magnetic flux density measured in fast and slow solar ... - IOPscience

The Astrophysical Journal, 753:130 (9pp), 2012 July 10 C 2012.

doi:10.1088/0004-637X/753/2/130

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

MAGNETIC FLUX DENSITY MEASURED IN FAST AND SLOW SOLAR WIND STREAMS 1

˝ 1 and A. Balogh2 G. Erdos Wigner Research Centre for Physics of the Hungarian Academy of Sciences, H-1525 Budapest, POB 49, Hungary; [email protected] 2 The Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 2012 January 23; accepted 2012 May 4; published 2012 June 21

ABSTRACT The radial component of the heliospheric magnetic field vector is used to estimate the open magnetic flux density of the Sun. This parameter has been calculated using observations from the Ulysses mission that covered heliolatitudes from 80◦ S to 80◦ N, from 1990 to 2009 and distances from 1 to 5.4 AU, the Advanced Composition Explorer mission at 1 AU from 1997 to 2010, the OMNI interplanetary database from 1971, and the Helios 1 and 2 missions that covered the distance range from 0.3 to 1 AU. The flux density was found to be much affected by fluctuations in the magnetic field which make its calculated value dependent on heliospheric location, type of solar wind (fast or slow), and the level of solar activity. However, fluctuations are distributed symmetrically perpendicular to the average Parker direction. Therefore, distributions of the field vector in the two-dimensional plane defined by the radial and azimuthal directions in heliospheric coordinates provide a way to reduce the effects of the fluctuations on the measurement of the flux density. This leads to a better defined flux density parameter; the distributions modified by removing the effects of fluctuations then allow a clearer assessment of the dependence of the flux density on heliospheric location, solar wind type, and solar activity. This assessment indicates that the flux density normalized to 1 AU is independent of location and solar wind type (fast or slow). However, there is a residual dependence on solar activity which can be studied using the modified flux density measurements. Key words: magnetic fields – solar wind – Sun: activity

distant interplanetary space (Smith 1993). However, this process may not be general and even its existence remains subject to disagreements. Further considerable topological complexities of the magnetic field lines are present in the heliosphere corresponding to coronal mass ejections (CMEs), due to the formation and evolution of single or multiple magnetic clouds. Assuming that the solar wind flows radially, the freezing-in condition yields a simple R−2 decay of the radial component of the magnetic field. Unlike the tangential component, the radial component is independent of the solar wind velocity, therefore the open magnetic flux density is best characterized by the quantity BR R2 (where R is the radial distance from the Sun, measured in AU). The open flux density defined above is relatively free of transient events in the heliosphere such as compressions, provided that the solar wind keeps its radial flow direction. The rationality of the normalization to 1 AU is that most interplanetary measurements have been made either by Earth-orbiting satellites or probes close to the L1 Lagrange point of the Sun–Earth system. The open magnetic flux of the Sun is an important quantity as far as both the heliosphere and solar physics are concerned. The magnetic state of the heliosphere is primarily determined by the magnetic field vectors on the source surface. The source surface magnetic field strength and polarity, as a function of the heliospheric longitude and latitude, have a determinant role in the modulation of cosmic rays. The interplanetary magnetic flux also largely affects the interaction of the solar wind with planets; in particular, it is the prime source of energy accumulated in magnetospheres. As for solar physics, the open magnetic flux has an apparent variation with solar cycle; its absolute value is largest during solar maximum (Wang et al. 2000; Wang 2009). Although the flux carried by the solar wind is a small fraction of the total magnetic flux on the surface of the Sun, the open flux may be an indicator of the development of the cycle. Some researchers argued, for instance (Nandy et al. 2011), that the smaller magnetic flux observed at the end of cycle 23 could be

1. INTRODUCTION Spectroscopic imaging of the solar disk is routinely carried out, providing us with magnetic field maps of the photosphere. According to these maps, the magnetic field lines are mostly concentrated in sunspots and active regions. The field lines usually form loops where the two endpoints of the field lines are anchored into the surface of the Sun, in particular to sunspots frequently appearing in pairs with opposite magnetic polarities. However, some field lines extend into the outer corona (cf. Schrijver & DeRosa 2003). One important source of the open magnetic field lines is coronal holes, in particular in the polar regions around sunspot minimum. The geometry and topology of the magnetic field lines in the lower corona are complex, however beyond a few solar radii (RS ), the coronal plasma becomes such a tenuous and highly conducting medium that the magnetic field lines are frozen into it and the solar wind carries the field lines out into the heliosphere according to Parker’s (1959) model. An alternative model, proposed by Fisk (1996), takes into account the differential rotation of the photosphere, combined with the rigid rotation of the coronal holes with their open magnetic field lines, to explain the transport of magnetic field lines from higher to lower heliolatitudes. A hypothetical sphere of 2–3 RS is customarily introduced, called the source surface of the solar wind, beyond which distance from the Sun the magnetic field freezing is reasonably well satisfied (Wang & Sheeley 2003). The open magnetic flux of the Sun, by definition, is the number of field lines crossing the source surface. Given that the magnetic field satisfies Maxwell’s equation divB = 0, the number of outward pointing magnetic field lines should be the same as the number of inward pointing ones. Note that all of the outward and inward pointing field lines should be joined somewhere, which may happen beyond the termination shock of the supersonic solar wind, where the Parker model is no longer valid. In principle, the field lines may close before the solar wind reaches the termination shock, causing a flux deficit in the 1

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The Astrophysical Journal, 753:130 (9pp), 2012 July 10

associated with the unusually long duration of the cycle. As for longer timescales, a possible increase of the open magnetic flux in the heliosphere in the course of the last century was found by Lockwood et al. (1999). The association of the long-term variations in the open magnetic flux with the evolution of the solar activity, and even perhaps climate changes, remains the subject of ongoing research. There are two independent ways to provide information on the magnetic flux in the heliosphere. One obvious possibility is the direct measurement of the field by magnetometers on board space probes which can provide reliable magnetic flux density values, as a function of time and location, but observations are limited to a single point. Further, apart from the unique Ulysses mission, the observations are generally restricted to the ecliptic plane. The other method is the extrapolation of photospheric magnetic field maps to the source surface. Once the magnetic field is known on the source surface, the heliospheric propagation of the flux is normally approximated by a simple ballistic motion of the plasma. Note that during disturbed periods such as CMEs or in corotating interaction regions, a much more sophisticated magnetohydrodynamical (MHD) treatment is necessary to describe the transport of the magnetic flux in the heliosphere. The most serious source of uncertainty comes from the extrapolation of the field from the photosphere to the source surface. MHD models and the potential field source surface models were summarized and compared by Riley et al. (2006). A modified version of the latter is the current sheet source surface model (CSSS) by Schatten (1971). Whatever model is used, the results are affected by the assumptions and approximations used in the calculations. Furthermore, the input parameters of the models, i.e., the photospheric measurements, are limited to about ±60◦ in latitude around the equator. Given the fact that only a fraction of the photospheric magnetic flux is extended to the source surface (Schrijver & DeRosa 2003), any small errors in the model calculations will give rise to large deviations in the open magnetic flux. Therefore, the results of the source surface models should be accepted with precautions. An apparent discrepancy between model calculations and heliospheric observations is that the models usually cannot predict the latitudinal independence of the magnetic flux density observed by Ulysses both near solar minimum and solar maximum (Smith & Balogh 1995, 2003). The explanation of the latitudinal independence of the heliospheric magnetic flux density is a challenge to modelers; a possible solution is to use the CSSS model and to place the source surface at a larger distance from the Sun, about 10 RS (Schüssler & Baumann 2006). The reason for extending the source surface is to allow the magnetic pressure to generate the uniform latitudinal distribution of the flux density at the distance which is the boundary between the sub-Alfvénic and Alfvénic regimes. This surface has been named the heliobase by Zhao & Hoeksema (2010) who have used the somewhat different horizontal current–current sheet–source surface (HCCSSS) model originally introduced by Zhao & Hoeksema (1995). As mentioned above, high-latitude magnetic field measurements by Ulysses have shown that the magnetic flux density (referred to 1 AU) tends to be uniform, at least in the fast, polar solar wind. This encourages us to assume that the magnetic flux density measured at a single point is a representative sample of the absolute value of the magnetic flux density everywhere in the heliosphere. If this assumption is correct, this would overcome the problem of the very limited spatial coverage of the in situ, direct measurements of the magnetic field (and flux density) provided by spacecraft. Recent investigation by Owens

et al. (2008) showed such a tendency of uniform magnetic flux density (referred to 1 AU) in the heliosphere. However, Ulysses observations pointed out a remarkable difference of the magnetic flux distribution between the fast and slow solar wind (Erd˝os & Balogh 2005). Another question is the existence of any change in the value of the magnetic flux during the solar wind outflow. A possible increase in magnetic flux with radial distance was reported by Lockwood et al. (2009a, 2009b). The calculation leading to this result has been questioned by Smith (2011). In this paper, we address the question of the evolution of the magnetic flux in the heliosphere. Statistical analysis of the magnetic field data, measured by spacecraft in various places in the heliosphere, has been performed. Data have been distinguished according to the type of solar wind (i.e., fast or slow) and according to the phase of the solar sunspot cycle (i.e., minimum or maximum). The prime source of our study is the Ulysses magnetic field data, which cover the heliosphere in three dimensions over almost two solar activity cycles. Another advantage of the Ulysses data is that coronal freezingin temperature data calculated from the oxygen ion charge ratio are available which help us to discern intervals of CME activities and to justify the selection of slow and fast solar wind intervals (von Steiger et al. 1995). An identical instrument is flown on board Advanced Composition Explorer (ACE) that has provided data for the studies of the magnetic flux at 1 AU in this paper. In addition, we have analyzed the OMNI data set (http://omniweb.gsfc.nasa.gov/). The latter has no coronal temperature data, but provides an extremely good statistics by covering a 40 year interval. As for the shorter distances from the Sun, the magnetic field data of Helios 1 and 2 have been analyzed in this paper. 2. MAGNETIC FLUX DENSITY OBSERVATIONS We have analyzed the open magnetic flux, measured by spacecraft in various locations in the heliosphere. The coverage of observations included the heliographic latitudinal range from 80◦ S to 80◦ N, and the distance range 0.3–5.4 AU from the Sun. In this paper, we use the RTN coordinate system, where R points radially away from the Sun, T is the cross product of the Sun’s rotation vector and R, and N completes the right-handed triad. The magnetic flux density was simply BR R2 , i.e., the radial component of the field vector, normalized to 1 AU. We have used 6 hr averages in all investigations presented in this paper. The observations were grouped according to the velocity of solar wind (slow or fast) and in some cases according to the sunspot number (solar minimum or maximum). 2.1. Magnetic Flux Density as a Function of Heliographic Latitude: Ulysses Observations As mentioned in the Introduction, the off-ecliptic orbit of the Ulysses spacecraft provided us with observations in a broad range of heliolatitudes, from 80◦ S to 80◦ N. One consequence of this is that the spacecraft spent a long time in the fast solar wind, originating from the polar coronal holes, close to sunspot minimum. Therefore, the fast solar wind observations are well represented, with very good statistical weight in the data covering the total lifetime of the mission. Another advantage of Ulysses is the presence of the SWICS instrument which measures the charge state distributions of heavy ions (von Steiger et al. 1995). The ratio of O7 + /O6 + gives the temperature 2

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Figure 1. Ulysses solar wind observations by the SWICS instrument (upper row) and magnetic field observations by the MAG instrument (lower row) from 1990 to 2008. Upper row: scatter plot and contour lines of the two-dimensional distribution of the solar wind velocity and coronal temperature (left panel); one-dimensional distribution of the solar wind velocity (middle) and coronal temperature (right panel). Green, red, and blue areas represent part of the distributions that classified CME, slow wind, and fast wind events, respectively (see Table 1). Lower row: distribution of the magnetic flux density for CME, slow wind, and fast wind time sections. Table 1 Classification of the Type of Solar Wind Type of Wind Slow wind Fast wind CME

Color Mark in Figures

Solar Wind Velocity (v SW )

Coronal Temperature (T)

Red Blue Green

v SW 500 km s−1 v SW 600 km s−1 v SW 600 km s−1

T 1.4 MK T 1.2 MK T 1.4 MK

of the lower corona where the charge state of oxygen ions is frozen into the expanding (collisionless) solar wind (Geiss et al. 1995). Low coronal temperature (∼1 MK) is typical for the solar wind originating from coronal holes. The top left-hand panel of Figure 1 shows a scatter plot of the velocity of solar wind alpha particles and the freezing-in oxygen coronal temperature, as measured by the SWICS instrument during the mission from the launch in 1990 to the loss of the high gain antenna in 2008. All data points represent 6 hr averages. This averaging interval is also used in all further investigations below in order to keep consistency when comparing the results. The contour plot of the density of data points is superimposed, which clearly shows two peaks, corresponding to the two types of solar wind. One is the slow solar wind, characterized by a most probable velocity of about 400 km s−1 , and by a most probable coronal temperature of about 1.5 MK. The other peak corresponds to the fast solar wind, having a most probable velocity of about 750 km s−1 and a most probable coronal temperature of about 1 MK. On the top row of Figure 1, the middle and right-hand panels show the distributions of the solar wind velocity and the coronal temperature, respectively. These distributions are simply the projections of the two-dimensional distribution (left-hand panel) to the horizontal and vertical axes, respectively. The slow and fast solar wind was distinguished on the basis of the measured solar wind velocity and coronal temperature.

Table 1 summarizes the selection criteria for the various types of solar wind. The coronal temperature data were only available for the Ulysses and ACE observations. Therefore, in the case of the OMNI and Helios data (discussed later), the fast and slow wind was identified by using the solar wind velocity observations alone. There are ranges of the solar wind parameters, the velocity, and coronal temperature, for which the solar wind was classified as neither fast nor slow and which were left out of the data analysis. Those intervals when the solar wind velocity was between 500 and 600 km s−1 were discarded. Similarly, a gap was introduced for the coronal temperatures between 1.2 and 1.4 MK. According to Figure 1, these gaps in the parameters look unnecessary because the fast and slow wind streams can be separated quite precisely in the Ulysses data. However, different ranges of the solar wind velocity and coronal temperature should have been discarded then for the other spacecraft data, as is clear for the ACE data in Figures 2 and 3. Therefore, in order to be consistent in all our investigations, we used the same gaps in the parameters, which yielded a common selection criterion according to which the solar wind can be regarded as fast or slow safely for all observations. Note that the magnetic flux density distributions showed little response against the modification of the selection criteria. The upper left-hand panel in Figure 1 confirms the earlier result by Zurbuchen et al. (2002) that the data points mostly occupy two quadrants (upper left and lower right) in the solar 3

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Figure 2. ACE observations of the solar wind (upper panels) and magnetic flux density (lower panels) during solar minimum (data collected from 1998 to 2009), in the same format as in Figure 1. (Note that R = 1 AU.)

Figure 3. ACE observations of the solar wind (upper panels) and magnetic flux density (lower panels) during solar maximum (data collected from 1998 to 2009), in the same format as in Figure 1.

may exclude time intervals with CME activity and keep only the fast solar wind streams that originate from low-temperature coronal holes. The lower row in Figure 1 shows the distribution of the 6 hr averages of the magnetic flux density for the whole Ulysses mission (from late 1990 until the loss of the X-band capability in 2008 January that had provided near-complete data coverage). As defined in Table 1, three categories of the data points were introduced: the CME, the fast wind, and the slow wind plasma. The lower right-hand panel of Figure 1 shows that the distribution of the magnetic flux, observed in fast solar wind

wind velocity versus coronal temperature parameter space. Although there are data points in the other two quadrants as well, we believe that those are mainly due to statistical scatterings. However, there are data points in the upper right quadrant which apparently cannot be attributed to statistical fluctuations. Those time intervals represent significantly high solar wind velocity and significantly high coronal temperature, which is typical of CME events (Lepri et al. 2001). CMEs originate from hightemperature coronal regions, while the velocity of the plasma can also be high due to subsequent acceleration in the corona. It is worth noting that by analyzing the coronal temperature, we 4

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streams, has two peaks. The height of the two peaks relative to each other is not really important, because it depends on the trajectory of Ulysses (length of time periods that the spacecraft has spent in opposite magnetic polarities). The shape of the two peaks, however, is roughly symmetric with respect to zero and shows that the average of the magnetic flux, normalized to 1 AU is about ±3 nT. As the data were collected during times when Ulysses was at various heliographic latitudes (from 80◦ S to 80◦ N), the two relatively sharp peaks in the distribution prove that the magnetic flux, observed in the fast solar wind through the whole Ulysses mission, was roughly independent of heliographic latitude and longitude. This is a confirmation of the earlier result by Forsyth et al. (1996) that the magnetic flux density was found independent of heliographic latitude during the declining phase of sunspot cycle 22. The present investigation (and also the paper by Erd˝os & Balogh 2005) extends the time interval of analysis to include solar cycle 23 as well. As for the distribution of the magnetic flux density in the slow solar wind, there is little or no indication of a bimodal distribution (lower row, middle panel in Figure 1). A similar result was published earlier by Erd˝os & Balogh (2005). Here, we have rejected those time intervals that included CME signatures by investigating the coronal temperature at the time of the observations as discussed above. The lower left-hand panel in Figure 1 is the magnetic flux density observed during intervals that were likely to contain signatures of CMRs. Unfortunately, for this data set the statistics is too poor to be conclusive. However, the observation that the distribution of the magnetic flux density has a single peak in the slow solar wind seems to be robust; the rejection of CMEs or modifications of the selection criteria has little effect on the results.

Figure 4. Distributions of the magnetic flux density at 1 AU, according to the OMNI data set from 1971 to 2009. Upper row: distribution in the slow and fast wind (left and right panels, respectively) during solar minimum. Lower row: same for solar maximum.

3328 6 hr intervals corresponded to high solar activity. These corresponded to 44.8% and 23.6% of the total data set that consisted of 14,106 6 hr intervals. Because of the large statistics, we could reject those data intervals during moderate activity without losing statistical significance. Inspecting the ACE observations during solar minimum and solar maximum in Figures 2 and 3, respectively, we can observe that the fast solar wind is much less represented than in the Ulysses data (Figure 1), even during solar minimum. Nevertheless, the distribution of the magnetic flux has two peaks at about ±3 nT, very similar to the Ulysses observations. Also, the magnetic flux in the slow solar wind has a single peak during solar minimum (Figure 2, middle lower panel). Therefore, we can conclude that the ACE observations during solar minimum are similar to the Ulysses observations for the whole interval of analysis, without distinguishing between solar minimum and solar maximum epochs. However, during solar maximum (Figure 3, lower middle panel), unlike for the Ulysses observations in the slow wind, the distribution of the magnetic flux density has two peaks, at about ±3 nT. Therefore, during solar maximum the magnetic flux density in the slow wind at 1 AU at ACE is similar to that in the fast wind, although the two peaks at ±3 nT are much broader in the slow wind than in the fast wind. We can further test the ACE observations at 1 AU, by comparing them to the OMNI data set covering a very long time interval from 1971 to 2008. Unfortunately, no coronal temperature data are available, therefore the selection of the fast and slow wind intervals was made on the basis of the plasma velocity data alone, as given in Table 1. The distributions of the magnetic flux density at 1 AU, as determined from the OMNI data, are shown in Figure 4. The upper and lower rows show the

2.2 Magnetic Flux Density at 1 AU: ACE and OMNI Data The basic question of this paper is to find the reason why the distributions of the magnetic flux density appear to be very different in the fast and slow solar wind streams. One reason could be a different evolution of the magnetic flux in the two types of solar wind during its propagation. Therefore, we have investigated the magnetic flux at smaller distances from the Sun, first at 1 AU. We started with the analysis of the observations by ACE. That spacecraft has the advantage that the identical SWICS instrument, as flown on Ulysses, has provided us with coronal temperature data for characterizing the solar wind measurements. Magnetic field and coronal temperature data of ACE were analyzed from 1998 February to 2009 February. From the data in this time interval covering a complete sunspot cycle, two data sets were separated, corresponding to solar minimum and solar maximum epochs. The classification was made by the monthly sunspot numbers. Figure 2 shows the distribution of the magnetic flux density observed by ACE during times when the sunspot number was less than 50, a representative number for low to moderate activity during cycle 23. As a contrast, Figure 3 shows the magnetic flux density observed by ACE during solar maximum, by selecting those time intervals when the monthly sunspot number was greater than 100 (again a representative number for high solar activity for cycle 23 and those preceding it). In order to be safe in the classification we have introduced a gap (from 50 to 100) in the monthly sunspot numbers, when the epoch was regarded as neither close to solar minimum nor near solar maximum. Overall, 6320 6 hr intervals were included in the data set corresponding to low solar activity, 5

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Figure 5. Measurements of the distribution of the magnetic flux density jointly by the Helios 1 and 2 spacecraft. The measurements are shown (uppermost and lowermost panels) in the two-dimensional plane of the BR and BT components as a function of heliocentric distance (from left to right), and solar wind speed (topmost row: slow wind, lowermost row: fast wind). In the two middle rows, the distributions of the flux density (BR R2 ) are plotted, again as a function of distance and wind speed.

propagation of the solar wind might be important. Therefore, we have investigated the flux density down to the closest distances from the Sun for which data are available at present. The combined data sets of Helios 1 and 2 were used to determine the magnetic flux density as a function of the heliospheric range from 0.3 to 1 AU. The results are shown in Figure 5. Again, the observations were distinguished according to the velocity of solar wind: the upper two and the lower two rows are for the slow and fast wind streams, respectively. The columns represent various distances from the Sun, which increase from the left to the right (see the labels on the top). Although Helios 1 and 2 made observations during both the solar minimum and maximum epochs, by covering the rising phase of cycle 21, we did not separate the data set into two intervals which would correspond to high and low solar activity. The reason was that the relatively short duration of the missions (from 1975 to 1980) did not provide good statistics for such a division, in particular because we were interested in the variations by intervals of heliospheric range.

observations for solar minimum and solar maximum epochs, respectively. The total number of 6 hr intervals in the analysis was 43,543; of these, 21,258 (48.8%) corresponded to sunspot numbers less than 50, while 11,790 (27.0%) corresponded to sunspot maximum. The left- and right-hand columns are the observations in the slow and fast solar wind, respectively. Again, during solar minimum, both in the slow and fast wind the distribution of the magnetic flux density is similar to the observations by Ulysses. The single, broad peak in the slow wind is clearly visible. However, during solar maximum the distribution of the magnetic flux has two peaks, both in the slow and fast wind intervals. 2.3. Magnetic Flux Density at Distances Less than 1 AU: Helios 1 and 2 Observations The differences between the observations of the magnetic flux density at 1 AU and observations at larger distances by Ulysses suggest that the evolution of the flux density during the 6

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The Helios 1–2 observations at 1 AU reproduce the OMNI and ACE (solar minimum) data showing that the distribution of the magnetic flux density has a single, broad peak in the slow solar wind (second row, right hand panel). However, for observations closer to the Sun, a double-peaked distribution appears, best observed closest to the Sun, at distances 0.3–0.5 AU (second row, left-hand panel). As for the fast solar wind data (third row), the double-peaked distribution is apparent, as it was in all studies discussed previously, with the difference that the peaks are narrower at smaller distances from the Sun. All of the characteristics of magnetic flux density observations described previously can be understood and a coherent picture emerges if we look at the two-dimensional distributions of the magnetic flux density at Helios. (Similar two-dimensional scatter plots have been shown by Roberts et al. 1990, with the objective to assess the evolution of fluctuations from Helios to Voyager.) These are shown in the upper row and the bottom row in Figure 5 for the slow and fast solar wind intervals, respectively. In those panels, scatter plots of the radial and tangential components of the magnetic field are given, together with contour lines of their two-dimensional distributions. The one-dimensional distributions of the radial component of the magnetic field, displayed in the second and third rows, are simply the projections of the two-dimensional distributions onto the horizontal axis. All two-dimensional distributions have two well-separated peaks, corresponding to the negative and positive magnetic sectors. The maxima of the distributions are expected to be along the average Parker field line. The present observations seem to support that, as was proved previously in many investigations (e.g., Forsyth et al. 2002). However, the two separated peaks tend to merge into a single peak with increasing distance from the Sun if we project the two-dimensional distribution to the horizontal axis, representing the radial component of the magnetic field. There are two reasons. One is that the width of the peaks in the two-dimensional distribution increases with radial distance, as clearly shown in the contour plots (upper row in Figure 5). This is due to the fact that the radial component of the magnetic field decays faster (as R−2 ) than the amplitude of the fluctuations around the averaged field vectors. For the decay of the amplitude of the fluctuations, observational evidence provides R−1.3 (Balogh et al. 1996) and R−1 (Smith 2011, see his Figure 5); both these results show a slower decay than would be expected from WKB, which is R−1.5 (e.g., Horbury & Balogh 2001). Whether the theoretical or observational values are used for the radial decay of the amplitudes of the fluctuations, the relative value of the fluctuations compared to the radial component of the magnetic field increases with radial distance. Another reason for the merging of the peaks is that the deviation of the Parker field lines from the radial direction increases with increasing distance from the Sun (from near-radial direction at 0.3 AU to about 45◦ at 1 AU). Therefore, if we move to 1 AU or beyond, the projections of two peaks along the Parker directions increasingly overlap when projected onto the radial direction. As a contrast to the slow wind data, the distribution of the open flux in the fast wind has two separate peaks in all the observations presented previously. Inspecting the bottom row of Figure 5, it is clear that the fluctuation of the magnetic field vectors around the average field is much less in the fast wind than that in the slow wind. Also, at a given distance from the Sun, the magnetic field lines in the fast wind remain closer to the radial direction than in the slow wind (as the spiral is less tightly wound). The combined effect of these two features is that the

Figure 6. Upper row: distributions of the radial component BR of the magnetic field from the OMNI data set for solar maximum and solar minimum conditions. Lower row: two-dimensional plots of the radial and azimuthal magnetic field components (BR and BT ) from the OMNI data set during maximum and minimum solar wind conditions. Note that the distributions in the upper row are simply the projections of all the vectors in the lower row onto the horizontal, BR axis. In the lower row, right-hand panel, the direction of the Parker spiral is shown, together with the direction normal to the Parker spiral.

positive and negative magnetic sectors are well separated in the distribution of the BR component of the magnetic field, even for the distant Ulysses observations in the fast wind intervals. 3. SOLAR CYCLE DEPENDENCE Incidentally, the 1 AU distance from the Sun seems to be a critical turning point, as far as the open magnetic flux is concerned. The Helios observations in Figure 5 show that it is at about 1 AU where the overlapping of the two peaks that correspond to the positive and negative magnetic sectors becomes apparent in the BR distributions. Therefore, the open magnetic flux should be studied carefully, in particular when using near Earth observations where most of the measurements of the flux density have been taken. Near Earth observations in the slow solar wind by ACE and the OMNI data set have shown that the distribution of the magnetic flux density is remarkably different between solar minimum and maximum epochs (Figures 2–4). The explanation is simple if we look at the two-dimensional distribution of the magnetic flux density in the slow solar wind using the OMNI data set, displayed in the bottom row of Figure 6. It is clear that the two peaks of the distributions, corresponding to the positive and negative magnetic sectors, are more separated during solar maximum (left-hand panel) than during solar minimum (right-hand panel). This agrees with earlier observations (Wang 2009) that the open magnetic flux is larger during solar maximum than during solar minimum. A consequence of the larger open flux during solar maximum is that the two peaks in the BR distribution are better separated, as opposed to the solar minimum case when the two peaks are merged. 7

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4. DETERMINATION OF THE OPEN MAGNETIC FLUX Observations of the distributions of the magnetic flux density show that they are broadened by fluctuations of the magnetic field. This leads to the overlapping, and even merging, of the positive and negative magnetic sectors in the distributions, in particular in the slow solar wind. The extent of the overlapping/ merging is different in the fast and slow solar wind, or during solar minimum and solar maximum. Therefore, there is a high danger of the misinterpretation of the observations. However, there is a way to reduce the effect of the fluctuations. The twodimensional distributions in Figure 6 show that the magnetic field vectors scatter symmetrically around the Parker field line. Therefore, we can neglect the B⊥ magnetic field component perpendicular to the Parker field line, because the error introduced in the BR component cancels out on statistical average. The distribution of the remaining B component has two well-separated peaks even during solar minimum (see the bottom right-hand panel in Figure 6). The Parker field line is locally parallel to the plasma velocity vector in a frame corotating with the Sun. As a first approximation, we may neglect the non-radial components of the solar wind velocity vector (in stationary frame). The angle α between the Parker field line and the radial directions is determined by the equation

Figure 7. Left-hand column: distribution of the modified magnetic flux density BR∗ . Right-hand column: distribution of the magnetic field component perpendicular to the Parker field line. OMNI data during solar minimum (upper row) and during solar maximum (lower row).

tan α = WR /2π ΩR cos Λ,

5. SUMMARY AND CONCLUSIONS

where WR is the radial component of the plasma velocity, Ω is the angular velocity of the Sun. R and Λ stand for the position of the observer (distance from the Sun and heliographic latitude, respectively). Note that the determination of the angle α is independent of the magnetic field measurements, but its value varies in time, mainly due to variations of the solar wind velocity. With the help of the B magnetic field component parallel to the Parker field line, we can define the modified component BR∗ as follows:

The in situ distribution of the magnetic flux density has been determined in different locations in the heliosphere—at different heliocentric distances between 0.3 and 5.4 AU and covering all heliolatitudes from 80◦ S to 80◦ N. The measure of the magnetic field flux density is the quantity BR R2 , or simply BR for R = 1 AU. Both solar maximum and solar minimum conditions have been covered. The data sets analyzed excluded intervals with CME signatures. The measurements show that the distribution of the magnetic flux density, when measured by the quantity BR R2 , appears to be strongly dependent on solar wind type at 1 AU and along the Ulysses orbit. 1. In the fast solar wind, determined by either the coronal freezing-in temperature or the velocity of the solar wind, the distributions at 1 AU (both for ACE and the much longer OMNI data set) and at Ulysses show a two-peaked shape. The maxima of these distributions are located at ∼±3 nT AU2 at solar minimum and at a somewhat greater value at solar maximum. 2. In the slow solar wind, the magnetic flux density at Ulysses is broadly distributed around zero. At 1 AU (ACE and OMNI data sets) the distribution in the slow solar wind is similar to that at Ulysses at solar minimum, but is twopeaked, but with very broad peaks at solar maximum. The Helios 1–2 observations are, however, remarkable, because the distributions show a strong evolution from two, wellformed peaks in the slow solar wind at 0.3–0.5 AU but a single broad peak at 0.9–1 AU in the slow wind. In the fast wind, a two-peaked distribution persists from 0.3 to 1 AU. However, the two-dimensional plot of the two in-ecliptic magnetic field components (BR and BT ) shows two peaks from 0.3 to 1 AU, with the two peaks placed approximately along the Parker spiral direction, and approximately symmetrically with respect to zero. This is the case for both slow and fast solar wind intervals, indicating that the distinction between these two epochs in the other

BR∗ = B|| cos α R 2 . The left-hand column of Figure 7 is the distribution of the corrected magnetic flux density, determined from the OMNI data for solar minimum and solar maximum epochs (top and bottom panels). It is clear that the distributions have two wellseparated peaks, even during solar minimum. Note that the value of the open flux is larger during solar maximum than during minimum. This result cannot be attributed to different levels of fluctuations; in fact, the levels of fluctuations around maximum activity are larger even if the increase cannot be simply quantified. The increase in the flux around solar maximum with respect to its value at solar minimum is related to the activity-dependent evolution of both the solar surface magnetic field and the coronal structures. Given the results in this paper concerning the accurate measurements of the magnetic flux density, the evolution of the open flux between solar minimum and maximum and its relationship to the evolution of the solar surface magnetic fields can be further investigated. The right-hand column gives the distributions of the B⊥ component. The distributions are symmetric around zero, as mentioned earlier. The mean value of B⊥ is 0.067 nT for solar minimum (upper right panel in Figure 7) and 0.006 nT for solar maximum (lower right panel in Figure 7), thus justifying the calculation. 8

˝ & Balogh Erdos


data sets is somewhat artificial. It has also been observed in the two-dimensional Helios plots that the distributions around the peaks widen with heliocentric distance, indicating a significant relative increase in the fluctuations around the means. However, the relative importance of the fluctuations appears to be smaller in the fast wind. The Helios observations point to the importance of taking into account the distortion that the fluctuations in the magnetic field introduce in the determination of the magnetic flux density. In fact when the two-dimensional distribution of the OMNI data set was plotted for solar minimum, two-peaked distributions appeared in both the slow and fast solar wind intervals, similar to those observed at Helios. However, due to the much broader peaks due to the fluctuations in the slow wind, when projected onto the horizontal (BR ) axis, the overlap in the distributions made the peaks disappear, giving rise to the single, broad peak observed when only the distribution of the BR R2 quantity was taken into account. A way to reduce the distortion introduced by the fluctuations was found by first projecting the twodimensional distribution onto the direction of the Parker spiral, thus effectively removing the fluctuations perpendicular to it. (This is justified by the highly symmetric distribution of the fluctuations perpendicular to the Parker spiral direction.) Then, the projection of the remaining peaks along the Parker direction onto the BR axis leads to a magnetic flux density distribution that shows two-peaked distributions also in the slow solar wind at solar minimum. The main conclusions of the paper are as follows.

dimensional distributions, it is possible to subtract them and provide a more conservative measure of the magnetic flux density. This procedure requires the determination of the average Parker direction that requires only the use of the solar wind speed and does not involve the magnetic field itself and can determine the magnetic flux density safely at 1 AU. In future work, this technique can be used for investigating, using the in situ measurement of the magnetic field, the dependence of the magnetic flux density on location and time, in particular for comparisons with solar magnetic source surface models. The authors are grateful for the wide availability of important magnetic field, solar wind velocity and solar wind composition data sets from the ACE, Helios, and Ulysses missions, as well as the well-maintained OMNI data set. We thank the referee who has identified several points in the original manuscript that have now benefited from clarifications. REFERENCES Balogh, A., Forsyth, R. J., Horbury, T. S., & Smith, E. J. 1996, in AIP Conf. Proc. 382, Proc. 8th Int. Solar Wind Conf. on Solar Wind Eight, ed. D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, & M. Neugebauer (Melville, NY: AIP), 221 Erd˝os, G., & Balogh, A. 2005, Adv. Space Res., 35, 625 Fisk, L.A. 1996, J. Geophys. Res., 101, 15547 Forsyth, R. J., Balogh, A., Horbury, T. S., et al. 1996, A&A, 316, 287 Forsyth, R. J., Balogh, A., & Smith, E. J. 2002, J. Geophys. Res., 107, 1405 Geiss, J., Gloeckler, G., & von Steiger, R. 1995, Space Sci. Rev., 72, 49 Horbury, T. S., & Balogh, A. 2001, J. Geophys. Res., 106, 15929 Lepri, S. T., Zurbuchen, T. H., Fisk, L. A., et al. 2001, J. Geophys. Res., 106, 29231 Lockwood, M., Owens, M., & Rouillard, A. P. 2009a, J. Geophys. Res., 114, 11103 Lockwood, M., Owens, M., & Rouillard, A. P. 2009b, J. Geophys. Res., 114, 11104 Lockwood, M., Stamper, R., & Wild, M. N. 1999, Nature, 399, 437 Nandy, D., Munoz-Jaramillo, A., & Martens, P. C. H. 2011, Nature, 471, 80 Owens, M. J., Arge, C. N., Crooker, N. U., Schwadron, N. A., & Horbury, T. S. 2008, J. Geophys. Res., 113, A12103 Parker, E. N. 1959, J. Geophys. Res., 64, 1675 Riley, P., Linker, J. A., Mikic, Z., et al. 2006, ApJ, 653, 1510 Roberts, D. A., Goldstein, M. L., & Klein, L. W. 1990, J. Geophys. Res., 95, 4203 Schatten, K. H. 1971, Cosmic Electrodyn., 2, 232 Schrijver, C. J., & DeRosa, M. L. 2003, Solar Phys., 212, 165 Schüssler, M., & Baurnann, I. 2006, A&A, 459, 945 Smith, E. J. 1993, Adv. Space Res., 13, 5 Smith, E. J. 2011, J. Geophys. Res. (Space Phys.), 116, 12101 Smith, E. J., & Balogh, A. 1995, Geophys. Res. Lett., 22, 3317 Smith, E. J., & Balogh, A. 2003, in AIP Conf. Proc. 679, Solar Wind Ten, ed. M. Velli, R. Bruno, & F. Malara (Melville, NY: AIP), 67 von Steiger, R., Wimmer-Schweingruber, R. F., Geiss, J., & Gloeckler, G. 1995, Adv. Space Res., 15, 7 Wang, Y.-M. 2009, Space Sci Rev., 144, 383 Wang, Y.-M., & Sheeley, N. R., Jr. 2003, ApJ, 599, 1404 Wang, Y.-M., Sheeley, N. R., Jr., & Lean, J. 2000, Geophys. Res. Lett., 27, 621 Zhao, X., & Hoeksema, J. T. 1995, Adv. Space Res., 16, 181 Zhao, X. P., & Hoeksema, J. T. 2010, Solar Phys., 266, 379 Zurbuchen, T. H., Fisk, L. A., Gloeckler, G., & von Steiger, R. 2002, Geophys. Res. Lett., 29, 1352

1. The distribution of the magnetic flux density appears to undergo a complex evolution as the solar wind propagates from its origin in the corona. The primary cause for the complex behavior is the importance of the fluctuations which broaden the distributions. This means that the flux density as measured does not appear to be a conservative quantity. Therefore, fluctuations should also be a manifestation of non-radial flows that, as the magnetic field is frozen into the plasma, leads to the breakdown of the conservation of the flux density locally. Alfvénic fluctuations are good candidates for such a scenario. 2. The measure of the distribution of the magnetic flux density by the quantity BR R2 is a complex function of location, type of solar wind, and solar activity epoch. However, the two-dimensional distributions in the BR –BT plane show the existence of distinct peaks in all conditions corresponding to the positive and negative magnetic sectors in the ecliptic. These two peaks, broadened by the fluctuations, lead to an overlap when projected onto the BR axis—this explains the complex behavior when only the distribution of BR R2 is used for the flux density. This overlap seems to be of particular concern for the ∼1 AU measurements of the flux density, as it can clearly lead to spurious dependences on location, solar wind type, and solar activity epoch. 3. As the fluctuations appear to be highly symmetric around the average Parker direction, as can be seen in the two-

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