William M. Arden^{1}, Aimee A. Norton^{2}, Xudong Sun^{2}

^{1} University of Southern Queensland, Toowoomba, QLD, Australia

^{2} W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA

The potential field source surface (PFSS) model is a well-established method for calculating open solar flux at a “source surface,” an imaginary spherical surface above the corona. PFSS is used to represent the large-scale geometry of the solar coronal magnetic fields. It is customary to take the height of the source surface to be 2.5 solar radii (Rs) in a heliocentric coordinate system^{1}. Nothing in the model, however, requires that this height be fixed, either spatially or in time; the height of the source surface can be taken as a free parameter. A non-spherical source surface height has been explored in the past ^{2}, for instance.

Recent work by Lee et al.^{3} examined the prospect of varying the source surface height over the duration of the solar cycle, suggesting that varying this height during periods of solar minimum yields better agreement between PFSS models and the measured magnitude of the IMF open flux at 1 AU — in other words, the source surface “breathes” in and out over the course of the solar cycle. We have built on that work to examine the evolution of open flux during all of cycle 23 and the first part of cycle 24 using photospheric magnetic field maps from the SOHO/MDI and SDO/HMI instruments^{4}.

Figure 1: Total unsigned open flux as calculated by LMSAL (blue) and Stanford (red) models, compared to IMF open flux from OMNI data (black).

We used two independent PFSS models. One is based on the SolarSoft collection from the Lockheed Martin Solar and Astrophysics Laboratory (LMSAL); this model assimilates surface flux estimates along with MDI or HMI data (depending on the availability of data from the two instruments). The second model was developed at Stanford’s HEPL. We began by comparing the two and determining the time periods in which each gave valid results. We used the LMSAL model, with MDI data, to cover the time period from mid-1996 to mid-2003 (when the SOHO spacecraft had to be rolled every six months to compensate for a failed antenna). The Stanford model needed a few solar rotations at the beginning of the cycle to converge to consistent results, but was suitable from mid-1999 to the end of cycle 24. Figure 1 shows total unsigned open flux calculated by both models, compared to IMF open flux from OMNI data.

We then determined the value of source surface height that provides a best fit to the IMF open flux at 1 AU (using the OMNI 2 data set) for the time period 1996—2012. For each model, total open flux was calculated for selected time periods over a range of source surface heights. The resulting value was compared to average IMF open flux at 1 AU during that time period, calculated using the method of Lockwood^{5} as F = 4πR12|Br|.

Figure 2: (top) IMF open flux compared to 30 day average values for source surfaces from 2.0 Rs to 4.25 Rs, predicted from the LMSAL PFSS model. (bottom) Optimum source surface height matching IMF open flux, based on LMSAL model.

What did we find? The canonical 2.5 Rs source surface matches the measured IMF open flux well during periods of solar maximum (despite concerns that PFSS models are generally less accurate at that phase of the solar cycle), but needs to be raised by approximately 15-30% in order to match the measured IMF open flux at the periods of solar minimum. Our results indicate that a source surface height as high as 2.9 Rs for the LMSAL model, and 3.3 for the Stanford model, best reproduce the IMF total open flux during the minimum at the end of cycle 23 from 2007 to mid-2009. Figures 2 & 3 show the calculated open fluxes as a function of source surface height, and the optimum source surface heights, for the LMSAL model (figure 2) and the Stanford model (figure 3).

In short, then, here are our results:

- Both PFSS models did a reasonable job of estimating IMF open flux over the time period of cycle 23 and the start of 24 using the typical source surface height of 2.5 Rs, with correlation coefficients of 0.73 (Stanford) and 0.91 (LMSAL).
- Differences between modeled (PFSS) open flux with a source surface height of 2.5 Rs and measured (IMF) open flux were generally on the order of the rms value of IMF open flux after 30-day smoothing and detrending.
- Altering the source surface height yielded more precise matching of the PFSS calculated open flux to measured IMF open flux. Calculating open flux over a range of source surface heights enabled the use of interpolation to find the best match over the solar cycle.
- Our results show that raising the source surface height 15-30% during solar minimum (depending on the model used) better reproduces the observed IMF open flux from OMNI. We used two different PFSS models and the MDI/HMI magnetograms as input. This finding agrees with the 20% rise in source surface height at solar minimum shown in Figure 6 of Lee, et al. using Mt. Wilson and MDI data in order to best reproduce coronal hole area observed in extreme ultraviolet images.

Figure 3: (top) IMF open flux compared to Stanford model predictions for source surfaces from 2.0 Rs to 3.0 Rs. Black curve = IMF at 1 AU; green = 2.0 Rs; red = 2.5 Rs; blue = 3.0 Rs. (bottom) Optimum source surface height matching IMF open flux, based on Stanford model, smoothed over three Carrington rotations. Maximum source surface height calculated was 4.0 Rs.

### References

[1] Hoeksema, J. T., Wilcox, J. M., Scherrer, P. H. 1983, *J. Geophys. Res.*, **88**, 9910

[2] Schulz, M., Frazier, E. N., Boucher, Jr., D. J. 1978, *Sol. Phys.*, **60**, 83

[3] Lee, C. O., Luhmann, J. G., Hoeksema, J. T., Sun, X., Arge, C. N., de Pater, I. 2011, *Sol. Phys.*, **269**, 367

[4] Arden, W. M., Norton, A. A., Sun, X. 2014, *J. Geophys. Res. Space Physics*, **119**, 1476

[5] Lockwood, M. 2013, *Liv. Rev. Sol. Phys.*, **10**, 4