ] . γ (u ) r c
(A3)
12
In Eq. (A3) the force is expressed explicitly via the line density of charges and their speed in the frame S and the speed u of the test charge. Note that the density of charges are included in (A3) in accordance to their charge signs ( − ρ and + σ ). But the movement direction of charges (in the direction of the x axis or against the x axis) is not selected. The quantity (σ − ρ ) is the net 2(σ − ρ ) charge line density and then we can consider = E as the electric field. The quantity r [σv+ − ρv− ] is the net current density and then 2 [σv + − ρv− ] = 2I = B we can consider as the rc rc magnetic field. Now we can rewrite (A3) in the form:
F = qE −
quB . c
(A4)
The minus sign of the second term in Eq. (A4) indicates a force directed towards a wire for positive q, u and I. The sign of the first term depends on the directions of v+ and v − and their values. Thus we find that the force acting on the test charge, moving along the long straight wire consists of the sum of the electric and magnetic forces. Both these forces are of relativistic origin and turn to zero in the case of nonrelativistic transformation of the charge velocities and if it is assumed that the charge line density does not depend on its speed. Consider now a typical metal long wire where the nuclei and bound electrons are at rest ( v + =0, σ = σ 0 ), and where the current is produced by the conduction electrons. The force Eq. (A3) acting on the test charge q is then
F=
ρuv 2q (σ 0 − ρ ) + 2 − . r c
(A5)
The wire is electrically neutral at zero current, σ 0 − ρ 0 = 0 . Since ρ = γ (v− ) ρ 0 = γ (v− )σ 0 and σ 0 = ρ / γ (v − ) we can exclude σ 0 from Eq. (A5) and obtain: F=
uv 2 qρ 1 − 1) + 2− . ( r γ (v − ) c
Using two first terms of the expansion of the quantity 1 / γ (v − ) = 1 −
(A6)
1 v−2 we can rewrite Eq. 2 c2
(A6) as: F =−
qρv−2
rc 2
+
2qρuv−
rc 2
.
(A7)
The explicit expressions for electric and magnetic fields in that case are: E=−
ρv−2 rc
2
; B=
2 ρuv − . rc
(A8)
13
Note that the electric field in Eq. (A8) depends on the square of the velocity v−2 of the conduction electrons, while the magnetic field is proportional to the first degree of v− . Therefore, if we change the direction of the electrons velocity to the opposite, then we will have:
F∗ = −
qρv −2 2qρuv− . − rc 2 rc 2
(A9)
The field near the conductor consisting of two strands with currents, opposite in direction (a "bifilar winding"), can be obtained by summarizing Eq. (A7) and Eq. (A9). In this case, the magnetic fields mutually compensate each other and the electric field doubles and becomes:
E=−
2 ρv−2
rc 2
.
(A10)
The electric fields of such origin, as it will be shown below, can exist in the heliospheric plasma. Let us select some volume element of space plasma in the solar wind and choose two inertial frames of reference – the frame S which is moving with the same velocity V as the solar wind (and the selected plasma volume element), and the space probe rest frame S' which is moving with the velocity u = –V. Suppose that the plasma is fully ionized hydrogen plasma with Maxwellian distribution functions and with the same temperature T of the electrons and protons. It is also assumed that the electric field E=0 in the frame S. In the case of Maxwellian velocity distribution functions the average thermal electron speed 2 in the x direction (parallel to the solar wind velocity V) is v xe
2 average thermal proton speed is v xp
1/ 2
1/ 2
= kT / me ≡ v − and the
= kT / m p ≡ v+ , where k is the Boltzmann constant
and m e and m p are the electron and proton masses, respectively (Bittencourt, 2004). Suppose now that in the selected plasma volume the one sixth of electrons and the one sixth of protons move with the averaged thermal velocities v − and v+ in the direction of the x axis and the same amount of particles move with the same velocities against the x axis. The movement of electrons and protons along the axis х we consider as an analogue of linear currents in long straight wires. Knowing the plasma temperature T we can calculate the averaged thermal velocities v − and v+ of the electrons and protons in the frame S and their ratio v − / v + = m p / me of ~43. Bearing in mind that v − >> v + , we will not take into account the thermal movement of the protons, putting the averaged thermal speed of the protons v + = 0. This has two consequences. The first is that the proton current density σv + = 0. The second is that the proton charge density σ = σ 0 , as the frame S is now the rest frame for the protons. As a result, we have got a situation similar to that for the wires with a bifilar winding, when two electron currents flow in opposite directions along the x axis. Now let's calculate the net charge density (A2) in the frame S', after defining all the required values – the velocities v+ , v− , v+' , v−' and the proton σ and electron ρ charge densities – from physical considerations and from the experiment. Consider separately two cases: a) v − >0 and b) v − <0.
14
In the case a) Lorentz factors of expression (A2) are γ (v + ) = 1 , γ (v − ) =
1 1 − v−2 / c 2
,
v− − u uv . = γ (u )γ (v− )1 − 2− where v −' = c 1 − uv− / c 2 1− v / c 1− v / c2 Then the net charge density in the frame S' is given by:
γ (v − ) =
1
2 −
2
, γ (v −' ) =
σ '− ρ ' = γ (u )σ − (1 −
1
' 2 −
)ρ . c
uv− 2
(A11)
In the same way as in a Purcell's problem, presented above, we get the electric field and the electric force acting on the test charge q in the frame S'. In the frame S the electric force Fa acting on the test charge is given by: Fa =
qEa' 2q uv = σ − (1 − 2− ) ρ . γ (u ) r c
(A12)
Case b) differs from the case of a) only in expressions for the Lorentz factor − v− − u 1 uv . The electric γ (v −' ) = = γ (u )γ (v− )1 + 2− and for the velocity v −' = 2 c 1 + uv − / c 2 1− v' / c2 −
force acting on the test charge q in case b) is: Fb =
qEb' 2q uv = σ − (1 + 2− ) ρ . γ (u ) r c
(A13)
The total force acting on the test charge q in the frame S' is equal to the sum 2q ⋅ 2(σ − ρ ) . Such a force can be created by the electric field F = Fa + Fb = r E=
4(σ − ρ ) , r
(A14)
where the charge line density is 2(σ − ρ ) . Thus, we have found that, if Te = T p , in the plasma the electric field is created in the same way as the electric field near a long straight wire. The source of this field is the thermal motion of the electrons, which creates two oppositely directed electron flows along the x axis. The magnetic fields of these electron flows mutually compensate each other. The above expressions for the electric field in plasma are similar to expressions for ordinary long wires. It is obvious that mathematical expressions for the fields in the plasma and the fields beside the usual conductors must somehow be different. The heliospheric plasma is a continuum with the distributed volume currents where the distance r from the point of observation to the x axis is meaningless. But this does not change anything in fact that the thermal motion of electrons with the averaged velocities v − exceeding the averaged thermal velocities v+ of protons will produce some net charge density 2(σ − ρ ) ≠ 0 in the plasma, and, according to Maxwell's equation 2(σ − ρ ) = ε 0 divE , generate the electric field ( ε 0 is the permittivity of free space).
15
Therefore, our basic assumption is, that, provided the v− > v+ , in the heliospheric plasma the uncompensated negative volume charge is formed which is the source of the electric field E. Because the heliospheric plasma has high conductivity and protons are free to move in response to that electric field, the proton current occurs in the plasma. As a result of that the density of the positive charges increases and the neutrality of the plasma is restored, so the condition E = 0 will be satisfied. The magnetic field appears simultaneously with the proton current. Using the condition σ − ρ = 0 which is equivalent to condition E = 0, we estimate now the speed of protons v p , sufficient to restore the neutrality of the plasma. Recall that there exist the following relations for the charge densities in the frame S : ρ = γ (v− ) ρ 0 , σ = σ 0 and the condition of the plasma neutrality in the absence of currents σ 0 − ρ 0 = 0 . After the protons will begin to move with the speed v p their charge density will be equal to σ = σ 0γ (v p ) . We present 2ρ0 , since the two third of electrons 3 3 move in directions transverse to the x axis and their charge densities are not changed by the Lorentz transformation. Then from σ − ρ = 0 it follows that
the electron charge density in the form of ρ =
σ 0 γ (v p ) −
ρ 0 γ (v− ) 3
−
ρ 0 γ (v − )
−
2ρ 0 = 0. 3
(A15)
Using the expansion of the Lorentz factors γ (v p ) and γ (v− ) we find
σ 0 (1 +
v 2p
ρ0
2c
3
)= 2
(3 +
v −2 ) = 0. 2c 2
Solving this equation for the proton velocity v p gives v p = v −2 / 3 = kT / 3me .
Directed motion of protons with the velocity v p produces the current I = σ v p S in the plasma volume element, where S is the cross section of the volume element. Current I in the direction of x axis creates the magnetic field: 2I 2 Sσ kT = = K ∗σ T (A16) B= rc rc 3me 2S where K ∗ = k / 3me . rc As it was said above the quantity r in Eq. (A16) can not be considered as a radial distance from the x axis to the point of observation. We assume that the expression Eq. (A16) just sets the proportionality between the magnetic field B and the density of positive charges σ and temperature T of plasma, where the coefficient K ∗ = f (r , S ) is some function of the spatial variables which has to be defined in more rigorous theory. The magnetic field B in Eq. (A16) is given in the frame of reference S, which is moving along with the plasma. The magnetic field B' measured in the spacecraft frame S' is B' = γ (u ) B = K ∗σγ (u ) T = K ∗σ ' T = K ∗ ne+ T , where σ ' = ne+ = σ γ (u ) is the proton charge density in the frame S', n is the proton number density and e+ is the proton electric charge. Denoting K ∗ e+ = K , we can write the expression for the magnetic field in the form B' = Kn T . Here the magnetic field is represented as a function of the plasma parameters n and T, which are directly measured in the spacecraft reference frame S'. By comparing the observed magnetic fields with the values B', we have found that a better description of the experimental data provides an expression B' = Kn T + B0 . We have used that expression to calculate the magnetic fields in this work.
16
References Bame, S.J., Goldstein, B.E., Gosling. J.T., Harvey, W., McComas, D.J., Neugebauer, M., Phillips, J.L., 1993. Ulysses observations of a recurrent high speed solar wind stream and the heliomagnetic streamer belt. Geophys. Res. Lett. 20(21), 2323–2326. Bittencourt, J.A., 2004. Fundamentals of Plasma Physics. 3rd Edition, Springer-Verlag, 583 p. Borovsky, J.E., Denton, M.H., 2006. Differences between CME-driven storms and CIR-driven storms, J. Geophys. Res. 111, A07S08: 1–17, doi: 10.1029/2005JA011447. Broiles, T.W., Desai, M.I., Lee, C.O, MacNeice, P.J., 2013. Radial evolution of the threedimensional structure in CIRs between Earth and Ulysses. J. Geophys. Res. 118, 4776–4792. Broiles, T.W., Desai, M.I., McComas, D.J., 2012. Formation, shape, and evolution of magnetic structures in CIRs at 1 AU. J. Geophys. Res. 117, A03102: 1-21, doi: 10.1029/2011JA017288 Burlaga, L.F., Wang, C., Richardson, J.D., Ness, N.F., 2003. Evolution of magnetic fields in corotating interaction regions from 1 to 95 AU: Order to chaos. Astrophys. J. 590, 554–566. Gosling, J.T., Bame, S.J., McComas, D.J., Phillips, J.L., Pizzo, V.J., Goldstein, B.E., Neugebauer, M., 1993. Latitudinal variation of solar wind corotating stream interaction regions: Ulysses. Geophys. Res. Lett. 20, 2789–2792. Gosling, J.T., Bame, S.J., McComas, D.J., Phillips, J.L., Pizzo, V.J., Goldstein, B.E., Neugebauer, M., 1995. Solar wind corotating stream interaction regions out of the ecliptic plane: Ulysses. Space Sci. Res. 72, 99–104. Gosling, J.T., Pizzo, V.J., 1999. Formation and evolution of corotating interaction regions and their three dimensional structure. Space Sci. Res. 89, 21–52. Hundhausen, A.J., 1973. Nonlinear model of high-speed solar wind streams. J. Geophys. Res. 78, 1528–1542. Jian, L.K., Russell, C.T., Luhmann, J.G., MacNeice, P.J., Odstrcil,·D.,·Riley, P., Linker, J.A., Skoug, R.M., Steinberget,·J.T., 2011. Comparison of observations at ACE and Ulysses with Enlil model results: Stream interaction regions during Carrington rotations 2016–2018. Sol. Phys. 273, 179–203. Lee, M.A., 2000 An analytical theory of the morphology, flows, and shock compressions at corotating interaction regions in the solar wind. J. Geophys. Res. 105, 10491–10500. Odstrcil, D., 2003. Modeling 3-D solar wind structure. Adv. Space Res. 32, 497–506, doi: 10.1016/S0273-1177(03)00332-6. Pizzo, V.J., Gosling, J.T., 1994. 3-D simulation of high-latitude interaction regions: Comparison with Ulysses results. Geophys. Res. Lett. 21, 2063–2066. Purcell, E.M., 1985. Electricity and Magnetism, 2nd Edition, McGraw-Hill Book Comp., 507 p. Riley, P., Gosling, J.T., Weiss, L.A., Pizzo, V.J., 1996. The tilts of corotating interaction regions at midheliographic latitudes. J. Geophys. Res. 101, 24349-24357. Svirzhevsky, N.S., Bazilevskaya, G.A., Svirzhevskaya, A.K., Stozhkov, Yu.I., 2014. The heliospheric magnetic field and its relation to the temperature, density, and velocity of solar plasma: Experimental evidence. Cosmic Research 52, 15-24. Yermolaev, Y.I., Nikolaeva, N.S., Lodkina, I.G., Yermolaev, M.Y., 2012. Geoeffectiveness and efficiency of CIR, sheath, and ICME in generation of magnetic storms. J. Geophys. Res. 117, A00L07:1-9, doi: 10.1029/2011JA017139.
17
Fig. 1. Comparison of the measured magnetic field BUlys (grey) and calculated magnetic field Bcalc (black) in the presence of fast solar wind streams from a coronal hole. The heliolatitude of the spacecraft is shown by a dashed line. Ulysses’ distances from the Sun at the beginning and end of the examined time interval and the correlation coefficient R between two sets are also shown.
Fig. 2. Comparison of the measured BUlys and calculated Bcalc magnetic fields during the 1996 solar activity minimum. Notations on the figure are the same, as in Figure 1. Fig. 3. Comparison of the measured and calculated values of magnetic fields during the crossing of the sector zone at the final orbit of Ulysses around the Sun. Fig. 4. Comparison of the observed BUlys magnetic field with the values B − B0 = Kn T , B0 = 0.2 nT in the first half of 2006. 5 Fig. 5. The dependence of the B0 on the distance to the Sun in October 1995-December 1997. Fig. 6. The scattering plot for the whole sets of the Ulysses data in 1990-2008. The regression equation and the correlation coefficient R are also shown. Fig. 7. a) The observed (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on August 27 - December 5 1992 at distances to the Sun 5.28-5.12 AU and south heliolatitudes 16.1°-21.4°. b) Plasma density (black) and temperature (grey) observed in the same place. Fig. 8. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 7-11 October 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shock, calculated by Riley et al. (1996), CIR 10. Time (in hours) is counted out since the beginning of 1992. Fig. 9. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1-5 November 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shock, calculated by Riley et al. (1996), CIR 11. Time (in hours) is counted out since the beginning of 1992. Fig. 10. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1-6 August 2005. b) The plasma density (black) and temperature (grey). Time (in hours) is counted out since the beginning of 2005. Fig. 11. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 24-29 February 2004. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of shock forward and reverse fronts, calculated by Broiles et al. (2013). Time (in hours) is counted out since the beginning of 2004.
18
S0273-1177(14)00521-3 http://dx.doi.org/10.1016/j.asr.2014.08.009 JASR 11914
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
3 February 2014 1 August 2014 9 August 2014
Please cite this article as: Svirzhevsky, N.S., Bazilevskaya, G.A., Svirzhevskaya, A.K., Stozhkov, Yu.I., Comparison of measured and calculated magnetic fields along the Ulysses orbit, Advances in Space Research (2014), doi: http:// dx.doi.org/10.1016/j.asr.2014.08.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Comparison of measured and calculated magnetic fields along the Ulysses orbit N.S. Svirzhevsky, G.A. Bazilevskaya, A.K. Svirzhevskaya, Yu.I. Stozhkov Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia
Abstract The existence of close relations between the temperature, density and velocity of the solar plasma and the heliospheric magnetic field (HMF) was shown along the space probe Ulysses orbit. A simple mathematical formula describing a relation between the HMF and the solar plasma temperature and density was introduced and the expected values of the HMF were calculated using daily and hourly Ulysses data. Correlation coefficients and regression equation between the values of the measured and calculated magnetic fields have been defined. An origin of the peaks in the magnetic field which are observed in the heliospheric sector zone near the corotating interaction regions is discussed as well as the specific role of plasma density and temperature in the formation of magnetic peaks. Keywords: heliospheric magnetic field; solar wind; plasma density; plasma temperature 1. Introduction In this work we have considered a possibility to express the magnetic field in the heliosphere by means of the local parameters of heliospheric plasma – temperature and density. The proposed relationship between the magnetic field and the plasma parameters is expressed in the following form (Svirzhevsky et al., 2014): Bcalc = Kn T + B0 ,
(1)
where n is the proton number density and T is the temperature of the heliospheric plasma, K and B0 are parameters, which were defined from comparison of measured and calculated magnetic fields along the Ulysses orbit. The measured magnetic fields, in dependence on the distance r to the Sun, can be described by two sets of K and B0 parameters, namely, at r = 1–2.5 AU K = 0.003 and B0 = 0.6 nT, and at r = 2.5–5.4 AU K = 0.006 and B0 = 0.2 nT. The derivation of Eq. (1) is given in the Appendix. The derivation is based on the assumption that the thermal movement of electrons with the average velocity exceeding the average thermal velocity of protons violates the neutrality of the plasma and causes the formation of the negative volume charge and the electric field in the heliospheric plasma. As a result of that the proton current occurs in the plasma which tends to restore its neutrality, as well as creates the magnetic field in it. The dependence of the magnetic field on the plasma density and temperature was obtained in a preliminary model which does not take into account some feature of the heliospheric magnetic field, in particular, its Parker spiral form. We also intend to discuss below the relationship between corotating interaction regions (CIRs) and peaks in the magnetic field which are observed in the heliospheric sector zones. The interactions between fast solar wind from coronal holes and slow solar wind from the belt of coronal streamers form regions of compressed plasma with enhanced density and temperature. ----------------------------------------------Corresponding author Email address: [email protected] Tel: +74954854100 1
These interactions, which lead to the formation of the CIRs, are considered as one of the primary processes shaping the structure of the interplanetary medium (Riley et al., 1996). Usually the CIRs are well developed at the distance of 1 AU (Hundhausen, 1973; Broiles et al., 2012). In some cases, observed at 1 AU CIRs may be identified as individual CIRs or/and as merged ones at a distance of about 5 AU (Broiles et al., 2013; Jian et al., 2011). At distances of 5–15 AU single CIRs merge into the corotating merged interaction region zones (CMIR zones), whose existence was confirmed by Voyager 2 observations (Burlaga et al., 2003). Many of the observed onboard the Ulysses spacecraft CIR characteristics were identified and examined in details in (Gosling et al., 1993; Bame et al., 1993; Gosling et al., 1995; Riley et al., 1996; Gosling and Pizzo, 1999; Broiles, et al., 2012). Various theoretical approaches to the modeling of corotating interaction regions were considered (Pizzo and Gosling, 1994; Lee, 2000; Odstrcil, 2003; Jian et al., 2011). We want to point out some features of CIRs, which can be set using the magnetic field strength, plasma density and temperature values. In particular, it will be shown that the forward and reverse shocks, bounding the CIRs, coincide with the forward and rear fronts of magnetic peaks. We can also note the difference between the magnetic peaks occurring in the interactions of fast and slow solar winds and created by shocks from coronal mass ejections. Different role of plasma density and temperature in the formation of magnetic peaks will also be shown. 2. Measured and calculated magnetic fields along the Ulysses orbit Ulysses was the first space probe which made observations of the magnetic fields and heliospheric plasma along a near-polar orbit around the Sun. The Ulysses measurements enable us to determine the relations between the HMF and the plasma parameters at different heliolatitudes and distances to the Sun. We can also compare the correlation between measured and calculated magnetic fields during the minimum and maximum of solar activity. Ulysses began to move to the south pole of the Sun after its encounter with Jupiter in February 1992 and to the middle of 1993 it was in the sector zone and regularly crossed fast solar wind streams. Comparison of the measured on board the Ulysses magnetic fields and calculated in accordance with Eq. (1) is shown in Fig. 1 for 1993. The Ulysses daily averaged data on magnetic fields and heliospheric plasma taken at website http://helio.estec.esa.nl/ulysses/archive are used in Fig. 1. The large magnetic peaks with a periodicity approximately that of the solar rotation are seen in Fig. 1. In Fig. 2 the HMF in the northern hemisphere is shown during the 1996 solar activity minimum. Moderate fluctuations are typical for the magnetic field, which is carried away by the solar wind from a coronal hole. Ulysses was located in the field of one polarity for more than a year and a half, and not a single considerable peak in the magnetic field was detected during this time. Only in the beginning of October 1996 Ulysses at 27° north heliolatitude entered a sector zone with an alternating sign of the HMF azimuth component and with large peaks of the magnetic field (exceeding the background field by a factor of 5–10). The observed magnetic field BUlys and calculated field Bcalc are quite consistent in this period, the coefficient of correlation between them is R = 0.884. Comparison of calculation with experimental data obtained during high solar activity of 2000 at high latitudes and in the area of the South Pole found that the correlation between the BUlys and Bcalc still remained high, the correlation coefficient is R = 0.832, despite the fact that the structure of the magnetic field during this period, with uncertain position of the heliospheric current sheet, has become more complex. Comparison of the measured and calculated values of magnetic fields during the crossing of the sector zone in 2007 is shown in Fig. 3. Unlike 1995, the sector zone in 2007 was located symmetrically relative to the helioequator. The coefficient of correlation between BUlys and Bcalc is R = 0.898.
2
В, nT
R=0.856
Heliolatitude, deg
4
0 5.07- 3.83 AU
3.5
B(Ulyss es ) Bcalc
-10
Heliolatitude
3
-20
2.5 2
-30
1.5
-40
1 -50
0.5 0 1993
-60 1993.2
1993.4
1993.6
1993.8
Years
1994
Fig. 1. Comparison of the measured magnetic field BUlys (grey) and calculated magnetic field
Bcalc (black) in the presence of fast solar wind streams from a coronal hole. The heliolatitude of the spacecraft is shown by a dashed line. Ulysses' distances from the Sun at the beginning and end of the examined time interval and the correlation coefficient R between two sets are also shown.
B, nT
R=0.884
Heliollatitude, deg
3
60 B(Ulysses)
3.06 - 4.71 AU.
Bcalc
2.5
50
Heliolatitude 2
40
1.5
30
1
20
0.5
10
0 1996
0 1996.2
1996.4
1996.6
1996.8
1997 Years
Fig. 2. Comparison of the measured BUlys and calculated Bcalc magnetic fields during the 1996 solar activity minimum. Notations in the Figure are the same, as in Fig. 1. Adopted from Svirzhevsky et al., 2014.
3
B, nT
1.67- 1.34 -1.55 AU
Heliolatitude, deg
10 9
60 B(Ulysses) Bcalc Heliolatitude
R=0.898
8
50 40 30
7
20
6
10
5
0
4
-10 -20
3
-30 2
-40
1
-50
0
-60
2007.4
2007.5
2007.6
2007.7
Years
2007.8
Fig. 3. Comparison of the measured and calculated values of magnetic fields during the crossing of the sector zone at the final orbit of Ulysses around the Sun. B, nT
R=0.893
2 B(Ulyss es )
4.47 - 3.69 AU
1.8
Bcalc - Bo 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 2006
2006.1
2006.2
2006.3
2006.4
2006.5 Years
Fig. 4. Comparison of the observed BUlys magnetic field with the values B − B0 = Kn T , B0 = 0.2 nT in the first half of 2006. Adopted from Svirzhevsky et al., 2014. In Fig. 4 the measured magnetic fields BUlys are compared with the values B − B0 = Kn T , where B0 = 0.2 nT. High correlation between BUlys and Kn T can be interpreted as an indication that the magnetic fields are formed by currents that flow in the same place where the measurements of plasma density and temperature are made. The origin of the parameter B0 included in the calculated magnetic field module B is not understood. The value of B0 = 0.2 nT used in calculations at distances r = 2.5–5.4 AU to the Sun is not negligibly small and it seems the B0 is decreasing with the distance more slowly then 1 / r 2 . For example, the dependence of 4
the B0 on the distance to the Sun in October 1995–December 1997 (65° N – the ecliptic) is shown in Fig. 5. Bo, nT
1 Oct 1995 - Dec 1997
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 2.5
3
3.5
4
4.5
5
r, AU
Fig. 5. The dependence of the B0 on the distance to the Sun in October 1995–December 1997. In Fig. 6 the BUlys – Bcalc scattering diagram constructed for the entire sets of the Ulysses daily data for 1990–2008 is shown. The regression equation is Bcalc = (0.16 ± 0.011) + (0.97 ± 0.0067) BUlys , with the correlation coefficient R = 0.874 ± 0.003 . Bcalc, nT
1990-2008
num ber of points 6576
16 14 12 10 8 y = 0.9711x + 0.1598 6
R=0.874
4 2 0 0
2
4
6
8
10
12
14 16 B(Ulysses), nT
Fig. 6. The scattering plot for the whole sets of the Ulysses data in the regression equation and the correlation coefficient R are also shown. Adopted from Svirzhevsky et al., 2014. In order to check the dependence of the regression and correlation coefficient R on points with the peak values of magnetic fields the regression equation was calculated under restrictions BUlys ≤ 4 nT, Bcalc ≤ 4 nT. It was obtained in that case Bcalc = (0.24 ± 0.008) + (0.87 ± 0.0063) BUlys , with
5
the correlation coefficient R = 0.867 ± 0.003 and number of points 6301. It means that high correlation between Bcalc and BUlys is not caused by the peak values of magnetic fields. The regression equations and correlation coefficients were also calculated using selected samples of hourly data. The result of calculations for the 3 year period from 2003 to 2005 without constraints on BUlys and Bcalc gives the regression equation: Bcalc = (0.14 ± 0.01) + (0.62 ± 0.01) BUlys , the correlation coefficient equals R = 0.696 ± 0.003, and the number of points is 26040. When the strengths of the magnetic fields intensities are limited as BUlys < 4 nT and Bcalc < 4 nT, the obtained regression equation is: Bcalc = (0.17 ± 0.01) + (0.56 ± 0.01) BUlys , with the correlation coefficient being R = 0.729 ±
0.003 at the number of points of 25822. After elimination of the peak values of Bcalc and BUlys the correlation coefficient increased. 5
1000
B(Ulysses)
4.5
Bcalc
a)
B, nT
4
900
K=0.007
800
Vsw solar wind
3.5
700
3
600
2.5
500
2
400
1.5
300
1
200
0.5
100
0
Vsw, km/s
1992
0 240
260
280
300
320
340
Day of year
1.4
500000 density
1.2
b)
temperature
1 300000
0.8 0.6
200000
0.4
Temperature, K
Density, 1/cm3
400000
100000 0.2 0 240
250
260
270
280
290
300
310
320
0 330 340 Day of year
Fig. 7. a) The observed (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on August 27 – December 5 1992 at distances to the Sun 5.28–5.12 AU and south heliolatitudes 16.1°–21.4°. b) Plasma density (black) and temperature (grey) observed in the same place. 6
3. Magnetic field peaks and corotating interaction regions
To discuss the relationship between magnetic fields, solar wind velocity, plasma density and temperature we consider in more details days 240–340 of 1992 (Fig. 7, a, b). In the Fig. 7a four large (2.5–3.5 nT) magnetic peaks (measured and calculated) are seen. Each peak in the magnetic field coincides with the leading front of the fast solar wind stream that is with the interaction region of fast and slow solar wind streams. As it is seen in Fig. 7 the variations of plasma density and temperature taken separately are not similar to the variations of magnetic fields. During the formation of magnetic peaks at first density increases, and temperature peaks appear by the time when the density are already decreasing. Variations of plasma density and temperature are considerably different in the case of small (nearly 1 nT) magnetic peaks formation. But the combination of plasma density and temperature in the form Kn T describes both high and low magnetic peaks with a quite good accuracy. B, nT
7-11 Oct 1992
Days 281-285
Vsw, km /s 800
3.5 a) 3
700 2.5 600
2 1.5
500
1 B(Ulys) Bcalc Vsw
0.5
400
0
300
6720
6744
n, 1/cm3 1.4 1.2
6768
7-11 Oct 1992
6792
6816
Days 281-285
6840 Tim e, hours
T, K 600000
density tem perature
b)
1
500000 400000
0.8 300000 0.6 200000
0.4
100000
0.2 0 6720
0 6744
6768
6792
6816 6840 Time, hours
Fig. 8. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 7–11 October 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shock, calculated by Riley et al. (1996), CIR 10. Time (in hours) is counted out since the beginning of 1992.
7
We have considered two magnetic peaks of Fig. 7 (days 282–285 and 305–310) using hourly Ulysses data (Figs 8 and 9). The CIRs in these days were analyzed by Riley et al. (1996) and, in particular, the time of crossing of shock forward and reverse fronts, bounded the CIRs, were identified. In Figs 8 and 9 we can see that the rear fronts of magnetic peaks (and the reverse shocks) coincide with the rapid and large decreases in temperature. The forward front of a magnetic peak arises usually after the density peak increase with the delay of several hours. There are, however, cases where the front of a magnetic peak and a density peak coincides, as in Fig. 9. In a few considered cases analyzed by Riley et al. (1996) shock fronts match up to 1 hour with the fronts of the magnetic peaks. In Figs 8 and 9 the typical behavior of solar wind velocity within the CIRs is also shown. The speed of the solar wind within the CIR increases, sometimes with small abrupt steps on the fronts of the magnetic peak. B, nT 4.5 4 3.5
1-5 Nov 1992
Days 306-310
B(Ulys) Bcalk Vs w
Vsw, km/s 900
a) 800
3
700
2.5 600 2 1.5
500
1
400
0.5 0 7320
300 7344
7368
7392
7416 7440 Time, hours
n, 1/cm 3
T, K 600000
2
b)
density tem perature
500000 400000
1.5
300000 1 200000 0.5
100000
0 7320
0 7344
7368
7392
7416
7440 Time, hours
Fig. 9. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1–5 November 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shocks, calculated by Riley et al. (1996), CIR 11. Time (in hours) is counted out since the beginning of 1992. Between the observed BUlys and calculated Bcalc according to Eq. (1) magnetic fields in almost every developed CIR there is a characteristic distinction, which is not visible on the averaged daily data. In the calculated magnetic field Bcalc there is a gap approximately in the center of the peak, whose origin is due to temporal behavior of the plasma density and temperature. The observed magnetic field BUlys at this point of the peak is usually enhanced. In 8
general, the consistency between the short-term details in the calculated and observed magnetic fields within the CIRs is not so good, although both magnetic field values match very well on the daily basis. In addition to magnetic peaks formed in the interactions of slow and fast solar wind streams, the Ulysses data contain magnetic peaks formed by coronal mass ejections (CMEs). In some cases, their identification is very simple, because the temperature of the plasma on the fronts of these peaks reaches coronal values. An example of the event from the coronal mass ejection is shown in Fig. 10. The behavior of the plasma density and temperature, as well as of the speed of the solar wind within such peaks is different from what we have seen in magnetic peaks of CIRs. 1 - 6 August 2005
B, nT
Days 213-218
Vs w, km/s
8
900
7
800 700
6
600
5
500
B(Ulyss es ) Bcalc solar wind
4 3
400 300
2
200
1
100
0
0
5089
5113
n, 1/cm3
5137
5161
5185
5209 Hours
5233
T, K
1 - 6 August 2005
3.5
1600000
3
1400000
Density
2.5
1200000
Tempetatute 1000000
2 800000 1.5 600000 1
400000
0.5 0 5089
200000
5113
5137
5161
5185
5209
0 5233
Hours
Fig. 10. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1–6 August 2005. b) The plasma density (black) and temperature (grey). Time (in hours) is counted out since the beginning of 2005. Identification of the CIR and CME events is not always easy, especially in the case of small CMEs (see for example Borovsky and Denton, 2006; Yermolaev et al., 2012). In Fig. 11, we present the event, which was analyzed by Broiles et al. (2013) and was identified as a CIR. However, according to the variations in the solar wind speed, temperature and density, it can be concluded that this event was formed by the CME in spite of the gap-structure was observed.
9
B, nT
24-29 Feb 2004
Days 55-60
Vsw, km/s
2.5
600 580
2
560 540
1.5
520 500
1
480 460
B(Ulys ) Bcalc Vsw
0.5
0 1296
1320
1344
1368
1392
440 420
1416
400 1440
Time, hours n, 1/cm3 0.8 0.7
24-29 Feb 2004
Days 55-60
T, K 250000
density tem perature
200000
0.6 0.5
150000
0.4 100000
0.3 0.2
50000
0.1 0 1296
0 1320
1344
1368
1392
1416 1440 Tim e, hours
Fig. 11. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 24–29 February 2004. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of the shock forward and reverse fronts, calculated by Broiles et al. (2013). Time (in hours) is counted out since the beginning of 2004. At the forefront of the shock from CMEs there is a sharp increase of the solar wind velocity and the temperature of plasma. Then the speed of the solar wind slowly decreases, whereas in the case of a CIR solar wind speed gradually increases. The plasma density increases significantly in the center of the event. Small increases in density there are also on the fronts of the peak. Note that in the theory of shocks there is a limit on the increase in density at the forefront of a wave, density may not be increased more than fourfold. Sometimes this rule was violated for the CIRs registered onboard the Ulysses spacecraft (Riley et al., 1996). The forward and rear fronts of the magnetic peak, associated with a CME, are produced by rapid changes in temperature. The central part of the peak is formed due to the density increase. The gap in the central part of the magnetic peak is absent. 4. Conclusions
The main purpose of this publication consists in demonstration of the possibility to describe the HMF along the orbit of Ulysses through local parameters of the heliospheric plasma – density and temperature. As it has been shown above the values of the calculated magnetic fields Bcalc = Kn T + B0 are consistent with the observed magnetic fields in the heliosphere at 10
distances of 1–5.4 AU and at various heliolatitudes with a good precision. The term Kn T in the above expression is responsible for magnetic field variations. This is clearly seen in Fig. 5, where both the observed magnetic field BUlys and Bcalc − B0 are presented. The high correlation between these two values can be considered as an indication that the observed magnetic field is generated by a current, whose density is proportional to n T and which flows at the same place where the plasma density and temperature are measured. The total magnitude of Bcalc includes the field B0 , whose origin is unclear. The value B0 = 0.2 nT, as seen in Figs 4 and 5, is not negligibly small in comparison with observed fields at distances 2.5–5.4 AU from the Sun and apparently it decreases with distance r slower then 1 / r 2 . We have also considered the relationship between the corotating interaction regions and peaks in the magnetic field. The regions of compressed plasma with enhanced density and temperature (CIRs) are formed in the case when the fast and slow wind streams in the heliosphere propagate in the same radial direction. As a magnetic field supposedly depends on plasma density and temperature, the formation of an interaction region is simultaneously the formation of a magnetic peak. Magnetic field peaks are generated in the heliosphere in the interactions between fast and slow solar wind streams. It was shown that theoretically calculated shock fronts match up to 1 hour with the fronts of the magnetic peaks, so the magnetic peaks can be considered as the magnetic images of the CIRs. It was shown (on daily and hourly data) that the plasma density and temperature in various ways are involved in the formation of magnetic peaks. A density peak on the leading part of a fast solar wind stream is connected with the forward shock in a CIR as well as with the leading edge of a magnetic peak. The trailing edge of the magnetic peak with its step-like temperature decrease may be a location of the intersection of the reverse shock by the Ulysses spacecraft. A gap between the density and temperature peaks corresponds to a rarefaction region. Spatial extents of the examined magnetic peaks are about 1 AU at solar wind velocity 700 km/s, which agrees with distance between forward and reverse shocks in a developed CIR. We have pointed out the possibility of using data on plasma density and temperature in the identification of the CIRs and CMEs. It was shown that the forward front of the CME magnetic pick is formed by the sharp increase in temperature, whereas the forward front of the CIR is formed by the sharp increase in plasma density. It was noted the different behavior of the solar wind velocity in the CIRs and CMEs and some other differences. We have also noted the difference between the magnetic peaks produced by the CIRs and by the CMEs. Nevertheless the forward and rear fronts of magnetic peaks coincide with the forward and reverse shocks of CMEs. It was proposed a possible explanation of the correlation between the measured magnetic field and plasma temperature and density. In the Appendix the derivation of Eq. (1) is given. The derivation is based on analogy between currents in long straight wires and currents in the plasma. Our basic assumption is that the thermal motion of electrons with the averaged velocities exceeding the averaged thermal velocities of protons will produce some uncompensated negative volume charge density in the plasma, which is the source of the electric field E ≠ 0 . Because the heliospheric plasma has high conductivity and protons are free to move in response to that electric field, the proton current occurs in the plasma. As a result of that the density of the positive charges increases and the neutrality of the plasma will be restored and the electric field turn to zero, E = 0. The magnetic field in the plasma appears simultaneously with the proton current. Acknowledgments
11
This work was partially supported by the Russian Foundation for Basic Research (grants 1402-00905a, 14-02-10006k) and by the "Fundamental Properties of Matter and Astrophysics" Program of the Presidium of the RAS. The authors are grateful to the scientific teams of Ulysses for the data on the magnetic fields and heliospheric plasma presented in Internet.
Appendix. Relationship between the heliospheric magnetic field and the plasma temperature and density
To derive the expression Bcalc = Kn T + B0 (A1) we used the method of determination of the magnetic field near a long straight wire carrying an electric current considered by E. Purcell in his textbook (1985). In Purcell's approach the force acting on the test charge q, moving along a wire, is calculated in the test charge rest frame. If the charge is at rest there is no magnetic force acting on it, and an electric field only needs to be taken into account. The electric force, calculated in the test charge rest frame, is then converted to a "laboratory" frame of reference and analyzed. In order to understand the origin of magnetic and electric fields around current carrying wires, consider two inertial frames − the "laboratory" S and the test charge rest frame S'. It is supposed that the coordinates of the two frames are oriented so that the velocity u of the charge q is parallel to the x axis. Suppose that the current in a wire, which is at rest in the frame S, comes from movement of both negative ρ and positive σ charges. The line charge densities are not Lorentz invariant. Denote the line density of the positive and negative charges in their rest frames as ρ 0 and σ 0 , respectively. Then the relation between ρ and ρ 0 can be expressed by the formula 2 2 ρ = ρ 0γ (v − ) where γ ( v− ) = 1 / 1 − v − / c , and v − is a relative velocity of the negative charge rest frame with respect to S (i.e. the drift velocity of the electrons in the wire). The relation ρ = ρ 0γ (v − ) is a consequence of the Lorentz transformations of a 4-vector current density. For the line charge density σ , as well as for the line charge densities ρ ' and σ ' in the primed frame S' we can find the same relations: σ = σ 0γ (v + ) , ρ ' = ρ 0γ (v' − ) , σ ' = σ 0γ (v' + ) , where v + , γ (v+ ) , v ' − , γ (v' − ) , v ' + and γ (v' + ) have the same sense as v − and γ (v− ) . Now we can write an expression for the net charge line density in the frame S' as:
γ (v+' ) γ (v −' ) ρ . σ− σ '− ρ ' = γ (v − ) γ (v+ )
(A2)
2(σ '− ρ ' ) , and the electric force acting on the test charge r' q is F ' = qE ' . In the frame S the charge q experiences the force:
Then the electric field is given by E ' =
F=
F' 2q u = (σ − ρ ) − 2 [σv+ − ρv − ] . γ (u ) r c
(A3)
12
In Eq. (A3) the force is expressed explicitly via the line density of charges and their speed in the frame S and the speed u of the test charge. Note that the density of charges are included in (A3) in accordance to their charge signs ( − ρ and + σ ). But the movement direction of charges (in the direction of the x axis or against the x axis) is not selected. The quantity (σ − ρ ) is the net 2(σ − ρ ) charge line density and then we can consider = E as the electric field. The quantity r [σv+ − ρv− ] is the net current density and then 2 [σv + − ρv− ] = 2I = B we can consider as the rc rc magnetic field. Now we can rewrite (A3) in the form:
F = qE −
quB . c
(A4)
The minus sign of the second term in Eq. (A4) indicates a force directed towards a wire for positive q, u and I. The sign of the first term depends on the directions of v+ and v − and their values. Thus we find that the force acting on the test charge, moving along the long straight wire consists of the sum of the electric and magnetic forces. Both these forces are of relativistic origin and turn to zero in the case of nonrelativistic transformation of the charge velocities and if it is assumed that the charge line density does not depend on its speed. Consider now a typical metal long wire where the nuclei and bound electrons are at rest ( v + =0, σ = σ 0 ), and where the current is produced by the conduction electrons. The force Eq. (A3) acting on the test charge q is then
F=
ρuv 2q (σ 0 − ρ ) + 2 − . r c
(A5)
The wire is electrically neutral at zero current, σ 0 − ρ 0 = 0 . Since ρ = γ (v− ) ρ 0 = γ (v− )σ 0 and σ 0 = ρ / γ (v − ) we can exclude σ 0 from Eq. (A5) and obtain: F=
uv 2 qρ 1 − 1) + 2− . ( r γ (v − ) c
Using two first terms of the expansion of the quantity 1 / γ (v − ) = 1 −
(A6)
1 v−2 we can rewrite Eq. 2 c2
(A6) as: F =−
qρv−2
rc 2
+
2qρuv−
rc 2
.
(A7)
The explicit expressions for electric and magnetic fields in that case are: E=−
ρv−2 rc
2
; B=
2 ρuv − . rc
(A8)
13
Note that the electric field in Eq. (A8) depends on the square of the velocity v−2 of the conduction electrons, while the magnetic field is proportional to the first degree of v− . Therefore, if we change the direction of the electrons velocity to the opposite, then we will have:
F∗ = −
qρv −2 2qρuv− . − rc 2 rc 2
(A9)
The field near the conductor consisting of two strands with currents, opposite in direction (a "bifilar winding"), can be obtained by summarizing Eq. (A7) and Eq. (A9). In this case, the magnetic fields mutually compensate each other and the electric field doubles and becomes:
E=−
2 ρv−2
rc 2
.
(A10)
The electric fields of such origin, as it will be shown below, can exist in the heliospheric plasma. Let us select some volume element of space plasma in the solar wind and choose two inertial frames of reference – the frame S which is moving with the same velocity V as the solar wind (and the selected plasma volume element), and the space probe rest frame S' which is moving with the velocity u = –V. Suppose that the plasma is fully ionized hydrogen plasma with Maxwellian distribution functions and with the same temperature T of the electrons and protons. It is also assumed that the electric field E=0 in the frame S. In the case of Maxwellian velocity distribution functions the average thermal electron speed 2 in the x direction (parallel to the solar wind velocity V) is v xe
2 average thermal proton speed is v xp
1/ 2
1/ 2
= kT / me ≡ v − and the
= kT / m p ≡ v+ , where k is the Boltzmann constant
and m e and m p are the electron and proton masses, respectively (Bittencourt, 2004). Suppose now that in the selected plasma volume the one sixth of electrons and the one sixth of protons move with the averaged thermal velocities v − and v+ in the direction of the x axis and the same amount of particles move with the same velocities against the x axis. The movement of electrons and protons along the axis х we consider as an analogue of linear currents in long straight wires. Knowing the plasma temperature T we can calculate the averaged thermal velocities v − and v+ of the electrons and protons in the frame S and their ratio v − / v + = m p / me of ~43. Bearing in mind that v − >> v + , we will not take into account the thermal movement of the protons, putting the averaged thermal speed of the protons v + = 0. This has two consequences. The first is that the proton current density σv + = 0. The second is that the proton charge density σ = σ 0 , as the frame S is now the rest frame for the protons. As a result, we have got a situation similar to that for the wires with a bifilar winding, when two electron currents flow in opposite directions along the x axis. Now let's calculate the net charge density (A2) in the frame S', after defining all the required values – the velocities v+ , v− , v+' , v−' and the proton σ and electron ρ charge densities – from physical considerations and from the experiment. Consider separately two cases: a) v − >0 and b) v − <0.
14
In the case a) Lorentz factors of expression (A2) are γ (v + ) = 1 , γ (v − ) =
1 1 − v−2 / c 2
,
v− − u uv . = γ (u )γ (v− )1 − 2− where v −' = c 1 − uv− / c 2 1− v / c 1− v / c2 Then the net charge density in the frame S' is given by:
γ (v − ) =
1
2 −
2
, γ (v −' ) =
σ '− ρ ' = γ (u )σ − (1 −
1
' 2 −
)ρ . c
uv− 2
(A11)
In the same way as in a Purcell's problem, presented above, we get the electric field and the electric force acting on the test charge q in the frame S'. In the frame S the electric force Fa acting on the test charge is given by: Fa =
qEa' 2q uv = σ − (1 − 2− ) ρ . γ (u ) r c
(A12)
Case b) differs from the case of a) only in expressions for the Lorentz factor − v− − u 1 uv . The electric γ (v −' ) = = γ (u )γ (v− )1 + 2− and for the velocity v −' = 2 c 1 + uv − / c 2 1− v' / c2 −
force acting on the test charge q in case b) is: Fb =
qEb' 2q uv = σ − (1 + 2− ) ρ . γ (u ) r c
(A13)
The total force acting on the test charge q in the frame S' is equal to the sum 2q ⋅ 2(σ − ρ ) . Such a force can be created by the electric field F = Fa + Fb = r E=
4(σ − ρ ) , r
(A14)
where the charge line density is 2(σ − ρ ) . Thus, we have found that, if Te = T p , in the plasma the electric field is created in the same way as the electric field near a long straight wire. The source of this field is the thermal motion of the electrons, which creates two oppositely directed electron flows along the x axis. The magnetic fields of these electron flows mutually compensate each other. The above expressions for the electric field in plasma are similar to expressions for ordinary long wires. It is obvious that mathematical expressions for the fields in the plasma and the fields beside the usual conductors must somehow be different. The heliospheric plasma is a continuum with the distributed volume currents where the distance r from the point of observation to the x axis is meaningless. But this does not change anything in fact that the thermal motion of electrons with the averaged velocities v − exceeding the averaged thermal velocities v+ of protons will produce some net charge density 2(σ − ρ ) ≠ 0 in the plasma, and, according to Maxwell's equation 2(σ − ρ ) = ε 0 divE , generate the electric field ( ε 0 is the permittivity of free space).
15
Therefore, our basic assumption is, that, provided the v− > v+ , in the heliospheric plasma the uncompensated negative volume charge is formed which is the source of the electric field E. Because the heliospheric plasma has high conductivity and protons are free to move in response to that electric field, the proton current occurs in the plasma. As a result of that the density of the positive charges increases and the neutrality of the plasma is restored, so the condition E = 0 will be satisfied. The magnetic field appears simultaneously with the proton current. Using the condition σ − ρ = 0 which is equivalent to condition E = 0, we estimate now the speed of protons v p , sufficient to restore the neutrality of the plasma. Recall that there exist the following relations for the charge densities in the frame S : ρ = γ (v− ) ρ 0 , σ = σ 0 and the condition of the plasma neutrality in the absence of currents σ 0 − ρ 0 = 0 . After the protons will begin to move with the speed v p their charge density will be equal to σ = σ 0γ (v p ) . We present 2ρ0 , since the two third of electrons 3 3 move in directions transverse to the x axis and their charge densities are not changed by the Lorentz transformation. Then from σ − ρ = 0 it follows that
the electron charge density in the form of ρ =
σ 0 γ (v p ) −
ρ 0 γ (v− ) 3
−
ρ 0 γ (v − )
−
2ρ 0 = 0. 3
(A15)
Using the expansion of the Lorentz factors γ (v p ) and γ (v− ) we find
σ 0 (1 +
v 2p
ρ0
2c
3
)= 2
(3 +
v −2 ) = 0. 2c 2
Solving this equation for the proton velocity v p gives v p = v −2 / 3 = kT / 3me .
Directed motion of protons with the velocity v p produces the current I = σ v p S in the plasma volume element, where S is the cross section of the volume element. Current I in the direction of x axis creates the magnetic field: 2I 2 Sσ kT = = K ∗σ T (A16) B= rc rc 3me 2S where K ∗ = k / 3me . rc As it was said above the quantity r in Eq. (A16) can not be considered as a radial distance from the x axis to the point of observation. We assume that the expression Eq. (A16) just sets the proportionality between the magnetic field B and the density of positive charges σ and temperature T of plasma, where the coefficient K ∗ = f (r , S ) is some function of the spatial variables which has to be defined in more rigorous theory. The magnetic field B in Eq. (A16) is given in the frame of reference S, which is moving along with the plasma. The magnetic field B' measured in the spacecraft frame S' is B' = γ (u ) B = K ∗σγ (u ) T = K ∗σ ' T = K ∗ ne+ T , where σ ' = ne+ = σ γ (u ) is the proton charge density in the frame S', n is the proton number density and e+ is the proton electric charge. Denoting K ∗ e+ = K , we can write the expression for the magnetic field in the form B' = Kn T . Here the magnetic field is represented as a function of the plasma parameters n and T, which are directly measured in the spacecraft reference frame S'. By comparing the observed magnetic fields with the values B', we have found that a better description of the experimental data provides an expression B' = Kn T + B0 . We have used that expression to calculate the magnetic fields in this work.
16
References Bame, S.J., Goldstein, B.E., Gosling. J.T., Harvey, W., McComas, D.J., Neugebauer, M., Phillips, J.L., 1993. Ulysses observations of a recurrent high speed solar wind stream and the heliomagnetic streamer belt. Geophys. Res. Lett. 20(21), 2323–2326. Bittencourt, J.A., 2004. Fundamentals of Plasma Physics. 3rd Edition, Springer-Verlag, 583 p. Borovsky, J.E., Denton, M.H., 2006. Differences between CME-driven storms and CIR-driven storms, J. Geophys. Res. 111, A07S08: 1–17, doi: 10.1029/2005JA011447. Broiles, T.W., Desai, M.I., Lee, C.O, MacNeice, P.J., 2013. Radial evolution of the threedimensional structure in CIRs between Earth and Ulysses. J. Geophys. Res. 118, 4776–4792. Broiles, T.W., Desai, M.I., McComas, D.J., 2012. Formation, shape, and evolution of magnetic structures in CIRs at 1 AU. J. Geophys. Res. 117, A03102: 1-21, doi: 10.1029/2011JA017288 Burlaga, L.F., Wang, C., Richardson, J.D., Ness, N.F., 2003. Evolution of magnetic fields in corotating interaction regions from 1 to 95 AU: Order to chaos. Astrophys. J. 590, 554–566. Gosling, J.T., Bame, S.J., McComas, D.J., Phillips, J.L., Pizzo, V.J., Goldstein, B.E., Neugebauer, M., 1993. Latitudinal variation of solar wind corotating stream interaction regions: Ulysses. Geophys. Res. Lett. 20, 2789–2792. Gosling, J.T., Bame, S.J., McComas, D.J., Phillips, J.L., Pizzo, V.J., Goldstein, B.E., Neugebauer, M., 1995. Solar wind corotating stream interaction regions out of the ecliptic plane: Ulysses. Space Sci. Res. 72, 99–104. Gosling, J.T., Pizzo, V.J., 1999. Formation and evolution of corotating interaction regions and their three dimensional structure. Space Sci. Res. 89, 21–52. Hundhausen, A.J., 1973. Nonlinear model of high-speed solar wind streams. J. Geophys. Res. 78, 1528–1542. Jian, L.K., Russell, C.T., Luhmann, J.G., MacNeice, P.J., Odstrcil,·D.,·Riley, P., Linker, J.A., Skoug, R.M., Steinberget,·J.T., 2011. Comparison of observations at ACE and Ulysses with Enlil model results: Stream interaction regions during Carrington rotations 2016–2018. Sol. Phys. 273, 179–203. Lee, M.A., 2000 An analytical theory of the morphology, flows, and shock compressions at corotating interaction regions in the solar wind. J. Geophys. Res. 105, 10491–10500. Odstrcil, D., 2003. Modeling 3-D solar wind structure. Adv. Space Res. 32, 497–506, doi: 10.1016/S0273-1177(03)00332-6. Pizzo, V.J., Gosling, J.T., 1994. 3-D simulation of high-latitude interaction regions: Comparison with Ulysses results. Geophys. Res. Lett. 21, 2063–2066. Purcell, E.M., 1985. Electricity and Magnetism, 2nd Edition, McGraw-Hill Book Comp., 507 p. Riley, P., Gosling, J.T., Weiss, L.A., Pizzo, V.J., 1996. The tilts of corotating interaction regions at midheliographic latitudes. J. Geophys. Res. 101, 24349-24357. Svirzhevsky, N.S., Bazilevskaya, G.A., Svirzhevskaya, A.K., Stozhkov, Yu.I., 2014. The heliospheric magnetic field and its relation to the temperature, density, and velocity of solar plasma: Experimental evidence. Cosmic Research 52, 15-24. Yermolaev, Y.I., Nikolaeva, N.S., Lodkina, I.G., Yermolaev, M.Y., 2012. Geoeffectiveness and efficiency of CIR, sheath, and ICME in generation of magnetic storms. J. Geophys. Res. 117, A00L07:1-9, doi: 10.1029/2011JA017139.
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Fig. 1. Comparison of the measured magnetic field BUlys (grey) and calculated magnetic field Bcalc (black) in the presence of fast solar wind streams from a coronal hole. The heliolatitude of the spacecraft is shown by a dashed line. Ulysses’ distances from the Sun at the beginning and end of the examined time interval and the correlation coefficient R between two sets are also shown.
Fig. 2. Comparison of the measured BUlys and calculated Bcalc magnetic fields during the 1996 solar activity minimum. Notations on the figure are the same, as in Figure 1. Fig. 3. Comparison of the measured and calculated values of magnetic fields during the crossing of the sector zone at the final orbit of Ulysses around the Sun. Fig. 4. Comparison of the observed BUlys magnetic field with the values B − B0 = Kn T , B0 = 0.2 nT in the first half of 2006. 5 Fig. 5. The dependence of the B0 on the distance to the Sun in October 1995-December 1997. Fig. 6. The scattering plot for the whole sets of the Ulysses data in 1990-2008. The regression equation and the correlation coefficient R are also shown. Fig. 7. a) The observed (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on August 27 - December 5 1992 at distances to the Sun 5.28-5.12 AU and south heliolatitudes 16.1°-21.4°. b) Plasma density (black) and temperature (grey) observed in the same place. Fig. 8. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 7-11 October 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shock, calculated by Riley et al. (1996), CIR 10. Time (in hours) is counted out since the beginning of 1992. Fig. 9. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1-5 November 1992. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of forward and reverse fronts of shock, calculated by Riley et al. (1996), CIR 11. Time (in hours) is counted out since the beginning of 1992. Fig. 10. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 1-6 August 2005. b) The plasma density (black) and temperature (grey). Time (in hours) is counted out since the beginning of 2005. Fig. 11. a) The measured (grey) and calculated (black) magnetic fields as well as the solar wind velocity (darkgrey) on 24-29 February 2004. b) The plasma density (black) and temperature (grey). Two vertical lines show the positions of shock forward and reverse fronts, calculated by Broiles et al. (2013). Time (in hours) is counted out since the beginning of 2004.
18