J. Todd Hoeksema^{[1]}, Yang Liu^{[1]}, Xudong Sun^{[2]}, Viacheslav Titov^{[3]}, & Zoran Mikič^{[3]}

1. W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA94305-4085

2. Institute for Astronomy, University of Hawaii at Manoa, Pukalani, HI 96768

3. Predictive Science Inc., San Diego, CA 92121

The topology of the coronal magnetic field is essential for understanding solar energetic events and properties of the solar wind. Q, a geometrical parameter that describes the *squashing factor* of elemental flux tubes, is useful for understanding coronal configurations relevant to space weather. Q characterizes the local divergence of nearby magnetic field lines.

Figure 1|For a given field line, an infinitesimal circle is mapped along field lines into an infinitesimal ellipse. The circle and ellipse are on boundary surfaces of a finite volume — for example, the photosphere and the source surface of a coronal field model. The aspect ratio of the ellipse defines the squashing factor: Q=N^{2}/δ, where N is the generalized norm and δ is the expansion factor^{[1,2]}. When N is large and/or δ is small, Q can become large. The quantity slogQ is especially convenient for analyzing the structure of magnetic configurations where:

slogQ = sign(Br) * log [Q/2 + (Q^{2}/4 – 1)^{0.5}]

Magnetic null points, separatrix and quasi-separatrix surfaces are likely to be preferential sites for magnetic reconnection. Q is large at separatrix surfaces and quasi-separatix layers, which mark boundaries between topologically distinct flux systems having different properties. Closed-field regions of similar field-line connectivity are enclosed by high-Q curves, as are open field regions with different photospheric sources. Q can be computed throughout the corona using a 3D model applied to the observed photospheric magnetic field.

**Synoptic Q-Maps for Solar Cycles 23 and 24**

We have computed Q-Maps using current observations from HMI (Helioseismic and Magnetic Imager) and archival observations from MDI (Michelson Doppler Imager). Input synoptic maps are corrected for the obscured polar field according to Sun et al.^{[3]}. We have calculated slogQ at ten heights for each rotation from 1996 to the present using the Potential-Field Source-Surface (PFSS) model. Horizontal slices of slogQ can be used to locate and track features from one height to another. Figure 2 shows results for CR 2099 in August 2010. Each of the four panels is a map of the whole Sun at some height at or above the photosphere. This figure is the frame of a movie at http://hmi.stanford.edu/QMap/summary_synop.mp4.

Figure 2|Sun and Corona in CR 2099. Upper left: smoothed radial magnetic field in the photosphere used to calculate the coronal field. Positive is white; negative black. The Sun was fairly quiet in mid 2010. Upper right: Foot points near the photosphere that open to the solar wind are shown in blue (positive) and red (negative). Lower left: slogQ at 1.10 Rs showing complex features. Dark colors are high values of Q with strong local divergence. Lower right: slogQ at 2.499 Rs shows simplified structure higher in the corona. The neutral line separating blue and red is the base of the heliospheric current sheet.

Two kinds of data series are now available at the SDO/HMI JSOC for of three synoptic series. One provides Q-maps and the other the coronal field. We first describe the **HMI synoptic series**.

**hmi.Q_synop** has three arrays for each rotation starting with CR 2097 in May 2010.

1. **slogQ** is a 3D data cube containing the slogQ values at 10 heights in the corona with quarter-degree resolution.

2. **chmap** is a 1441*721 map computed just above the photosphere (1.001 Rs) indicating the polarity of open-field foot-points.

3. **Brq** is just like slogQ, but with maps of the radial field computed at the 10 heights.

** hmi.pfss_synop ** has six segments.

1. ** Br, Bt, Bp** are the radial, theta (southward) and phi (westward) field components computed with one-degree resolution at 51 heights from the photosphere to the PFSS source surface at 2.5 Rs.

2. ** Brq ** is the same as in hmi.Q_synop.

3. ** Br0 ** is the smoothed one-degree photospheric map of inferred Br used in the computation.

4. ** gh ** lists the PFSS spherical harmonic coefficients up to order 121.

The ** MDI synoptic series ** (mdi.Q_synop and mdi.pfss_synop) are just like the HMI series. MDI observed from mid-1996 (CR 1911) until 2011. The instruments overlap from CR 2097 – 2107. The MDI Q-maps movie is at http://hmi.stanford.edu/QMap/summary_synop_mdi.mp4 .

Figure 3|The colorful figure shows the Q-map for August 25, 2018 at 1.001 Rs. The map is 360 degrees wide and extends from pole to pole. The left 120 degrees of the input map is from a magnetogram taken at 12 UT on that day. The slogQ color bar is at the bottom, with blue (yellow) representing strongly diverging negative (positive) features.

** Daily HMI Q-maps ** (hmi.q_synframe and hmi.pfss_synframe) are computed using a better measure of the global magnetic field. Standard synoptic maps collected over 27 days observe each longitude at a different time. * The synoptic frame * uses 120 degrees of longitude from a * single magnetogram * to replace the leftmost edge of the usual synoptic map. Daily Q-maps can better reflect dynamic conditions as they evolve at the photosphere and in the corona. Figure 3 shows a daily Q-map for August 25, 2018.

Annual daily-Q-maps movies can be found in http://hmi.stanford.edu/QMap/YearFrame.

Q-maps are computed using a data cube of coronal field values, which can be determined using any model. Contact the HMI Magnetic Team for information about alternative models.

** Additional information can be found at http://hmi.stanford.edu/QMap. **

This work was supported by NASA Grant NNX15AN49G to Stanford University.

### References

[1] Titov, V. S., Z. Mikic, J.A. Linker, and R. Lionello, 2008, *ApJ*, **675**, 1614.

[2] Titov, V. S., Z. Mikic, J.A. Linker, R. Lionello, and S.K. Antiochos, 2011, *ApJ*, **731**, 111.

[3] Sun, X., Y. Liu, J.T. Hoeksema, K. Hayashi, and X. Zhao, 2011, *Solar Phys.*, **270**, 9