Expert: Philip A. Stahl - 11/19/2007

Question
Under what conditions might a loss-cone instability lead to a solar flare?

Answer
Hello again,

First, let's go back and review a bit about what a loss cone is and then loss-cone instability.

We define what is called the “loss cone angle”:

sin (THETA)_L  =   ± [B_min/  B_max]^1/2

where B_min is the minimum magnetic field strength or induction at the apex, say, of a loop magnetic mirror system. Then B_max represents the maximum value, usually at the loop ends or "foot points" as they are called.

A special condition obtains which applies to the angle - for which the electrons will be TRAPPED, provided:

THETA (min) >  (THETA)_L

Thus, THETA (min) =  (THETA)_L

is said to be the "loss cone" of the system or machine.

Now, the criterion for the hydrodynamic loss cone instability requires that the particular condition for the ratio of untrapped to trapped particles:

n/ no  >   2  OMEGA e /  (pi) w_p  =  0.1

where OMEGA_e is the electron cyclotron frequency, and w_p is the electron plasma frequency.

If such a condition were to apply, say to a solar coronal loop, it could elicit an "inverse population" in the transverse velocities (e.g. v_perp) for electrons in the loop.

(Recall we looked at such velocities last year and their ratio to the parallel velocities expressed as a function of the constant of the motion, u)

If the loss cone instability specified by the preceding condition then occurs in electrostatic waves near the upper hybrid frequency, we can get an anisotropic distribution of the electrons in velocity space. This can give rise to an electron particle beam.

More importantly, energy to support the beam - and perhaps trigger a flare- can arise from the two -stream instability which you also asked about last year.  To reiterate that:

In finding conditions under which it operates, one considers treating a dispersion relation for plasma waves such that:

 F(x, y) = (me/mi)/ x^2 + 1/ (x^2 - y^2)

 where (me/mi) denotes the electron to ion mass ratio, and we define the variables x, y as follows:

 x = w/ w_e   or the ratio of the plasma frequency to electron plasma frequency

 y = k V_o/ w_e  or the ratio of the product of the wave number k by the electron thermal velocity (V_o) to the electron plasma frequency

 Plotting the graph on the axes:

 F(x, y)
 !
 !
 !- 1
 !
 !
 !
 !-----------------!---------->x
 0                       x=y

 Will yield a bifurcated graph with 4 roots. It will always feature a local minimum F_m such that:

 0 <=  F_m < =  x=y

 When:

 F(x_m, y) = F_m <  1  there will be four real roots

 When F_m >  1 there will be two real and two complex roots

 with suitable approximations, e.g. k^2 V-o^2 <=  w_e^2

 the complex roots are found to be:

 w/w_e =  1/2 [1 +  i(3)^1/2]

 w/w_e = 1/2 [1 -  i(3)^1/2]

 These will give the limits for the instability for when F_m > 1

Once this arises, and bear in mind it is the reverse of Landau damping, the conditions may be sufficient to trigger a solar flare - especially if the beam is capable of generating a field -aligned current, say in the vicinity of a "separatrix" between differing field configurations on or within the loop.

This seems to have been exactly what transpired with one flare - an optical class 1B and X-ray class M-4 on Nov. 5, 1980. In this case, the ratio of untrapped to trapped particles was ~  0.5 or a factor five beyond what was required for the the hydrodynamic loss cone instability.

A number of investigations disclosed that the localized loop’s (designated BC) particle velocity ratio (v _perp  / v ||) was found to increase from 2.5 near loss-cone half angles ˝ THETA_ L (B)  ~ 0.68,  ˝ THETA_ L (C)  ~ 0.47 to (v_perp  / v ||) > 12 for ˝ THETA_ L (B, C)  = 0.1.  This condition is also responsible for creating an inverse population in the transverse velocities, wherein it is possible to have instability in the electrostatic waves near the upper hybrid frequency ( w_ UH)

Examination of the electron, electron beam Maxwellians for the in the loop also appears to show a distribution with a "bump on the tail" - the key sign for a two -stream instability.   

Hope this helps, and remember if any of it is not 100% clear, please ask a follow-up to clarify.

I would much prefer that as opposed to assigning an "8" for clarity rating and not even bothering to pursue the follow-up! Nothing gets my dander up more than that!