# Radiation Belt Response to Fast Reverse Shock at Geosynchronous Orbit

Ankush Bhaskar NASA/Goddard Space Flight Center, Greenbelt, Maryland, United
States  University Coperation of Atmospheric Research, Boulder, United States
Space Physics Laboratory, Vikram Sarabhai Space Centre, Thiruvananthapuram,
India  David Sibeck  NASA/Goddard Space Flight Center, Greenbelt, Maryland,
United States  Shrikanth G. Kanekal  NASA/Goddard Space Flight Center,
Greenbelt, Maryland, United States  Howard J. Singer  NOAA-Space Weather
Prediction Center,Colorado, United States  Geoffrey Reeves  Los Alamos
National Laboratory,New Mexico, United States  Denny M. Oliveira  NASA/Goddard
Space Flight Center, Greenbelt, Maryland, United States  Goddard Planetary
Heliophysics Institute, University of Maryland, Baltimore, United States  Suk-
Bin Kang  NASA/Goddard Space Flight Center, Greenbelt, Maryland, United States
Colin Komar  NASA/Goddard Space Flight Center, Greenbelt, Maryland, United
States

###### Abstract

Fast reverse shocks (FRSs) cause the magnetosphere to expand, by contrast to
the well-known compressions caused by the impact of fast forward shocks (FFS).
Usually, FFSs are more geoeffective than FRSs, and consequently the inner
magnetosphere dynamic responses to both shock types can be quite different. In
this study, we investigate for the first time the radiation belt response to
an FRS impact using multi-satellite observations and numerical simulations.
Spacecraft on the dayside observed decreases in magnetic field strength and
energetic ($\sim$40-475 keV) particle fluxes. Timing analysis shows that the
magnetic field signature propagated from the dayside to the nightside
magnetosphere. Particles with different energies vary simultaneously at each
spacecraft, implying a non-dispersive particle response to the shock.
Spacecraft located at lower L-shells did not record any significant
signatures. The observations indicate a local time dependence of the response
associated with the shock inclination, with the clearest signatures being
observed in the dusk-midnight sector. Simulations underestimate the amplitude
of the magnetic field variations observed on the nightside. The observed
decreases in the electron intensities result from a combination of radial
gradient and adiabatic effects. The radial gradients in the spectral index
appear to be the dominant contributor to the observed variations of electrons
seen on the dayside (near noon and dusk) and on the nightside (near midnight).
This study shows that even an FRS can affect the radiation belts significantly
and provides an opportunity to understand their dynamic response to a sudden
expansion of the magnetosphere.

Interplanetary shock, Radiation belt, magnetosphere, particle acceleration

##  1 Introduction

Interplanetary (IP) shocks are a recurrent feature of the solar wind (Burlaga,
1971; Richter et al., 1985). Fast mode IP shocks occur when the relative speed
between the ambient solar wind and the shock speed is larger than the local
magnetosonic speed (Landau & Lifshitz, 1960; Richter et al., 1985; Tsurutani
et al., 2011; Oliveira, 2017). Fast IP shocks that are move away from the Sun
are classified as fast forward shocks (FFSs), while fast shocks that propagate
towards the Sun are classified as fast reverse shocks (FRSs) (Burlaga, 1995;
Tsurutani et al., 2011; Oliveira, 2017). Similarly to FFSs, FRSs are carried
antisunward by the continuous solar wind flow as seen from a reference frame
defined as a spacecraft in the IP space or the Earth itself (Richter et al.,
1985; Burlaga, 1995; Tsurutani et al., 2011; Oliveira, 2017; Oliveira &
Samsonov, 2018). FFSs are more numerous during solar maxima, while FRS
occurrence rates have no clear correlation with solar activity(Echer et al.,
2003; Kilpua et al., 2015; Cavus et al., 2019). Kilpua et al. (2015) showed
that the occurrence rates of FFSs are higher than the occurrence rates of FRSs
during all solar phases, except during solar minima.

The steepening conditions across the fronts of FFSs and FRSs are different. In
the case of FFSs, all plasma parameters (particle number density, thermal
temperature, and velocity), along with the interplanetary magnetic field (IMF)
increase. Conversely, in the case of FRSs, all solar wind and IMF parameters
decrease, except the solar wind plasma velocity (Landau & Lifshitz, 1960;
Burlaga, 1971; Richter et al., 1985; Burlaga, 1995; Burguess, 1995; Tsurutani
et al., 2011). See Figure 2 of Oliveira (2017) for comparisons between
schematic profiles of FFSs and FRSs. These distinct shock dynamics are
responsible for different magnetospheric responses to the impacts of FFSs and
FRSs.

FFSs are known to be the most geoeffective class of IP shocks (Echer et al.,
2004; Alves et al., 2011; Tsurutani et al., 2011; Oliveira & Raeder, 2015;
Oliveira & Samsonov, 2018). The first and most dramatic magnetospheric
response to the impact of FFSs are characterized by positive sudden impulses
(SI+) in the horizontal component of the geomagnetic field measured by
magnetometers located in the space (Patel & Coleman, 1970; Huttunen et al.,
2005; Wang et al., 2010; Su et al., 2015; Rudd et al., 2019) and on the ground
(Siscoe et al., 1968; Smith et al., 1986; Takeuchi et al., 2002b; Rudd et al.,
2019), as a consequence of a sudden increase of the solar wind dynamic
pressure (Russell et al., 1994a, b; Echer et al., 2005; Wang et al., 2010;
Rudd et al., 2019). On the other hand, in the case of FRSs, the sudden impulse
response is negative (SI-) as geospace and ground magnetometers observe
negative excursions in the horizontal component of the geomagnetic field
(Akasofu, 1964; Takeuchi et al., 2002a; Andrioli et al., 2006; Zhang et al.,
2010). This is due to magnetospheric expansions caused by solar wind dynamic
pressure decreases such as those associated with FRS/solar wind discontinuity
impacts (Joselyn & Tsurutani, 1990; Araki, 1994; Zhang et al., 2010; Vichare
et al., 2014).

The impact of strong shocks can inject and energize particles into the
radiation belts resulting in transient variation of the particle intensities.
The classic example of such events is the strongest SI+ recorded which
occurred on 24 March 1991 (Araki et al., 1997). The shock associated with this
SI+ formed a double-peaked inner radiation belt which lasted for years (Mullen
et al., 1991; Looper et al., 1994). The radiation belt electron and proton
responses to shocks have been simulated by Li et al. (1993) and Hudson et al.
(1995). Their simulation results showed that electrons of few MeV were
transported from L $>6$ to L $\sim 2.5$ by an inductive electric field which
energizes them up to 40 MeV. A study by Kanekal et al. (2016) showed that
ultrarelativistic electrons with energies $>6$ MeV were injected deep into the
magnetosphere at L $\sim$ 3 within $\sim$ 2 min following the 17 March 2015 IP
shock impact. Schiller et al. (2016) carried out a statistical study of highly
relativistic electron injections caused by IP shock interactions with the
magnetosphere. They observed that not all their shocks show signatures in MeV
electrons; around 25% of IP shocks are associated with prompt MeV electron
energization, and 14% are associated with prompt MeV electron depletions.
Moreover, Zhang et al. (2018) studied shock‐induced electric fields caused by
IP shocks and suggested implications of these impulsive electric fields to the
acceleration and transport of radiation belt electrons. Note that, the
processes responsible for the enhanced/depleted intensities of energetic
particles within the magnetosphere following shock impacts can be understood
in terms of some combination of (i) enhanced particle source populations or
their radial transport and (ii) localized acceleration. Although intense
shocks are relatively rare, their impact on the inner radiation belt can last
for years or even decades. Therefore, they can not be ignored in space weather
and space technology related studies.

As can be seen, there has been considerable interest in studying the
magnetospheric effects of SI+s (caused by FFSs), but FRS interactions with
radiation belt have received less attention. FRSs are generally not strong
shocks however Cattell et al. (2017) showed that even weak FFS can disturbed
the magnetosphere significantly, thus the study of FRS impact on magnetosphere
is important. Just as a compression of the magnetosphere due to FFSs results
in various transient processes, a sudden rarefaction due to FRSs or solar wind
discontinuities can also generate interesting dynamics like generation of
transient currents in magnetosphere and ionosphere (Vichare et al., 2014; Shen
et al., 2017). To the best of our knowledge, there are various studies
reporting on the geomagnetic field response to SI-; however, there is no
comprehensive study of the radiation belt response to a FRSs.

This gap in understanding motivates the present study to investigate an FRS
impact on the outer radiation belt in detail. Although shock impact has been
studied extensively using spacecraft observations and simulations, there are
at least three important problems which still need attention: (i) Do reverse
shocks affect the radiation belts significantly?, (ii) Do the impacts of
reverse shocks on the inner and outer radiation belts simply mirror those of a
forward shocks?, (iii) What are the primary drivers (or causes) responsible
for transient variations in particle fluxes during the shocks? To address
these questions, the present study carries out multi-point spacecraft
observations to evaluate and compare the outer radiation belt responses to the
rarefaction of the magnetosphere during the transit of a reverse shock across
the Earth’s magnetosphere. The observed variations in electron and proton
fluxes are interpreted based on our current understanding of particle dynamics
in the magnetosphere.

Arrival of FRS at the Earth could make the radiation environment more harmful
during and after shock transit. Intensification of high energy particle fluxes
in radiation belt affect orbiting communication and navigation satellites
adversely. If a high-intensity shock similar to the event of 24 March 1991
occurs in near future, its impact on space technology will be severe. The
current study shows that even FRSs can affect radiation belt particles even
though they are generally weaker than FFSs. The results from this study will
assist in the better understanding of underlying physical processes and
prediction of particle variations induced by FRS interactions with the Earth’
magnetosphere. The study has further implications to shock-induced processes
in other planetary magnetosphere of the solar system and exoplanets.

##  2 Data and Methods

We examined the interplanetary shock lists available at
(http://ipshocks.fi/database) and
(https://www.cfa.harvard.edu/shocks/ac_master_data/ac_master_2014.html) to
identify FRSs from 2013-2016, an interval when high quality continuous data
are available from the Van Allen Probes and other spacecraft.

There were a total of 29 FRSs during this period. Here we present a detailed
case study of the single event which strongest event on the list. This FRS was
observed by the Wind spacecraft on 6 December 2014, at 21:09 UT, the
spacecraft was located upstream in the solar wind at the time of the shock
impact. To study the magnetosphere response GOES-13, GOES-15, LANL-97A,
THEMIS, and Cluster spacecraft are used.

The Wind Magnetic Field Investigation (MFI) instrument, consisting of two
fluxgate magnetometers, was used for IMF observations (Lepping et al., 1995).
The Wind Solar Wind Experiment (SWE) Faraday Cup - Ion Data was used to obtain
information on solar wind plasma conditions (Ogilvie et al., 1995). We use the
MAGNETOMETER (MAG) and MAGnetospheric Electron Detector (MAGED) instrument on
GOES-13 and 15 to understand the response at geosynchronous orbit (Hanser,
2011; Rodriguez et al., 2014). MAGED is a set of nine collimated solid state
detectors, data are provided as differential electron fluxes for 40, 75, 150,
275, and 475 keV midpoint energies from nine telescopes. The Synchronous Orbit
Particle Analyser (SOPA) on the Los Alamos space environment monitor LANL-97A
in geosynchronous orbit ($\sim 6.6R_{E}$), is used to study electron and
proton intensities in the nightside geosynchronous orbit. SOPA measures
electrons in nine electron channels from $\sim$ 0.05-1.5 MeV and protons from
$\sim$ 0.2-2.5 MeV. Note that neither LANL-97A nor any other LANL space
environment monitor carries any magnetometers (Taylor et al., 2004). The
THEMIS Solid State Telescope (SST) records ions and electrons within the
energy range from 25 keV to 6 MeV. The Flux Gate Magnetometer (FGM) on THEMIS
measures the background magnetic field in the near-Earth space environment
(Angelopoulos, 2009; Auster et al., 2008). The measurements of the IMF, solar
wind parameters, and magnetospheric particle and field with one-minute cadence
were utilized for the current study.

Figure 1:  Solar wind parameters observed by Wind during the FRS on 06
December 2014. The onset of the shock at Wind is shown by left vertical line
and the right vertical line shows the onset of the shock at the ground as seen
in the SYM-H index.

Figure 1 shows IMF and solar wind parameters for the studied event. The top
panel in the figure shows the total IMF strength (IMF, $B_{t}$) and its
components (Bx, By, Bz). At the 21:09 UT shock onset, IMF Bz, thermal speed
(Vth) and the solar wind particle density (Np) decreased sharply, whereas the
solar wind bulk (Vsw) shows the increase. These observed changes in the solar
wind parameters clearly indicate the shock is a FRS. The Rakine-Hugoniot
equations, based on assumptions of energy and momentum conservation across the
shock front (Landau & Lifshitz, 1960; Tsurutani et al., 2011; Oliveira, 2017)
yield the values 590 km/s and $\sim 10^{\circ}$ for the shock speed and shock
impact angle, respectively. Here, shock impact angle is the angle made by IP
shock normal with the Sun-Earth line. The bottom panel of the figure shows
SYM-H index, a measure of the global averaged; low latitude magnetic field
response on the ground. The SYM-H index shows a decrease of about 20 nT around
21:54 UT which is consistent with the expected solar wind flow time $\sim 45$
min from Wind’s upstream location ($X_{GSE}=204.5Re$, $Y_{GSE}=-89.5Re$,
$Z_{GSE}=-0.05Re$) to the Earth.

Figure 2: (a)Spacecraft positions during the FRS event of 06 December 2014 at
21:50 UT. The magnetopause shapes just prior to (blue dashed line) shock
arrival and at the shock onset (black solid line) are shown. Dotted circle
shows geosynchronous orbit. (b) Magnetic field variation caused by the FRS
event of 06 December 2014. the dashed vertical lines represent onset of the
FRS as seen at the respective spacecraft. Total magnetic field of GOES is
shown in top two panels. Third panel from top shows y component of magnetic
field as recorder by THEMIS-A. Note that the y-component of the magnetic field
(Hy) for THEMIS-A, is shown as other components did not show any clear
variation. At the bottom global low latitude average geomagnetic field
response is shown by the SYM-H index.

Figure 2a shows magnetospheric spacecraft positions for the event. GOES-15 was
located near noon, GOES-13 and THEMIS-A were located near dusk, and LANL-97A
was near midnight. The figure also shows the magnetopause boundary estimated
by the Shue et al. (1998) magnetopause model just before (blue dashed curve)
and just after (black solid curve) the shock impacted the magnetosphere. The
shock wavefront is shown as inclined dashed black lines. The left and right
lines indicate the location of the shock front at the time of magnetopause
impact and at the time of the maximum depression observed in SYM-H associated
with the shock i.e after 10 min of shock onset, respectively. Note that the
magnetosphere expands in response to the FRS impact, as opposed to the
compression caused by the impact of an FFS, as shown in figure 2 of Rudd et
al. (2019).

##  3 Observations and Results

###  3.1 Magnetic field response

Figure 2b shows the space-based and averaged global magnetic field response to
this shock. The dashed vertical line indicates the onset of the FRS as seen in
the magnetic field at GOES-15, GOES-13, THEMIS-A spacecraft and in SYM-H on
the ground. A clear decrease is seen at all spacecraft and on the ground. The
onset of the shock is seen earlier ($\sim$21:50:00) at GOES-15, later at
GOES-13 ($\sim$ 21:52:00), and still later on the ground ($\sim$ 21:53:00).
Timing analysis using GOES-15, GOES-13, and SYM-H index data indicates that
the rarefaction front moved through the magnetosphere with a antisunward speed
of $\sim$ $209-232km/s$. The observed speed is consistent with the reported
speed of $\sim 300km/s$ for an SI- by (Araki & Nagano, 1988). THEMIS-A shows a
gradual decrease in the magnetic field strength but the onset time is not very
clear. After the onset of the shock observed at GOES-15 the magnetic field
reaches a minimum in 10 minutes and after that almost remains steady. Unlike
GOES-15, GOES-13 shows a steady decrease in magnetic field strength after the
onset and the magnetic field continued to decrease. A clear local time
dependence can be observed by comparison of the magnetic field profiles of
GOES-15 and GOES-13. Note that around 21:30 UT, prior to the FRS onset as seen
at GOES-13,15 and SYM-H, there is a sharp enhancement in the magnetic field
which could be due to a small density jump around 20:30 UT seen at Wind.

###  3.2 GOES Particle flux response

Figure 3: Electron flux response to the FRS as observed by GOES-13,15. The
vertical line indicate the onset of FRS signature as seen in magnetic field
variations measured by GOES-15. There is steady decreases in magnetic field at
GOES-13 unlike GOES-15. The low energy particles show FRS signature whereas
there is no clear signature a higher energies at both the spacecraft.

Figure 3 shows GOES-15, 13 electron fluxes. The vertical line indicates the
onset of the shock at each spacecraft as identified in Figure 2. A decrease in
electron flux for selected channels follows the rarefaction-induced magnetic
field for the FRS duration. The following general characteristics can be
readily noted based on observed signatures: (1) An almost instantaneous
decrease is observed in most of the electron flux channels. (2) There is no
indication of any dispersive effect over the entire spectral range as all the
channels which show the signature show a decrease at the same time. (3) low
energy (40-75 keV ) particle fluxes follow the magnetic field variations quite
well.

However, close observation reveals strong energy dependent effects leading to
significantly different signatures in the various energy channels. The lower
energy channels show a rapid decrease as compared to the higher energy
channels, which show weak or almost unidentifiable signatures. The signals at
higher energies could be smeared out in time due to the fast drift rate
superposing the time and spatial-variation. Note that all electron fluxes show
sharp changes around 21:30 UT prior to the FRS onset which could be associated
with a small pressure jump around 20:30 UT observed by Wind. This is also
evident in the GOES magnetic field and it is manifested as a sharp increase in
SYM-H around 21:35 UT. Curiously, the flux profile seen in the 75 keV channel
of GOES-13 is very well correlated with the magnetic field variation seen by
GOES-15. The percentage change in intensity for the different energy channels
varies. For the 75 keV channel more than a 70% decrease was recorded by
GOES-13. The proton data were absent for both the spacecraft.

Figure 4: Field-aligned (PA$\sim 30^{o}$) and perpendicular (PA$\sim 90^{o}$)
pitch angle Electron flux response to the FRS as observed by GOES-13,15 for
the 75 keV channel

Pitch angle distributions at various energies were examined. Intensities at
all pitch angles decrease after the shock impact at both spacecraft. Figure 4
shows near field-aligned ($\sim 30^{o}$) and perpendicular ($\sim 90^{o}$)
electron fluxes of 75 keV electrons. Just prior to the shock arrival field-
aligned intensities were greater than those for the equatorially trapped
particles, in contrast to the nominal distribution. The anisotropy index (A)
was calculated for both spacecraft, where
$A=[j(90^{o})-j(30^{o})]/[j(90^{o})+j(30^{o})]$. The value of $A$ was negative
prior to the onset of the FRS, and as an immediate response became less
negative or even positive. After the shock $A$ remained close to zero, i.e
almost isotropic distributions were observed. All other energies show a
similar effect (not shown here). This implies that the particle population
residing radially inward had an almost isotropic pitch angle distribution and
that these particle populations moved to the spacecraft due to an expansion of
the magnetosphere. Asymmetric or butterfly distributions are more commonly
observed in the outer magnetosphere or higher L-shells than in the inner
magnetosphere (Ni et al., 2016). Therefore, during the expansion of the
magnetosphere caused by the FRS, a spacecraft could observe a decrease in
anisotropy as the spacecraft encounters particles from the inner magnetosphere
i.e as the distribution shifts from butterfly-like to one that is more
isotropic.

###  3.3 LANL Particle intensities response

Figure 5: Electron (Left panel) and Proton (right panel) flux response to FRS
as observed by LANL-97A. The vertical line indicates the onset of the FRS
signature as seen in particle intensities which coincide with the onset
observed in the SYM-H index. Energy spectra of high energy electrons observed
by LANL-97A (bottom panel) in night sector before and after the shock onset.

Figure 5 shows the electron and proton fluxes observed by the LANL-97A
spacecraft, located near midnight. A very strong and clear response is
observed at this spacecraft in both the electron ($\sim$ 1-9 MeV) and proton
($\sim$0.2-0.9 MeV) fluxes. The observed decrease is non-dispersive as the
onset is seen simultaneously in all the energy channels. Interestingly, even
MeV electrons show the decrease in intensity at the onset time. The overall
percentage of the decrease is $\sim 70\%$ for each electron channel and $\sim
47\%$ for the each proton channel which showed the signature.

As shown in Figure 5 bottom panel, before and after electron spectra of
electrons show that the spectral slope does change after the onset of the
shock. The observed spectral slope for electrons is $\sim 3.6$ prior to the
shock but has steepen to $\sim 4.1$ after the shock. The change in slope value
is more than the error, indicating that the observed change is real and
significant. The implications of this will be discussed in the Section 5.

##  4 Model comparisons

We use magnetospheric models to simulate the global response of the
magnetosphere to the FRS and for comparison with the spacecraft observations.
They can provide magnetic field responses at locations where no spacecraft
observations exist.

The Block-Adaptive-Tree-Solarwind-Roe-Upwind-Scheme BATS-R-US (Ridley &
Liemohn, 2002; Tóth et al., 2005, 2012) was used in the present study. This
model was developed by the Computational Magnetohydrodynamics (MHD) Group at
the University of Michigan and is now available as part of the Space Weather
Modeling Framework (SWMF). The model run for the shock event was carried out
at the Community Coordinated Modeling Center (CCMC). BATS-R-US has been used
to investigate the interaction of IP shocks with the Earth’s magnetosphere by
many studies (Koval et al., 2006; Ridley et al., 2006; Samsonov et al., 2007;
Samsonov & Sibeck, 2013; Samsonov et al., 2015).

Solar wind plasma parameters (density, velocity, temperature) and IMF observed
by the Wind spacecraft are inputs to the model. The solar wind data is
transformed into GSM coordinates and propagated from Wind’s position to the
sunward boundary of the simulation domain. The Earth’s magnetic field is
approximated by a dipole and the dipole orientation updated during the
simulation period. Outputs of the run are magnetospheric plasma parameters
including density, pressure, velocity, magnetic field, and electric currents.

Figure 6: BATS-R-US model outputs for the studied event. Top panels are
magnetosphere density profiles in the equatorial plane. Bottom panels are time
series outputs of pressure and magnetic field at GOES-15 and LANL-97A
locations.

Figure 6 shows simulated densities in the equatorial plane of the
magnetosphere before (left top panel) and after (right top panel) the shock
arrival. The simulation reproduces the clear reduction in particle number
density at geosynchronous orbit, an expected FRS impact signature. The model
time series for the magnetic field (B) and pressure (P) variations at the
location of GOES-15 and LANL-97A are shown in the lower panels. On the dayside
the GOES-15 modeled-pressure shows a clear step like pressure drop, whereas
the pressure decrease at LANL-97A is gradual. The model predicts a step-like
decrease of $\Delta B\sim 25nT$ at GOES-15, with the magnetic field remaining
low after the shock. Overall the model fields agree with GOES-15 observations
to a reasonable degree. The modeled magnetic field at LANL-97A shows a
decrease in $B$ of just $6nT$, very small compared to that at GOES-15. There
are no magnetic field measurements available from LANL-97A. So this model
predication can not be validated using LANL-97A. Thus to validate the
nightside model fields, we have comparedits predictions to the Cluster
spacecraft observations (not shown here) which were taken in the nightside.
The decrease of $\Delta B\sim 15nT$ is observed by all four spacecraft.
However, the BATS-R-US model predicts only $\sim 6$ nT variation at this
location. This indicates that the model underestimates magnetic field values
on the nightside. Therefore, it is likely that it also under-predicts magnetic
field variations at LANL-97A and thus under estimates the adiabatic effect.
The following section tries to quantify the contribution of adiabatic and
other effects to the observed particle intensity variations.

##  5 Contribution of adiabatic effects and radial gradients

The variations in the particle intensities attending the shock arrival could
result from any combination of (1) advected radial gradients, (2) adiabatic
effects, (3) spectral changes, and other (4) non adiabatic processes. From
Wilken et al. (1986) the observed change in the flux (j) to first order
estimate can be expressed by the following equation, the first term represents
the change due to adiabatic processes, whereas the second term represents the
changes associated with advected radial gradients in the particle intensities.

$\Delta j(E=const)=\frac{\delta j}{\delta B}\Delta B+\frac{\delta j}{\delta
R}\Delta R\,,$ (1)

where $\Delta B$ and $\Delta R$ represent the changes in magnetic field
strength and radial position seen by the magnetosphere plasma population. If
we assume the first invariant and phase space density are conserved during the
transient rarefaction of the magnetosphere, this relationship can be expressed
as

$\frac{\Delta j}{j(E)}=\frac{\Delta B}{B_{0}}(\gamma_{0}+1)+\frac{\Delta
k}{k_{0}}+\frac{\Delta\gamma}{\gamma_{0}}ln\Bigg{(}\frac{j_{0}}{k_{0}}\bigg{)}$
(2)

Here, the first term represents the change due to adiabatic processes, whereas
the second term represents the changes associated with radial gradients and
third due to change in spectral properties of the particle population. We have
attempted to evaluate each term of this expression using THEMIS-A particle
data. The orbit of THEMIS-A crossed a wide range of L-shells which enables
estimation of gradients in particle intensity and spectral index. These
gradients are estimated around geosynchronous orbit prior to the onset of the
shock. Figure 7a shows the electron flux data in nine energy channels utilized
for the estimates. Note that the higher fluxes are observed at lower L-shells
for these energies and locations. The flux measurements are used to calculate
radial gradients in flux and spectral index.

Thus, the various terms of Equation 2 are calculated based on these
measurements. Since there are no magnetic field measurements onboard LANL-97A
to estimate the adiabatic term, magnetic field observations by Cluster
spacecraft $\Delta B=15nT$ and $B_{0}=100nT$ were used. By using the TS04
model the magnetic foot points are estimated for pre-shock and post-shock
solar wind dynamic pressure conditions to deduce a $\sim 0.5R_{E}$ expansion
of the magnetosphere at geosynchronous orbit, thus $\Delta R$ was taken as
$0.5R_{E}$.

Figure 7: (a)The L-shell variation of electron fluxes as observed by THEMIS-A
during inbound 11:03 to 11:13 UT on 6th December 2014. (b) Estimated
contribution of various terms associated with radial gradients and adiabatic
effect in the observed decrease using THEMIS-A data. The red squares represent
the net contribution of all three terms

Figure 7b shows values for each term in Eq. 2 for several energies. The total
fractional change ($\frac{\Delta j}{j_{0}}$) is also presented to see the net
estimate of the decrease in the various energy channels. The adiabatic
($\frac{\Delta B}{B_{0}}(\gamma_{0}+1)$) and spectral gradient
($\frac{\Delta\gamma}{\gamma_{0}}ln\big{(}\frac{j_{0}}{k_{0}}\big{)}$) terms
are the dominant contribution to the observed decrease. Advection of the
negative radial gradients ($\frac{\Delta k}{k_{0}}$) in intensity make only
negligible positive contributions to the particle intensity. The calculated
total fractional decrease ($\frac{\Delta j}{j_{0}}$) for $100$ keV and above
energies is $<-0.5$ reaches almost $-0.8$ for $700$ keV energy. This estimate
is comparable to that observed. The observed fractional flux decrease at
LANL-97 for $\sim 1122$ keV is $-0.75$. Whereas, at GOES-13 for 75 keV
electrons the observed fractional decrease is $-0.71$ which is very close to
the estimate of $-0.69$. Note that these estimates are done using observed
magnetic field, spectral slope and particle intensity gradients and remarkably
we do get a close match with the observed fractional changes in the particle
intensities.

##  6 Role of shock impact angle

Van Allen Probes were located deep within the post midnight magnetosphere and
did not see any significant variation in particle intensities or magnetic
field. This could be due to a subdued response of magnetosphere to the shock
at lower L shells. Also, the magnetosphere might respond asymmetrically to an
interplanetary shock depending on the angle of shock impact (Oliveira &
Raeder, 2015; Vichare et al., 2014; Oliveira & Samsonov, 2018; Rudd et al.,
2019). The shock was slightly inclined, and first impacted the magnetosphere
on the afternoon side which also could contribute to a weak or absent of shock
impact as the Van Allen Probes location .

Figure 8: Local time SMR indices during the event. The vertical dotted line
represents the onset of the FRS. whereas, the curves denote the local time SMR
index at various local times. A strong local time dependence of the magnetic
field decrease is associated with FRS.

In order to infer the shock inclination influence we have used the Partial
Ring current local time SMR index. SuperMAG partial ring current index (SMR)
is similar to the SYM-H index but computed by a higher number of ground
magnetometers (Newell & Gjerloev, 2012). Figure 8 shows the various local time
SMR indices. Four equally sized local time sectors are defined with centers at
00, 06, 12, and 18 MLT to compute the local time SMR which are shown in the
figure. The maximum decrease is observed in the noon sector, whereas the
minimum is seen in the dawn sector. The second strongest response is seen near
the dusk-midnight sector. This implies that there is a strong local time
dependence of the ground-based magnetic field response due to an asymmetric
rarefaction of the magnetosphere. The Van Allen Probes were located in the
dawnside magnetosphere where the shock impact was weaker due to its
inclination. Thus an asymmetric expansion of the magnetosphere could explain
the observed varied response of the particles to the shoversus local time .

##  7 Discussion and conclusions

The present study shows that not only FFSs, but also FRS, generate interesting
dynamics. The geosynchronous response is investigated using GOES-13, 15 and
LANL-97A spacecraft data. The magnetic field strength decreases at
geosynchronous orbit and on the ground in response to the inclined FRS. The
SYM-H index dropped by $\sim 20nT$. The timing analysis shows that the
rarefaction wave front propagated antisunward at $209-232km/s$. The particle
response also seems to be global in nature as most electron energy channels
from 10s of keV to MeV energies show a decrease in intensity. However, not all
the energy channels show the same relative decrease or variation in magnitude.
It is possible that gradient and betatron effects counterbalance each other
resulting in the absence or small size of the signature or its absence in
different energy channels which we have attempted to address here. The shock
impact not only caused a decrease in particle intensities but also in pitch-
angle anisotropy. This implies that those particle populations which advected
outward through geosynchronous orbit due to the shock-induced magnetospheric
expansion had an almost isotropic pitch angle distribution.

To identify the contribution of radial gradient and adiabatic effects to the
observed decrease we followed the formalism of (Wilken et al., 1986).
Moreover, the required magnetic field to carry out this calculation was
inferred using the BATS-R-US model at the LANL-97 A location. However, when we
validated the model outputs using the Cluster spacecraft magnetometer data we
found that the model underestimated perturbations in the nightside magnetic
field. Our first order estimate following Wilken et al. (1986), and using
THEMIS-A showed that the radial gradient makes only a small contribution. The
work by West et al. (1973) and West Jr (1979) shows that the radial gradient
of fluxes at constant energy tends to assume small values for low energy
($<200keV$) electrons and increase monotonically for higher energies. Our
first order estimate following the Wilken et al. (1986) approach agrees with
this, i.e the radial gradient values are very small ($\sim~{}0$) for keV
electrons. Further, we find that the adiabatic and spectral terms
significantly contribute to the observed decrease. The calculated contribution
are consistent with the observed decrease in electron intensities at
GOES-15,13 and LANL-97A spacecraft.

The shock impact on the magnetosphere was slightly inclined, resulting in an
asymmetric expansion of the magnetosphere. This is well shown by variations in
the SMR index at different local times. Spacecraft to spacecraft variations in
the response at different local times could be explained by the inclined
incidence of the shock. This highlights the importance of the shock impact
angle during shock interaction with the magnetosphere and is consistent with
the earlier reports, as shown by simulations (Guo et al., 2005; Oliveira &
Raeder, 2014; Samsonov et al., 2015) and observations (Takeuchi et al., 2002b;
Wang et al., 2006; Oliveira et al., 2016, 2018).

The adiabatic effect underlying the observed decrease in relativistic electron
intensities can be understood as follows. The onset of the FRS causes the
magnetosphere to expand, relativistic electrons conserving their third
invariant move radially outward to compensate for the reduced local magnetic
field strength. Thus LANL-97 which is almost located at fixed radial distance
then measures the intensity of electrons previously present at lower L-shells.
The outward moving electrons experience a weaker magnetic field and to
conserve the first adiabatic invariant ($\mu$), their perpendicular energy
must decrease. Also, the electrons will now see longer field lines which
reduces their parallel energy. Therefore, the distribution of electrons shifts
to lower energy. A spacecraft measuring flux at fixed energy, $E_{0}$ and
fixed radial distance measures the flux of electrons initially at a lower
L-shell shifted to lower energies. Thus generally this outward movement
results in a decrease of electron intensity at fixed energy for a decrease in
magnetic field strength. In addition to this effect, if the pre-shock
magnetosphere had a strong gradient in spectral slope, even this also can
contribute to the decrease. The dominance of the estimated spectral term
supports this. These two effects are primarily responsible for the observed
decrease in particle fluxes caused by the FRS on 06 December 2014.

Note that the magnetosphere responds differently to interplanetary FFSs and
FRSs. The FFS compresses the magnetosphere, whereas FRS causes an expansion of
the magnetosphere. The present study shows that the reverse shock
predominantly caused the decrease in particle intensities. However, particle
responses depend on magnetospheric conditions prior to the onset of the shock
such as radial gradients in intensities and spectral index and could manifest
as no effect or even enhancements during either fast forward or fast reverse
shock. Even though FRS are rare, a statistical study would shed light on the
general response of magnetosphere particles to FRS. Moreover, the findings of
the study have implications to understand the interaction of shocks with the
other planetary magnetosphere.

The solar wind parameters, interplanetary magnetic field and geomagnetic
indices used in this study are obtained from CDAWEB
(https://cdaweb.gsfc.nasa.gov/) and SUPERMAG (http://supermag.jhuapl.edu/). We
thank Wind, GOES, THEMIS, Cluster teams for providing the data. The modeling
part of the work was carried out using the SWMF/BATSRUS tools developed at The
University of Michigan Center for Space Environment Modeling (CSEM) and made
available through the NASA Community Coordinated Modeling Center (CCMC). Also,
we thank Community Coordinated Modeling Center (CCMC, Run Number
Ankush_Bhaskar_051118_1) for running the required models for this work. We
thank Vassilis Angelopoulos, UCLA and Brian Kress, NOAA for valuable
discussions. This work is carried under NASA Living With a Star Jack Eddy
Postdoctoral Fellowship Program, administered by UCAR’s Cooperative Programs
for the Advancement of Earth System Science (CPAESS). A.B thanks UCAR, CPAESS
and NASA/GSFC for providing excellent facilities and environment for the work.
Some work at NASA/GSFC was funded by the Van Allen Probes mission.

## References

  * Akasofu (1964) Akasofu, S.-I. 1964, Planetary and Space Science, 12, 573, doi: 10.1016/0032-0633(64)90004-2
  * Alves et al. (2011) Alves, M. V., Echer, E., & Gonzalez, W. D. 2011, Journal of Atmospheric and Solar-Terrestrial Physics, 73, 1380, doi: 10.1016/j.jastp.2010.07.024
  * Andrioli et al. (2006) Andrioli, V. F., Echer, E., Savian, J. F., & Schuch, N. J. 2006, Revista Brasileira de Geofísica, 25, 175, doi: 10.1590/S0102-261X2007000600021
  * Angelopoulos (2009) Angelopoulos, V. 2009, in The THEMIS mission (Springer), 5–34
  * Araki (1994) Araki, T. 1994, in Solar Wind Sources of Magnetospheric Ultra-Low-Frequency Waves, Geophysical Monograph Series, ed. M. J. Engebretson, K. Takahashi, & M. Scholer, Vol. 81 (Washington, D.C.: American Geophysical Union), 183–200, doi: 10.1029/GM081p0183
  * Araki & Nagano (1988) Araki, T., & Nagano, H. 1988, Journal of Geophysical Research: Space Physics, 93, 3983
  * Araki et al. (1997) Araki, T., Fujitani, S., Yumoto, K., et al. 1997, Journal of geophysical research, 102, 14075
  * Auster et al. (2008) Auster, H., Glassmeier, K., Magnes, W., et al. 2008, Space Science Reviews, 141, 235
  * Burguess (1995) Burguess, D. 1995, in Introduction to Space Plasma Physics, ed. M. G. Kivelson & C. T. Russell (Cambridge, United Kingdom: Cambridge University Press)
  * Burlaga (1971) Burlaga, L. F. 1971, Space Science Reviews, 12, 600, doi: 10.1007/BF00173345
  * Burlaga (1995) —. 1995, Interplanetary Magnetohydrodynamics (New York, NY: Oxford University Press)
  * Cattell et al. (2017) Cattell, C., Breneman, A., Colpitts, C., et al. 2017, Geophysical research letters, 44, 8712
  * Cavus et al. (2019) Cavus, H., Araz, G., Caglar Coban, G., Rahee, A., & Karafistan, A. I. 2019, Advances in Space Research, 63, doi: 10.1016/j.asr.2019.09.056
  * Echer et al. (2004) Echer, E., Alves, M. V., & Gonzalez, W. D. 2004, Solar Physics, 221, 361, doi: 10.1023/B:SOLA.0000035045.65224.f3
  * Echer et al. (2005) Echer, E., Gonzalez, W. D., Dal Lago, A., et al. 2005, Advances in Space Research, 36, 2313, doi: 10.1016/j.asr.2005.04.030
  * Echer et al. (2003) Echer, E., Gonzalez, W. D., Vieira, L. E. A., et al. 2003, Brazilian Journal of Physics, 33, 115, doi: 10.1590/S0103-97332003000100010
  * Guo et al. (2005) Guo, X.-C., Hu, Y.-Q., & Wang, C. 2005, Chinese Physics Letters, 22, 3221, doi: 10.1088/0256-307X/22/12/067
  * Hanser (2011) Hanser, F. 2011, EPS/HEPAD calibration and data handbook (Tech. Rep. GOESN-ENG-048D). Carlisle, MA: Assurance Technology Corporation
  * Hudson et al. (1995) Hudson, M., Kotelnikov, A., Li, X., et al. 1995, Geophysical research letters, 22, 291
  * Huttunen et al. (2005) Huttunen, K. E. J., Slavin, J., Collier, M., et al. 2005, Annales Geophysicae, 23, 609, doi: 10.5194/angeo-23-609-2005
  * Joselyn & Tsurutani (1990) Joselyn, J. A., & Tsurutani, B. T. 1990, Eos Transactions AGU, 71, 1808, doi: 10.1029/90EO00350
  * Kanekal et al. (2016) Kanekal, S., Baker, D., Fennell, J., et al. 2016, Journal of Geophysical Research: Space Physics
  * Kilpua et al. (2015) Kilpua, E. K. J., Lumme, E. Andréeová, K., Isavnin, A., & Koskinen, H. E. J. 2015, Journal of Geophysical Research: Space Physics, 120, 4112, doi: 10.1002/2015JA021138
  * Koval et al. (2006) Koval, A., S̆afránková, J., Nĕmec̆ek, Z., et al. 2006, Geophysical Research Letters, 33, doi: 10.1029/2006GL025707
  * Landau & Lifshitz (1960) Landau, L. D., & Lifshitz, E. M. 1960, Electrodynamics of continuous media (Oxford, England: Pergamon Press)
  * Lepping et al. (1995) Lepping, R., Acũna, M., Burlaga, L., et al. 1995, Space Science Reviews, 71, 207
  * Li et al. (1993) Li, X., Roth, I., Temerin, M., et al. 1993, Geophysical Research Letters, 20, 2423
  * Looper et al. (1994) Looper, M., Blake, J., Mewaldt, R., Cummings, J., & Baker, D. 1994, Geophysical research letters, 21, 2079
  * Mullen et al. (1991) Mullen, E., Gussenhoven, M., Ray, K., & Violet, M. 1991, IEEE transactions on nuclear science, 38, 1713
  * Newell & Gjerloev (2012) Newell, P., & Gjerloev, J. 2012, Journal of Geophysical Research: Space Physics, 117
  * Ni et al. (2016) Ni, B., Zou, Z., Li, X., et al. 2016, Geophysical Research Letters, 43, 5644
  * Ogilvie et al. (1995) Ogilvie, K. W., Chornay, D. J., Fritzenreiter, R. J., et al. 1995, Space Science Reviews, 71, 55, doi: 10.1007/BF00751326
  * Oliveira (2017) Oliveira, D. M. 2017, Brazilian Journal of Physics, 47, 81, doi: 10.1007/s13538-016-0472-x
  * Oliveira & Raeder (2014) Oliveira, D. M., & Raeder, J. 2014, Journal of Geophysical Research: Space Physics, 119, 8188, doi: 10.1002/2014JA020275
  * Oliveira & Raeder (2015) —. 2015, Journal of Geophysical Research: Space Physics, 120, 4313, doi: 10.1002/2015JA021147
  * Oliveira et al. (2016) Oliveira, D. M., Raeder, J., Tsurutani, B. T., & Gjerloev, J. W. 2016, Brazilian Journal of Physics, 46, 97, doi: 10.1007/s13538-015-0389-9
  * Oliveira & Samsonov (2018) Oliveira, D. M., & Samsonov, A. A. 2018, Advances in Space Research, 61, 1, doi: 10.1016/j.asr.2017.10.006
  * Oliveira et al. (2018) Oliveira, D. M., Arel, D., Raeder, J., et al. 2018, Space Weather, 16, 636, doi: 10.1029/2018SW001880
  * Patel & Coleman (1970) Patel, V. L., & Coleman, P. J. 1970, Journal of Geophysical Research, 75, 7255, doi: 10.1029/JA075i034p07255
  * Richter et al. (1985) Richter, A. K., Hsieh, K. C., Luttrell, A. H., Marsch, E., & Schwenn, R. 1985, in Collisionless Shocks in the Heliosphere: Reviews of Current Research, Geophysical Monograph Series, ed. B. T. Tsurutani & R. G. Stone, Vol. 35 (Washington, D.C.: American Geophysical Union), 33–50, doi: 10.1029/GM035p0033
  * Ridley & Liemohn (2002) Ridley, A., & Liemohn, M. 2002, Journal of Geophysical Research: Space Physics, 107, SMP
  * Ridley et al. (2006) Ridley, A. J., De Zeeuw, D. L., Manchester, W. B., & Hansen, K. C. 2006, Advances in Space Research, 38, 263, doi: 10.1016/j.asr.2006.06.010
  * Rodriguez et al. (2014) Rodriguez, J., Krosschell, J., & Green, J. 2014, Space Weather, 12, 92
  * Rudd et al. (2019) Rudd, J. T., Oliveira, D. M., Bhaskar, A., & Halford, A. J. 2019, Advances in Space Research, 63, 317, doi: 10.1016/j.asr.2018.09.013
  * Russell et al. (1994a) Russell, C. T., Ginskey, M., & Petrinec, S. M. 1994a, Journal of Geophysical Research, 99, 253, doi: 10.1029/93JA02288
  * Russell et al. (1994b) —. 1994b, Journal of Geophysical Research, 99, 13,403, doi: 10.1029/94JA00549
  * Samsonov et al. (2015) Samsonov, A. A., Sergeev, V. A., Kuznetsova, M. M., & Sibeck, D. G. 2015, Geophysical Research Letters, 42, 4716, doi: 10.1002/2015GL064294
  * Samsonov & Sibeck (2013) Samsonov, A. A., & Sibeck, D. G. 2013, Journal of Geophysical Research, 118, 3055, doi: 10.1002/jgra.50329
  * Samsonov et al. (2007) Samsonov, A. A., Sibeck, D. G., & Imber, J. 2007, Journal of Geophysical Research, 112, 1, doi: 10.1029/2007JA012627
  * Schiller et al. (2016) Schiller, Q., Kanekal, S., Jian, L., et al. 2016, Geophysical Research Letters, 43, 12
  * Shen et al. (2017) Shen, X., Shi, Q., Zong, Q.-G., et al. 2017, Journal of Geophysical Research: Space Physics, 122, 1658
  * Shue et al. (1998) Shue, J.-H., Song, P., Russell, C., et al. 1998, Journal of Geophysical Research: Space Physics, 103, 17691
  * Siscoe et al. (1968) Siscoe, G. L., Formisano, V., & Lazarus, A. J. 1968, Journal of Geophysical Research, 73, 4869, doi: 10.1029/JA073i015p04869
  * Smith et al. (1986) Smith, E. J., Slavin, J. A., Zwickl, R. D., & Bame, S. J. 1986, in Solar Wind and Magnetosphere Coupling, ed. Y. Kamide & J. A. Slavin (Tokyo, Japan: Terra Scientific), 345
  * Su et al. (2015) Su, Z., Zhu, H., Xiao, F., et al. 2015, Geophysical Research Letters, 42, 3129, doi: 10.1002/2015GL063906
  * Takeuchi et al. (2002a) Takeuchi, T., Araki, T., & Viljanen, A. 2002a, Journal of Geophysical Research, 107, 1096, doi: 10.1029/20001JA900152
  * Takeuchi et al. (2002b) Takeuchi, T., Russell, C. T., & Araki, T. 2002b, Journal of Geophysical Research, 107, SMP 6, doi: 10.1029/2002JA009597
  * Taylor et al. (2004) Taylor, M., Friedel, R., Reeves, G., et al. 2004, Journal of Geophysical Research: Space Physics, 109
  * Tóth et al. (2005) Tóth, G., Sokolov, I. V., Gombosi, T. I., et al. 2005, Journal of Geophysical Research: Space Physics, 110
  * Tóth et al. (2012) Tóth, G., Van der Holst, B., Sokolov, I. V., et al. 2012, Journal of Computational Physics, 231, 870
  * Tsurutani et al. (2011) Tsurutani, B. T., Lakhina, G. S., Verkhoglyadova, O. P., et al. 2011, Journal of Atmospheric and Solar-Terrestrial Physics, 73, 5, doi: 10.1016/j.jastp.2010.04.001
  * Vichare et al. (2014) Vichare, G., Rawat, R., Bhaskar, A., & Pathan, B. M. 2014, Earth, Planets and Space, 66, 1, doi: 10.1186/1880-5981-66-92
  * Wang et al. (2006) Wang, C., Li, C. X., Huang, Z. H., & Richardson, J. D. 2006, Geophysical Research Letters, 33, 1, doi: 10.1029/2006GL025966
  * Wang et al. (2010) Wang, C., Li, H., Richardson, J. D., & Kan, J. R. 2010, Journal of Geophysical Research, 115, doi: 10.1029/2009JA014833
  * West et al. (1973) West, H., Buck, R., & Walton, J. 1973, Journal of Geophysical Research, 78, 1064
  * West Jr (1979) West Jr, H. I. 1979, Quantitative Modeling of Magnetospheric Processes, Geophys. Monogr. Ser, 21, 150
  * Wilken et al. (1986) Wilken, B., Baker, D., Higbie, P., et al. 1986, Journal of Geophysical Research: Space Physics, 91, 1459
  * Zhang et al. (2018) Zhang, D., Liu, W., Li, X., et al. 2018, Geophysical Research Letters, 45, 7287
  * Zhang et al. (2010) Zhang, X. Y., Zong, Q.-G., Wang, Y. F., et al. 2010, Journal of Geophysical Research, 115, doi: 10.1029/2009JA015016