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We studied the magnetic flux density carried by solar wind to various locations in the ... between the in-ecliptic magnetic flux density at 1 AU (OMNI data) and the ...

The Astrophysical Journal, 781:50 (12pp), 2014 January 20  C 2014.

doi:10.1088/0004-637X/781/1/50

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

MAGNETIC FLUX DENSITY IN THE HELIOSPHERE THROUGH SEVERAL SOLAR CYCLES 1

˝ 1 and A. Balogh2 G. Erdos Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary; [email protected] 2 The Blackett Laboratory, Imperial College London, London SW7 2BZ, UK Received 2013 October 25; accepted 2013 December 1; published 2014 January 7

ABSTRACT We studied the magnetic flux density carried by solar wind to various locations in the heliosphere, covering a heliospheric distance range of 0.3–5.4 AU and a heliolatitudinal range from 80◦ south to 80◦ north. Distributions of the radial component of the magnetic field, BR , were determined over long intervals from the Helios, ACE, STEREO, and Ulysses missions, as well as from using the 1 AU OMNI data set. We show that at larger distances from the Sun, the fluctuations of the magnetic field around the average Parker field line distort the distribution of BR to such an extent that the determination of the unsigned, open solar magnetic flux density from the average |BR | is no longer justified. We analyze in detail two methods for reducing the effect of fluctuations. The two methods are tested using magnetic field and plasma velocity measurements in the OMNI database and in the Ulysses observations, normalized to 1 AU. It is shown that without such corrections for the fluctuations, the magnetic flux density measured by Ulysses around the aphelion phase of the orbit is significantly overestimated. However, the matching between the in-ecliptic magnetic flux density at 1 AU (OMNI data) and the off-ecliptic, more distant, normalized flux density by Ulysses is remarkably good if corrections are made for the fluctuations using either method. The main finding of the analysis is that the magnetic flux density in the heliosphere is fairly uniform, with no significant variations having been observed either in heliocentric distance or heliographic latitude. Key words: solar wind – Sun: activity – Sun: heliosphere – Sun: magnetic fields

where BR is the radial component of the magnetic field at the source surface as a function of heliolatitude L, (Carrington) longitude φ, and time t. In the case of the application of potential field models, the time is defined as the epoch associated with each Carrington rotation, as the photospheric map of the magnetic field used as the input is completed once per solar rotation. Maps used are from either the Wilcox Solar Observatory (WSO) or from the Mount Wilson Observatory. Routine PFSS maps and data at RSS = 2.5 R are published by the WSO (http://wso.stanford.edu); Wang & Sheeley (2002) use photospheric data from both observatories. The definition given in Equation (1) is, in principle, more general and could be used to include the contribution of CMEs if, at the same time, a suitable knowledge of the whole surface magnetic field map were available. In that case, magnetic field lines that open up through the source surface as the CME loops expand beyond it could, in principle, be also accounted for in the total flux (Riley 2007). It is difficult to estimate the contribution of CMEs to the total unsigned flux in interplanetary space; it clearly depends on the rate of CMEs that reach into interplanetary space—the so-called ICMEs—but also on heliocentric distance. Using a correlation between sunspot numbers and the rate of CMEs and in turn between the rate of ICMEs and CMEs, and further making assumptions about the average contribution of each ICME to the unsigned “open” magnetic flux at 1 AU, Riley (2007) found that the magnetic flux is dependent on the solar cycle (as is the rate of CMEs), and that it is significant around solar maximum when the rate of CMEs is highest. The open magnetic flux of the Sun is one of the most important quantities characterizing the solar cycle (see, e.g., Wang & Sheeley 2002). One of the relevant measures of the unusual long duration of the last solar cycle is the low value of the open magnetic flux at its solar minimum (Wang et al. 2009). As far as the physics of the heliosphere is concerned,

1. INTRODUCTION The heliospheric magnetic field (HMF) originates from regions of the solar corona that are magnetically open (statically or dynamically) to the heliosphere. The inner boundary of the heliosphere is defined as a Sun-centered closed surface (called the source surface) that is generally taken to be associated with the origin of the solar wind. Most of the solar wind and the HMF originate from generally long-lasting coronal structures, coronal holes, in which magnetic field lines rooted in the photosphere remain open through the source surface. A dynamic fraction of the solar wind and the HFM, variable with the phase of the solar cycle and observed at 1 AU and beyond, is due to coronal mass ejections (CMEs) when material and magnetic fields originally in closed coronal structures are ejected into the heliosphere. In the widely used potential field source surface (PFSS) model of the solar corona (Schatten et al. 1969; Hoeksema et al. 1982; Wang & Sheeley 1992) the measured photospheric magnetic fields are used to calculate the magnetic field in the corona, assuming that it is current-free and on the outer boundary condition that magnetic field lines that cross a spherical source surface of radius RSS = 2.5 R are radially oriented. Magnetic field lines crossing the source surface are considered to be open and define the “open” magnetic regions in the corona and the open magnetic flux of the Sun. This definition of the open regions and open flux has been extensively tested against other solar and coronal parameters (see, e.g., Schrijver & DeRosa 2003). The areas defined by the coronal holes were found to closely match the magnetically open regions calculated using the PFSS technique (Wang et al. 1996). The total open magnetic flux at the source surface as given by, e.g., Wang et al. (2006) is  2 Φopen (t) = RSS

|BR (RSS , L, ϕ, t)|dΩ,

(1) 1

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The Astrophysical Journal, 781:50 (12pp), 2014 January 20

the open magnetic flux is also relevant. The magnetic state of the heliosphere, which is dependent on the solar cycle, largely determines the physical processes taking place in it, including the propagation of charged particles. In particular, the modulation of galactic cosmic rays is directly affected by the magnetic field magnitude (Cliver et al. 2013); one possible reason for this is the scattering mean free path of particles scales with their gyroradius. Motivated by both the solar and heliospheric aspects, the open flux is routinely determined form the source surface magnetic maps from the times when photospheric magnetic maps are available. However, the determination of the open flux involves uncertainties, partly because the photospheric magnetic measurements are not reliable at high heliographic latitudes. The extrapolation of the photospheric field to the source surface by model calculations also involves assumptions, although these are not always fully justified. On the other hand, the open magnetic flux is carried out into interplanetary space by the solar wind, where in situ magnetic measurements by space probes serve as good reference points for the open magnetic flux densities. Therefore, the interplanetary magnetic field measurements, at least, provide vantage points where the source surface magnetic field calculations can be tested. Furthermore, the magnetic flux density was proven to be relatively independent of heliospheric latitude (Smith & Balogh 1995; Owens et al. 2008), which means that even a single-point magnetic measurement provides a good representation of the total open magnetic flux of the Sun. The motivation of our article is to further test the level of uniformity of the magnetic flux density in the heliosphere. Assuming radial solar wind outflow, the conservation of the magnetic flux yields a decay of the radial magnetic field component BR by R−2 . The density of the open magnetic flux is characterized by the BR component of the magnetic field, normalized to 1 AU. In accordance with Equation (1) the unsigned magnetic flux density is calculated from the interplanetary measurements by simply taking the average |BR | R2 . Observations have shown that the distribution of BR is a complex function of heliospheric location, solar cycle epoch, and type of solar wind (slow or fast). This complexity is largely due to the magnetic field fluctuations around the Parker field lines (Erd˝os & Balogh 2012). The radial component of the magnetic field decreases faster than the amplitude of the fluctuations as a function of heliospheric distance (Smith 2011a). Therefore, the deviations of the actual magnetic field line from the Parker field line increase with increasing distance from the Sun at the same time as the Parker field lines become more azimuthal in the outer heliosphere. The increase in the Parker angle, together with the magnetic field fluctuations around the Parker field line results in an overlap of the positive and negative magnetic sectors when the field vectors are projected onto the radial axis. This means that a false magnetic sector may appear where the magnetic field line is turned back with respect to the radial line. Taking the absolute value of the BR component is no longer justified if the study aims at the comparison of the in situ heliospheric measurements with solar observations. The reason is that the magnetic flux freezing into the solar wind is valid for the signed BR component but not for its absolute value. Without the applicability of that conservation law, the evolution of the |BR | quantity with the solar wind flow becomes so complex that it becomes practically impossible to interpret. It has been demonstrated that the effect of fluctuations can be reduced significantly by exploiting the fact that the fluctuations are symmetric around the Parker field lines (Erd˝os & Balogh 2012). In this article we further investigate the distribution of the

Table 1 Spacecraft and Time Coverage Used in the Study Spacecraft

Start Date (Day of Year)

Stop Date (Day of Year)

OMNI Helios-1 Helios-2 Ulysses ACE STEREO-A STEREO-B

1963 Nov 27 (331) 1974 Dec 12 (346) 1976 Jan 17 (17) 1990 Nov 18 (322) 1998 Sep 2 (245) 2007 Feb 14 (45) 2007 Mar 1 (60)

2013 Feb 21 (52) 1980 Nov 23 (328) 1980 Mar 4 (64) 2009 Feb 7 (38) 2010 Oct 3 (276) 2012 Nov 30 (335) 2012 Nov 30 (335)

Note. Only those time sections were used when both the solar wind velocity and magnetic field vector data were available.

radial component of the magnetic field observed by space probes in various places in the heliosphere. The study covers a heliocentric distance range of 0.3–5.4 AU and a latitudinal range of 80◦ south to 80◦ north. We introduce two methods to reduce the effect of magnetic field fluctuations. These methods are compared to each other and tested on Ulysses and OMNI observations. In the second half of the article, we analyze the (in)dependence of the magnetic flux density on heliospheric location. 2. METHODS OF THE DETERMINATION OF THE MAGNETIC FLUX DENSITY The open magnetic flux of the Sun is defined as an unsigned quantity (see Equation (1)). However, the freezing, in theory, is applicable to the signed magnetic flux. Close to the Sun at the source surface, using the absolute value of the BR magnetic field component is not a concern because the field is close to radial; therefore the sign of BR indicates the outward and inward magnetic polarities with great confidence. However, at larger distances from the Sun the fluctuations of the field introduce false magnetic sectors where using the absolute value of the BR component is not justified. In this section we expose the problems of determining the magnetic flux density from the BR component of the magnetic field, measured by space probes, at and beyond 1 AU in particular. We also introduce two methods to overcome the difficulties. In the analysis presented in this article, 6 hr averages of the magnetic field vector and solar wind velocity data were used, obtained from Ulysses, ACE, OMNI, Helios 1–2, and STEREO A and B. We have explored the heliosphere in the latitude range of 80◦ south to 80◦ north and the distance from the Sun from 0.3 AU to 5.4 AU. The time interval analyzed covered practically the lifetime of the missions, when both the magnetic field and plasma velocity data were available. However, further restrictions were imposed in specific cases when selection of the data was introduced according to the phase of the solar cycle, velocity of plasma, and location of the spacecraft, the details of which will be given for such specific cases. Table 1 summarizes the time intervals covered in the study. If we assume that the magnetic field is frozen into the solar wind that propagates radially outward, the BR component of the field decays by R−2 . Therefore, the observations of the BR distributions taken at various heliocentric distances can be easily compared if normalization is performed to 1 AU by the factor of R2 . Figure 1 shows such normalized BR distributions at various heliographic distances. As far as the effect of fluctuations is concerned, the slow wind conditions and the minimum epochs of the solar sunspot cycle are the most critical ones. One reason is that the fluctuations are larger, and the Parker angle is also larger 2

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The Astrophysical Journal, 781:50 (12pp), 2014 January 20

Figure 1. Distribution of the magnetic flux density in the slow solar wind (v SW < 600 km s−1 ) during solar minimum (sunspot number < 50). Left column: flux density distributions near 1 AU: OMNI, ACE, and STEREO A and B data. Right column: distributions of the magnetic flux, normalized to 1 AU at various heliocentric distances. Top panel: Helios 1 and 2 observations at close distances from the Sun (R < 0.5 AU). Lower panel: Ulysses observations at large heliocentric distances (R > 3 AU).

in the slow solar wind. The other reason is that the magnitude of the open magnetic flux is smaller during solar minimum. These effects combine to increase the overlap between the peaks corresponding to the positive and negative magnetic sectors. This is why we required the velocity of solar wind to be less than 600 km s −1 , and the sunspot number to be less than 50, in the selection of the data. The left-hand plots show observations at 1 AU by the OMNI database, ACE, and STEREO A and B spacecraft, respectively, for the three panels from the top to the bottom. The right-hand top panel shows observations in the range of 0.3–0.5 AU, taken by Helios 1–2. The right bottom panel is the measurement by Ulysses in the heliocentric range of 3–5.4 AU. We can observe a general trend that the distributions broaden with increasing distance from the Sun.

Only the distribution in the closest distance (Helios 1–2) retains the bimodal character. Such a bimodal distribution is expected in all cases due to the alternation of positive and negative magnetic sectors during the measurements. However, the two peaks corresponding to the opposite magnetic sectors broaden and join to a single peak around zero with increasing distance from the Sun, partly due to the increase of the Parker angle (the angle between the Parker field line and radial direction) and partly because of the increase of magnetic field fluctuations relative to the averaged field (Smith 2011a; Erd˝os & Balogh 2012). The overlap of the two peaks corresponding to the positive and negative magnetic sectors is a concern if the |BR | average is calculated to characterize the open magnetic flux density. 3

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Figure 2. Normalized magnetic flux densities observed by Ulysses during four polar passes in sunspot cycles 22 and 23 (left and right panels, respectively). Top panels: trajectory of Ulysses, shown by the distance from the solar rotation axis (X) and distance from the equator (Z), during the four time intervals (given in decimal years). Middle panels: magnetic flux density, normalized to 1 AU, as a function of the distance from the Sun. Bottom panels: distribution of the normalized magnetic flux density, observed by Ulysses in the fast solar wind in cycles 22 and 23 (left and right, respectively).

22, the northern hemisphere had a positive polarity with a flux density of also about 3 nT. In the next solar cycle the magnetic flux was also uniform in time; see the middle right-hand panel. One obvious difference is that the polarity was reversed in the two hemispheres, as expected. What is more interesting, the value of the normalized flux density was significantly smaller, about 2 nT and −2 nT, respectively, in the southern and northern polar passes, which is in agreement with Smith (2011b). The left and right bottom panels in Figure 2 jointly show the southern and northern distributions of the magnetic flux density during cycles 22 and 23, respectively. The bimodal character in the magnetic flux density is apparent even though the spacecraft traveled to larger distances from the Sun, up to about 4.5 AU. As mentioned earlier, and also shown in Figure 1, the magnetic flux in the slow solar wind is more complicated. The evolution of the magnetic flux with the solar wind outflow is best understood by considering the two-dimensional (2D) distribution of the magnetic field vectors in the plane of BR and BT (radial and azimuthal components of the field, respectively). The centers of Figures 3–5 display these 2D distributions for the Helios 1–2, OMNI, and Ulysses observations. Starting first with the Helios 1–2 observations at 0.3–0.5 AU, we can see on the 2D scatter plot that the magnetic field vectors form two well-defined

One way of overcoming the problem is offered by the Ulysses measurements in the unipolar, high latitude regions in the fast polar solar wind (Smith 2011a). In that case we do not need to take the absolute value of BR , because we know which magnetic polarity we are in, and the averaging of the BR values can be done separately for the positive and negative sectors. Furthermore, the fluctuations of the field in the fast polar solar wind is much less than in the slow wind. Figure 2 shows four polar passes of Ulysses during the first and third off-ecliptic orbits, with both orbits corresponding to the minimum phase of the solar sunspot cycles. The top panels give the position of Ulysses in coordinates of the distance from the solar equator (Z) versus the distance from the rotation axis of the Sun (X). During the time intervals as marked in the figure, Ulysses observed high speed solar wind with uniform magnetic polarity, the sign of which depended on the hemisphere (southern or northern hemisphere) and solar cycle number (even or odd). We note the uniform magnetic polarity, and we can also identify the magnetic polarity on the middle panels, giving the magnetic flux density and normalized to 1 AU as a function of the distance from the Sun. During the southern polar pass in cycle 22 (marked by epoch 1) the magnetic flux density was about −3 nT, independent of the time of observations through one and a half years. Later in cycle 4

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Figure 4. Magnetic flux density at 1 AU in the slow solar wind (v sw < 600 km s−1 ) during solar minimum (sunspot number < 50), with data taken from the OMNI database. One-dimensional distribution of the BR , Bpar , Bper components and the 2D distribution in the R–T plane, in the same format as in Figure 3. Figure 3. Helios 1–2 observations of the normalized (to 1 AU) magnetic flux density in the slow solar wind (v SW < 600 km s−1 ) at close (R < 0.5 AU) distance from the Sun. Middle panel: scatter plot of the BR and BT components of the field vectors (radial and azimuth components). Also shown are the contour lines of the 2D distribution of the magnetic flux density in the R–T plane. The magenta line is the averaged Parker field line direction as calculated from the solar wind velocity measured onboard (perpendicular to the green line). The tilted panel on the left shows the distribution of the magnetic field component perpendicular to the Parker field line (Bper ), and the right panel is the distribution of the magnetic field component parallel to the Parker line (Bpar ). The magnetic sectors were identified by the sign of Bpar ; see the blue and red lines and dots for the negative and positive sectors, respectively. The bottom panel shows the distribution of the BR radial component of the magnetic field (black line), which is decomposed into the negative and positive magnetic sectors (blue and red lines, respectively, but hardly visible except at BR ≈ 0).

the 2D scatter plot. In this way, we can illustrate the projection of the field vectors to the BR axis separately for the negative and positive magnetic sectors (blue and red lines on the bottom panel). Also shown in Figures 3–5 is the distribution of the magnetic field component BPer perpendicular to the Parker line (left tilted panel). The distribution of the BR component has two peaks, which is in agreement with the right top panel of Figure 1. However, if the solar wind travels to larger distance, the scatter of the magnetic field vectors increases, and the Parker angle also increases to a less favorable direction for the projection onto the BR axis. Figure 4 shows the magnetic field distributions for the complete OMNI data set. The 2D distribution shown in the center gives firm evidence that the positive and negative sector data is still visible and discernable, but when the field vectors are projected to the BR axis the positive and negative sectors join to form a single peak around zero. This is not the case for the distribution of the BPar component (right panel), which serves as the basis of the corrections for the fluctuations. We offer two methods as follows.

peaks along the Parker direction. The Parker angle was calculated for each magnetic field vector by using simultaneous solar wind measurement; therefore that angle varied according to the velocity. The average Parker line is shown in the figure (magenta line) just for illustration. The 2D plot also shows the contour lines of the 2D distributions of the magnetic field. The projection of the 2D distribution to the horizontal line illustrates the distribution of the BR component. As the peaks in the 2D distribution are quite narrow, the projection also shows two well-defined peaks; see the black line in the bottom panel. We may also make projections to other coordinate axes. In particular, we can determine the one-dimensional (1D) distribution of the BPar component (the component parallel to the instantaneous Parker line), as shown on the tilted panel on the right of Figure 3; this projection will have more importance in the subsequent figures (Figures 4 and 5), but here we display and discuss it for completeness. According to the sign of the BPar component, we can subdivide our data in negative and positive magnetic polarities, respectively, shown by blue and red dots on

1. Sector-sensitive calculation of the flux density. As mentioned already, the sign of the BPar component provides a possibility to identify the negative and positive magnetic sectors. Therefore, we can subdivide the BR distribution into two parts: BR + (red line) and BR − (blue line). We can then calculate the average of the BR flux density separately for the positive and negative sector data. Mathematically, we substitute the commonly used |BR | average with BR +  − BR − . This is the generalization of the absolute value of the BR flux, used previously, to account correctly for the positive and negative sector data. The advantage of the new method is that it works even for such cases when the 5

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Figure 5. Normalized magnetic flux density at large distances (R < 3AU) from the Sun, observed by Ulysses in the slow solar wind (v sw < 600 km s−1 ) during solar minimum (sunspot number < 50). The format and color code is the same as in Figures 3 and 4, but the plot is extended by the 1D distribution of the BT component as well (left panel).

BR values, corresponding to the dominant polarity of one sector, extend to the opposite polarity, as can be seen in Figure 4. The problem is that if we take the absolute value of the BR + flux, then the negative part of the distribution is mirrored to the positive side, which increases the average of the magnetic flux in the positive sector. The situation is the same in the negative sectors. Therefore, using the absolute value of the BR component overestimates the magnetic flux. This is a serious problem that is largely reduced with this new method. Note that a similar method was suggested by Smith (2011a) but with the more stringent criteria that the magnetic field vectors with small |BPar | values should be rejected from the analysis, considering them as uncertain data for sector identification. We also note that, at least in principle, we may improve the suggested method by introducing other considerations for identifying the magnetic sectors, such as electron heat flux, propagation direction of Alfv´en waves, visual inspection of the field time series, etc. 2. Neglecting the fluctuations of the magnetic field perpendicular to the Parker line. We can see that the distribution of the BPer magnetic field component (left tilted panel in Figure 4) is highly symmetric around zero. Therefore we suggest that, for the determination of the magnetic flux density, the BPer component can be neglected; in other words, the 2D distribution can be compressed to the Parker line. Note that the compressed 2D distribution will still have a finite thickness due to variations in the solar wind velocity. The rationality of this procedure is as follows. Let us select a small domain on the 2D plane. When we project the magnetic field vectors in that domain to the Parker line (by

neglecting the BPer component), we introduce some error in the BR component. Now we can mirror the selected domain with respect to the Parker line. When we project the field vectors to the Parker line in the mirrored domain, then the error of the BR component will be the same as in the original domain but with the opposite sign. However, the number of field vectors in the original and mirrored domain are statistically the same, and therefore the errors introduced by the projection of the field vectors to the Parker line will cancel each other out if long-term averaging is performed. Technically the procedure means that we substitute the BR distribution with the distribution of the cos(α)BPar parameter, where α is the Parker angle. We call this 1D distribution the corrected BR∗ distribution (distribution along the Parker line but with rescaling according to the Parker angle). The right tilted panel of Figure 4 proves that the corrected BR∗ distribution is bimodal with only a small overlap between the negative and positive sector data. This ensures that during the determination of the flux density, taking the absolute value of BR∗ introduces much less error than in the case of the uncorrected BR distribution. Although the effect of the fluctuations is not very large at 1 AU, we must correct it by using one of the two methods above. The reason for this is that the extent of the overestimation of the magnetic flux by the use of the absolute value of BR depends on many parameters in a complex way, such as the phase of the sunspot cycle or the velocity of the solar wind. Without corrections for the fluctuations, a long time series of the magnetic flux could be contaminated by such an error, 6

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Figure 6. Magnetic flux density at 1 AU in the slow solar wind during solar minimum, as determined from the OMNI database. Top row: distribution of the BR component (left); distribution of |BR | (right). We call the mean of |BR | the “uncorrected flux.” Middle row: distributions of B− and B + (radial component in the negative and positive magnetic sector, respectively; left); joint distribution of B + and −B− (right). We call the mean value the “sector-sensitive flux.” Bottom row: distribution of BR∗ = BP cosα, where α is the Parker angle (left). Distribution of |BR∗ | (right). We call the main of |BR∗ | the “corrected flux.”

which involves spurious solar cycle variations. Beyond 1 AU the situation is worse, as the error introduced by the use of the absolute value of BR increases rapidly with the distance from the Sun, which leads to the problem of “flux excess” in the outer heliosphere (Lockwood et al. 2009a, 2009b). Figure 5 shows the 2D and 1D distributions of the magnetic field components, observed by Ulysses in the heliospheric range of 3–5.4 AU during slow solar wind conditions. We can observe that the distribution of the BR component (bottom panel, black line) is very broad without any sign of the sector structure. However, we can see the traces of the magnetic sectors on the 2D distribution as well as on the distribution of the BPar component, although the peak corresponding to the positive

magnetic polarity dominates over the negative polarity. This asymmetry is presumably connected to the trajectory and time of observations of the Ulysses mission. Also shown on the left is the distribution of the BT component, which confirms the sector structure of the magnetic field. Let us compare the methods for the corrections of the fluctuations in a quantitative way. Figure 6 shows the magnetic field data at 1 AU, determined by the OMNI data in the slow solar wind during sunspot minimum. The top row displays the distribution of the radial magnetic field component BR (left panel) and the distribution of |BR | (right panel). The mean value of |BR | = 2.43 nT is the uncorrected magnetic flux density (see the thin vertical line in the figure). The middle 7

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The Astrophysical Journal, 781:50 (12pp), 2014 January 20

Figure 7. Magnetic flux density at large distances (R < 3 AU) from the Sun in the slow solar wind during solar minimum, as observed by Ulysses. One-dimensional normalized magnetic flux densities and the unsigned flux (top row), sector-sensitive flux (middle row), and the corrected flux (bottom row), in the same format as in Figure 6.

row illustrates the sector-sensitive calculation of the flux. The left panel shows the distribution of the BR component in the negative and positive magnetic sectors separately (BR − and BR + , respectively). The right panel shows the joint distribution of B + (positive sector) and −B− (mirrored negative sector), with the mean value calculated from the average of B + −B−  = 2.18 nT. This average is called the sector-sensitive magnetic flux density. The left panel in the bottom row is the distribution of the corrected radial component of the magnetic field, BR∗ . As the “corrected” values for the positive and negative magnetic sectors are less overlapping than in the “uncorrected” case (compare the bottom row to the top row), we may take the distribution of the |BR∗ | to calculate the corrected magnetic flux density, |BR∗ | =

2.23 nT. Figure 6 tells us that the uncorrected flux is about 10% larger than both the sector-sensitive flux and the corrected flux at 1 AU. Furthermore, the sector-sensitive and corrected flux values are reasonably close to each other. In the outer heliosphere the effect of the fluctuations is harder to handle. Figure 7 gives the Ulysses observations in the heliospheric range of 3–5.4 AU in the slow solar wind during solar minimum. The format is the same as in Figure 6. The magnetic flux densities are as follows: uncorrected flux = 5.54 nT (top row), sector-sensitive flux = 2.31 nT, corrected flux = 2.42 nT. Here the uncorrected flux is larger, by about a factor of two, than both the sector-sensitive and the corrected flux densities. Although the difference between the 8

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Figure 8. Time variations of the normalized unsigned magnetic flux density, averaged for Carrington rotations. Comparison of Ulysses (red line) and OMNI (blue line) observations. The bottom panel is for the distance of Ulysses from the Sun (magenta line, left scale) and the heliographic latitude of Ulysses (green line, right scale).

increasing distance from the Sun. Figures 9 and 10 show what happens if we remove the effect of the magnetic fluctuations with the two methods presented above. Figure 9 gives the sectorsensitive calculation of the magnetic flux density observed by Ulysses and OMNI. The scatter of the flux values, in particular for Ulysses, is large, but the averages of the Ulysses and OMNI data are generally close to each other. The scatter of Ulysses sector-sensitive flux data at large heliocentric distances is due to the large level of fluctuations combined with the increase of the Parker angle, as already described. The negative values for Ulysses are extreme cases of the scatter due to use of shorter Carrington rotation averages. We can conclude that the large flux excess present in the uncorrected magnetic flux of the Ulysses data near aphelion disappears when the flux is calculated with the sector-sensitive method. Using the alternative data analysis method, if the magnetic field fluctuations perpendicular to the local Parker direction is neglected, then the match between the Ulysses and OMNI data is even better. Figure 10 shows the corrected magnetic flux density averaged for Carrington rotations, observed by Ulysses and near-Earth satellites. The agreement between the Ulysses and OMNI data is remarkable given the fact that the two observations were carried out in completely different places of the heliosphere (at different latitudes and different heliospheric distances—see the bottom panel of Figure 10—as well as at different longitudes). Notice that the Ulysses magnetic flux was normalized to 1 AU, as in all cases throughout this article. The uniform distribution of the magnetic flux density in the heliosphere was further investigated in a more quantitative way.

sector-sensitive and corrected flux is larger than that at the 1 AU observations, we may consider that the match is still reasonable given the fact that the corrections for the fluctuations is the most critical in the case presented in Figure 7 (slow solar wind, sunspot minimum, large distance from the Sun). 3. COMPARISON OF THE MAGNETIC FLUX DENSITY AT VARIOUS PLACES IN THE HELIOSPHERE In this section we compare the Ulysses magnetic field observations to the OMNI data set. We present the observations in a time series, where each data point corresponds to the determination of the magnetic flux during one Carrington rotation. This time resolution is acceptable as far as the solar cycle variations and orbital variation of Ulysses are concerned. Figure 8 compares the magnetic flux observed by Ulysses (red line) to the OMNI magnetic flux when no corrections are made for the magnetic field fluctuations. There are large discrepancies between the two flux values; the Ulysses observations are periodically larger than the flux measured close to the earth by a factor of two. These large deviations are connected to the orbit of Ulysses, which can be seen if we compare the variations of the flux values to the plots in the bottom panel representing the heliographic range and latitude of the spacecraft. It is clear that the largest deviations are around the aphelion section of the trajectory of Ulysses. As we have argued (see also Smith 2011a), the “flux excess” is due to the folding back of the magnetic field lines (with respect to the radial line), which appears more frequently with 9

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Figure 9. Comparison of the Ulysses and OMNI normalized sector-sensitive magnetic flux density.

magnetic flux is a confirmation of the early Ulysses finding that the magnetic flux is constant in the fast polar solar wind (Smith & Balogh 1995). Later, Owens et al. (2008) showed the independence of the magnetic flux by heliospheric latitude and longitude, although their study was affected by the flux excess problem. Our study provides evidence that the open magnetic flux is highly independent of heliographic distance (right bottom panel) and latitude (right top panel) if proper corrections are made for the effect of magnetic field fluctuations.

Figure 11 shows the ratio of the Ulysses and OMNI normalized magnetic flux density as a function of the heliographic latitude of Ulysses (upper row) and as a function of the distance of Ulysses from the Sun (lower row). Since the magnetic flux, observed by Ulysses at various places in the heliosphere, is compared to the OMNI data taken at a fixed position (inecliptic, 1 AU), time variations connected to the solar cycle are canceled out. The left-hand panels show what happens if no correction is made to take into account the effect of the magnetic field fluctuations. With this commonly used practice, the magnetic flux appears to increase considerably with solar distance (left bottom panel). This radial dependence of the flux was interpreted earlier as excess flux in the outer heliosphere (Lockwood et al. 2009a, 2009b), a result that was criticized by Smith (2011a). Also, without corrections for the fluctuations, the open magnetic flux of the Sun seems to decrease toward the poles (see the top left panel in Figure 11). These variations by heliospheric location are due to different time percentages of “false” magnetic sectors (i.e., times when the magnetic field line is folded back with respect to the radial direction). In particular, in the polar region the Parker angle is close to radial, and in the fast solar wind (typical for high heliolatitudes) fluctuations have smaller amplitudes. During those conditions the false magnetic sectors are less frequent, resulting in a decrease of the unsigned magnetic flux relative to the in-ecliptic value, as visible on the top left-hand panel in Figure 11. However, if corrections are made for the effect of fluctuations, the variation by solar distance and heliographic latitude disappears, as shown in the right-hand panels in Figure 11. The latitudinal independence of the open

4. SUMMARY In this article we have investigated the open magnetic flux density of the Sun, as measured for long time periods by magnetometers onboard space probes at various locations in the heliosphere. Distributions of the radial component of the magnetic field, normalized to 1 AU, were determined. It was shown that bimodal distributions are typical close to the Sun as a consequence of the alternating positive and negative magnetic sectors. However, at larger distances from the Sun, fluctuations of the field vectors around the Parker spiral broaden the two peaks of the distributions to such an extent that the two magnetic sectors are not discernable in the radial component of the magnetic field. We have pointed out that this creates a problem when the unsigned open magnetic flux is determined by simply using the absolute value of the radial component of the magnetic field. Long-time interval studies have also been performed for the 2D distribution of the radial and azimuthal components 10

˝ & Balogh Erdos

The Astrophysical Journal, 781:50 (12pp), 2014 January 20

Figure 10. Comparison of the Ulysses and OMNI normalized corrected magnetic flux density.

Figure 11. Latitudinal and solar distance (in)dependence of the normalized magnetic flux (top and bottom row, respectively) as determined from the ratio of the Ulysses and OMNI observations. The horizontal axes mark the position of Ulysses (heliographic latitude and range). Each circle covers one Carrington rotation average. Left column: unsigned flux; right column: corrected flux.

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˝ & Balogh Erdos

The Astrophysical Journal, 781:50 (12pp), 2014 January 20

of the magnetic field vectors. The investigations covered the heliospheric range of 0.3–5 AU by using Helios 1–2, OMNI, and Ulysses data. It has been shown that even at large heliocentric distances the sector structure of the magnetic field is still visible in the 2D distributions. In the 1D BR distributions the merging of the two peaks, corresponding to the positive and negative magnetic sectors, is caused by the joint effect of the fluctuations of the field vectors around the average Parker field line and the increase of the Parker angle with increasing distance from the Sun. We have introduced two methods to correct for the effect of fluctuations. One is based on the separation of the data according to the magnetic sectors. The identification of the sectors was performed according to the sign of the cosine of the angle between the measured magnetic field vector and the Parker field line. In the other method the magnetic field component perpendicular to the Parker field line was simply neglected to remove the effect of fluctuations. Notice that the Parker field line was determined from the solar wind velocity measured onboard (i.e., from measurements independent of the magnetic field). The magnetic flux density observed by Ulysses throughout its lifetime was compared to the near-Earth magnetic flux density obtained from the OMNI database. With the commonly used practice that the unsigned flux is calculated from the absolute value of the BR component, the flux measured by Ulysses would seem significantly larger than that given by the OMNI data. The largest deviations, even by a factor of two, are typically close to the aphelion of Ulysses, which was interpreted earlier as a “flux excess” in the outer heliosphere. We strongly argue that the flux excess is an artifact caused by the fluctuations of the magnetic field vectors. Our analysis shows that the OMNI and Ulysses flux density determinations (normalized to 1 AU) are remarkably close to each other when proper care is taken to reduce of the effect of magnetic field fluctuations, in spite

of the fact that the two sets of observations were carried out at completely different locations (in particular, latitude) in the heliosphere. This proves indirectly that the magnetic flux is uniformly distributed in the heliosphere. As a consequence, a single spacecraft measurement of the magnetic field in the heliosphere provides information on the total solar unsigned magnetic flux. Our method for the correction of the magnetic fluctuations opens new possibility to make comparisons of the heliospheric magnetic flux to source surface model results and also provides a new opportunity to study the long-term solar cycle variations of the magnetic field carried by the solar wind. The authors are grateful for the wide availability of important magnetic field and solar wind velocity data sets from the ACE, STEREO, Helios, and Ulysses missions, as well as the wellmaintained OMNI data set. REFERENCES Cliver, E. W., Richardson, I. G., & Ling, A. G. 2013, SSRv, 176, 3 Erd˝os, G., & Balogh, A. 2012, ApJ, 753, 130 Hoeksema, J. T., Wilcox, J. M., & Scherrer, P. H. 1982, JGR, 87, 10331 Lockwood, M., Owens, M., & Rouillard, A. P. 2009a, JGR, 114, 11103 Lockwood, M., Owens, M., & Rouillard, A. P. 2009b, JGR, 114, 11104 Owens, M. J., Arge, C. N., Crooker, N. U., Schwadron, N. A., & Horbury, T. S. 2008, JGR, 113, A12103 Riley, P. 2007, ApJL, 667, L97 Schatten, K. H., Wilcox, J. W., & Ness, N. F. 1969, SoPh, 4, 442 Schrijver, C. J., & DeRosa, M. L. 2003, SoPh, 212, 165 Smith, E. J. 2011a, JGRA, 116, 12101 Smith, E. J. 2011b, JASTP, 73, 277 Smith, E. J., & Balogh, A. 1995, GeoRL, 22, 3317 Wang, Y.-M., Hawley, S. H., & Sheeley, N. R., Jr. 1996, Sci, 271, 464 Wang, Y.-M., Robbrecht, E., & Sheeley, N. R., Jr. 2009, ApJ, 707, 1372 Wang, Y.-M., & Sheeley, N. R., Jr. 1992, ApJ, 392, 310 Wang, Y.-M., & Sheeley, N. R., Jr. 2002, JGRA, 107, 1302 Wang, Y.-M., Sheeley, N. R., Jr., & Rouillard, A. P. 2006, ApJ, 644, 638

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