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2. Creating a Dataset of Phase Space Density and Identifying Errors
2.0 Introduction
Appendix A demonstrates how to avoid time periods when the HIST electron flux
measurements are anomalous. This chapter will also focus on removing unwanted signal
from the data. Specifically, we discuss how to transform HIST electron flux measured as
a function of energy and position to phase space density as a function of the three
adiabatic invariants. The transformation removes signal generated by the “adiabatic
effect” also known as the “Dst effect”. This effect causes large changes in electron flux
measured at fixed energy and position but cannot account for flux enhancements that
occur during the recovery phase of some storms [Kim and Chan, 1997; Li et al., 1997].
Since the goal of this study is to understand storm time flux enhancements, we must
remove flux changes due to the adiabatic effect from our data. This chapter describes the
adiabatic effect, how it generates large flux changes, and how we remove these changes
from our data. The chapter concludes with a discussion of potential errors introduced by
the transformation to phase space density. The chapter gives methods for estimating
errors and discusses how errors can affect results.
2.1 The adiabatic effect
The adiabatic effect was first introduced by Dessler and Karplus [1961] to
explain the relativistic electron flux changes during geomagnetic storms. Surprisingly,
high energy relativistic electrons behave differently during geomagnetic storms than low
energy electrons. During a storm, relativistic electron flux is correlated with the Dst index
while low energy particle flux is anti-correlated [Lyons and Williams, 1976; Williams,
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1981]. The flux of low energy electrons in the inner magnetosphere increases during the
storm main phase. The enhanced particle flux increases currents that contribute to the
decrease in Dst [Lui et al., 1987]. During the storm recovery phase, low energy particle
flux decreases and Dst increases back to its pre-storm levels. Typically, relativistic
electron flux correlates to Dst in the opposite manner. Both relativistic electron flux and
Dst decrease during the storm main phase and increase during the recovery phase.
Dessler and Karplus [1961] proposed that the relativistic electron flux changes were
caused by electrons moving to conserve all three adiabatic invariants, even the third
invariant, in response to changes in Dst.
The theory, commonly referred to as the Dst effect, applies only to high energy
electrons because low energy electrons do not conserve the third invariant during storms.
The third invariant is conserved when time variations of the magnetic field are slow
compared to the drift period of a particle. Electrons conserving the third invariant satisfy
the following inequality [Roederer, 1970]:
 drift dB
B dt
 1
(1)
where B is the field magnitude, dB/dt is the time rate of change of the field magnitude,
and drift is the drift period of the particle. Rapid magnetic field changes occur during the
main phase of a storm at a typical rate of ~20nT/hour. Using this value in the above
equation shows that at L=7 only particles with drift periods <~30 minutes conserve their
third invariant. A drift period of 30 minutes corresponds to an electron with energy ~.7
MeV. Therefore, during a storm electrons with energy >.7 MeV are controlled by
adiabatic motion. Assuming no other loss processes, relativistic electrons conserve the
third invariant throughout the storm and react adiabatically to changes in Dst.
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Relativistic electron flux variations occur in response to Dst changes as follows.
As Dst decreases during the main phase of the storm, electrons conserving their third
invariant move outward to compensate for the reduction of field magnitude and still
enclose the same magnetic flux in a drift orbit. A spacecraft at fixed radial distance then
measures the flux of electrons previously present at lower L. The outward moving
electrons are now in a weaker magnetic field and to conserve the first adiabatic invariant,
, their energy must decrease. The distribution of electrons shifts to lower energy as
illustrated schematically in Figure 1. A spacecraft measuring flux at fixed energy, E0, and
fixed radial distance measures the flux of electrons initially at a lower L value shifted to
lower energies. Generally, the outward movement results in a decrease of electron flux at
fixed energy during the main phase of the storm. Any change in electron flux due to the
adiabatic response is reversible assuming no loss or addition of new electrons. As the
magnetic field returns to its pre-storm value the electrons move back to their original
position and the fluxes recover to pre-storm levels.
2.2 Calculating phase space density
We can account for adiabatic flux changes by converting HIST flux


measurements, j ( E , x ,  , t ) , measured as a function of energy (E), position ( x ), pitch
angle () and time to phase space density as a function of the adiabatic invariants and
time, f(,K,L*,t). Liouville’s theorem states that phase space density of electrons with
fixed adiabatic invariants is unaffected by slow temporal changes of the magnetic field
and thus is insensitive to the Dst effect . Time changes of f(,K,L*,t) result only from
new sources or losses of electrons. Electrons accelerated by violation of one or more
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adiabatic invariants constitute a new source of electrons. It is these new sources and
acceleration processes that we hope to understand and that will be illuminated in the
phase space density dataset.

Converting the data from j ( E , x ,  , t ) to f(u,K,L*,t) requires multiple steps and
computer intensive calculations. Briefly, the method restricts measurements to only those
electrons with energy and pitch angle corresponding to fixed values of , K, and L. The
transformation requires calculating the three invariants, , K, and L* defined in the
introduction along each point of the Polar orbit. The method is broken into the four basic
steps outlined below and followed with complete descriptions.
Step 1: Find the particle pitch angles corresponding to constant K along the Polar orbit by
tracing particle motion in a model magnetic field.
Step2: Find the particle energy corresponding to constant  at each point of the orbit.
Step 3: Find the phase space density at the pitch angle and energy of constant K and by
interpolation.
Step 4: Determine L* by calculating the magnetic flux enclosed in the drift path of a
particle with pitch angle corresponding to constant K. The electron drift path and the flux
enclosed are calculated in a model magnetic field.
2.2.1 Step 1:
Our method of transforming the data begins with the calculation of K. The
invariant K depends on the mirror magnetic field and so varies with particle pitch angle.
To measure phase space density of electrons at fixed K throughout the magnetosphere
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means to measure particles with only specific pitch angles at the observation location. To
find the pitch angles corresponding to fixed K, we calculate K values for local pitch
angles from 5o to 90o at 5o intervals along the orbit. We use the Tsyganenko 96 (T96)
field model [Tsyganenko, 1996] and the UNILIB code provided by the Belgian Institute
for Space Aeronomy available on the web at http://www.magnet.oma.be/home/unilib to
do the calculations. We create a table containing 18 K values at 20 second time steps
along the orbit. Interpolation is done between these values to find the pitch angles
corresponding to a given fixed K.
The T96 model used in the calculations varies depending on the value of required
input parameters. These input parameters are solar wind By, Bz, dynamic pressure, and
Dst. We use the omni dataset as input to the model. The model is valid for only a
specified range of solar wind and Dst values. These ranges are 0.5 to10 nPa for dynamic
pressure, -100 to 20 nT for Dst, -10 to10 nT for IMF By, and –10 to10 nT for IMF Bz. The
measured solar wind or Dst values may fall outside these limits especially during the
main phase of a storm,. When this happens we use the maximum or minimum allowed
value instead of the measured value. The parameters fall outside of the valid model range
only 4% of the time.
2.2.2 Step 2:
The second step is to find the particle energy that corresponds to fixed . This is
done using the equation for  to solve for the momentum p:.
p2 
2m0 B
sin 2  k
(2)
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Here  is chosen to be a fixed value and k is the pitch angle of fixed K found in step 1.
The relativistic momentum can be written in terms of energy as,
p2=(E2+2moc2E)/c2
(3)
where E is kinetic energy and c is the speed of light. We solve for the kinetic energy by
combining the two equations. The energy of a particle with fixed  depends on magnetic
field strength. We use the magnetic field strength measured by the MFE instrument
[Russell et al., 1995] onboard Polar to determine the energy.
2.2.3 Step 3:
We obtain phase space density from the HIST flux measurements using the


following relation, f ( E, x, t ,  )  j ( E, x, t ,  ) / p 2 [Schulz and Lanzerotti, 1974]. Phase
space density of electrons at fixed and  is found by determining f of electrons with
pitch angle and energy calculated in steps 1 and 2. Because the HIST instrument
measures discrete pitch angles and energies interpolation is necessary. We find phase
space density measured at the pitch angle calculated in step 1 by fitting the pitch angle
distribution of each energy channel using a downhill simplex minimization routine to the
following function.
j ( )  C0 sin   C1 sin C2 
(4)
This form was used because it fits both butterfly and highly peaked distributions. From
the fitted data we obtain phase space density of electrons with pitch angle of constant K,

f ( E , x , K , t ) ,at 14 discrete energies. We fit the phase space density as a function of
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energy to an exponential function. The fit is used to find the phase space density of
electrons at the energy of constant determined in step 2.
2.2.4 Step 4.
Lastly, we calculate the third adiabatic invariant, L*. This computationally
intensive calculation requires tracing the drift of an electron around the entire
magnetosphere. This drift is slightly dependent on the pitch angle of the particle. We use
the pitch angle determined for constant K to do the calculation. As with the second
invariant calculations we use the T96 model and the BISA UNILIB code.
2.3 Errors.
Errors in our calculation of phase space density arise from two sources: poor data
fits and imperfect magnetic field models. Poor data fits (step 3) occur because our chosen
functional forms may not always accurately represent the data. In addition, during low
count periods uncertainties in the measurements become large and result in poor fits.
Phase space density errors introduced from poor data fits are quantified by the standard
error of the fits to the data. Imperfect magnetic field models result in inaccurate estimates
of L* and K. Our goal is to understand how incorrect estimates of L* and K affect the
calculated phase space density.
2.2.1 Errors of Poor Fits
Step 3 in our method requires that we fit our measured data to functional forms.
We quantify the error in each of these fits by the standard error given below
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s

n

1
2
 (d fit  d measured )
n 1
(5)
where s is the standard error, n is the number of data points,  is the standard deviation,
dfit is the estimate of the fit, and dmeasured is the measured data.
2.2.1 L* errors.
L* depends inversely on , the total magnetic flux enclosed by the drift orbit of
an electron, and so is affected by the global accuracy of the magnetic field model. An
imperfect field model will change L* in predictable ways. If the field model
underestimates , the calculated L* will be larger than expected. Likewise, if the field
model overestimates , L* will be smaller than expected. Therefore, errors in L* simply
shift the calculated phase space density radially as shown by the cartoon of Figure 2.
The cartoon depicts three scenarios. In the first scenario, the field model
systematically underestimates  and the calculated phase space density versus L* profile
shifts to larger L*. In the second scenario, the field model systematically overestimates 
and the calculated phase space density versus L* profile shifts to smaller L*. In the final
scenario the model overestimates  at small radial distances and underestimates it at
large radial distances. This type of error stretches the phase space density versus L*
profile. Inaccurate calculation of L* does not change the value of phase space density.
The errors only shift the value radially.
We estimate the error by analyzing how L* changes as the field model is
modified. Changing the input parameters modifies the field model. Over the range of
distances relevant to this study, the L* parameter is mostly controlled by the Dst input to
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the model. Therefore we test the dependence on the field model by calculating L* for
electrons at a range of radial distance and input Dst values. Figure 3 plots L* values
calculated for electrons at X=-8-8, Y=0, and Z=0 in magnetic coordinates using Dst
values from 20nT to –100 nT at 10nT intervals. Circles mark L* values for electrons with
90o equatorial pitch angle and asterisks mark L* values for electrons with 20o equatorial
pitch angle. The colors of the symbols shows the Dst value used as input to the model.
Blue symbols mark L* values calculated using the minimum input value of –100 nT and
red symbols mark L* values calculated using the maximum input value of 20 nT.
Before discussing the relevance of the plot to calculating errors we describe some
expected features of the plot. These features are worth noting because they clarify the
meaning of the L* parameter and its dependence on particle position and pitch angle. For
example, the plot shows that calculated L* values decrease with decreasing Dst at all
radial distances. This feature is expected because the L* parameter gives the radial
distance of an electron when all external magnetic fields are turned off leaving only the
dipole field. If the magnetic field measured by Dst were turned off, electrons would move
radially inward to conserve their third invariant. The more negative the initial Dst
perturbation the farther inward an electron moves and the smaller its L* parameter is. The
opposite is true for positive Dst perturbations.
Another noticeable feature is that inside of X=4, L* is not highly dependent on
Dst. This is expected because at small radial distance the dipole field of the earth
dominates over any perturbations measured by Dst.
One final interesting feature is that L* calculated for 90o and 20o pitch angle
particles differ systematically. The plot shows that on the dayside, L* of a 90o pitch angle
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particle is smaller than L* of a 20o pitch angle particle for the same input Dst value. This
feature results from the compression of the magnetic field on the dayside and stretching
of the tail on the nightside that causes particle motion known as drift shell splitting
[Roederer, 1967]. Drift shell splitting predicts that electrons with 90o pitch angle move
radially outward in the compressed field region to remain at constant magnetic field
strength and conserve the first invariant. The particles with 20o pitch angle, on the other
hand, move radially inward to conserve the second invariant. If the dayside magnetic
field compression is turned off leaving only the dipole field, the 90o electron moves
radially inward and the 20o particle moves outward. Therefore the L* parameter of the
90o particle is smaller. On the nightside of the magnetosphere particles are affected by
drift shell splitting in exactly the opposite manner. Here the equatorial field strength
decreases. Particles with 90o pitch angle move inward and those with 20o pitch angle
move outward. The plot shows L* of 90o particles is larger than that of 20o particles as
expected.
We now discuss how to use the information plotted in Figure 3 to estimate errors
in L*. Figure 3 shows that the largest range of L* values at any radial position is
L*=2.5. As an estimate of the error we assume a range equal to 60% of this maximum.
In other words, our final values of L* are given as L*  0.8. The error in phase space
density is the minimum and maximum value calculated within this range of L* values.
2.2.2 K errors.
Calculating K requires integration along a magnetic field line. The integral
depends on the length of the field line and the strength of the magnetic field along the
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field line. The K calculation is compromised by an imperfect field model. The phase
space density is affected by an inaccurate estimate of K because inaccurate estimates of K
will change the pitch angle determined in step 2 and used in step 3 to find phase space
density. Figure 4 demonstrates how erroneous estimates of K affect the phase space
density calculations. Panel A is a schematic showing phase space density as a function of
K. Large K values correspond to small pitch angle values. Therefore, phase space density
typically decreases as K increases as shown. If the model overestimates K values, the
phase space density trace shifts to the right. The phase space density of a chosen fixed K
will be overestimated. Likewise, if the model underestimates K the phase space density
trace shifts left. The phase space density of fixed K will be underestimated. K and phase
space density will be overestimated when the stretching of the model is too large.
Likewise K will be underestimated when the stretching is too small.
With this understanding we estimate a range of probable phase space density
values by modifying the field model and analyzing the change in the calculation of pitch
angles of constant K. It is not intuitively clear which input parameters will most affect the
calculation of K. We modify each input parameter separately to determine which input
parameter has the greatest effect.
We begin by describing how different Dst inputs to the model affect the
calculation of pitch angles of constant K. To understand the effect we calculate pitch
angles of fixed K for electrons with positions Y=0, Z=0 and X=-8-8 for the full range of
Dst inputs from 20nT to –100nT at 10 nT intervals. The other four input parameters are
kept at the following constant values, Bz=0 nT By=0 nT and dynamic pressure=.5 nT.
Figure 5 plots the results for three values of K and shows several notable features. First,
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the dependence on Dst is most apparent at large radial distance and hardly noticeable in
the inner magnetosphere. This feature is expected because the earth’s dipole field
dominates the inner magnetosphere. Also, the effect of Dst is most noticeable at small K
values. This feature is understood because changes in Dst predominately modify the
magnetic field at the equator. Particles with small K reside near the equator and are most
likely to be affected. In addition, there are differences between the day and night side
calculations. The calculations on the night side vary more with changes in Dst.
Figure 4 demonstrates that the pitch angle corresponding to constant K depends
significantly on Dst especially for small K values. The pitch angles on the nightside at
X=-7 and K=100 range from 20o when Dst=-120 to 70o when Dst=20. However, Polar
spends much of its time off the equator measuring only particles with large K values. The
bulk of the analysis in this thesis relies on particles with K>1000. At these K values the
maximum range of pitch angles for models with different Dst input values is ~20 degrees.
Next we look at the effect of Bz on the model field and calculation of pitch angles
of fixed K. Figure 5 shows the effect of Bz on the pitch angles corresponding to constant
K and is plotted using the same format as Figure 4 discussed above. For these
calculations we vary the Bz input value from –10nT to 10nT while holding the other input
variables constant at By=0 nT, Dst=0 and dynamic pressure=0.5 nPa. Comparing Figure 5
to Figure 4 reveals that the pitch angles are less dependent on the Bz input than the Dst
input. The maximum range of pitch angles at a given L* in Figure 5 is only 8o.
Varying By has an even smaller effect on the calculation of pitch angles of fixed
K as shown by Figure 6. In this figure By varies from –10 to 10 nT at 2 nT intervals while
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the other input parameters are held constant at Bz=0, Dst=0 and dynamic pressure=0.5
nPa. Varying By only changes the pitch angle by 2o.
Lastly we analyze changes in the pitch angle corresponding to fixed K relative to
variations of dynamic pressure. Figure 7 plots the dependence of the pitch angle on
dynamic pressure. In this figure dynamic pressure varies from 0.5-10 nPa. The maximum
range of pitch angle calculated using different input dynamic pressures is 10o. The change
is less than that observed when varying Dst. It is interesting to note that increasing the
dynamic pressure affects the day and night side pitch angles of constant K in an opposite
manner. On the dayside, increasing the dynamic pressure causes the pitch angle of
constant K to also increase. On the nightside, increasing dynamic pressure causes the
pitch angle of constant K to decrease. The differences are explained as follows. A crude
approximation of K is K=[Bmirror-Beq] 1/2 s where Bmirror is the mirror point magnetic field,
Beq is the equatorial magnetic field and s is the field line length between mirror points.
Changing dynamic pressure changes the field magnitude. Assuming the field strength
changes nearly uniformly along the field line yields
Kcompressed=C[Bmirror-Beq] 1/2 s
(6)
where C is the compression factor. However, changing the field uniformly does not
change particle pitch angles, eq=asin(Bmirror/Beq). Therefore, particles with the same
equatorial pitch angles now have Kcompressed=CKuncompressed. On the dayside, the
compressed field increases and C>1. For these particles Kcompressed is greater than
Kuncompressed. Therefore the pitch angle of constant K also increases. On the nightside,
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compressing the field has the opposite affect. In this region the field decreases, C<1, and
Kcompressed is less than Kcompressed. Here the pitch angle of constant K decreases.
Dst is the input parameter that most affects the calculation of pitch angles
corresponding to fixed K. To analyze how incorrect estimates of K affect our results we
calculate pitch angles corresponding to constant K and the corresponding phase space
density using different input values to the T96 field model. We calculate pitch angles
corresponding to constant K using the T96 model with four different sets of input values.
We calculate phase space density using the measured solar wind and Dst values as input
to the model. This value of phase space density gives the most likely value. In addition
we calculate phase space density using Dstmeasured +40 nT and Dstmeasured –40nT as inputs
to the model. We chose to use Dst +/-40nT because it covers 60% of the range of
allowable input values. Lastly we calculate phase space density using input values of Dst
and dynamic pressure that minimize the difference between the magnetic field measured
locally at Polar and the model predicted magnetic field. We find the best parameters by
stepping through all allowable Dst and dynamic pressure values from –100 to 20 nT and
0.5-10 nPa.
2.4 Application of Error Analysis Throughout
We have identified three sources of errors in our development of the phase space
density dataset; errors from poor data fits, errors from the calculation of L*, and errors
from the calculation of K. Errors from poor data fits are quantified by the standard error
of the measured data to the fit data. Errors in L* and K are most dependent on the Dst
input to the model. Changing the other input parameters, By, Bz, and dynamic pressure
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cause less significant changes to phase space density. Phase space density errors will be
most important when analyzing the changing phase space density versus L* profile as is
done in chapter 4.
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References
Dessler, A.J., R. Karplus, Some effect of the diamagnetic ring currents on Van Allen
Radiation, J. Geophys. Res., 66, 2289, 1961.
Kim, H.-J. Chan, A. A., Fully adiabatic changes in storm time relativistic electron fluxes,
J. Geophys. Res., 102, 22107, 1997.
Li ,X., D. N. Baker, T. E. Cayton, G. D. Reeves, R.A. Christensen, J. B. Blake, M. D.
Looper, R. Nakamura, S. G. Kanekal, Multisatellite observations of the outer zone
electron variation during the Noveber 3-4, 1993, magnetic storm, J. Geophys. Res., 24,
923, 1997.
Lui, A.T.Y, R.W. McEntire, S. M. Krimigis, Evolution of the ring current during two
geomagnetic storms, J. Geophys. Res., 92, 7459, 1987.
Lyons, L. R., D. J. Williams, Storm-associated variations of equatorially mirroring rin
current protons, 1-800 keV, at constant first adiabatic invariant, J. Geophys. Res., 81,
216, 1976.
Roederer, , J. G., On the adiabatic motion of energetic particles in a model
magnetosphere, J. Geophys. Res., 72, 981, 1967.
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Roederer, J. G., Dynamics of Geomagnetically Trapped Radiation, 166 pp., Cambridge
Univ. Press, New York, 1970.
Russell, C. T., R. C. Snare, J. D. Means, D. Pierc, D. Dearborn, M. Larson, G. Barr, G.
Le, The GGS/Polar magnetic fields investigation, Space Sc. Rev. 71, 563, 1995.
Schulz, M., L. J. Lanzerotti, Particle Diffusion in the Radiation Belts, vol. 7 SpringerVerlag, New York, 1974.
Tsyganenko, N.A., and D. P. Stern, Modeling the global magnetic field of the large-scale
Birkeland current systems, J. Geophys. Res., 101, 187, 1996.
Williams, D. J., Phase space variations of near equatorially mirroring ring current ions, J.
Geophys. Res., 86, 189, 1981.
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Figure Captions
Figure 1.Schematic showing how the Dst effect changes flux measured at fixed energy
and position. The black trace shows electron flux as a function of energy prior to the
storm. The gray trace shows the electron flux during the main phase of a storm as
electrons move outward to conserve all three adiabatic invariants. The adiabatic motion
generally causes a decrease of flux measured at fixed position and energy, E0.
Figure 2. Schematic showing how imperfect field models affect the calculation of L*.
Panel A shows the affect on phase space density when the model gives an underestimate
of the third invariant . Panel B shows the affect on phase space density when the model
overestimated . Panel C shows the affect on phase space density when the model
overestimates  at small L* and underestimates  at large L*.
Figure 3. Plot demonstrating how changing the input Dst parameter to the T96 model
affects the calculation of L*. The L* value is calculated over a range of radial distances
from X=-8 to 8 in magnetic coordinates. The L* values are color coded by the Dst input
value used in the calculation. Circles mark L* values calculated assuming an equatorial
pitch angle of 90o and asteriks mark the L* values calculated assuming an equatorial pitch
angle of 20o.
Figure 4. Schematics demonstrating how an over- or underestimate of K affects the phase
space density calculations. Panel A shows the affect of overestimating K. The black trace
shows a typical profile of phase space density as a function of K. If the model
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overestimates K the profile shifts to higher K as shown by the gray trace. Phase space
density of constant K is overestimated by the model predictions. Panel B shows the affect
of underestimating K. The black trace shows a typical profile of phase space density as a
function of K. If the model underestimates K the profile shifts to lower K as shown by the
gray trace. Phase space density of constant K is overestimated by the model predictions.
Figure 5. Three plots demonstrating how changing the Dst input parameter to the T96
model affects the pitch angle of constant K. Pitch angles are calculated for three different
K values over a range of radial distances from X=-8 to 8 in magnetic coordinates. The top
plot shows pitch angles of K=10000 G 1/2 km. The middle plot shows pitch angles of
K=1000 G 1/2 km and the bottom plot shows pitch angles of K=100 G 1/2 km. The colors
identify the Dst value used as input to the model.
Figure 6. Three plots demonstrating how changing the Bz(nT) solar wind input parameter
to the T96 model affects the pitch angle of constant K. Pitch angles are calculated for
three different K values over a range of radial distances from X=-8 to 8 in magnetic
coordinates. The top plot shows pitch angles of K=10000 G 1/2 km. The middle plot
shows pitch angles of K=1000 G 1/2 km and the bottom plot shows pitch angles of K=100
G 1/2 km. The colors identify the Bz(nT) value used as input to the model.
Figure 7. Three plots demonstrating how changing the By (nT) solar wind input
parameter to the T96 model affects the pitch angle of constant K. Pitch angles are
calculated for three different K values over a range of radial distances from X=-8 to 8 in
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magnetic coordinates. The top plot shows pitch angles of K=10000 G 1/2 km. The middle
plot shows pitch angles of K=1000 G 1/2 km and the bottom plot shows pitch angles of
K=100 G 1/2 km. The colors identify the By (nT) value used as input to the model.
Figure 8. Three plots demonstrating how changing the solar wind dynamic pressure (nPa)
input parameter to the T96 model affects the pitch angle of constant K. Pitch angles are
calculated for three different K values over a range of radial distances from X=-8 to 8 in
magnetic coordinates. The top plot shows pitch angles of K=10000 G 1/2 km. The middle
plot shows pitch angles of K=1000 G 1/2 km and the bottom plot shows pitch angles of
K=100 G 1/2 km. The colors identify the dynamic pressure (nPa) value used as input to
the model.
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