1. Conclusions Using the Diffusive Equilibrium Mapping Technique we have connected a starting point of a field line on the photosphere with its final location in the heliosphere. The radial component of the heliospheric field shows no latitudinal gradient, as is consistent with spacecraft observations. Expansion factors calculated with this method show very little variation with latitude. Efforts to relate the PFSS magnetic field expansion factors to the speed of the solar wind are insufficient as those models cannot account for open magnetic flux located outside of coronal holes or active regions. When magnetic flux is included outside of coronal holes and active regions on the photosphere, it will affect the locations of coronal- hole boundaries in the heliosphere. If the slow solar wind originates from regions outside of coronal holes, a significant amount of open flux must be located there. Observations show that over 60% of the area of the heliosphere contains slow wind during solar maximum (See the poster of L. Zhao). This corresponds to over 70% of the total open flux having foot points located outside of coronal holes and active regions. References Altschuler, M. D., and G. Newkirk (1969), Sol. Phys., 9, 131-149. Fisk, L. A. and N. A. Schwadron (2001), Astrophys. J., 560, 425-438. Fisk, L. A., et al. (1999), Astrophys. J., 521, 868-877. Fisk, L. A. (2005), Astrophys. J., 626, 563-573. Gilbert, J. A., T. H. Zurbuchen, and L. A. Fisk (2006), Astrophys. J., (submitted) Munro, R. H., and B. V. Jackson (1977), Astrophys. J., 213, 874-886. Schrijver, C. J., and A. M. Title (2003), Astrophys. J., 597, L165-L168. Smith, E. J., and A. Balogh (1995), Geophys. Res. Lett., 22, 3317-3320. Abstract Open magnetic flux is unevenly distributed at the solar surface, but observations indicate that at some distance in the outer corona it becomes uniformly distributed and approximately radial. Using a new technique for mapping open magnetic flux, which leads to a uniform distribution in the outer corona, magnetic expansion factors are calculated and compared to the results from Potential-Field Source- Surface models. The technique is used in conjunction with heliospheric observations to determine the amount of open flux located outside of coronal holes. This is demonstrated for both solar maximum and solar minimum conditions. A Conceptual Description Open magnetic flux expands from the photosphere, where it is unevenly distributed, into the corona, where it equilibrates, leading to an approximately constant radial field. If we assume that the field is essentially radial both at the photosphere and at some distance from the photosphere (the source surface), we can map field lines between them. Assume that the frozen-in field lines move around with some velocity on a surface. The velocity will displace the field lines until the field is in pressure equilibrium (Fig. A and B). We change the frame of reference and assume that the field lines are static and anchored in the photosphere while a spherical surface expands radially outward. In this frame, the field lines appear to be moving on the expanding spherical surface with an apparent velocity, which is the expansion velocity (Fig. C). Hence, the velocity field connects magnetic-field foot points in the photosphere to their counterparts in the corona. We calculate this expansion based on two assumptions: a) The expansion of all field lines occurs without an intertwining of the field; b) The current sheet, calculated under potential-field assumptions, remains fixed in its location. With these assumptions, the displacement of the field lines from the original location to their equilibrium position is unique. Without shear motions each field line will remain next to its neighbor, so if the field lines at the current sheet are fixed, the final equilibrium positions are fixed as well. (For details, see the Mathematical Mapping Technique box on the top right and Gilbert et al., 2006). C When results from the mapping technique are compared to mappings from a potential-field source-surface model, significant differences become apparent. Fig. 1 and 2 show field configurations during solar minimum and maximum, with the WSA-PFSS model traced in black and the equilibrated mapping shown in red. Note: only the endpoints of the red traces have physical significance - the trace between endpoints is a mathematical construct. Fig. 3 shows expansion factors from Fig. 1, with the solid and dashed lines corresponding to the PFSS model and our mapping technique, respectively. In an equilibrated heliospheric field, the expansion factors do not vary greatly with latitude, as should also be expected from MHD models. The Mathematical Mapping Technique The expansion velocity is always normal to the interface between coronal holes and other fields. Since it is also perpendicular to the radial direction, we have labeled it u. When equilibrium is achieved no further motions occur, thus: when we must have. The radial component of B is related to u by the following: Plug this velocity into the MHD induction equation to get a diffusion equation: With the current sheet as a fixed boundary through which no open flux can pass, and the locations of open flux within coronal holes on the photosphere, we have sufficient boundary conditions for the diffusion. At each time step, we calculate the velocity field u and determine the displacement d of each field line from: This mapping technique can also map transient open flux located outside of coronal holes and active regions, as is expected in many models of slow solar wind. Fig. 4 shows an example in which open flux is distributed outside of the boundaries of polar coronal holes. Wind associated with the coronal holes no longer extends all the way to the current sheet, which is consistent with heliospheric observations. This tool allows calculations of heliospheric flux expansions with a broad range of assumptions on the photospheric flux distribution. This is well beyond traditional potential field models, which predict that all solar wind originates from coronal holes, in apparent contradiction to heliospheric observations of fundamental differences between fast and slow wind. Applications of the Diffusive Equilibrium Mapping Technique J.A. Gilbert, T.H. Zurbuchen, L.A. Fisk Department of AOSS, University of Michigan Conclusions Using the Diffusive Equilibrium Mapping Technique we calculate magnetic-field connections between the photosphere and the open corona. The radial component of the heliospheric field is assumed to have no latitudinal gradient, consistent with spacecraft observations. Expansion factors calculated with this method show very little variation with latitude, as expected from MHD calculations. This is inconsistent with PFSS models, which calculate expansion factors with little physical basis. When magnetic flux associated with slow solar wind is included outside of coronal holes in the photosphere, we are able to calculate the locations of coronal-hole/slow wind boundaries in the heliosphere. Using our technique, we show that if the slow solar wind originates from regions outside of coronal holes, a significant amount of open flux must be located there. Observations show that over 60% of the area of the heliosphere contains slow wind during solar maximum (See the poster of L. Zhao). This corresponds to over 70% of the total open flux having foot points located outside of coronal holes and active regions. The three figures below apply this mapping technique using observations of the solar wind from ACE and Ulysses during CR 1964 (13 June - 10 July, 2000). Fig. 5 shows the locations of open magnetic flux on the photosphere from the WSA-PFSS model, with red indicating positive polarity. Fig. 6 gives the locations of the coronal hole boundaries in the heliosphere. The labels on the contour lines in Fig. 6 indicate the percentage of the total flux that was placed outside of the “open coronal hole” regions shown in Fig. 5. As the amount of open flux in these “closed” regions is increased, the solid-angle expansion of the coronal holes is reduced, as expected. Fig. 7 shows comparison with data from ACE and Ulysses, overlayed on a source surface plot from WSO. The green lines indicate detection of slow solar wind (<450 km/s), the red vertical lines are fast wind, and the yellow lines are detection of a CME event. Slow solar wind is assumed to be associated with the distributed (and likely transient) flux from “closed regions”. In order to have the coronal-hole boundaries only contain areas of fast wind or CMEs in the data, it was necessary to place 70% of the total flux in “closed” field regions in the model. By solid angle, the amount of the heliosphere that contains slow wind would be ~50% according to the simulation in Fig. 7. The data in Fig. 7 suggests that slow solar wind occupies over 60% of the solid-angle area of the heliosphere. See the poster of Liang Zhao for a more detailed analysis of the solar-wind data. 6 7 4 3 2 1 Note: the expression “coronal hole” is used as common in the heliospheric community: “Coronal holes” are sources only of fast wind with cool charge states, and not of all solar wind. 5 Suess, S. T., and E. J. Smith (1996), Geophys. Res. Lett., 23, 3267-3270. Wang, Y. M., and N. R. Sheeley (1992), Astrophys. J., 392, 310-319. PFSS data was provided by C. N. Arge, and derived from the Mount Wilson Solar Observatory