1 Introduction

Solar magnetic fields are swept into interplanetary space by solar wind flows. Solar fields dominate the structure and dynamics of the heliosphere. The heliosphere varies spatially and temporally, following the magnetic field variation. The regularity in magnetic field space and time distribution at different distances from the Sun and during different solar cycle phases are not well known. This is largely due to the weakness of coronal magnetic fields, especially at distances greater than 5 \(R_{\mathrm{s}}\) (solar radii). To analyze different processes in the solar corona and interplanetary space, the magnetic field distribution and cycle evolution must therefore be ascertained.

Currently, several different methods are used to measure the coronal magnetic field at different distances from the Sun (Akhmedov et al., 1982; Lin, Penn, and Tomczyk, 2000, Lin, Kuhn, and Coulter, 2004, Bogod and Yasnov, 2016; Gelfreikh, Peterova, and Riabov, 1987). Some measurements of coronal magnetic fields were made at different wavelengths (Lin, Penn, and Tomczyk, 2000; Lin, Kuhn, and Coulter, 2004; Raouafi et al., 2016). The Faraday rotation measurement technique is commonly used to estimate the coronal magnetic field strength within 10 \(R_{\mathrm{s}}\) (Pätzold et al., 1987; Sakurai and Spangler, 1994; Spangler, 2005; Ingleby, Spangler, and Whiting, 2007). Pätzold et al. (1987) found that the coronal magnetic field at 5 \(R_{\mathrm{s}}\) is around \(100 \pm 50~\mbox{mG}\), Sakurai and Spangler (1994) derived magnetic field values at 9 \(R_{\mathrm{s}}\) of \(12.5 \pm 2.3~\mbox{mG}\), and Spangler (2005) found a value of 39 mG at 6.2 \(R_{\mathrm{s}}\). Ingleby, Spangler, and Whiting (2007) measured the Faraday rotation with the Very Large Array (VLA) at frequencies of 1465 and 1665 MHz and found that the coronal magnetic field is in the range of 46 – 120 mG at a heliocentric distance of 5 \(R_{\mathrm{s}}\). Xiong et al. (2013) used coordinated observations in polarized white light and Faraday rotation measurements to determine the spatial position and magnetic field of an interplanetary sheath. Magnetic field values can also be derived from measurements of the solar wind plasma using pulsars (Ord, Johnston, and Sarkissian, 2007; You et al., 2012).

Several methods of magnetic field determination were developed using solar radio emission. Using data from UltraViolet Coronagraph Spectrometer (UVCS) Solar and Heliospheric Observatory (SOHO) and radio spectrograph observations, Mancuso et al. (2003) estimated magnetic field and plasma properties in the active region corona. The results show that the magnetic field is expressed by \(B(r) \le (0.6 \pm 0.3)(r -1)^{-1.2}\) G, which is valid in the range \(1.5 \le r \le 2.3\) \(R_{\mathrm{s}}\) (Mancuso et al., 2003). The magnetic field strength can be derived from the band splittings in type II radio bursts if the coronal density distribution is given (Vrs̆nak et al., 2002; Cho et al., 2007; Hariharan et al., 2014). Using band splitting of coronal type II radio bursts, Cho et al. (2007) obtained a coronal magnetic field strength of 1.3 – 0.4 G in the height range of 1.5 – 2 \(R_{\mathrm{s}}\).

Recent investigations demonstrate that shock waves propagating into the corona and interplanetary space, associated with major solar eruptions, can be used to derive the strength of magnetic fields over a very long interval of heliocentric distances and latitudes. The so-called ‘standoff-distance’ method was developed and is often used now (Gopalswamy and Yashiro, 2011; Gopalswamy et al., 2012; Kim et al., 2012; Bemporad et al., 2016; Schmidt et al., 2016). In this method, a coronal mass ejection (CME) and CME-driving shock dynamics are analyzed. The shock standoff distance, speed, and the radius of the CME flux rope curvature are measured. Alfvén speed and Mach number can be derived using the measured data. Then the magnetic field can be derived by applying a coronal density model. Using this method, Gopalswamy and Yashiro (2011) have found that the magnetic field declines from 48 to 8 mG in the distance range from 6 to 23 \(R_{\mathrm{s}}\). Following the standoff-distance method and using data from the Coronagraph 2 and Heliospheric Imager I instruments on board the Solar Terrestrial Relations Observatory (STEREO), Gopalswamy et al. (2012) have found that the radial magnetic field strength decreases from 28 mG at 6 \(R_{\mathrm{s}}\) to 0.17 mG at 120 \(R_{\mathrm{s}}\). They also noted that the radial profile of the magnetic field strength can be described by a power law. Kim et al. (2012) performed a statistical study by applying this method to ten fast (≥ 1000 km s−1) limb CMEs (Large Angle Spectroscopic Coronagraph (LASCO) data) to measure the magnetic field strength in the solar corona in the height range 3 – 15 \(R_{ \mathrm{s}}\). They found that the magnetic field strength is in the range 6 – 105 mG. They showed that the magnetic field values derived with the standoff-distance method are consistent with other estimates in a similar distance range.

Despite the increasing number of space missions and the progress made recently in the solar and interplanetary space observations, reliable measurements of the coronal and interplanetary magnetic field (IMF) strength and orientation at different distances and cycle phases do not exist, however. At present, magnetic fields are routinely measured at the photospheric level, besides the National Solar Observatory (NSO) Synoptic Optical Long-term Investigations of the Sun (SOLIS)/Vector Spectromagnetograph (VSM) also observes the full-disk chromospheric field using the Ca II 8542 nm line, but not in the solar corona and IMF. The magnetic field in the solar corona and interplanetary space is therefore estimated from the observed photospheric fields using different extrapolation techniques into the solar corona. Coronal magnetic fields are investigated and modeled at different distances from the Sun up to several solar radii. Some models at coronal heights are based on the magnetic fields in active regions (Brosius and White, 2006; Bogod and Yasnov, 2008, Bogod, Stupishin, and Yasnov, 2012, Kaltman et al., 2012). Different models are developed both analytically (Banaszkiewicz, Axford, and McKenzie, 1998) and numerically, including magnetohydrodynamics (MHD), magnetohydrostatic, force-free, or potential-field models (Gibson and Bagenal, 1995; Wiegelmann, 2004; Wiegelmann, Petrie, and Riley, 2017, and references therein).

It should also be noted that most models only describe the magnetic field distribution in the radial direction from the Sun. As a rule, in such models the measurements of the magnetic field obtained at different times are summarized in one curve, regardless of a cycle or a cycle phase. However, it is well known that the magnetic field in the quiet corona during sunspot minimum is much lower than the field determined for an average sunspot maximum, and spherical symmetry is not observed. Furthermore, at solar maximum, the magnetic field is different in different solar cycles as a result of different levels of activity. Below the height of ≈ 3 \(R_{\mathrm{s}}\), the magnetic field is governed by the active-region fields. Above a height of ≈ 3 \(R_{\mathrm{s}}\), the radial field, decreasing as \(R^{-2}\), becomes dominant. It should be noted that all coronal magnetic field models that describe magnetic fields above ≈ 3 \(R_{\mathrm{s}}\) give only one value for the distance required, regardless the cycle phase. It is necessary to use more realistic models for the radial distribution of magnetic fields that take into account the variations in the solar magnetic field cycle. Our attention thus has been directed to a detailed description of the observed solar magnetic field distribution and cycle variation, to create a model of the magnetic field radial distribution and cycle variations from 1 \(R_{\mathrm{s}}\) to 1 AU. We then compare the magnetic fields derived using our model with the field measured at different times and distances, as well as with magnetic field profiles from previous models. The article will summarize the most recent models and results on magnetic field measurements.

The article is organized as follows. The data are described in Section 2. In Section 3, the photospheric and interplanetary magnetic field distribution and cycle evolution are presented, and a new model of the magnetic field calculation at different distances from the Sun with the consideration of solar cycle variations is suggested. The magnetic fields measured by different methods are compared with the field calculated using our model in Section 4. The comparison of our model results with those derived using previous models is made in Section 5. The results are discussed in Section 6. The main results are listed in Section 7.

2 Data

Data on the large-scale photospheric magnetic field from the Wilcox Solar Observatory (WSO) were used for 1976 – 2015. Full-disk synoptic maps span a full Carrington rotation (1 CR = 27.2753 days). They are assembled from individual magnetograms that were observed during a solar rotation. The WSO synoptic maps only represent the radial component of the photospheric field (derived from observations of the line-of-sight field component by assuming the field to be approximately radial). The entire dataset consists of 530 synoptic maps and covers CRs 1642 – 2172 (June 1976 – December 2015). Synoptic map magnetic-field data consist of 30 data points in equal steps of sine latitude from \(+70^{\circ }\) to \(-70^{\circ }\). As the solar magnetic fields are measured from Earth, the field above \(70^{\circ }\) in the north and south hemispheres is not resolved. The longitude is presented in \(5^{\circ }\) intervals (Duvall et al., 1977; Hoeksema and Scherrer, 1986).

WSO polar field observations were used in this study. The Sun’s polar magnetic field strength is measured in the polemost 3’ apertures each day at the WSO in the north and south hemispheres. The line-of-sight magnetic field between about \({\pm}\, 55^{\circ }\) and the pole in the corresponding hemisphere is measured. The daily polar field measurements are averaged every 10 days in a centered 30-day window. The solar coordinates of the apertures shift and the square aperture at the pole is oriented differently on the Sun during each measurement due to the Earth motion above and below the solar equator each year.

Data on the IMF were obtained from the database called multi-source Operating Missions as a Node on the Internet (OMNI 2: King and Papitashvili, 2005). From the OMNI 2 database, the hourly mean values of the IMF measured by various spacecraft near the Earth orbit were taken. Only the radial component, \(B_{\mathrm{X}}\), of the OMNI 2 IMF was used in this study.

3 Large-Scale Photospheric and Coronal Magnetic Field Variations in Cycles 21 – 24

To study the variations in the magnetic field distribution in the solar corona, we have used the direct observations of the large-scale non-polar (\({\pm}\, 55^{\circ }\)) and polar (from \(55^{\circ }\) N to \(90^{\circ }\) N and from \(55^{\circ }\) S to \(90^{\circ }\) S) photospheric magnetic fields, which are the visible manifestations of the cyclic changes in the toroidal and poloidal components of the global magnetic field of the Sun. In addition to the general magnetic field distribution, occasional displacements and oscillations appear in localized regions of the solar corona due to the magnetic fields caused by CMEs, flows from coronal holes, flares, etc. Therefore, CR-averaged data were used.

To describe the non-polar magnetic field cycle variations, we created diagrams of the photospheric magnetic field distribution based on the observed large-scale photospheric magnetic fields from \(55^{\circ }\) S to \(55^{\circ }\) N latitude through Cycles 21 – 24. Figure 1a shows the longitudinal time-space distribution of the large-scale photospheric magnetic fields. The smoothed photospheric magnetic field longitudinal distribution is presented in Figure 1b. The CR-averaged distribution of positive- and negative-polarity IMF at Earth’s orbit is shown in Figure 1c. The longitudinal diagrams were created in a CR-rotation system. The \(x\)-axes denote the date of 0∘ CR longitude at the central meridian, and the \(y\)-axes denote the longitude in each CR magnetic-field diagram. A detailed description of the magnetic field longitudinal diagram creation and solar global magnetic field evolution is given in Bilenko (2012) and Bilenko (2014). The maxima and minima of the cycles are marked at the top of Figure 1. Figure 1 demonstrates the close connection of the magnetic field direction and strength variations of the IMF at Earth’s orbit and the photospheric magnetic fields.

Figure 1
figure 1

Longitudinal diagrams. (a) Large-scale photospheric magnetic fields. (b) Large-scale photospheric magnetic fields smoothed in \(7\times7\) CRs. (c) IMF at 1 AU. Red denotes the positive-polarity magnetic fields, blue denotes the negative-polarity magnetic fields. Black denotes the missing data. The maxima and minima of Cycles 21 – 24 are marked at the top.

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Figure 2a shows the cycle changes in the photospheric large-scale magnetic field obtained from the longitudinal distribution of the photospheric magnetic fields by averaging over the latitude from 55 S to 55 N for every CR displayed in Figure 1a. Black denotes the positive- and negative-polarity photospheric magnetic fields, and light blue denotes the sum of their absolute values. Thin lines show changes in the CR-averaged magnetic fields, and thick lines correspond to the seven CR-averaged data. The maxima and minima of Cycles 21 – 24 are marked at the top of Figure 2. Examination of Figures 1 and 2 shows that the magnetic field does not change smoothly from the minimum of solar activity to the maximum, but in the form of some impulses. These changes reflect cyclical changes in the structure and strength of the solar global magnetic field (Bilenko, 2012, 2014; Bilenko and Tavastsherna, 2016). The maximum of the magnetic field magnitudes decreased, and it was highest in Cycle 21 and lowest in Cycle 24. The magnetic field strength decrease can be given by

$$ B_{\mathrm{max}}(t) = 317.82 - 0.30 \times t_{\mathrm{\mathtt{CR}}}, $$
(1)

where \(t_{\mathrm{\mathtt{CR}}}\) is the time scale for the magnetic field calculation (the \(x\) axis units are in CRs) for the magnetic field calculation.

Figure 2
figure 2

(a) Photospheric non-polar magnetic fields from the longitudinal diagram in Figure 1a. Black denotes the positive- and negative-polarity photospheric magnetic fields, and light blue denotes the sum of their absolute values. (b) CR-averaged polar magnetic field cycle variations at the north (N) and south (S) poles, and the sum of their absolute values (black line). (c) Sum of the positive- and negative-polarity photospheric (non-polar) magnetic fields (light blue line) from panel (a) and the sum of the north and south pole magnetic fields (black line) from panel (b). The red line denotes the sum of polar and non-polar magnetic fields. (d) IMF measured at 1 AU. Black denotes the CR-averaged positive- and negative-polarity magnetic fields, and green denotes the sum of their absolute values. (e) The red line denotes the magnetic field at 1 AU calculated from Equation 4 as the sum the non-polar and polar magnetic fields. The green line denotes half of the sum of the CR-averaged positive- and negative-polarity IMF at 1 AU. In panels (a), (c), (d), and (e) the thin lines correspond to CR-averaged and the thick lines to seven CR-averaged data. The maxima and minima of Cycles 21 – 24 are marked at the top. The straight lines in panel c are described in the text.

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The minimum values of the magnetic field during solar activity minima also decreased from Cycle 21 to Cycle 24, and it was also highest in Cycle 21 and lowest in Cycle 24. We find a regression line

$$ B_{\mathrm{min}}(t)= 26.17 - 0.47 \times t_{\mathrm{\mathtt{CR}}}. $$
(2)

Such cyclic changes in magnetic fields (Figure 2a) reflect the time evolution of the toroidal component of the solar global magnetic field. This means that the toroidal component decreases from Cycle 21 to 24.

Magnetic field cycle variations at the solar poles are the observational manifestation of the poloidal component of the solar global magnetic field. Figure 2b shows CR-averaged magnetic field variations in the north (blue line) and south (red line) poles. The black line denotes the sum of their absolute values. The polar magnetic fields at the north and south poles diminished from Cycle 21 to 24. The cyclic changes in magnetic field (Figure 2b) reflect the time evolution of the poloidal component of the solar global magnetic field from Cycle 21 to 24. This means that both components have decreased from Cycle 21 to 24. The decrease in the polar magnetic field can be expressed by

$$ B_{\mathrm{pol}}(t) = 161.04 - 0.25 \times t_{\mathrm{\mathtt{CR}}}. $$
(3)

The magnetic field in the ecliptic plane can be modeled as the superposition of the toroidal and poloidal component cycle variations. The toroidal component dominates at solar maxima, and the influence of the poloidal component increases during solar minima. The toroidal component increases during solar maxima because of the enhancement of the active region magnetic fields. At solar minima, about 60% of the solar sphere above 2 \(R_{\mathrm{s}}\) is dominated by polar coronal holes. The total magnetic field strength can therefore be determined as the sum of the non-polar photospheric magnetic fields (toroidal component of the solar global magnetic field, Figure 2a) and the polar magnetic field (poloidal component) obtained from Figure 2b. The non-polar (light blue line) and polar (black line) magnetic field components and their sum (red line) are shown in Figure 2c. We detect the magnetic field components that directly influence the measured IMF at Earth’s orbit. It is assumed that the corona is in a steady state from 1 \(R_{\mathrm{s}}\) to 1 AU. The non-polar and polar fields are observed in the ecliptic plane from Earth, therefore we can derive the radial component (\(B_{\mathrm{x}}\)) of the IMF from the observed non-polar and polar photospheric magnetic field strength in the ecliptic plane using the relation

$$ B(t,r) = \biggl(\frac{B_{\mathrm{phot+}}(t) + B_{\mathrm{phot-}}(t)}{2} + \frac{B_{ \mathrm{pol}}(t)}{2} \biggr) \times \biggl( \frac{1}{r} \biggr) ^{2}, $$
(4)

where \(B_{\mathrm{phot+}}\) and \(B_{\mathrm{phot-}}\) are averaged over the latitude from 55 S to 55 N. The absolute values of the positive- and negative-polarity magnetic fields at the photosphere derived from Figure 1a and shown in Figure 2a for each CR represent the toroidal component of the solar magnetic field. \(B_{\mathrm{pol}}\) is the average of the sum of the observed unsigned north and south polar magnetic fields calculated for each CR (Figure 2b), and this represents the poloidal component of the solar global magnetic field. \(r\) is the distance from the center of the Sun in units of the solar radius. The different behavior of polar and non-polar magnetic fields is also known from the cycle evolution of active region (Zharkov, Gavryuseva, and Zharkova, 2007) and coronal holes (Lowder, Qiu, and Leamon, 2017; Bilenko and Tavastsherna, 2016). Polar and non-polar magnetic fields represent the different components of the solar global magnetic field. Their cycle evolution is different. Therefore, they were included in Equation 4 as separate components.

As noted above, the magnitudes of the toroidal and poloidal magnetic field components of the solar global magnetic field decreased from Cycle 21 to Cycle 24. We find a regression line for our model (Equation 4) magnetic field cycle maxima

$$ B_{\mathrm{m_{max}}}(t) = 339.93 - 0.32 \times t_{ \mathrm{\mathtt{CR}}}, $$
(5)

while the expression for the minima can be written as

$$ B_{\mathrm{m_{min}}}(t) = 77.67 - 0.12 \times t_{\mathrm{\mathtt{CR}}}. $$
(6)

In Figure 2c the dashed lines are the regression lines derived in Equations 1, 2, 3, 5, and 6.

Figure 2d shows the variation of the observed \(B_{\mathrm{x}}\) component of the IMF at Earth’s orbit. Black denotes the CR-averaged positive- and negative-polarity magnetic fields, and green denotes the sum of their absolute values. Thin lines show the CR-averaged IMF, and thick lines correspond to the seven CR-averaged data.

In Figure 2e the magnetic field derived at 1 AU using our model (Equation 4) is shown in red. The green line denotes the half of the sum of CR-averaged positive- and negative-polarity IMFs at 1 AU in Cycles 21 to 24. CR-averaged magnetic fields were compared. The magnetic fields retrieved from the photospheric and interplanetary magnetic fields measured at Earth’s orbit were calculated the same way. In Equation 4, the magnetic fields were used as half of the sum of the absolute values of positive- and negative-polarity non-polar and polar magnetic fields averaged during each CR. Likewise, the IMF is also the sum of CR-averaged of the absolute values of the positive- and negative-polarity \(B_{\mathrm{x}}\) component of the IMF divided by 2. Figure 2e shows that our model fits the \(B_{\mathrm{x}}\) component of the IMF at 1 AU very well, particularly during Cycles 23 and 24. From a comparison of the magnetic fields as observed and calculated from our model, we conclude that our model magnetic field strength variation adequately explains the main features of the IMF changes during Cycles 21 to 24. However, there are certain differences. The difference is greater during the maximum phase in Cycle 21, but the mismatch is significantly weaker in the maxima of Cycles 22 and 24 (Figure 2e). Some peaks in magnetic fields derived using our model (Equation 4) coincide with peaks in the IMF, and some peaks follow the peaks in the IMF. The coincidence between the IMF and our model-predicted magnetic fields at 1 AU is more pronounced during the declining phases. The reduction of the misalignment between the interplanetary magnetic field and the field predicted using our model observed from Cycle 21 to Cycle 24 can be explained by the fact that the quality of observations and measurements of both IMF and solar magnetic fields has significantly improved by now. We have also carried out a correlation analysis of our results in order to quantify the agreement. The correlation coefficients between the observed IMF and magnetic fields at 1 AU predicted by our model were calculated. The modeled CR-averaged magnetic fields correlate with the IMF at Earth’s orbit with a coefficient of 0.691. The correlation between seven CR-averaged calculated magnetic fields and seven CR-averaged \(B_{\mathrm{x}}\) components of the IMF reaches 0.81. For Cycles 21 to 24, the modeled CR-averaged magnetic field correlates with the CR-averaged \(B_{\mathrm{x}}\) component of the IMF at Earth’s orbit with coefficients of 0.42, 0,70, 0.74, and 0.75 and that for seven CR-averaged values with coefficients 0.52, 0,88, 0.85, and 0.89. Again, this implies a very good correlation. Our model, Equation 4, is shown to be a good representation of the measured radial, \(B_{\mathrm{x}}\), component of the IMF in the ecliptic plane.

Figures 3 and 4 show some examples of the modeled magnetic field evolution from Cycles 21 to 24 calculated at distances of 3 \(R_{\mathrm{s}}\), 5 \(R_{\mathrm{s}}\), 6 \(R_{\mathrm{s}}\), 7 \(R_{\mathrm{s}}\), 10 \(R_{\mathrm{s}}\), and 12 \(R_{\mathrm{s}}\) (Figure 3), and 75 \(R_{\mathrm{s}}\), 80 \(R_{\mathrm{s}}\), 85 \(R_{\mathrm{s}}\), 90 \(R_{\mathrm{s}}\), 95 \(R_{\mathrm{s}}\), and 100 \(R_{\mathrm{s}}\) (Figure 4). In Figures 3 and 4 we see that the magnetic field declines from \({\approx}\,20~\upmu \mbox{T}\) to ≈ 20 nT in the distance range from 3 \(R_{\mathrm{s}}\) to 100 \(R_{\mathrm{s}}\). The magnetic field cycle variations derived for different distances and illustrated in Figures 3 and 4 should be regarded as a good approximation of the cycle behavior of the radial magnetic field profiles in the heliosphere. Thus, this method for calculating the magnetic field is a useful tool for describing the evolution of the solar corona and interplanetary large-scale magnetic field cycle at different distances from the Sun.

Figure 3
figure 3

Model calculated magnetic fields at 3 \(R_{\mathrm{s}}\), 5 \(R_{\mathrm{s}}\), 6 \(R_{\mathrm{s}}\), 7 \(R_{\mathrm{s}}\), 10 \(R_{\mathrm{s}}\), and 12 \(R_{\mathrm{s}}\). Red points denote the directly measured IMF (from Table 1) close to these distances. The labels of the points correspond to their number in Table 1. The maxima and minima of Cycles 21 to 24 are marked at the top.

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Figure 4
figure 4

Modeled magnetic fields at 75 \(R_{\mathrm{s}}\), 80 \(R_{\mathrm{s}}\), 85 \(R_{\mathrm{s}}\), 90 \(R_{\mathrm{s}}\), 95 \(R_{\mathrm{s}}\), and 100 \(R_{\mathrm{s}}\). Red points denote the IMF measured by the Helios 1 and Helios 2 spacecraft from Table 2. The labels of the points correspond to their numbers in Table 2. The maxima and minima from Cycles 21 to 24 are marked at the top.

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With our model, we can also derive the radial magnetic field strength profiles for different solar activity phases and for different distances from the Sun. Figure 5 shows the magnetic field radial profiles from 1 \(R_{\mathrm{s}}\) to 1 AU. The profiles were derived using Equation 4 as a function of distance for different phases of Cycles 21 to 24. The magnetic field profiles were calculated for different corona activity states: i) blue lines denote the solar activity minimum corona, ii) red lines correspond to the solar activity maximum corona, iii) light blue lines denote the solar corona at the rising phases, and iv) green lines denote the solar corona at the declining phases. In each panel, some CRs are shown for different phases of solar activity.

Figure 5
figure 5

Magnetic field distributions calculated using our model (Equation 4) for different activity states of the corona. In each panel, some CRs are shown for different phases of solar activity. Blue curves denote the solar activity minimum corona, red curves correspond to the solar activity maximum corona, light blue curves denote solar corona at the rising phases, and green curves denote the solar corona at the declining phases.

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4 Coronal Magnetic Field Measurements in Cycles 21 to 24

Table 1 summarizes the observed magnetic fields (column 6) determined by different authors at different distances (column 4) from the Sun and those calculated using our model, Equation 4, (column 5) at the same distances and the same time (columns 2 and 3). The magnetic field measurements were taken from the articles cited in column 7.

Table 1 Observed magnetic fields and fields calculated using our model at the same distances from the Sun and the same time. \(R\) is the distance from the Sun.
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Table 1 shows that the derived magnetic field values are consistent with other estimates in a similar distance range. In Figure 3 the calculated magnetic fields at 3 \(R_{ \mathrm{s}}\), 5 \(R_{\mathrm{s}}\), 6 \(R_{\mathrm{s}}\), 7 \(R_{ \mathrm{s}}\), 10 \(R_{\mathrm{s}}\), and 12 \(R_{\mathrm{s}}\) are shown and compared to those measured. Red points denote the directly measured IMF (from Table 1) that is close to these distances. We conclude from this that our model adequately describes the observed magnetic fields. The comparison of the calculated magnetic field with the observations is complicated by the fact that many of the observations were made using different methods and using different solar corona plasma density models. Moreover, large-scale and small-scale irregularities, as well as flows of dense and rarefied plasma, move with different speeds in the solar corona and interplanetary space, creating regions of high and sparse density at different distances from the Sun. Even a cursory examination of the measured magnetic fields (Table 1, column 6) and cataloged in the literature (Table 1, column 7) shows a wide disparity between the determinations of various authors. In spite of the large scatter in the observations, we conclude that they are adequately reproduced by our model. Such a difference can also result from either calibration differences or the influence of different solar activity events. Observed and modeled magnetic fields differ also because it is hard to distinguish the enhanced magnetic fields, for example in a CME or a streamer region, from the background magnetic field. Our model determines the background radial distribution of the magnetic field strength and its cycle evolution. The difference in the measured magnetic fields can also result from the dependence of the magnetic field on the selected place of the observed events. Hariharan et al. (2014) estimated the magnetic field strength in the solar corona ahead of and behind the MHD shock front associated with a CME at a distance of ≈ 2 \(R_{\mathrm{s}}\). They found magnetic fields of \({\approx} \,(0.7\,\mbox{--}\,1.4) \pm 0.2~\mbox{G}\) and \({\approx}\,(1.4\,\mbox{--}\,2.8) \pm 0.1~\mbox{G}\), respectively. Bemporad and Mancuso (2010) determined the plasma parameters of a fast CME-driven shock associated with the solar eruption of 22 March 2002. According to their study, the magnetic field undergoes a compression from a pre-shock value of ≈ 0.02 G up to a post-shock magnetic field of ≈ 0.04 G.

The Helios mission provides very good magnetic field observations. The observations performed by the Helios 1 and Helios 2 spacecraft allow monitoring the IMF conditions in the inner heliosphere from 0.3 to 0.6 AU. We also compare the calculated magnetic field strength (Equation 4) with in situ measurements made by the Helios 1 and Helios 2 spacecraft (Villante, Mariani, and Cirone, 1982).

In Table 2 we list the hourly averages of the observed unsigned IMF radial component measured by the Helios 1 (H1) and Helios 2 (H2) (column 2) spacecraft (column 8, from Villante, Mariani, and Cirone, 1982). Magnetic fields in column 7 were calculated using our model (Equation 4) at the same distances (columns 5, 6) and time (columns 3, 4) as in Villante, Mariani, and Cirone (1982). The calculated magnetic fields at 75 \(R_{\mathrm{s}}\), 80 \(R_{\mathrm{s}}\), 85 \(R_{\mathrm{s}}\), 90 \(R_{\mathrm{s}}\), 95 \(R_{\mathrm{s}}\), and 100 \(R_{\mathrm{s}}\) are shown in Figure 4. Red points denote the IMF measured by the Helios 1 and Helios 2 spacecraft from Table 2. The agreement between our model predictions and those derived from direct in situ measurements is quite satisfactory. From a comparison of Figures 2, 3, 4, and 5 and Tables 1 and 2 of the different magnetic field values as observed and calculated from our model (Equation 4), we conclude that our model adequately explains the main features of the magnetic field at different distances from the Sun during different cycle phases. Therefore, the comparison of the observed magnetic fields with those predicted by our model shows that the magnetic field in the heliosphere is determined by the cycle variations of the sum of the poloidal and toroidal components of the solar global magnetic field.

Table 2 Magnetic fields observed with Helios 1 (H1) and Helios 2 (H2), and those calculated using our model at the same distances from the Sun and the same time.
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5 Coronal Magnetic Field Models

Magnetic field measurements are important for testing coronal magnetic field models. Magnetic field models have played a very important role in the interpretation of different solar activity phenomena and in the study of outward-flowing coronal material into interplanetary space. In this section, we first summarize various models for deriving the magnetic field radial distribution. The models discussed below are presented in the form of functions in Figure 6. When we assume conservation of the magnetic flux in interplanetary space, the magnetic field can be continued to arbitrary radial distances, \(r\), from the Sun (Parker, 1958) by

$$ B(r) = B_{\mathrm{s}} \times \biggl( \frac{R_{\mathrm{s}}}{r} \biggr) ^{2}, $$
(7)

where \(B_{\mathrm{s}}\) is the magnetic field at the photosphere, obtained using some measurements and assumptions (e.g. Mann et al., 1999), or is composed of the field of an active region and that of the quiet Sun (Warmuth and Mann, 2005), \(R_{\mathrm{s}}\) is the radius of the Sun, and \(r\) is the distance from the center of the Sun in units of the solar radius.

Figure 6
figure 6

Magnetic field distributions from 1 \(R_{\mathrm{s}}\) to 1 AU calculated for different models (thin black profiles). Thick black lines denote the part of the profiles marking the distances for which the models were developed by their authors. Red profiles denote the distribution of magnetic fields using our model for the maximum of Cycle 21 (CR 1712). Blue profiles denote the distribution of magnetic fields using our model for the minimum of Cycle 24 (CR 2079).

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The radial profile of the magnetic field can also be represented by an empirical formula of the form

$$ B(r) = K \times r^{-\alpha }, $$
(8)

where \(K\) and \(\alpha\) are coefficients derived from some approximation of different measurements or models, and \(r\) is the distance from the center of the Sun in units of the solar radius. A significant number of models were created based on Equation 8 (e.g. Dulk and McLean, 1978; Pätzold et al., 1987; Mancuso et al., 2003; Gopalswamy and Yashiro, 2011). Many models have used Equation 8 formally, as a mathematical approximation only. The magnetic field values are not used in these equations.

Traditionally, the problem of a radial magnetic field distribution has been solved by combining the data obtained using different observations of magnetic fields measured at different times with different methods and at different distances from the Sun with instrumentation onboard different spacecraft. Then, the observations are summarized in one curve and fitted by a function like Equations 7 and 8. Such models only depend on \(r\). They do not reflect the actual distribution and cycle evolution of magnetic fields.

Figure 6 summarizes the radial magnetic field distributions calculated using different models. The profiles were computed for radial distances from 1 \(R_{\mathrm{s}}\) to 1 AU (thin black profiles). Thick black lines denote the part of the profiles that marks the distances for which the models were developed by their authors. For comparison, the radial magnetic field profiles calculated using our model are also shown in Figure 6. The magnetic field radial profiles corresponding to the maximum and minimum solar cycle phases are included. Red lines denote the radial distribution of magnetic fields calculated using our model for the maximum of Cycle 21 (CR 1712). Blue lines denote the radial distribution of magnetic fields calculated using our model for the minimum of Cycle 24 (CR 2079). These modeled profiles were calculated from the observed photospheric magnetic fields for the highest and the lowest solar activity. Therefore, they determine the range of profiles that are possible at different levels of the solar activity, and they can be used for the estimation of different models.

Using measurements of the IMF from several different spacecraft, Behannon (1976) showed that the radial component of the magnetic field between 0.5 AU and 5 AU can be fitted by the function

$$ B(r) = 3.0 \times 10^{-5} \biggl( \frac{r}{216} \biggr) ^{-2}, $$
(9)

where \(B(r)\) is in G and \(r\) is the distance from the center of the Sun in units of the solar radius (Figure 6a).

Musmann, Neubauer, and Lammers (1977) derived a similar expression for the radial variation of the IMF between 0.3 AU and 1.0 AU from Helios 1 data during solar minimum

$$ B(r) = 1.18 \times 10^{-4} r^{-2}, $$
(10)

where \(B(r)\) is in T, and \(r\) is in units of the solar radius (Figure 6b).

One of the most frequently used models is the empirical formula proposed by Dulk and McLean (1978). These authors concentrated their attention on the magnetic field above active regions. Using different techniques and different observational data, they proposed an empirical single-parameter formula to calculate the radial profile of the magnetic field from 1.02 to 10 \(r/R_{\mathrm{s}}\)

$$ B(r) = 0.5 \times \biggl( \frac{r}{R_{\mathrm{s}}} - 1 \biggr) ^{-1.5}, $$
(11)

where \(B(r)\) is in G, \(R_{\mathrm{s}}\) is the solar radius, and \(r\) is in units of the solar radius (Figure 6c). They pointed out that the magnetic field in the corona can vary from one active region to another by an order of magnitude and that Equation 11 is consistent with the different data used within a factor of about three.

Pätzold et al. (1987) derived the mean coronal magnetic field from Faraday rotation measurements for \(3 \le r \le 10\) \(R_{\mathrm{s}}\) during solar minimum in 1975 – 1976 (Figure 6d, Equation 12),

$$ B(r) = 7.9 \times 10^{-4} r^{-2.7}. $$
(12)

The magnetic field profile derived using a fit to the Faraday rotation data with a dipole term and an interplanetary term is given by (Figure 6d, Equation 13)

$$ B(r) = \bigl(6r^{-3} + 1.18r^{-2}\bigr)\times 10^{-4}, $$
(13)

where \(B(r)\) is in T, and \(r\) is in units of the solar radius. Clearly, our model profile is closer to that of Pätzold et al. (1987) in Equation 13 than to that of Pätzold et al. (1987) in Equation 12. The profile of Pätzold et al. (1987) in Equation 12 agrees well with our model for solar maximum at shorter distances, but it is lower than our model for solar minima.

Mancuso and Spangler (2000) and Spangler (2005) proposed a method for deriving the strength and spatial structure of the solar coronal magnetic field using observations of the Faraday rotation of radio sources (radio galaxy) occulted by the solar corona at heliocentric distances of 6 – 10 \(R_{\mathrm{s}}\) (Figure 6f)

$$ B(r) = 0.06 \biggl( \frac{r}{R_{\mathrm{1}}} \biggr) ^{-3} + 3.1 \biggl( \frac{r}{R_{\mathrm{1}}} \biggr) ^{-2}, $$
(14)

where \(B(r)\) is in nanoTesla, \(R_{\mathrm{1}}\) is 1 AU, and \(r\) is the distance from the Sun in units of the solar radius. In this model, the field changes polarity at the coronal neutral line (Spangler, 2005).

Several models of the magnetic field radial distribution were based on CME observations. Gopalswamy and Yashiro (2011) determined the coronal magnetic field strength using white-light coronagraph measurements of the shock standoff distance and the radius of curvature of the flux rope during the 25 March 2008 CME. They showed that the radial profile of the magnetic field can be represented by a power law of the form

$$ B(r) = p \times r^{-q}, $$
(15)

where \(B(r)\) is in G, \(r\) in units of \(R_{\mathrm{s}}\), and \(p\) and \(q\) are the coefficients that depend on the plasma density model. Using the Saito (1977) density model with \(\gamma =4/3\) and for distances \({>}\,9~R_{\mathrm{s}}\), they obtained \(p = 0.377\) and \(q = 1.25\) (Figure 6g). Using the Leblanc, Dulk, and Bougeret (1998) density radial-distribution model, they obtained \(p = 0.409\) and \(q = 1.3\) (Figure 6h). The Gopalswamy and Yashiro (2011) model extrapolation results in a slightly flatter magnetic field profile than that from our model. The modeled profiles reported by Gopalswamy and Yashiro (2011) agree with our model profiles at shorter distances, but the difference increases with distance. They determined the coronal magnetic field strength at a heliocentric range of 6 – 23 solar radii.

Following the standoff-distance method, Poomvises et al. (2012) derived the radial magnetic field strength in the heliocentric distance range from 6 to 120 \(R_{\mathrm{s}}\) using data from the Coronagraph 2 and Heliospheric Imager I instruments on board the STEREO spacecraft. They found that the radial magnetic field strength decreases from 28 mG at 6 \(R_{\mathrm{s}}\) to 0.17 mG at 120 \(R_{\mathrm{s}}\). They derived magnetic field profiles in the form of Equation 15. Using the Saito (1977) density model and \(\gamma =4/3\), they obtained \(p = 845.870\) and \(q = 1.59\). Using the Leblanc, Dulk, and Bougeret (1998) density radial distribution model, they obtained \(p = 706.383\) and \(q = 1.54\). The coefficients are close in value, so that the resulting profiles coincide (Figure 6i).

Several models of the magnetic field radial distribution are based on solar radio emission analysis. Mann et al. (1999) have shown that coronal transient (or EIT) waves discovered by the Extreme Ultraviolet Imaging Telescope (EIT) on board the SOHO and coronal shock waves associated with type II radio bursts can be used to determine the magnetic field in the solar corona. They proposed a model of the radial magnetic field distribution for quiet regions at solar minimum

$$ B(r) = 2.2 \times \biggl( \frac{R_{\mathrm{s}}}{r} \biggr) ^{2}, $$
(16)

where \(B(r)\) is in G and \(r\) is in units of \(R_{\mathrm{s}}\) (Figure 6e). They determined the magnetic field strength between 1.1 – 2.1 solar radii. The model profile matches our estimates for the coronal magnetic field at solar maximum well.

Mancuso et al. (2003) studied the coronal plasma by analyzing type II radio bursts and SOHO/UVCS observations. The data sample comprises 37 metric type II radio bursts observed by ground-based radio spectrographs in 1999, during the rising phase of Cycle 23. The shock speeds were used to set upper limits to the magnetic field above active regions. An average functional form of the magnetic field estimates can be represented by the following radial profile, valid between about 1.5 and 2.3 \(R_{\mathrm{s}}\),

$$ B(r) = (0.6\pm 0.3) \times (r-1)^{-1.2}, $$
(17)

where \(B(r)\) is in G and \(r\) is in units of \(R_{\mathrm{s}}\) (Figure 6j).

The coronal magnetic field can be inferred from band splitting of type II radio bursts (Vrs̆nak et al., 2002). Mancuso and Garzelli (2013a) have also analyzed the band splitting of type II radio bursts to determine the coronal magnetic field strength in the heliocentric distance range ≈ 1.8 – 2.9 \(R_{\mathrm{s}}\). The same radial profile was obtained higher up in the corona by Mancuso and Garzelli (2013b) based on Faraday rotation measurements of extragalactic radio sources occulted by the solar corona as

$$ B(r) = 3.76 \times r^{-2.29}, $$
(18)

where \(B(r)\) is in G and \(r\) is in units of \(R_{\mathrm{s}}\) (Figure 6j). The profiles derived describe the magnetic field in a range of heliocentric distances from 1.8 \(R_{\mathrm{s}}\) to 14 \(R_{\mathrm{s}}\). Clearly, our model profile is more in line with the results of Mancuso and Garzelli (2013a) than with those of Mancuso et al. (2003).

The comparison of models developed by Behannon (1976), Musmann, Neubauer, and Lammers (1977), Pätzold et al. (1987) (Equation 13), Spangler (2005), and Mancuso and Garzelli (2013a) (Figure 6a, b, d (Equation 13), f, j) shows that the radial profiles derived from these models and those predicted by our model for the maximum and minimum corona states agree very well. The agreement is quite satisfactory, although it is impossible, based on the profiles, to prefer one model above the other.

Schmidt et al. (2016) derived the profile of the strength of the magnetic field in front of a CME-driving shock based on white-light images and the standoff-distance method. They also simulate the CME and its driven shock with a 3D MHD code. They found good agreement between the two profiles (within ± 30%) between 1.8 and 10 \(R_{ \mathrm{s}}\). The authors noticed that in their model the magnetic field profile is decreasing stronger than a monopolar (wind-like) magnetic field profile \({\approx}\, r^{-2}\) and a dipolar profile \({\approx}\, r^{-3}\) for an ideal spherically symmetric system. The magnetic field strength profile derived can be represented as

$$ \lg \bigl(B_{\mathrm{calc}}(r)\bigr) = (-3.32 \pm 0.5) \times \lg (r) + (3.72 \pm 0.5) $$
(19)

and that simulated with the 3D MHD code

$$ \lg \bigl(B_{\mathrm{sim}}(r)\bigr) = (-2.47 \pm 0.5) \times \lg (r) + (3.19 \pm 0.5), $$
(20)

where \(B(r)\) is in mG and \(r\) is in units of \(R_{\mathrm{s}}\) (Figure 6k). The profiles are similar to our model minimum profile at shorter distances, but the difference is growing rapidly with increasing distance.

Using radio and white-light observations, Kumari et al. (2017) have shown that a single power-law fit is sufficient to describe magnetic fields in the heliocentric distance range ≈ 2.5 – 4.5 \(R_{\mathrm{s}}\),

$$ B(r) = 6.7 \times r^{-2.6}, $$
(21)

where \(B(r)\) is in G and \(r\) is in units of \(R_{\mathrm{s}}\) (Figure 6l). The resulting profile matches our model profile for the magnetic field maximum at shorter distances and is close to our model profile for the magnetic field minimum at large distances.

Figure 6 shows that magnetic field profiles derived using different models are different. The difference in the magnetic field variation at 1 AU is one order of magnitude. The main difference between our model and the models described above is that they give only one value for one point at a certain distance from the Sun. They do not take into consideration the solar cycle magnetic field evolution. Our model gives more realistic magnetic field radial distributions that take into account the cycle variations of the solar toroidal and poloidal magnetic fields.

6 Discussion

In this article, we presented a new model for the radial \(B_{\mathrm{x}}\) component of the IMF at various distances from the Sun during different solar cycle phases. The results show a rather good match between the measured \(B_{\mathrm{x}}\) component of the IMF and the model predictions. The magnetic fields from the WSO were used in the study. However, it is known that the magnetic field measurements at different observatories are different. To compare data from different observatories, synoptic maps from the WSO, the Mount Wilson Observatory (MWO), Kitt Peak (KP), SOLIS, the SOHO/Michelson Doppler Imager (MDI), and the Solar Dynamics Observatory (SDO)/Helioseismic and Magnetic Imager (HMI) measurements of the photospheric field were analyzed by Riley et al. (2014), Virtanen and Mursula (2016, 2017). The comparison has shown that while there is a general qualitative agreement in the measured data, there are also some significant differences (Riley et al., 2014). The observatories give a similar overall view of the solar magnetic field and the heliospheric current sheet evolution over the four last cycles. However, there are some periods when the data disagree (Virtanen and Mursula, 2016). The differences between the datasets can be due to instrument problems, the choice of the spectral lines that may be formed at different heights and may not measure the same magnetic field, and the measurement and treatment of polar fields. Furthermore for ground-based instruments, atmospheric turbulence can significantly degrade the image quality. Another problem can be caused by different algorithms used to create synoptic maps (Riley et al., 2014). This shows that the assumption that high-resolution maps from one observatory can be transformed into a lower-resolution map of another by simple averaging is strictly not true. Scaling factors are needed when comparing synoptic maps from different observatories. Therefore, conversion factors were computed by Riley et al. (2014) that relate measurements of different observatories using both synoptic map pixel-by-pixel and histogram-equating techniques. Virtanen and Mursula (2017) also proposed a method for scaling the photospheric magnetic field measurements based on a harmonic expansion. The benefit of the harmonic scaling method is that it can be used for datasets with different resolutions.

The WSO provides the longest and most homogeneous magnetic field observations; many investigations are based on WSO data. However, WSO provides the lowest values for the photospheric magnetic field data, i.e. a weaker coronal magnetic field strength than MWO, KP, SOLIS, SOHO/MDI, and SDO/HMI (Riley et al., 2014). The size of a WSO pixel is 3 arcmin in sky coordinates, or 180 arcsec, which at disk center represents about 126 Mm, or about 10 deg of heliographic longitude. Such large pixels cannot resolve active regions, therefore most active region flux is not detected by WSO. This shows that the very low spatial resolution of WSO might explain part of the good match, because the origin of the IMF is the large-scale photospheric magnetic field (Ness and Wilcox, 1964, 1965, 1966; Severny et al., 1970). The IMF evolves in response to the solar coronal and photospheric magnetic field at its base. The source of the interplanetary sector structure is associated with large-scale photospheric magnetic field patterns, i.e. the patterns of weak photospheric background fields (Wilcox, 1968, Wilcox and Ness, 1967) with an area equal to about one-fourth of the area of the solar disk (Scherrer and Wilcox, 1972); furthermore, according to Plyusnina (1985), the size of the unipolar photospheric magnetic field pattern is on average larger than \(40^{\circ }\). Therefore, such a good coincidence between the measured magnetic field and that predicted by our model largely arises because of the low-resolution measurements of the WSO. The total flux of the active region is unresolved. The magnetic field measured at the WSO is the large-scale field that forms and governs the IMF.

The photospheric magnetic field and the IMF are measured from Earth and at Earth’s orbit. Only the flux from different latitudes that is detected at Earth is considered. This does not imply that approximately the same proportion of low- and high-latitude fields contribute to the IMF \(B_{\mathrm{x}}\) component. We do not see the real radial polar magnetic field from Earth. As the value of the polar magnetic field observed from Earth is used, not the full magnetic flux of the polar regions is really measured, but only the tangential component of the polar magnetic field of the visible polar area. The contribution of polar regions is far less than that of low-latitude fields, especially during maximum phases, when the polar region influence is negligible, despite the fact that the polar flux is much more unipolar than the low-latitude flux over most of the cycle. The role of polar fields increases during minimum phases. The central part of the visible solar disk contributes with a great amount of magnetic flux to the magnetic field measured at Earth. Scherrer et al. (1977) have shown that about half the contribution to the mean (Sun-as-a-star) field comes from the central 35% of the disk area.

The influence of active region magnetic fields on the IMF is insignificant beyond the source surface. It was found that the IMF originates in the large-scale photospheric magnetic fields. The source of the interplanetary sector structure is associated with the pattern of weak photospheric background fields (Wilcox, 1968). The background solar magnetic field is represented predominantly by a radial field (as seen from vector magnetograms and EUV images). We stress that it does not show significant variations with latitude. The independence of latitude was also observed by Ulysses in the radial component of the IMF (Forsyth et al., 1996; Smith and Balogh, 1995; Balogh et al., 1995). As described above, however, both the photospheric magnetic field and the IMF are measured from Earth and Earth’s orbit. Hence, the contribution of the unipolar regions to the total magnetic flux that reaches Earth also mainly comes from the central solar region.

Finally, we note that the photospheric magnetic field must be accurately determined as it is used to calculate the coronal magnetic field and the IMF. The magnetic field measurements from other observatories can therefore be used with an appropriate scaling factor to calculate the IMF at various distances from the Sun during different cycle phases.

7 Conclusion

A new model has been proposed for the determination of the magnetic field at different distances from the Sun throughout solar cycles. The model depends on the observed large-scale photospheric magnetic fields. Direct observations of the large-scale non-polar photospheric (\({\pm}\, 55^{\circ }\)) and polar (from \(55^{\circ }\) N to \(90^{\circ }\) N and from \(55^{\circ }\) S to \(90^{\circ }\) S) magnetic fields were used, which are the visible manifestations of cyclic changes in the toroidal and poloidal components of the solar global magnetic field. The model magnetic field is determined as the sum of the non-polar photospheric magnetic field (toroidal component of the solar global magnetic field) and the polar magnetic field (poloidal component) cycle variations.

The agreement between our model predictions and magnetic fields derived from direct in situ measurements at different distances from the Sun and at different times is quite satisfactory. From a comparison of the magnetic fields as observed and as calculated from our model at 1 AU, we also conclude that the modeled variation in magnetic field strength adequately explains the main features of the IMF \(B_{\mathrm{x}}\) component cycle evolution at Earth’s orbit. The CR-averaged calculated magnetic fields correlate with the CR-averaged IMF \(B_{\mathrm{x}}\) component at Earth’s orbit with a coefficient of 0.691. The correlation between seven CR-averaged calculated magnetic fields with the IMF \(B_{\mathrm{x}}\) component reaches 0.81. For Cycles 21 to 24, the model CR-averaged calculated magnetic fields correlate with a CR-averaged IMF \(B_{\mathrm{x}}\) component at Earth’s orbit with coefficients of 0.42, 0,70, 0.74, and 0.75, and the fields for seven CR-averaged values with coefficients of 0.52, 0,88, 0.85, and 0.89.

The variations in the magnetic field with the cycle that we derived for different distances should be regarded as a good approximation of the radial cycle behavior of the magnetic fields in the heliosphere. Thus, this method of magnetic field calculation is a useful tool for describing the solar corona and interplanetary large-scale magnetic field cycle evolution at different distances from the Sun. The model should be regarded as a good approximation of the cycle behavior of the magnetic field in the heliosphere.

The magnetic field profiles derived from our model are similar to those of empirical models and previous estimates. The main difference between our model and the models described in Section 5 is that they give only one value for one point at a certain distance from the Sun. They do not take into consideration the solar cycle evolution. Our model gives more realistic magnetic field radial distributions that take into account the cycle variations of the solar toroidal and poloidal magnetic fields.

A particularly interesting finding is the decrease in the maximum of the photospheric magnetic field magnitude from Cycle 21 to Cycle 24. We note that the minimum values of the magnetic field in the solar activity minima have also decreased from Cycle 21 to Cycle 24. The polar magnetic field also decreased. Such changes in magnetic fields reflect the time evolution of the toroidal and poloidal components of the solar global magnetic field. This means that both components, and therefore, the solar global magnetic field, decreased from Cycle 21 to Cycle 24.