Predicted Electron Precipitation

Published in Geophysical Research Letters, volume 25, number 1, pages 13-16, January 1, 1998.

PREDICTED ELECTRON PRECIPITATION FOR A FILLED LOSS CONE AT THE TOP OF THE AURORAL ELECTRON ACCELERATION REGION

W. Calvert
University of Massachusetts Lowell
Lowell, Massachusetts

Abstract. The electron precipitation flux and energy flux for a filled loss cone at the top of the auroral electron acceleration region have been calculated for the actual height for the top of the acceleration region. The resulting flux and energy flux are thus found to be more than sufficient to account for the aurora, thereby contradicting Knight's well-known theory for electron precipitation during an auroral substorm.

Introduction

In order to explain the aurora during a substorm, it is often assumed that the loss cone for electron precipitation into the ionosphere must be uniformly filled at the top of the auroral electron accelera tion region. According to Knight [1973], this model for the aurora can account for the electric current which accompanies the aurora, under the assumption that the top of the acceleration region occurs at an altitude of approximately17 Earth radii, in the region of the magnetosphere which is currently referred to as the X-point reconnection region. It has however since been learned that the aurora during a substorm occurs on closed field lines which do not connect to the X-point reconnection region, and that the electron acceleration region does not extend nearly this far out into the tail of the Earth's magnetosphere. This well-known model for the aurora therefore needs to be updated in order to compare the predicted electron precipitation during a substorm with the observed electric potential of the auroral electron acceleration region.

Knight's Theory

The assumptions of this theory are as follows: (1) It is assumed that the loss cone for electron precipitation into the ionosphere is determined by the conservation of electron energy and magnetic moment. (2) It is also assumed that the initial electron velocity distribution at the top of the accelera tion region is an isotropic Maxwell-Boltzmann velocity distribution, and (3) It is assumed that the electric potential of the acceleration region decreases monotonically with altitude above the auroral zone. These three assumptions are identical to the assumptions of Knight [1973].

As measured by Reiff et al. [1993], the top of the acceleration region is assumed to occur at an altitude of approximately 15,000 km above the auroral zone. This represents a major departure from Knight's theory, since the ratio of the magnetic field strength in the ionosphere to the top of the accel eration region is about 33, instead of the 3000 to 6000 which had been assumed in Knight's theory. As discussed below, the effect of this difference is to reduce the predicted electron precipitation flux for large negative electric potentials with respect to the ionosphere at the top of the electron accelera tion region.

The electron density at the top of the acceleration region can also be estimated using the wave measurements of the CRRES satellite, which was ideally located along auroral field lines in the near- Earth plasma sheet region of the Earth's magnetosphere. From these measurements it will be as sumed that the electron density at the top of the acceleration region is approximately 1 electron/cm³, instead of the 0.1 to 1 electron/cm³ which was assumed by Knight [1973]. This change, however, turns out to have negligible effect on the predicted electron precipitation, since Knight appears to have used the larger of these two values in his calculations.

Although the results are quite different, since I have calculated the electron precipitation flux and energy flux instead of the electric current that was assumed to be the cause of the aurora at the time of Knight's calculations, the calculation procedure is identical to that which was used by Knight [1973]. This procedure involves integrating the number flux and energy flux in velocity space, per unit area in the ionosphere, and transforming that integral to the top of the acceleration region using Louiville's theorem. For the details of this calculation, see Knight [1973]. As shown in Figure 1, the resulting electron precipitation flux turns out to be

Equation (1)

where phi is the predicted auroral electron precipitation flux per unit area perpendicular to the magnetic field in the ionosphere, n and W_o are the density and average energy of the electrons at the top of the acceleration region, B_i is the magnetic field strength in the ionosphere, B_top is the magnetic field strength at the top of the acceleration region, e is the electron charge, m is the electron mass, and V is the electric potential with respect to the ionosphere at the top of the acceleration region. As shown in Figure 2, the corresponding electron energy flux also turns out to be

Equation (2)

where psi is the total power per unit area in the ionosphere, and all of the other quantities in Equation (2) are the same as in Equation (1). Equations (1) and (2) thus represent the predicted electron precipitation flux and energy flux for this model for the aurora. Figures 1 and 2 have also been checked by a numerical integration of the predicted electron flux and energy flux inside the loss cone as a function of altitude above the auroral zone. Figures 1 and 2 can then be used to check the validity of Knight's theory.

As shown in Figure 1, a region of linear increase in electron precipitation flux as a function of the electric potential of the acceleration region is found to occur for initial electron energies which are greater than about 500 eV. This theory therefore predicts that the electron precipitation flux ought to increase approximately proportional to the electric potential of the acceleration region, as pre viously pointed out by Calvert and Hardy [1997].

Figure 1. Predicted electron precipitation flux in the ionosphere for a density of 1 electron/cm3 at the top of the auroral electron acceleration region, where Wo is the initial electron energy above the top of the acceleration region.

As shown in Figure 2, for an electric potential which is greater than about 2 kV, it is also found that the predicted energy flux does not depend upon the electron energy at the top of the acceleration region. This result, which was not pointed out in Knight's theory, occurs because of the higher electron precipitation flux at lower initial electron energy in Figure 1. As a consequence, since the brightness of the aurora depends upon the energy flux, and not the electron number flux which is shown in Figure 1, local energization of electrons in the near-Earth plasma sheet cannot be invoked, according to this model, to account for the observed structure of the aurora.

Fifure 2

Figure 2. Predicted electron energy flux in the ionosphere for a density of 1 electron/cm3 at the top of the acceleration region, where Wo is the initial electron energy above the top of the acceleration region.

As indicated by Equations (1) and (2), it is also clear that a change in the height of the bottom of the acceleration region, as depicted in Figure 1 of Louarn et al. [1990], has no effect on the predicted electron precipitation. Equation (1), on the other hand, shows that the increase in the electron precipitation flux for a given potential at the top of the acceleration region is limited by the ratio of the magnetic field strength in the ionosphere to the magnetic field strength at the top of the acceleration region. In Knight's Figure 3 this increase was assumed to be nearly three orders of magnitude, whereas in Figure 1 it is found to be only a factor of 33. As shown in Figure 2, however, this limit turns out to be insignificant for most theories of the aurora, since the effect of this limit does not show up for the electron energies and electric potentials which are usually assumed in these theories.

Discussion

The mechanism that maintains the potential drop along an auroral field line has been discussed by Alfvén and Fälthammar [1963], Block and Fälthammar [1976], Lennartsson [1976], Whipple [1977], and Chiu and Schulz [1978]. The process that fills the loss cone has also been discussed by Lennartsson [1976] and Calvert [1995]. According to this model for the aurora, as subsequently discussed by Calvert [1997b], the diffuse and discrete aurora during a substorm can be attributed to scattering into the loss cone inside the electron acceleration region. It is also justified to propose a static model for the electric potential of the acceleration region, since there is no reason to assume that induction plays a significant role at these altitudes above the auroral zone.

Calvert [1997a] and Calvert and Hardy [1997] have also compared Knight's theory to recent ob servations of the electron precipitation flux at low altitudes above the auroral zone during an auroral substorm, with the conclusion that Knight's model for the aurora cannot account for the structure of the discrete aurora during a substorm. Other relevant comparisons of the electron precipitation flux and the electric potential of the acceleration region have been discussed by Lin and Hoffman [1982] and Louarn et al.[1990], although neither of these studies took advantage of the fact that their mea surements could have been used to test the validity of Knight's theory.

According to Knight's theory, which had been developed before it was certain that an auroral substorm occurred on closed field lines, a filled loss cone at the top of the acceleration region could be attributed to an isotropic electron velocity distribution in the central plasma sheet region in the distant tail of the Earth's magnetosphere. Since this region corresponds to the polar cap at the high- latitude edge of the auroral zone, it is therefore necessary to assume that the electrons which cause the aurora convect inward as a result of the electric field across the tail of the Earth's magnetosphere. This process has been analyzed by Calvert [1997b], with the conclusion that the loss cone on auroral field lines ought to be almost completely empty as a result of electron precipitation into the ionosphere at higher latitudes in the auroral zone. In order to circumvent this difficulty, others have assumed that the loss cone ought to be refilled as a result of scattering into the loss cone above the electron acceleration region. This theory, which relies upon Knight's predictions without justifying his assumptions, is referred to as the "filled-loss-cone theory of the aurora." The problem with the filled-loss-cone theory is that this amounts to acknowledging the problem without supplying a solution, since nobody has identified the wave mode or wave instability which supplies the loss-cone electrons which this model requires.

Although it may come as a surprise to those who have assumed that Knight's theory adequately accounts for the aurora, the predicted flux and energy flux which are shown in Figures 1 and 2 turn out to be an order of magnitude or more greater than the actual electron precipitation flux and energy flux of the diffuse and discrete aurora during a substorm. For a potential of 3 kV to 10 kV, which is typical of the inverted-V electron events which are found to accompany the aurora during a substorm, Knight's theory predicts an electron precipitation flux of 2-5 x 10⁹ electrons /cm² sec and an electron energy flux of 15-100 erg/cm² sec, whereas the measured flux and energy flux during a substorm are found to be in the range of 10⁷ to 10⁹ electrons/cm² sec, and only about 1-2.5 erg/cm² sec [see Calvert and Hardy, 1997; Hardy et al.,1985].

This means that the loss cone for electron precipitation into the ionosphere must be only partially filled at the top of the acceleration region, thereby contradicting the one and only significant assumption of Knight's theory. Moreover, since a partially-filled loss cone implies electron precipitation into the conjugate auroral zone, the observation of a partially-filled loss cone also implies that the loss cone ought to be almost completely empty as a result of electron precipitation into the conjugate auroral zone. The only feasible explanation for a partially filled loss cone is therefore an empty loss cone which is partially filled as a result of scattering into the loss cone somewhere along an auroral field line. Although this is consistent with the filled-loss-cone theory, unless there is some other reason to assume that the rate of scattering into the loss cone ought to be proportional to the size of the loss cone, Equation (1) no longer applies and Knight's theory is no longer an adequate explanation for the aurora.

Conclusion

The electron precipitation flux and energy flux which are predicted by Knight's theory are found to be roughly an order of magnitude more than is sufficient to account for the aurora during an auroral substorm, thereby contradicting Knight's well-known and widely-accepted theory for the electron precipitation during a substorm.

Acknowlegements. This work was supported in part by the Radio Atmospheric Science Center, Kyoto University, Uji, Japan, and by NASA contract NAS5-96020. The services of D. S. Evans and seven other reviewers at the Journal of Geophysical Research and the Journal of Atmospheric and Terrestrial Physics are also acknowledged.

References

Alfvén, H., and C.-G. Fälthammar, Cosmical Electrodynamics, Oxford Univ. Press, New York, 1963.

Block, L. P., and C.-G. Fälthammar, Mechanisms that may support magnetic-field-aligned electric fields in the magnetosphere, Ann. Geophys., 32, 1926, 1976.

Calvert, W., An explanation for auroral structure and the triggering of auroral kilometric radiation, J. Geophys. Res., 100, 14,887-14,894, 1995.

Calvert, W., Do localized electric fields cause the structure of the aurora?, Eos Trans. AGU, 78, 309- 310, 1997a.

Calvert, W., Uji Lectures on the Aurora, RASC, Kyoto University, Uji, Japan, 1997b.

Calvert, W., and D. A. Hardy, Local increases in auroral electron precipitation which were not accompanied by a corresponding increase in the electric potential of the auroral electron accel eration region, Geophys. Res. Lett., 24, 2933-2936, 1997.

Chiu, Y. T., and M. Schulz, Self-consistent particle and parallel electrostatic field distributions in the magnetospheric-ionospheric auroral region, J. Geophys. Res., 83, 629-642, 1978.

Knight, S., Parallel electric fields, Planet. Space Sci., 21, 741-750, 1973.

Lennartsson, W., On the magnetic mirroring as the basic cause of parallel electric fields, J. Geophys. Res., 81, 5583-5586, 1976.

Lin, C. S., and R. A. Hoffman, Narrow bursts of intense electron precipitation fluxes within inverted- V events, Geophys. Res. Lett., 9, 211-214, 1982.

Louarn, P., A. Roux, H. de Féraudy, and D. LeQuéau, Trapped electrons as a free energy source for the auroral kilometric radiation, J. Geophys. Res., 95, 5983-5995, 1990.

Reiff, P. H., G. Lu, J. L. Burch, J. D. Winningham, L. A. Frank, J. D. Craven, W. K. Peterson, and R. A. Heelis, On the high- and low-altitude limits of the auroral electric field region, in Auroral Plasma Dynamics, Geophys. Monogr. Ser., Vol. 80, edited by R. L. Lysak, pp. 143-154, AGU, Washington, D. C., 1994.

Whipple, E. C., The signature of parallel electric fields in a collisionless plasma, J. Geophys. Res., 82, 1525-1531, 1977.

(Received: July 14, 1997; revised: November 5, 1997; accepted: November 13, 1997.)

W. Calvert, 219 Friendship Street, Iowa City, Iowa, 52245.