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

2. What is the integrated vertical Lorentz force on a volume in the solar atmosphere? To get the total vertical 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

3. The only surface that contributes significantly is the photosphere: where the -1 in front comes from the fact that that the normal unit vector at the photosphere is in the –z direction. 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 vertical 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

4. The change in the Lorentz force drives vertical 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 x can be approximated as δB x =∂B x /∂t δt (similarly, the other components of B), and δ|B|/|B| is small, the change of the vertical Lorentz force δFz 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, 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

5. Upward acceleration should therefore occur if the magnetic field at the photosphere becomes more horizontal after a flare. A recent analysis of a number of 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. 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. What about the force acting downward (inward) on the photosphere and solar interior? 5

6. Newton’s third law requires the net change in the upward force be balanced by an equal and opposite change in the downward force: This is the estimate for the force exerted on the photosphere that is given in Hudson, Fisher & Welsch. (2008, ASP Conf. Series 383, 221). If flare-driven helioseismic disturbances are driven by this inertial reaction from the atmospheric Lorentz-force, it suggests that the helioseismic observation of flare-driven acoustic disturbances that propagate in the solar convection zone may provide a useful diagnostic of upward momentum driven in the early stages of eruptive flares. But this presumes that the Lorentz-force dominates other dynamic flare effects, such as gasdynamic pressure increases from electron beam-driven chromospheric evaporation (e.g. Kosovichev & Zharkova, 1998, Nature, 393, 317), or radiative backwarming (Donea et al, 2006, Sol Phys 239, 113). Can we systematically evaluate the effectiveness of the Lorentz force versus gasdynamic and backwarming mechanisms for disturbing the solar interior? 6

7. Comparing forces at photosphere from different models of the flare disturbance According to Hudson et al, typical values of 1/A δF z downward at the photosphere from Lorentz force are -2.5x 10 3 dyne cm −2, where the – sign means the force is downward. This is roughly consistent with force amplitudes estimated in Wang et al. Pressure increase due to electron-beam heating: Direct electron heating at photosphere is too small to be effective. If beam heating effects are important, they are due to (1) big pressure increases in the chromosphere due to chromospheric evaporation, and (2) propagation of this pressure disturbance to the photosphere. Pressure increase due to radiative backwarming: Most likely this is due to beam heating in the flare chromosphere deposited below the flare transition region. The flare chromosphere heats up and radiates the energy in UV/EUV line radiation, and to some extent optical and UV continuum. Half goes out, but half goes down to the photosphere, absorbed where it is absorbed and heats the photosphere. 7

8. Explosive Evaporation Driven by non- thermal Electron Heating 8

9. To explore explosive evaporation parametrically, use an approximate analytical model (ApJ 317, 502) What is the maximum pressure P max that the explosively heated region can achieve? Δz, ΔN Avg Heating rate per particle = Q Plasma Using this model, it is easy to calculate the maximum chromospheric pressure as a function of flare heating parameters and preflare atmospheric models. 9

10. How effectively can gas pressure disturbances in the chromosphere propagate to the photosphere? Approach 1: Compute the time-evolution of the downward-moving “chromospheric condensation” toward the photosphere and evaluate whether this disturbance can propagate that far. Result: The chromospheric downflow stops when the pre-flare gas pressure ahead of the moving front becomes equal to the initial pressure in the disturbance. For intense flare heating parameters, P max ~ 100-1000 dyne cm -2. This means the condensation can propagate to a column depth of roughly P max /mg ~ 10 21 – 10 22 cm -2 – far short of the 10 24 ~ 10 25 cm -2 column depth needed to get to the photosphere. Approach 2: Use analytical acoustic wave solutions, with initial conditions determined by pressures or downflow velocities from the flare-driven condensations, to estimate the amplitude of disturbances at the photosphere. Since these wave solutions are dissipation-free, they are overestimates of the photospheric effects. These solutions were derived in a 1989 discussion with Doug Braun about flare-driven acoustic waves at the photosphere and below. 10

11. Acoustic wave amplitudes die off like exp[-z/(2Λ)], where z is distance downward measured from the chromospheric disturbance… 11

12. This means exponential attenuation of the chromospheric pressure pulse at the pressure scale height length scale as the pulse propagates down to the photosphere Thus, if explosive evaporation can generate a 100- 1000 dyne cm -2 pressure in the flare chromosphere, the perturbed pressure will be reduced by factors of at least ~ exp(-2) ~ 0.1 by the time the pulse reaches the photosphere. Compare this to the Lorentz force estimates of ~ 2.5 x 10 3 dyne cm -2 at the photosphere! 12

13. Therefore, it appears that acoustic disturbances at the photosphere and below driven by flare gas-dynamic effects are far less effective than Lorentz forces… What about the effectiveness of radiative backwarming? Flare chromosphere Photosphere Energetic particles UV, EUV lines, b-f continua photons 13

14. How much flare energy flux is available for radiative backwarming? Essentially all of the energy being deposited below the flare transition is balanced by UV and EUV line radiation, as well as H b-f continua. Half of that energy is radiated downward. Therefore, an upper limit to the radiative backwarming flux is half the energy in energetic particles deposited below the flare transition region: where Q fl (N) is the column-depth dependence of the flare energetic particle heating rate, and N ftr is the column depth of the flare transition region. 14

15. How can we estimate how much the photospheric pressure increases due to backwarming? Δz, ΔN Avg Heating rate per particle = Q Plasma Try using the same analytical model as used for explosive evaporation! The difference is that now, Q is the average heating rate per particle in the photospheric layer, Δz,ΔN are the thickness and column thickness of the layer absorbing the radiation. Caveats: Model invalid once the temperature exceeds the blackbody limit for the sum of solar luminosity plus the backwarming flux, and ignoring gravity in the model is a poor assumption for Δz exceeding a pressure scale height at the photosphere. Result: P max ~4x10 4 dyne cm -2, for an assumed electron heat flux of 10 11 erg cm -2 s -1. This is comparable to the estimated perturbed force from the Lorentz force. 15

16. 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 that makes a significant downward force at the photosphere Gasdynamic forces from chromospheric evaporation cannot significantly affect conditions at the photosphere Radiative backwarming and the Lorentz Force may produce comparable downward forces at the photosphere. Further work on this mechanism is needed. 16