GEOPHYSICAL RESEARCH LETTERS
Robert Winglee, Space Sciences Editor
American Geophysical Union
2000 Florida Avenue, N.W.
Washington, D.C. 20009
Editor's Assistant: email@example.com
October 6, 1997
Dr. Robert B. Sheldon
Center for Space Physics
725 Commonwealth Ave
Boston, MA 02215
Re: "On the physical origin of outer radiation belt 1-10 MeV
Dear Dr. Sheldon:
We have sent your paper to two referees, whose reports are
enclosed. As you can see, both referees are favorable towards the
results and particle tracking components, but both have major concerns
about the section on cusp acceleration, which is preventing
publication at this stage. If you were to increase emphasis on the
results of the first two sections, while keeping the pertinent physics
in the theory and removing speculations, we might be able to move in a
more positive direction.
The paper is not, in its present form, acceptable for publication in
Geophysical Research Letters. However, if you wish to submit a
revised paper that may be sent out for further review, you should
respond specifically to the referee's questions and comments in your
Please send four copies of your revised paper to Maura T. C. Wright,
Editor's Asistant at the above address by October 29, 1997. If
you cannot respond within this time frame, or if you do not plan to
submit a revision, we would appreciate you letting us know.
for Robert Winglee
Space Sciences Editor
via Maura T. C. Wright
Review of "On the Physical Origin and Prediction of Killer Electron
Storms" by R. B. Sheldon et al.
In my original review, I raised 4 main objections. I find that the
authors have satisfactorily addressed only half of these in the
revised manuscript. Below I comment on these objections, assigning
the same number to them as in the original review.
1. The style of this manuscript has improved considerably. The
inappropriate sensationalism that permeated the original version has
been removed. I no longer have any objections to the style of the
2. The reference to the Telstar failure has been removed. The authors
now focus on presenting their POLAR observations and theory.
3 and 4. In the first three sections of the manuscript, the authors
present observations and simulations which support an electron
population being trapped in the outer cusp. Aside from a few new
comments that I explain later, I do not have any major objections
about the first three sections of the paper. In the cusp acceleration
theory section, the authors present a very simplistic argument that
the electrons trapped in the cusp can get energized to ~1 MeV and can
escape the trap to diffuse into the radiation belts. The section is
simply a discussion of a hypothesis. Any one can hypothesize or
theorize anything they want. However, if one wants others to publish
their hypothesis or theory in a scientific journal, one must provide
reasonable evidence for it. I am not asking for irrefutable proof. i
am asking for further analysis either in the form of
observations, theoretical analysis, or numerical simulations that
shows that this energization and untrapping process can happen under
the reasonable magnetospheric parameters. Since the authors invoked
simulations of electron trajectories in the Tsyganenko 1996 magnetic
field model, using simulations to show the energization and untrapping
would be a logical approach. That is why I suggested it. However,
other approaches are acceptable.
Thus, my original objections here still stand. There is a lack of
systematic and serious analysis presented in this manuscript ot
support their cusp acceleration theory. However, half the focus of
this paper is on their cusp acceleration theory. One suggestion is to
shift the focus of the paper toward more observational anlysis of the
electron population in the cusp region. Another suggestion is to
include further analysis (theory, simulations, or data analysis) that
shows what population of electrons (range of initial energies and
initial pitch angles) can gain sufficient energy in the cusp to escape
the trap to populate the outer radiation belt. Until a serious
revision such as suggested is made, I do not recommend publication of
this manuscript in Geophysical Research Letters.
1. Introduction, paragraph 3, lines 1-3: Your explanation of Selesnick
and Blake's  paper in your reply was better and more fair than
what you wrote here. You need to state here that the phase space
density at constant magnetic moment uniformly rises toward higher
L-shells during quiet times (but not during active times). You
left out that key point.
2. Data Analysis Section, paragraph 2, lines 7-8: In Figure 1 the
radiation belt fluxes look more isotropic than the cusp fluxes. This
Report to Authors
The most troublesome part of the paper is the section on "A Theory of
Cusp Acceleration". I believe that the argument described in the first
paragraph and upon which the section depends is incorrect. If a charged
particle moves in static electric and magnetic fields while experiencing
elastic scattering, its energy after moving from r0 to
r1 is E(r1)=E(r0) +
q[V(r0) - V(r1)], where V(r) is
the electrostatic potential at r. This result is independent of the
path, number and sequence of collisions, etc. In the example of
"magnetic pumping" given, the particle will indeed gain energy in the
first 180 deg drift. This energy will be lost on the return trip
ascending the potential. If scattering reduces the pitch angle and
therefore the magnetic moment, the particle will not be able to
climb the potential to the starting point. It will return to a lower
potential point near the starting position. If the magnetic moment after
scattering is too small, the particle will not complete a loop but
will E X B drift out of the region. In any case a particle
cannot cumulatively gain energy by recirculating as described. The
energy will always be given by the above expression no matter how many
cycles are made. The magnetic pumping process referred to by the authors'
(such as described by Alfven and Falthammer) involves quite different
circumstances, i.e. changing magnetic or electric fields.
1) In the abstract it is stated that POLAR observed increases in cusp
electrons which preceded electrons seen at geosynchronous altitude. This
point is not discussed in the paper except on page 7--8 the authors
say they will make this comparison in a future study.
2) Figure 1 needs a caption. I was unable to make the phase space
comparisons as my copy was not in color. I assume the comparisons
described in the text are valid.
4) Table 1. "MeV" should be "keV". Also I suspect mu is not relativistically
correct as it assumes p2 is proportional to Energy. I am
surprised the bounce period does not depend on local time.
5) On page 7 the mirror equation should have sin's squared.
6) Page 3. Selescick and Blake did not see phase space densities
uniformely rising with L. Because of the sporadic substorm injections,
their data show a variety of shapes including phase space density
maxima at L~5.
Dr. Robert Sheldon
Center for Space Physics
725 Commonwealth Av
Boston, MA 02215
Geophysical Research Letters
University of Washington
Dear Dr. Winglee,
I have carefully read the referee comments, and am encouraged that
they find no fault with the observational first half of the paper. I
am also quite flattered to have achieved an informal reporting style,
it has been quite difficult to accomplish. However I am troubled that
they believe a pure observational result is of greater value than one
having both observations and explanatory discussion. (I regret having
called such speculation a theory, it is more properly simply a
discussion.) I believe these results are only marginally valuable
unless the discussion and data are presented together. Thus I will do
my best to satisfy Referee A that the speculation is much further
developed than he suspects, and I will plead with Referee B to
overlook my faux pas in conflating two separate acceleration
mechanisms into one, and to regard the discussion section as still
relevant. Clearly the resistance to this work centers on the discussion
section, which we address at length below, however I will address the
issues in the order in which they were presented in the referee
Old 3+4: ``I am asking for further analysis...that shows that this
energization and untrapping process can happen under the reasonable
The process of untrapping can be divided into 2 categories, static and
dynamic. Static untrapping occurs when adiabatic particle drifts put
the particle in a location where one of the adiabatic invariants
breaks down. The drift loss cone of Roederer 1970 is such an example.
Dynamic untrapping occurs when a time-variable system destroys one of
the adiabatic invariants, pitch-angle scattering into the loss cone is
an example. We consider both in detail below:
Our mapping of particle trajectories in a 1996 Tsyganenko magnetic
field shows that the radial limit of stable trapping shrinks with
energy. Defining this radial limit as the cusp ``L-shell'', the
distance of the ``cusp equator'' (defined in the text as the tangency
of the field line with the curves of constant |B|-magnitude) from the
center of the cusp (defined as the minimum |B| point near the
magnetopause) we can show that the ``hole'' in the center of the cusp,
where particles are untrapped due to the extreme curvature of the
field line passing near the center of the cusp, expands with
energy. Simultaneously the largest cusp L-shell decreases with energy,
so that the annulus of trapped cusp L-shells shrinks in area with
increasing energy, eventually shrinking to zero area at around 5
MeV. The reason for this is obvious, as the gyromotion of the
particles becomes comparable to the cusp size, it no longer is trapped
by the cusp gradients.
I have generated several gigabytes of trajectory files (which were
used to make figure 2.) and from which I have obtained this result.
It was gratifying to see that limit of E>5 MeV untrapping corresponds
very well with the 2 MeV cutoff visible in the data. However the
presentation of these results appears to be too lengthy for the space
available in this GRL manuscript. If I can beg the referee to accept
it on faith, I will elaborate in much greater detail in a second
Invoking pitch-angle scattering as an untrapping mechanism is always
possible, particularly in the cusp with its large broadband
electromagnetic noise spectrum. This is particularly valid because a
particle with a 60 degree pitchangle in the cusp will have a much
more trapped ~90 degree pitchangle in the tail, owing to the large
increase in magnetic field seen between the cusp and the drift-orbit
connected location in the tail. That is, the cusp-loss cone is the
dipole-trap, not the dipole-loss cone. However we prefer to discuss
the dynamic untrapping of the mirror-point magnetic field. That is,
the effect of weakening the magnetic field strength near the nose of
the magnetosphere. The first mechanism would produce a steady drizzle
of energetic particles into the quasi-trapping regions near the nose
of the magnetosphere. The second mechanism would produce a veritable
The first mechanism might explain the radial profiles of the radiation
belts during ``quiet'' periods. The second would explain the sudden
increases typified by the January 10, 1997 storm. Since the existence
of pitch-angle scattering is well accepted, I will suppose that the
referee will grant me a steady drizzle. Turning my attention then to
the flood, we need to consider carefully what happens to a trapped particle
when the boundary conditions are changed rapidly, i.e., do the particles
stay fixed in space and the field-lines change their mapping, or do the
particles follow the rapidly moving field line?
We can sidestep some of this difficulty by recognizing that a particle
undergoing a ``bounce'' is following an L-shaped trajectory that is
both perpendicular and parallel to the magnetopause. The magnetopause
responds to a sudden decrease in SW pressure by retreating outwards
and simultaneously decreasing its Chapman-Ferraro currents. Now when
the trapped particle is closest to the earth, it is on a field line
essentially perpendicular to the magnetopause. Thus cusp currents do not to
first order affect the field line on which the particle is
gyrating. Rather, the circular currents on the magnetopause form a
dipolar field that decays rapidly with distance and to second order
imperceptibly weakens the |B|-magnitude of the field-line on which the
particle is gyrating. Thus we are safe in saying that the sudden decrease
in SW pressure to first order has no effect on this class of particles.
Now as this particle moves along its field line out toward the cusp,
it finds that the cusp has retreated, but nonetheless passes the cusp
with slightly smaller pitchangle and moves outward along the
magnetopause (see the lily-orbit plot). However the weaker C-F
currents in the magnetopause, (which create the dayside maximum to first
order), make too small an increase in the field to force this particle
to mirror, and so it is lost to the cusp and rejoins the dipole trap
particles. Thus motions of the magnetopause can untrap particles on a
bounce timescale, leading to a flood of exiting particles.
One hardly needs to run the T96 code to make the discovery that the
dayside maximum in the |B|-magnitude is dependent on SW pressure. Thus
I have merely claimed this result without plotting it.
The evidence that energization is occurring is inescapable, with phase
space densities in all regions surrounding the cusp lower than the
cusp, it is clear that the cusp must energize the particles (or
violate diffusion). The ability for fluctuations to energize particles
is well accepted in the trapped dipolar regions of the magnetosphere
(e.g. radiation belt diffusion), so I would suppose that the referees
have no difficulty in accepting standard resonant acceleration
mechanisms once they accept the existence of the three adiabatic
invariants. On the other hand, the proposition of chaotic acceleration
requires some further elaboration.
I have shown a table of adiabatic periods of the motion in and around
the cusp. We have also plotted (but not displayed) the magnetic power
spectra from the POLAR/MFE magnetometer taken during cusp passes that
show both large power and peaks (~100 mHz) in this frequency
range. However I find this evidence inconclusive since it is well
known that the cusp is a region having broadband noise (one of the
best ways to identify the cusp before POLAR/PWI failed). Nor is there
a well-established lower limit on the size of the fluctuations needed
to drive the diffusion. Nor have their been correlations between the
degree with which the resonances overlap with the magnitude of the
fluctuations needed to drive the orbits chaotic. In short, I cannot
present a complete theory of chaotic acceleration in four pages, nor,
I believe in 11 pages of JGR, but I CAN show that the conditions in
the cusp are completely applicable to the theory.
Since there is a shortage of alternative theories of ORBE
acceleration, it would appear to me that the burden of proof is
considerably lightened E.g., an initial discovery paper need not be
the final word on the subject, only that it present an observationally
consistent theory that in relation to its competition, deserves
mention. Thus my goal is merely to show that the speculation is
observationally consistent. In view of the Referee's objection, I
have attempted to indicate that this is a speculative agreement.
I have modified the text to be in better agreement with both Selesnick's
paper and my deductions.
I concluded that this was difficult to show from my data presentation and
completely extraneous to the argument. So I have deleted it.
0) Magnetic Pumping?
The Referee is entirely correct in rejecting my previous draft. I had
never intended to include a section on ``resonant acceleration
mechanisms'' but being prompted by the referee I added this section
too quickly, erroneously conflating a Bohm-diffusion mechanism with a
magnetic pumping mechanism, which I had thought was applicable to
both. Please overlook my previous explanation, and try to give the
present explanation a fresh reading. The sole purpose of this
paragraph is to demonstrate by way of example a type of resonant
acceleration, since resonant processes are commonly applied to trapped
particles. They are also well accepted mechanisms for violating the
three adiabatic invariants.
( beginning of parenthetical remarks)
For the sake of my pride, the mechanism I described though not
``magnetic pumping'' works as follows:
(end of parenthetical remarks)
1) Three adiabatic invariants, e.g., Closed drift orbits are present.
2) Large scale electric field (scalar potential) is present.
Then the corollaries to this are:
3) the total energy can be represented by Ho = mu Bm + q U
where mu is the first invariant, q is the charge, Bm is the mirror
point magnetic field strength, and U is the electrostatic scalar potential.
4) Then drift from point P to point Q, say, into a region of smaller U
requires larger Bm such that mu Bm(P) + qU(P) = mu Bm(Q) = qU(Q)
5) Pitch-angle scattering changes mu but in such a way that
mu * Bm = constant
Now let us use this information in an example:
1) Let point P and Q be two locations in a trap (dipole or cusp)
with a linear potential field and circular |B| potentials.
2) Let Q be such that it is a tangency point between constant
|B|-surfaces and constant U surfaces. Travel from the tangency
point along surfaces of constant U leads to weaker |B| in the
case of a dipole, but stronger |B| in the case of a cusp trap.
3) Let the particle begin with Bm=B0 e.g. a 90 degree pitchangle
at point P with higher U and drift toward Q at lower U. U(P)>U(Q).
4) Motion from higher U(P) to lower U(Q) requires Bm(Q) to be
larger. In the cusp, this means that the cusp L-shell is
increasing. E.g., the particle spirals outward.
5) At point Q, let pitch-angle scattering reduces mu to an
arbitrarily small value, increasing Bm to a very large
value. (e.g., converting E-perpendicular into E-parallel.)
6) Let the particle then drift from point Q to point R.
7) From the energy equation, one can see that for very small
values of mu, the particles essentially remain at constant U.
At point Q, this implies that the particle must drift to a
region of stronger equatorial |B|-magnitude. |B_0(R)| > |B_0(Q)|
8) Let pitch-angle scattering then change the pitch-angle entirely
back to 90deg particles. We argue that mu Bm(R) > mu Bm(Q).
1b) Now let the particle drift from point R to S, where S is a
tangency point like Q. The system can then continue back at
step 4, tracing out a zig-zag path as it diffuses in L-shell and
In this way, a particle can zig-zag toward larger Bm, without the need for
a fluctuating electric field which spirals the particles outward, picking
up energy in the process. The purpose of this demonstration was to show
that violation of any of the adiabatic invariants can be used to
1) Cusp preceding belt?
The observations of cusp particle enhancements preceding belt
enhancements in the abstract are valid, though clearly require
additional work in the text. They were observed using movies of the
cusp and belts, which is quite difficult to reproduce in the text. I
have used multiple frames from the movie to make this
point, but at a cost of a lot of real estate in an already crowded
2) Caption to figure 1?
There IS a caption, which refers to the text. Since the description of this
data is crucial to the flow of the argument, it made more sense to include
it in the body of the text. For the sake of brevity, it was not duplicated
in the caption.
The error is mine. keV was meant. The energies were corrected for rest
mass, e.g., the particle trajectories were traced with a proper
relativistic velocity. Mu was taken to be the ``natural''
non-relativistic value, e.g., the perpendicular component of energy as
would be measured by a solid-state detector. This was not corrected
for relativistic mass changes. It would be easy to redefine it, though
for illustrative purposes, its exact value is of lesser interest than
It has been corrected.
The comments here are similar to those of Referee A. Please read the
referee A comments first.
Referee B gives an interpretation that does not exist in the Selesnick
paper itself. Referee B interprets the ``variety of shapes'' to be
``because of the sporadic substorm injections''. The connection of
substorms to outer radiation belt injections is not made in the paper
itself, Selesnick carefully avoids using the word ``substorm''
anywhere. I hope that the Referee merely meant ``particle
injections'', in which case I am in complete agreement with the
referee. I have tried to clarify my position: a time-variable
external source, which I reiterate, is in complete agreement with
Selesnick and Blake.
Let me restate this. If one posits an internal source (a local
acceleration that generates a peak in the trapping region, 210. Since decay is constantly occurring, the
quiet-time signatures must represent a steady source and not just a
transient decay signature, since the steady-state for a sourceless
population is zero flux. Therefore, if one posits an internal
acceleration mechanism for storms, one must posit a second, external
acceleration mechanism to explain the quiet-time fluxes.
A single external, time-variable source can account for both storms
and quiet periods. Therefore by Occam's Razor (what can be done with
fewer assumptions is done in vain by more), a single source appears
preferable to a double source.