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Astron. Astrophys. 331, 1147-1156 (1998)

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2. Electric field generation by a neutral beam in an unresponsive plasma

2.1. Infinite homogeneous beam case

In this case the background plasma only exerts a collisional drag on the beam particles so that electrodynamic effects are confined to the beam. Since the system is infinite and homogeneous and the beam is purely [FORMULA], any [FORMULA] arising must be uniform in space so that [FORMULA] so that no magnetic field [FORMULA] exists if [FORMULA] at [FORMULA]. Even then, this purely temporal problem is complicated by the large range of timescales involved - from the electron plasma period up to the proton collision time. Since the proton collision time is long compared to the electron collision time and extremely long compared to the electron plasma period, we first simplify the problem by considering the proton speed to remain constant and the electron speed to vary by only a small fraction of the initial beam (and proton) speed so that the electron collision term remains constant.

That is, we start from the equations (in a mean particle description of plasma processes):

[EQUATION]

[EQUATION]

[EQUATION]

where n is the beam density, [FORMULA] and [FORMULA] are electron and proton velocities, and [FORMULA] and [FORMULA] are collisional drag forces on electrons and protons due to the surrounding dense gas. Throughout this paper, primed quantities refer to the background plasma particles whereas unprimed quantities refer to the beam particles. The subscripts e and p are exclusively used to denote electron and proton respectively.

Based on our previous remarks, these equations simplify to:

[EQUATION]

[EQUATION]

where [FORMULA] is the constant proton speed.

These equations can be rewritten into one for the (oscillatory) motion of the relative proton-electron velocity u = [FORMULA] - [FORMULA], viz.

[EQUATION]

the solution of which is

[EQUATION]

where [FORMULA] is the angular beam electron plasma frequency with [FORMULA] and [FORMULA] since [FORMULA]. The corresponding electric field can be expressed as

[EQUATION]

It follows that the electrons here are fully dragged by the protons due to the mean electric field E = - [FORMULA] as stated by Simnett and Haines (1990), though oscillating about them at angular frequency [FORMULA] with very tiny space and velocity amplitudes which can be written

[EQUATION]

where l is the nonthermal beam Debye length [FORMULA] and we have used

[EQUATION]

Here [FORMULA] is the Coulomb logarithm, [FORMULA] is the density of the background gas, e is the charge, [FORMULA] is the electron mass and [FORMULA] the beam electron velocity. The factor 12 in this expression accounts for both the effect of the electron-electron collisions (8 parts) and for the electron-proton collisions (4 parts).

A more complete numerical solution of the full analytic mean particle equations, including both [FORMULA] and [FORMULA] and their dependences on velocity, shows exactly the same behaviour with the electrons oscillating about the protons as they decelerate together due to the combined dragged force [FORMULA], the electric field increasing as the velocities fall and the Cs increase.

2.2. Semi-infinite beam case

We now consider the more realistic case of injection of a semi-infinite neutral beam rather than an infinite one `launched' homogeneously. When such a neutral beam is injected and penetrates into a dense gas then, due to collisions, its electron component is again decelerated first. This deceleration force is equivalent to that of an electric field of magnitude [FORMULA] given by

[EQUATION]

The beam electron velocity initially (before E becomes significant) declines with time due to collisions according to [FORMULA], where [FORMULA] is the initial electron velocity, and [FORMULA] so that E increases as the electrons slow down. Due to the relative deceleration of electrons, a positively charged layer forms at the head of the finite neutral beam and at the tail, a negatively charged layer forms so that an electric field is generated along the whole beam. In terms of the initially growing charges in the widening end layers this field can be expressed as

[EQUATION]

where [FORMULA] is the neutral beam density initially equal to electron and proton beam densities everywhere, and the thickness of the charged layers is [FORMULA]. This electric field continues to grow, with the charge layer thickness d, until E reaches the value [FORMULA], when d takes the tiny value [FORMULA] given in Eq.  9above, and is strong enough to balance the differential collisional drag force. Thereafter the beam electrons are dragged along on average at the proton velocity, the two species slowing down together as proton collisional losses progress, but with electron oscillations superposed on this motion. Note that in this analysis, the beam width is taken to be large compared to its length (cf Oss and van der Oord 1995 and Miller 1982).

2.3. Acceleration of electrons

Though the charge separation induced by differential collisional drag is very small, the resulting E field along the beam is sufficient to offset the collisional drag on beam electrons and will also act to accelerate some of them which deviate randomly from the exact equilibrium speed given by [FORMULA]. But, as can be seen from a comparison of the electric field E and the collisional force which depends on electron velocities, the acceleration is possible only if an electron has a velocity greater than that of the monoenergetic beam average. If a test electron has a velocity exactly equal to the beam velocity then forces are in equilibrium, and the test electron is moving with the velocity of beam. For initial velocities smaller than the beam velocity, electrons are decelerated.

In Simnett and Haines (1990), a calculation was performed to find the maximum possible energy of an accelerated runaway electron. The first step was to find [FORMULA], the potential due to the assumed double layer. This step is questionable given their subsequent analysis since, as we shall show, this potential cannot remain unaffected if significant runaway occurs. A second less fundamental objection is that the [FORMULA] in the denominator of the collision term in their Eq. (8) should actually be [FORMULA]. The consequence of this last error is to reduce the maximum possible electron energy to 75% of the proton energy rather than 100%, as they stated.

Runaway electrons with high energies are in principle possible in an unresponsive plasma, if a particle has a speed greater than the average beam speed. The key issue now is how many electrons can be involved in this. Till now we considered only one test electron, or more generally a number sufficiently small not to affect the electrodynamic equations we used to find the neutral beam generated E field. But clearly, with an increasing number of runaway electrons their electrodynamic effect must be taken into account. Here we make a rough estimate of the upper limit to the number of runaway electrons which can be produced before their effect on E is significant. According to the previous scenario, protons of the neutral beam effectively pull electrons by a "charge separation" electric field

[EQUATION]

Any runaway electrons accelerated in this field would effectively result in a double layer of charges with associated electric field

[EQUATION]

where [FORMULA] is the density of runaway electrons, and [FORMULA] is the length of the runaway electron pulse. In order for the runaways not to modify the E field accelerating them we require that [FORMULA]. This gives the following constraint on the fraction of electrons that can runaway from a beam of length L:

[EQUATION]

where [FORMULA] is the beam `plasma parameter' (number of beam electrons in a Debye sphere). Since l is a small length, of the order of cm in solar conditions, and since N is always very large (typically of the order of [FORMULA]) compared to [FORMULA] ([FORMULA]) the runaway fraction is constrained to be extremely small.

We conclude, therefore, that runaway acceleration of electrons, whatever their origin, in the E field of a neutral beam, for the case of an unresponsive background plasma, has no relevance to the generation of flare hard X-ray emission. Such emission would require (from the basic bremsstrahlung equation - Brown 1972) very large electron fluxes, carrying a total electron beam power of near the total impulsive impulsive flare power (Brown 1971, Hoyng et al 1976 and others).

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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