Global Forces in Eruptive Solar Flares: The Role of the Lorentz Force George H. Fisher, Benjamin J. Lynch, David J. Bercik, Brian T. Welsch, & Hugh S. Hudson Space Sciences Lab, UC Berkeley

What is the integrated Lorentz force on a volume in the solar atmosphere? To get the total force, integrate the force per unit volume over the volume that goes from the photosphere out to infinity, and in area, around the strong field in an active region: Use the Maxwell stress-tensor to compute the vertical Lorentz force per unit volume: 2

The only surface that contributes significantly is the photosphere: Here the area integral is computed over the photospheric surface where the magnetic field is measured. Now, suppose that, as the result of a major flare, the magnetic field undergoes a rapid change. Then the net change of the Lorentz force within the atmospheric volume can be related solely to the change in the magnetic field measured at the plane of the photosphere: 3

The change in the Lorentz force can drive upward acceleration in the solar atmosphere Assuming that a rapid change of the magnetic field vector occurs over a time δt, and that the change in B r 2 can be approximated as δB r 2 =∂B r 2 /∂t δt (similarly, the other components of B) the change of the vertical Lorentz force δF r over the course of the flare is Can a rapid change in the Lorentz force be balanced by any other forces within the atmospheric volume? The only other significant forces in the solar atmosphere are from gas pressure, gravity, and inertia. In a low-β active region, we presume Lorentz forces will dominate gas pressure. If the mass in the volume stays invariant, gravitational forces won’t change much. Therefore, the change in the vertical Lorentz force will be matched primarily by a change in the upward inertia (i.e. the momentum of the erupting plasma). 4

Do photospheric magnetic fields really change significantly during solar flares? Yes they do! (example from Petrie & Sudol, ApJ 724, 1218 [2010], showing GONG measurements of line-of-sight field changes). Typical time-scale for changes: δt ~ 10m.

How big are the magnetic field changes? Table 3 from Petrie & Sudol, ApJ 724, 1218 (2010).

Upward acceleration should occur if the magnetic field at the photosphere becomes more horizontal as the result of a flare. A recent analysis of eleven vector and line-of-sight magnetograms by Wang et al. (2010, ApJ Letters 716, L195) taken before and after the occurrence of large eruptive flares shows that in nearly all the cases examined, it appeared that the magnetic field did indeed become more horizontal after the flare. Similar results have been found from changes in the line-of-sight fields from statistical studies of many large flares (Petrie & Sudol, 2010 ApJ 724, 1218). Is there an observational relationship between the upward momentum in an eruptive flare or CME and the measured change in the vector magnetic fields in a flare? The analysis here suggests there should be, but as far as we know, this has not yet been tested. 7

Lorentz-driven impulse on the solar atmosphere: Implications? To escape from the Sun, assuming the plasma is initially at rest, the increase in radial velocity δv r must exceed the escape velocity v e = (2GM  /R  ) 1/2. This results in an upper limit to CME mass: Observationally testable.

What about the forces acting on the solar interior? Balance of interior and atmospheric Lorentz forces requires the change in upward atmospheric force be balanced by opposite change in the solar interior force: The 2 nd expression is the estimate for the force exerted on the photosphere we gave in Hudson, Fisher & Welsch. (2008, ASP Conf. Series 383, 221). If the Lorentz force is the cause of sunquakes, then perhaps helioseismology can reveal some new information about eruptive flares and CMEs.

Feb X2.2 Flare results in a peak upward force of 4x10 22 dynes acting on the outer solar atmosphere, and an equal but opposite downward force acting on the solar interior.

The evolution of the magnetic field as described here has also been reported in a number of recent publications: Sun et al (ApJ 748, 77, 2012) Wang et al (ApJL 745 L17, 2012) Petrie (arxiv , 2012) The global Lorentz force behavior and surface Lorentz- force distributions described here have been confirmed in the study of Petrie (arxiv , 2012).

How do we interpret the observed Lorentz-force surface density pattern?

Lorentz-force surface density distribution from ARMS eruption simulation

Conclusions The Lorentz force in the solar atmosphere is upward for photospheric magnetic field configurations that become more horizontal after a flare. There should be an observational relationship between the initial upward momentum in an eruptive flare and the observed changes in magnetic fields at the photosphere There is a back-reaction to the Lorentz force in the upper atmosphere that makes an equal and opposite downward force in the plasma below the photosphere The Lorentz-force surface distribution of the Feb flare (a positive core surrounded by a weak negative halo) can be understood in terms of the magnetic field dynamics in the conventional eruptive flare geometry 14

Acknowledgements This work was supported by the NASA Heliophysics Theory Program (NNX11AJ65G), the NASA LWS TR&T Program (NNX11AQ56G), the AFOSR Young Investigator’s Program (FA ), and the NSF AGS program (AGS ).