7 Open Solar Flux, PFSS, and the Ulysses Result

7.1 Open solar flux

The open solar flux, FS, is the magnetic flux leaving the top of the solar atmosphere and entering the heliosphere. A notional surface, the “coronal source surface” is envisaged at the top of the corona, which is everywhere perpendicular to the field, and FS is the flux threading that surface and so is also called the “coronal source flux”. This review (unless otherwise stated) considers the “signed” open flux which means that only the flux of one polarity (inward or outward) is quantified. If we assume that Maxwell’s equations hold such that there are not significant numbers of magnetic monopoles inside the coronal source surface, then the inward and outward fluxes are equal and the “unsigned” open flux is simply twice the signed value. The source surface is usually taken to be a heliocentric sphere of radius 2.5 R ⊙, where R ⊙ is a mean solar radius (Arge and Pizzo, 2000).

7.2 The potential field source surface (PFSS) method

Prior to the Ulysses mission, FS could only be evaluated from solar magnetograms using “potential field source surface” (PFSS) modelling (Schatten et al., 1969; Altschuler and Newkirk, 1969; Schatten, 1999). In this method, photospheric magnetic fields observed by a magnetograph are mapped through the solar corona to the source surface with a number of assumptions. This involves solving Laplace’s equation within an annular volume above the photosphere in terms of a spherical harmonic expansion, the coefficients of which are derived from Carrington maps of the photospheric magnetic field (i.e., maps assembled over an entire solar rotation from magnetograms recorded by magnetographs on Earth’s surface or on board a spacecraft in orbit around the L1 Lagrange point). Two major assumptions are that there are no temporal variations within the 27 days taken to build up the map and that there are no currents in the corona (these are neglected so as to allow unique solutions in closed form). To eliminate the possibility that such simple harmonic expansions would result in all of the magnetic field lines returning to the Sun within a small heliospheric distance, the coronal field was required to become radial at the outer boundary, the source surface. Despite its many assumptions and obvious limitations, PFSS has been very successful in the study of a wide range of solar and heliospheric phenomena, including: coronal structure as seen during eclipses (e.g., Smith and Schatten, 1970), end-to-end modelling of Earth-impacting coronal mass ejections (CMEs, e.g., Luhmann et al., 2004), coronal null points and CME release (e.g., Cook et al., 2009), interplanetary magnetic fields (e.g., Burlaga et al., 1978), heliospheric current sheet structure (e.g., Hoeksema et al., 1982), waves in the corona (e.g., Uchida et al., 1973), solar wind acceleration (e.g., Neugebauer et al., 1998; Marsch, 1999), stellar coronal fields (e.g., Jardine et al., 2002), coronal hole and fast solar wind stream evolution (e.g., Wang and Sheeley Jr, 1990Jump To The Next Citation Point), co-rotating interaction regions and associated cosmic ray modulation (e.g., Rouillard et al., 2007Jump To The Next Citation Point), solar wind speed prediction (e.g., Arge et al., 2002), solar wind density structure (e.g., Rouillard et al., 2010), pseudostreamers (e.g., Owens et al., 2013Jump To The Next Citation Point), and quantifying the open solar flux (discussed below). The method has also generated results that compare well with images that reveal field line structure in the corona and with the results of MHD modelling (see the Living Review by Mackay and Yeates, 2012).

7.3 Ulysses observations

The Ulysses spacecraft is the first to carry out a comprehensive survey the magnetic field in heliosphere outside the ecliptic plane: its orbit covers a wide range of heliospheric distances, r and an almost full range of heliographic latitudes, Λ. This mission has generated a vitally important result in that it found that the average radial field of the heliospheric field was independent of Λ. This result was first found to apply as the satellite passed from the ecliptic plane to over the southern solar pole (Smith and Balogh, 1995Jump To The Next Citation Point; Balogh et al., 1995). Subsequently, the result was confirmed by the pole-to-pole “fast latitude scans” during the perihelion passes of the spacecraft (Smith and Balogh, 1995; Smith et al., 2001; Smith and Balogh, 2003; Smith et al., 2003). As pointed out by Smith (2011Jump To The Next Citation Point, 2013Jump To The Next Citation Point), this “Ulysses result” was initially derived by averaging the radial field over the inferred Toward and Away sectors of the heliospheric field; however, Lockwood et al. (1999b), Lockwood (2004Jump To The Next Citation Point), Lockwood et al. (2004Jump To The Next Citation Point), and Lockwood et al. (2009bJump To The Next Citation Point) have shown that the invariance with Λ was also found if the modulus of the radial field was employed over fixed averaging intervals. The differences between, and complementarity of, these two approaches are discussed in Section 7.5.

Figure 17View Image illustrates the Ulysses result by showing data from the third perihelion pass of Ulysses and compares them to simultaneous data from the ACE spacecraft. The top panel clearly defines the two large polar coronal holes seen by Ulysses (where VSW is large), separated by a single streamer belt around perihelion (minimum r, see the black line in the bottom panel) where VSW is lower. This latitudinal structure is characteristic of the sunspot minimum conditions prevailing during this pass and Ulysses passed from 80° in the south to 80° in the north (black line, panel f). Note that perihelion (minimum rU, panel g) is within the streamer belt at rU ≈ 1.4 AU > r1 = 1 AU, so when both are in the streamer belt, Ulysses is at greater r than ACE by about 0.4 AU. The panel b emphasises the variability of VSW is greater within the streamer belt. Panel c shows the dominant field orientation changes from away to toward as the streamer belt is crossed and panel d shows the modulus of the radial field seen by the two craft. Note that in this plot, the influence of the timeconstant T (on which the modulus is taken) has been eliminated by using a range of T down to 1 second and fitting the results so the asymptotic value at T → 0 can be calculated (see Figure 2 of Lockwood et al., 2009bJump To The Next Citation Point): this means that there is no cancellation of toward and away field in these values. The data have been normalised to r = r1 using an r-squared dependence (i.e., the Ulysses values have been multiplied by (rU∕r1)2). It can be seen from Figure 17View Image(d) that Ulysses observed almost the same range-normalised radial field as seen simultaneously by the ACE craft: there is a weak downward trend in both, owing to the pass taking place while the solar cycle was still declining slightly.

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Figure 17: A comparison of Ulysses and ACE data during the third perihelion pass of Ulysses, which took place between day 36 of 2007 and day 13 of 2008. In all panels, black lines are data from Ulysses while green lines are from ACE, which is in a halo orbit around the L1 Lagrange point. From top to bottom: (a) daily means of the solar wind speed, VSW; (b) The difference between successive daily VSW values, ΔV; (c) The 27-day running means of the away and toward (red and blue lines, respectively) radial field components Brp observed at Ulysses and averaged separately; (d) 27-day running means of the normalized absolute radial field magnitude at Ulysses, 2 (rU∕r1) ⟨|BrU|T⟩27d (black line), and ACE, ⟨|BrA |T⟩27d (green line); (e) The flux excess at Ulysses, defined as 2 2 ΔF = 2π × [rU ⟨|BrU |T ⟩ − r1⟨|BrA |T ⟩27d] (grey shaded area); (f) The heliographic latitude of Ulysses (ΛU, black line) and ACE (ΛA, green line); (g) The heliocentric distance of Ulysses (rU, black line) and ACE (rA = r1, green line). In all cases the radial field values are the asymptotic values for the averaging timescale T approaching zero. Image reproduced by permission from Lockwood et al. (2009bJump To The Next Citation Point), copyright by AGU.

The first and third Ulysses perihelion passes were at sunspot minimum, but the second was close to sunspot maximum. That the Ulysses result applied in all three (Lockwood et al., 2004, 2009bJump To The Next Citation Point) is an important demonstration of its generality. It is explained by the low plasma beta of the solar wind just after leaving the coronal source surface, i.e., the total magnetoplasma pressure is dominated by the magnetic pressure. This results in slightly non-radial flow close to the Sun, which smoothes out differences in the tangential pressure and, as this is dominated by 2 |Br |∕(2μo ), this renders the magnitude of the radial magnetic field, |Br | constant in latitude (Suess and Smith, 1996; Suess et al., 1996, 1998). It means that the signed (of one radial field polarity) open solar flux threading a heliocentric sphere of radius r, F (r) can be computed using

2 F (r) = (4πr )⟨|Br(r)|T⟩27 d∕2. (6 )
Averaging over 27 days ensures that any variations with Carrington longitude are averaged out.

7.4 Excess flux

In Figure 17View Image there is a slight difference between the derived flux threading the sphere on which ACE sits and that threading the sphere on which Ulysses sits and this is shown by the grey area in panel (e). Lockwood et al. (2009bJump To The Next Citation Point,cJump To The Next Citation Point) have termed this difference the “excess flux”, defined for general r as:

ΔF = F(r) − F (r ) = 2 π × [r2⟨|B (r)| ⟩ − r2⟨|B (r )| ⟩ ] (7 ) 1 U r T 1 r 1 T 27d

Note that in Figure 17View Image, the timescale on which the modulus is taken is T → 0. From comparison of Figures 17View Image(e) and 17View Image(a), it can be seen that ΔF is larger in the streamer belt than within the polar coronal holes.

Owens et al. (2008aJump To The Next Citation Point) surveyed Carrington-rotation means of magnetic field data recorded in the heliosphere (from 14 different spacecraft in different orbits) and compared them to the data taken simultaneously in near-Earth space. In their survey the modulus of Br was taken of averages over T = 1 h intervals (i.e., |Br |1h). The ΔF values, computed using Equation (7View Equation) showed no variation with heliographic latitude, Λ (as expected from the Ulysses result) but did increase with heliocentric distance, r, as shown in Figure 18View Image. This rise means that there are two components of the flux values computed using Equation (6View Equation): the coronal source flux, FS (which is the value of F (r) for r = 2.5R ⊙) and a component which arises in the heliosphere (and grows with r). Note that Figure 18View Image shows that this rise is present at r < r1 as well as r > r1, such that ΔF is negative at r < r1 for the definition given in Equation (7View Equation). This increase can arise from a number of physical phenomena, including Alfvén wave growth, transients such as CMEs, kinematic effects of solar wind flow structure on the frozen-in field, and the outward propagating structures, such as plasmoids and folded flux tubes, generated by near-Sun reconnection of flux. All these effects can give Toward and Away field structure in the heliosphere within a single coronal source sector (i.e., a region of constant unipolar radial field polarity at the source surface): and this heliospherically-imposed toward-and-away structure at general r has a range of characteristic spatial scales such that it will only partially be averaged out by taking the modulus of a mean over a given timescale T (Lockwood et al., 2006aJump To The Next Citation Point). Thus, ΔF varies with T and although most Alfvén waves are averaged out at relatively small T, the other phenomena are not (Lockwood and Owens, 2013Jump To The Next Citation Point). The problem is that using too large a T results in toward-and-away field that does map back to the coronal source structure also being cancelled. (Taken to its extreme, use of T = 27 days would result in F (r) ≈ 0 as toward and away sector flux cancel). There is no T at which the heliospheric effects are completely averaged out but none of the source sector structure is averaged out and so using a fixed T to eliminate this effect must be a form of compromise. Lockwood et al. (2009cJump To The Next Citation Point) devised a method to use the measurements of the tangential field and flow to map from a general r to r1 and thereby evaluate how much radial field structure has been amplified by longitudinal flow structure in the heliosphere. Such effects are known to take place, for example, prolonged intervals a near-radial heliospheric field (Jones et al., 1998) have been explained by Riley and Gosling (2007) in terms of this effect. Because this was based on the frozen-in flux theorem and simple kinematics, Lockwood et al. (2009cJump To The Next Citation Point) termed this the “kinematic correction”. It is a simple, single correction that has to account for the variety of effects discussed above; however, because for all of these phenomena the frozen-in theorem applies, it does have the potential to make allowance for them. Short period Alfvén waves, and any other small-scale Br polarity structure, on timescales shorter than 1 h was averaged out by using T = 1 h. The lines in Figure 18View Image give the distribution of the variations in ΔF predicted using the kinematic correction for each Carrington rotation from the mean variability in solar wind speed, ΔV: it can be seen that these fit the distribution of observed values very well.

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Figure 18: A comparison of ΔF values, as a function of r∕r1 (where r1 = 1AU), computed using Equation (7View Equation) for T = 1 h and averages over Carrington rotations (i.e., ⟨|Br(r)|1h⟩27 d is used for both the heliospheric and simultaneous near-Earth data) from 14 heliospheric spacecraft: Pioneer 6, Pioneer 7, Pioneer 10, Pioneer 11, Helios 1, Helios 2, Pioneer Venus Orbiter (PVO), International Sun-Earth Explorer ISEE3 (later renamed International Cometary Explorer, ICE), Ulysses, Near-Earth Asteroid Rendezvous (NEAR), Solar Terrestrial Relations Observatory (STEREO) A, STEREO B, Voyager 1, and Voyager 2. The lines show the distribution of predicted ΔF from the “kinematic correction” calculated for each sample (Lockwood et al., 2009cJump To The Next Citation Point), which are exceeded a fraction p of the time, where p is 0.1 (for black line), 0.25, 0.5, 0.75, and 0.9 (light gray line). Image reproduced by permission from Lockwood et al. (2009cJump To The Next Citation Point), copyright by AGU.

Figure 19View Image compares the raw data shown in Figure 17View Image (for T → 0 top panel) with results obtained using T = 1 day (middle panel) and using T = 1 h with the kinematic correction of Lockwood et al. (2009bJump To The Next Citation Point) (bottom panel). In all cases the thick black line is the Ulysses data, normalised to r = r1 by multiplying by (rU∕r1 )2, and the thin line is from the simultaneous ACE data in the ecliptic plane. The middle panel shows that using an averaging timescale T = 1 day before taking the modulus has reduced all the ACE data, which is expected because ACE is within the streamer belt and observed both Toward and Away field which cancels to some degree when T = 1 day is used. The Ulysses values are also reduced in the same way, particularly while it is within the streamer belt, but also to a lesser extent while the spacecraft is within the polar coronal holes: hence although good agreement is now obtained when both craft are within the streamer belt, this is not the case when Ulysses is in the polar coronal holes. The third panel shows the effect of applying the kinematic correction computed from the solar wind variability ΔV using the equations of Lockwood et al. (2009bJump To The Next Citation Point). With this correction, the Ulysses data are no longer showing enhanced values where ΔV is enhanced and good agreement is obtained between the Ulysses and ACE data throughout the pass.

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Figure 19: Application of the “kinematic correction” to the third Ulysses perihelion pass shown in Figure 17View Image. All three panels show 27-day running means of various estimates of the absolute radial field (normalized to r = r1 = 1AU) from measurements by Ulysses (thick black lines) and ACE (thin black lines). Ulysses data from the top panel are shown by the area shaded grey in all three panels to facilitate comparisons. a) the asymptotic limit of the timescale T → 0, ⟨|Br (r1)|T→0 ⟩27 d (for ACE data, thin black line) and (rU∕r1)2⟨|Br(rU)|T→0 ⟩27d (for Ulysses, thick black line); (b) the same for T = 1 day, ⟨|B (r )| ⟩ r 1 1d 27 d (thin black line) and (r ∕r )2⟨|B (r )| ⟩ U 1 r U 1d 27 d (thick black line); and (c) data for T = 1 h with the kinematic correction applied to the Ulysses data to allow for heliospheric effects between rU and r1, ⟨|Br(r1)|1h⟩27d (thin line) and (rU∕r1)2⟨|Br(rU)|1h⟩27d − ΔF ∕(2πr2U) (thick black line). Image reproduced by permission from Lockwood et al. (2009b), copyright by AGU.

As noted above, the open solar flux FS is the value of F (r) for r = 2.5R ⊙, inserting this into Equation (7View Equation) and re-arranging, we get an equation that allows us to use near-Earth IMF data to compute the open solar flux:

FS = 2πr21⟨|Br(r1)|T ⟩27d − ΔFS , (8 )
where ΔF S is the excess flux between the coronal source surface and near-Earth space (defined by Equation (7View Equation) because ΔF = ΔFS for r = 2.5R ⊙). Lockwood and Owens (2009) surveyed all the Ulysses data and showed that using T = 1 h and the kinematic correction to evaluate ΔFS reduced the overall error in FS, computed using Equation (8View Equation), to just ±2.5%.

As mentioned above, the PFSS method has also been used to compute open solar flux. In particular, Wang and Sheeley Jr (1995Jump To The Next Citation Point, 2003aJump To The Next Citation Point) and Wang et al. (2000) used the Ulysses result to compare the value derived from near-Earth in-situ observations with PFSS-derived values and obtained good agreement. (Note that these authors actually compared ⟨|Br |T⟩ rather than FS). However, two caveats to this comparison should be noted. First, the magnetogram data required re-processing using a latitude-dependent instrument saturation factor (Wang and Sheeley Jr, 1995Jump To The Next Citation Point) which has long been the subject of some debate (Svalgaard et al., 1978; Ulrich, 1992; Riley et al., 2013). Secondly, these authors used an averaging timescale of T = 1 day on the in-situ Br data before taking the modulus. This means, effectively, that they used Equation (8View Equation) with T = 1 day, for which ΔFS ≈ 0. The middle panel of Figure 19View Image shows that this works reasonably well within the streamer belt. Lockwood et al. (2006b) showed explicitly the effect of the value of T on this comparison and that T = 1 day is the optimum value on average. However, although it makes ΔFS go to zero on average, it is an approximation. Effectively, it is chosen as a compromise that averages out much of the heliospherically-generated component in ⟨|Br|T⟩, without averaging out too much of the source sector structure.

Figure 20View Image shows that the difference between the PFSS-derived values of open solar flux by Wang and Sheeley Jr (1995, 2003a) and the mean value of F (r1) (for T = 1 h) is consistent with the rise in F (r) values with r found using a wide variety of spacecraft in the heliosphere, as found by Owens et al. (2008a) (also using T = 1 h) and shown in Figure 18View Image. Hence the PFSS data also strongly support the kinematic correction. The degree to which the PFSS data and the near-Earth data are consistent (if the kinematic correction is deployed) is underlined by the variation of derived annual means shown in Figure 21View Image. The agreement is very good and better than using the T = 1 d, ΔFS = 0 approximation.

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Figure 20: Top: The excess flux ΔF from Equation (8View Equation) as a function of r ∕r1. The data points show the mean values from the spacecraft data shown in Figure 18View Image, averaged over bins in r that are 0.1 AU wide, with error bars of plus and minus one standard deviation. The lines are the same as shown in Figure 18View Image. The data point at the lowest r∕r1 is the entire PFSS data set for the coronal source surface at r = 2.5R ⊙. Bottom: The number of full Carrington rotation averages n contributing to these means. The open triangle shows the number in the OMNI-2 data set from r = r1. Note that n is shown on a logarithmic scale. Image reproduced by permission from Lockwood et al. (2009cJump To The Next Citation Point), copyright by AGU.
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Figure 21: The variation of various open solar flux estimates during the space age. The gray-shaded areas show annual means of the signed open solar flux from the OMNI-2 data derived using ΔF = 0 S and the absolute value of the radial field taken of means on timescale T: the light gray area bounded by the blue line is for T = 1 h, and successively darker gray areas are for T = 1, 2, and 3 days. The green line is the variation from solar magnetograms using the PFSS method. The red line shows the OMNI-2 values for T = 1 h, minus the correction term for kinematic effects, ΔFS . Image reproduced by permission from Lockwood et al. (2009cJump To The Next Citation Point), copyright by AGU.

7.5 The use of the modulus of the radial field

Smith (2011Jump To The Next Citation Point, 2013Jump To The Next Citation Point) argues that the use of the modulus of the radial field Br causes the excess flux and that the use of the kinematic correction is unnecessary. The first point is a somewhat semantic one but does have some validity. However, the second point does not follow from the first because it assumes that there is a problem-free and error-free alternative, which is not true. In fact the word “causes” is somewhat misleading: what taking the modulus (after averaging over an interval T) really does is cancel opposite polarity Br within the interval T. The alternative advocated by Smith (2011, 2013) is that B r be averaged over the durations of toward and away source field sectors. This is indeed, in principle, absolutely correct; however, it pre-supposes that in implementation one knows where the source sector boundaries lie. We here call this method “the variable-T method”, because it is identical to employing the modulus, except that the interval durations used T are not fixed but are varied to cover the source sector durations. It is important to stress here that it is the sector boundaries at the coronal source surface, not at the observing spacecraft, that matter: Lockwood and Owens (2013Jump To The Next Citation Point) show that if the polarity changes at the spacecraft are used to define the sectors then the result is mathematically identical to that obtained by taking the modulus. Therefore, to implement the variable-T method, decisions are required about where the source sector boundaries are and where they map to on the satellite orbit, and what is a genuine source sector rather than heliospherically-imposed Br polarity structure.

It is not adequate or acceptable to sidestep this issue by saying that source sector boundary crossings can be readily identified. For example, using electrons with energies > 2 keV to determine the connectivity between near-Earth space and the source surface, Kahler et al. (1996) found clear examples of opposite polarity field within well-defined source sectors and also found that in all cases they showed bidirectional electron streaming which they associated with CMEs on a wide range of spatial scales. For CMEs within an inferred source sector, the flux would be cancelled out by using the variable-T method, but would thread the source surface and hence CMEs are one source of potential error. The net heat flux from the electron data show regions where the heat flux is towards the Sun. These unambiguously reveal “folded flux” (in which source toward/away field is folded so it points away from/toward the Sun at greater r (e.g., Owens et al., 2013Jump To The Next Citation Point). Folded flux revealed by the suprathermal electron flows is often found in the vicinity of sector boundaries (Kahler et al., 1998; Crooker et al., 2004Jump To The Next Citation Point) along with seemingly plasmoidal structures (Foullon et al., 2011Jump To The Next Citation Point) (which may, in some cases, actually be folded flux that has latitudinal structure). In addition, Owens et al. (2013Jump To The Next Citation Point) have recently shown folded flux is also associated with pseudostreamers embedded within IMF polarity sectors. As a result of folded flux, Kahler and Lin (1995) deduced that some sector boundaries did not show local field reversals at the spacecraft and some field reversals were seen away from the true sector boundaries. Thus defining where the source sector crossing point is from the associated polarity change in Br at the spacecraft is not straightforward and would contribute to an unknown error to the derived open solar flux.

Thus, for the derivations of open solar flux from interplanetary craft to be repeatable, a full catalogue of assumed source sector crossings would be needed for the variable-T method and the error introduced into open solar flux estimates by inaccuracies in that catalogue would not be known. On the other hand, using the modulus with kinematic correction (or the easier-to-implement T = 1 day method) is a repeatable algorithm that does not depend on a catalogue of sector crossings. Another advantage of the modulus approach is that it makes the issue of heliospherically-produced radial field structure explicit and does not hide it in the choice of the intervals to average over in the variable-T method.

Lockwood and Owens (2013Jump To The Next Citation Point) argue that the variable-T and modulus approaches are actually complementary, each with its own strengths and weaknesses, and that it is neither correct nor helpful to advocate the use of one over the other: rather, it is important to fully understand the strengths, limitations and applicability of both. What is most interesting is that the two approaches generate similar answers in several important respects (for example, both give the Ulysses result of the latitudinal constancy of the radial field.) In addition, Lockwood and Owens (2013) point out that both methods average along a sector of the spacecraft orbit and so both are subject to errors (that are different in the two cases) associated with latitudinal variations in folded flux tubes.

It is here worth recording that there is another approach discussed in the literature which allows values of F S from PFSS to be matched to the values derived from in-situ spacecraft measurements. Erdős and Balogh (2012) analysed the field in a reference frame aligned to the Parker spiral direction, as computed from the frozen-in flux theorem using the observed solar wind flow speed (see Equation 5View Equation). Like the fixed-T (modulus) methods it has the advantage of being repeatable without implementing a catalogue of source sector boundary crossings. However, it should be noted that disconnected and folded flux is often well-aligned with the Parker spiral (Crooker et al., 2004; Foullon et al., 2011; Owens et al., 2013) and that the authors used an averaging interval of T = 6 h to match the open solar flux estimates derived from the satellite to those from the PFSS values, and hence there is an element of fixed-T averaging in this method also.

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