ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-35-239-2017The generation of Ganymede's diffuse aurora through pitch angle scatteringTripathiArvind K.aktrip2001@yahoo.co.inSinghalRajendra P.Singh IIOnkar N.Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005 (UP), IndiaArvind K. Tripathi (aktrip2001@yahoo.co.in)22February201735223925214August20167November20166January2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/35/239/2017/angeo-35-239-2017.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/35/239/2017/angeo-35-239-2017.pdf
Diffuse auroral intensities of neutral atomic oxygen OI λ1356 Å emission on
Ganymede due to whistler mode waves are estimated. Pitch angle diffusion of
magnetospheric electrons into the loss cone due to resonant wave–particle
interaction of whistler mode waves is considered, and the resulting electron
precipitation flux is calculated. The analytical yield spectrum approach is used
for determining the energy deposition of electrons precipitating into the
atmosphere of Ganymede. It is found that the intensities (4–30 R)
calculated from the precipitation of magnetospheric electrons observed near
Ganymede are inadequate to account for the observational intensities (≤ 100 R). This is in agreement with the conclusions reached in previous works.
Some acceleration mechanism is required to energize the magnetospheric
electrons. In the present work we consider the heating and acceleration of
magnetospheric electrons by electrostatic waves. Two particle distribution
functions (Maxwellian and kappa distribution) are used to simulate heating
and acceleration of electrons. Precipitation of a Maxwellian distribution of
electrons can produce about 70 R intensities of OI λ1356 Å emission for electron temperature of 150 eV. A kappa distribution can also
yield a diffuse auroral intensity of similar magnitude for a characteristic
energy of about 100 eV. The maximum contribution to the estimated intensity
results from the dissociative excitation of O2. Contributions from the direct excitation
of atomic oxygen and cascading in atomic oxygen are estimated to be only about 1
and 2 % of the total calculated intensity, respectively. The findings of
this work are relevant for the present JUNO and future JUICE missions to
Jupiter. These missions will provide new data on electron densities,
electron temperature and whistler mode wave amplitudes in the magnetosphere
of Jupiter near Ganymede.
Ganymede is a satellite of Jupiter and is the largest moon in our solar
system. It has a radius of 2634 km. It is larger than Mercury and Pluto and
three-quarters the size of Mars. It orbits Jupiter at about 1.070 million km
(∼ 15 RJ, RJ is the radius of Jupiter, i.e., 71 496 km). Its
orbit is very slightly eccentric and inclined to the Jovian equator, with
the eccentricity and inclination changing quasiperiodically due to solar
and planetary gravitational perturbations on a timescale of centuries.
These orbital variations cause the axial tilt (the angle between rotational
and orbital axes) to vary between 0 and 0.33∘ (Susanna et al., 2002).
Several probes flying by or orbiting Jupiter have explored Ganymede more
closely, including four flybys in the 1970s, and multiple passes in the
1990s to 2000s. Ganymede has been explored by the Pioneer 10 and 11 probes
(Mead, 1974) and the Voyager 1 and 2 probes, and they returned information about the
satellite (Scarf et al., 1979; Gurnett et al., 1979). The Galileo spacecraft
entered orbit around Jupiter and made six close flybys to explore Ganymede
(Gurnett et al., 1996). The spacecraft had a suite of instruments which
included a magnetometer, energetic particle detector and plasma wave
spectrometer. In addition, the Hubble Space Telescope (HST) also explored
Ganymede and provided evidence for a tenuous oxygen atmosphere (exosphere)
on Ganymede. The most recent close observations of Ganymede were made by New
Horizons (Grundy et al., 2007).
Data obtained from Galileo encounters have provided a large amount of new
information about the moon. These include the discovery and verification of
Ganymede's magnetic field, its magnetosphere, its trapped particle
population, its interaction with the Jovian environment and plasma waves
associated with Ganymede (Gurnett et al., 1996; Kivelson et al., 1996, 1997,
1998; Frank et al., 1997; Williams et al., 1997a, b, 1998; Williams and Mauk,
1997). The presence of a global magnetic field at
Ganymede was inferred from the detection of the electromagnetic and
electrostatic waves and radio emissions as the Galileo spacecraft approached
Ganymede (Gurnett et al., 1996) and was later confirmed by Galileo's
magnetometer data during closer flybys of the moon. These data have shown
that Ganymede has an intrinsic magnetic field strong enough to generate a
mini-magnetosphere (diameter 4–5 RG, RG is radius of Ganymede = 2631 km)
embedded within the Jovian magnetosphere (Kivelson et al.,
1996, 1997, 1998). A model with a fixed Ganymede-centered dipole
superimposed on the Jovian ambient field provided a good first-order match
to the data and suggested equatorial and polar field strengths at Ganymede's
surface of 750 and 1200 nT, respectively. These values are 6 to 10 times the
120 nT ambient Jovian field strength at Ganymede's orbit. According to this
model, magnetic field lines emanating from Ganymede's poles connect to
Jupiter, whereas lines closer to Ganymede's equator intersect Ganymede's
surface at both ends. The value of the moon's permanent magnetic moment is
about 1.3 × 1013T-m3, which is 3 times larger than
the magnetic moment of Mercury. The data acquired during four close flybys
of Galileo past Ganymede are consistent with a Ganymede-centered magnetic
dipole tilted by 10∘ from the spin axis towards 200∘ Ganymede east
longitude (Kivelson et al., 1996). The model of Ganymede's magnetic moment
was further refined after the last two flybys: G28 and G29 (Kivelson et al.,
1998). The current estimate is that the magnetic pole is tiled 4∘ from
the spin axis and points towards 156∘ W in the north and 336∘ W in
the south. The interaction between the Ganymedian magnetosphere and Jovian
plasma is similar in some respects to that of the solar wind and terrestrial
magnetosphere.
No atmosphere was revealed by the Voyager data. Evidence for a tenuous
oxygen atmosphere on Ganymede was found by the HST (Hall et al., 1998). The HST
actually observed the airglow of atomic oxygen in the far-ultraviolet range at the
wave lengths 130.4 and 135.6 nm. Airglow emissions are characterized by
the flux ratio F (1356 Å) /F (1304 Å) of roughly 1–2, which
suggests the dissociative electron impact excitation of O2. Inferred
vertical column densities are in the range of (1–10) × 1014 cm-2. The observed double-peaked profile of the neutral atomic oxygen (OI) 1356 Å feature
indicated a non uniform spatial emission distribution that suggested two
distinct and spatially confined emission regions, consistent with the
satellite's north and south poles. Additional evidence of the oxygen
atmosphere comes from the detection of various gases trapped in the ice on
Ganymede (Calvin and Spencer, 1994; Spencer et al., 1995; Calvin et al.,
1996). The evidence consisted of the spectroscopic detection of ozone
(O3), as well as absorption features that indicated the presence of O2.
The discovery of an intrinsic magnetic field associated with Ganymede was
reinforced by the HST observations of atomic oxygen emission associated with
the polar regions of the satellite (Hall et al., 1998) and by the subsequent
ultraviolet images obtained in 1998 that revealed unambiguous polar auroral
emission from Ganymede with a brightness of up to 300 Rayleigh (R) in localized
spots (Feldman et al., 2000). These images also show a background emission
above the detection limit of 50 R but not exceeding 100 R across the rest of
the disk of the satellite (Eviatar et al., 2001b). The oxygen emission is
thought to be produced primarily by electron dissociative excitation of the
molecular oxygen that dominates in Ganymede's tenuous atmosphere,
although there is also likely a lesser contribution from electron excitation
of the atomic oxygen component of the atmosphere. The OI emissions appear in
both hemispheres, at latitudes above |40∘|, in
accordance with Galileo magnetometer data that indicate the presence of an
intrinsic magnetic field such that Jovian magnetic field lines are linked to
the surface of Ganymede only at high latitudes. Both the brightness and
relative north–south intensity of emission vary considerably during the
5.5 h of observation, presumably because of the changing Jovian plasma
environment at Ganymede.
McGrath et al. (2013) present the ultraviolet images of Ganymede acquired
with HST from 1998 to 2007, all of which show auroral emission from electron
excited atomic oxygen. Ultraviolet emission at 1356 Å is brightest at
relatively high latitudes in the orbital trailing (upstream plasma)
hemisphere and in an auroral oval that extends to as low as ∼ 10∘ N
latitude in the orbital leading (downstream plasma) hemisphere. The
overall emission morphology appears to be driven primarily by the strong
Jovian magnetospheric plasma interaction with Ganymede. At any given
longitude, the latitude of the brightest emission does not change
significantly, but its brightness sometimes does. Ganymede's auroral
emission is characterized by localized bright regions with a peak brightness
of ∼ 100–400 R. The peak emission intensity at the oval is the
region that receives the maximum particle precipitation. The correspondence
between the boundaries of the UV oval emission and electron precipitation is significant. The background emissions of an
intensity of 50–100 R are called the
diffuse aurora at Ganymede. We study the diffuse emissions produced by
Jovian magnetospheric electron precipitation into the atmosphere of
Ganymede.
The diffuse aurora at Earth is explained as the result of the pitch angle
diffusion of electrons into the loss cone and subsequent precipitation in
the atmosphere by plasma waves. Two important wave modes which are being
considered are electrostatic electron cyclotron harmonic (ECH) and whistler
mode waves. In this paper, we are mainly concerned with diffusion process by
whistler mode waves. The whistler mode is one of the modes of the propagation
of electromagnetic waves below the electron cyclotron frequency, and the waves tend to
propagate parallel to the ambient magnetic field. On 27 June 1996 the
Galileo spacecraft made the first close flyby G1 of Ganymede. Intense plasma
waves were detected over a region of space nearly 4 times Ganymede's
diameter (Gurnett et al., 1996). The types of waves detected are whistler
mode emissions, upper hybrid waves, electrostatic electron cyclotron waves
and escaping radio emission. Electromagnetic and electrostatic plasma waves
have also been observed in the middle (10–20 RJ) magnetosphere of
Jupiter (Scarf et al., 1979; Gurnett et al., 1979, 1996; Stone et al., 1992). Plasma waves in the middle magnetosphere are not
limited to the region around Ganymede. Observations at Jupiter obtained by
the Plasma Wave Instrument on board the Galileo spacecraft indicate that
whistler mode chorus emissions with frequency-integrated power levels of
10-8 V2 m-2 or greater are observed commonly in the Jovian
magnetosphere near the magnetic equator (Menietti et al., 2008) in the
frequency range 400 Hz < f < 8 kHz. The emissions are
relatively narrow-banded.
The planetary magnetospheres produce various plasma instabilities which lead
to the emission of plasma waves propagating in various modes. Most of these
instabilities are due to anisotropic electron distribution, such as beam, a loss-cone feature and temperature anisotropy. If the pitch angle
distribution is anisotropic with more energy perpendicular than parallel to
the magnetic field, a particle distribution is unstable. There is free
energy to cause wave instability. Electron pitch angle anisotropy produced
by the atmospheric loss cone may produce electrostatic ECH and whistler mode
instability in a magnetosphere. The loss-cone anisotropy is produced when
particles moving within a cone of directions along the magnetic field strike
the planetary atmospheric surface and are lost from the system (Kennel and
Petschek, 1966). Both ECH and whistler mode waves can cause pitch angle
diffusion of electrons which results in particle precipitation into the
atmosphere producing a diffuse aurora.
In addition to causing pitch angle diffusion, electrostatic waves can also
heat and accelerate the ambient electrons (Swift, 1970). This can produce a
non-Maxwellian and suprathermal tail of energetic electrons in the magnetosphere
at the magnetic equator. Such non-thermal distributions, with overabundances
of fast particles, can be better fitted for superthermal velocities by
generalized Lorentzian or kappa distributions. In this distribution
function, characterization is done by real values of “spectral index
(κ)”, which assumes different shapes (Summers and Thorne, 1991;
Summers et al., 1994). At high velocities, the distribution has an inverse
power law tail in energy with the exponent (κ+ 1).
In a recent work Singhal et al. (2016) have studied the diffuse aurora on
Ganymede due to pitch angle diffusion of electrons by ECH waves. In the
present study we have extended this work and calculated the diffuse auroral
intensities from pitch angle diffusion of magnetospheric electrons by
whistler mode waves. In Sect. 2 we present the details on the method of
calculations. The potential results of the study are discussed in Sect. 3, and finally the concluding remarks of the present work have been provided in
Sect. 4.
Calculation detailsLoss cone at Ganymede
Due to the unique location of Ganymede in the Jovian magnetosphere, the
field lines emanating from Ganymede are connected to Jupiter above a latitude
of about |λ| > 40∘. There are two
mirror points on this field line: one near Ganymede and the other near
Jupiter. The minimum magnetic field on this field line defines the magnetic
equator. The Ganymede side loss cone can be calculated from the conservation
of the first adiabatic invariant, i.e.,
Sin2α/B=Constant,
where α is the electron pitch angle and B is the magnetic field.
Using Eq. (1) we obtain
Sin2αLC=Beq/BA,
where αLC is the loss-cone angle, Beq is the field at the
magnetic equator and BA is the field at the top of the atmosphere of
Ganymede. The density of O2 becomes 1/e (e= 2.71828…) of
its value at the surface at a radial distance of 1.005 RG (Eviatar et
al., 2001b). The top of the atmosphere is assumed at 1.005 RG . For
calculating the B fields, we trace the field line connecting Ganymede and
Jupiter using the equations
1/rdr/dθ=Br/Bθandsinθdφ/dθ=Bφ/Bθ,
where r is the distance from center of Jupiter, θ and φ are
colatitudes and longitude measured in a Jupiter-centered spherical
coordinate system, and Br , Bθ and Bφ are the
field components. The magnetic field of the Ganymede–Jupiter system is calculated,
assuming the static superposition, from (Tripathi et al., 2013, 2014)
B=-∇V+b+bG.
The magnetic field of Jupiter is a sum of contributions from internal and
external sources. The internal field is derivable from a scalar potential V.
The VIP4 model (Connerney et al., 1998) is used, in which V is expressed as a
spherical harmonic expansion to degree and order 4. The external field b is due
to a thin disc-shaped azimuthal current sheet. It is calculated using the
analytical expressions given by Connerney et al. (1981, 1982) and Acuna et al. (1983).
The magnetic field of Ganymede bG is modeled by a
Ganymede-centered dipole (Kivelson et al., 1998). Loss-cone angles
calculated in the present work are αLC≈ 10.8–13.6∘. It may be noted that the magnetic field model of
Ganymede given by Kivelson et al. (1996) has been used in the work of
Tripathi et al. (2013). In this work we have used the refined model
suggested by Kivelson et al. (1998).
Whistler mode wave growth rate at the magnetic equator
The temporal growth rate for whistler mode waves is calculated using the
appropriate dispersion relation. For parallel propagating R mode, the
dispersion relation is written as (Kennel and Petschek, 1966)
D(k,ω)= 1-c2k2ω2-π∑αωpα2ω∫0∞v⊥2dv⊥∫-∞+∞dv||×∂fα∂v⊥-kωv||∂fα∂v⊥-v⊥∂fα∂v||1kv||-ω+Ωα= 0,
where α denotes species, k is the parallel wave propagation vector,
ω is complex frequency (ω=ωr+iγ) and v=v||+v⊥ is the velocity in which
v|| and v⊥ are components parallel and perpendicular
to the ambient magnetic field. ωα and Ωα are plasma and gyrofrequency, respectively. Parameter fα is
the electron distribution function. The distribution is a combination of
Maxwellian (cold) and kappa loss-cone (hot) distributions (α=c,h). These are given by
fM=1π3/2vc3exp(-v⊥2/vc2-v‖2/vc2)vc2=(2Tc/me)andfκ=C111+EκEo(κ+1)(sinα)2s,
where C1 is the normalization constant. Here, α is the pitch
angle, Tc is the cold electron temperature, nc is the cold
electron density, me is the mass of electron, s is the loss-cone index,
κ is the spectral index and Eo is the characteristic energy. In
the case of the Maxwellian distribution, nc= 12.5 cm-3 and Tc= 18.6 eV; for the kappa loss-cone
distribution, κ= 1.5 and Eo= 500 eV are taken. Hot electron density nh= 0.1 nc, the index
s= 0.5 and ambient magnetic field Bo= 55 nT at the magnetic
equator are considered. The parameters nc and Tc are obtained at
the magnetic equator using the analytical expressions presented by Divine
and Garrett (1983). The parameters κ, Eo and nh are taken
from the work of Paranicas et al. (1999). Details about solving the
dispersion relation (Eq. 5) are described in Tripathi et al. (2014). The
temporal growth rate profile for whistler mode waves, calculated in the
present work, is presented in Fig. 1. The temporal growth rate profile is used
to represent the whistler mode wave spectral intensity in the calculation of
pitch angle diffusion coefficients.
Normalized temporal growth rate γ‾=γ/Ωeversus normalized real frequency ω‾r=ωr/Ωe for whistler mode
waves.
Pitch angle diffusion coefficients
The scattering of geomagnetically trapped particles is predominantly
controlled by plasma waves that are Doppler shifted in frequency to some
integral multiple of the particle cyclotron frequency, i.e.,
ω-k||v||=nΩe/γ,n= 0,±1,±2,……,
where γ=(1-v2/c2)1/2. The Landau (n= 0) resonance simply involves energy transfer between waves
and particle. For the cyclotron (n≠ 0) resonances, the diffusion
occurs predominantly in pitch angle (v= const).
We have calculated pitch angle diffusion coefficients (Dαα)
due to whistler mode waves using the expressions given by Lyons (1974) (also
Singhal and Tripathi, 2006) under the high-density approximation ((ωpe/Ωe)2≫ω/Ωe). This approximation simplifies the dispersion relation for whistler
mode waves and the equations for the diffusion coefficients. We may write
Dαα=∑n=-∞∞∫0xmaxxdxDααnx,
where x= tan η and η is the wave normal angle (the angle
between Bo and the wave propagation vector k). Dααnx is given by
Dααnx=πcos5ηΩe(-sin2α-nΩe/ωk)2⋅|Φn,k|22C1ψ3/2|1+nΩe/ωk|3I(ωk)×f(ω)gω(x)⋅1-1v||∂ωk∂k||xωkΩe=ωkΩeres-1×Bwave2Bo2×Ωe.α is the pitch angle, v is electron speed and Bwave is the wave
magnetic field. f(ω) is the wave spectral density and gω(x) gives the wave normal distribution. For gω(x) we use
gω(x)∝exp(-x2)forx≤1,0forx≥1.
It is assumed that the wave energy is proportional to the linear temporal
growth rate. The constants of proportionality do not appear in the
calculations of diffusion coefficients. The temporal growth rate profiles,
therefore, represent the distribution of wave energy with frequency. For
wave spectral density we have used the temporal growth rate profile shown in
Fig. 1. In Eq. (10), C1=∫f(ω‾)dω‾, (ω‾=ωr/Ωc) and
1-1v||∂ωk∂k||x= 1- 2ψ1+nΩe/ωk×2ψ+ 2ω2ΩpΩe-ω2Ωp2(1-M)2×(1+x2)ψ-1+ω2ΩpΩe+x22-1-1,|Φn,k|2=Dμ2-S2μ2sin2η-Pμ22+Pcosημ22-1×μ2sin2η-P2μ21+Dμ2-SJn+1+μ2sin2η-P2μ2×1-Dμ2-SJn-1+cotαsinηcosηJn2,I(ω)=∫0∞gω(x)x{(1+x2)ψ}-3/21+1ψω2ΩpΩe-12ω2Ωp2(1-M)2×(1+x2)ψ-1+ω2ΩpΩe+x22-1dx,
where the argument of Bessel function Jn=Jnxtanα-ωkΩe-n.
μ2=ωpe2Ωe21+MMψ-1,M=me/mp,meandmp
are the mass of electrons and protons, respectively.
ψ= 1-ωk2ΩpΩe-sin2η2+sin4η4+ωkΩp2(1-M)2cos2η1/2,P=-ωpe2Ωe2Ωe2ωk2(1+M),S=12(R+L),D=12(R-L),RL=±ωpe2Ωe2Ωeωk1+M1-M∓(ωk/Ωe-Ωp/ωk)
Here, Ωp is proton gyrofrequency. Bounce-averaged diffusion
coefficients are obtained from Lyons et al. (1972):
<Dαα>=1τb∫Dαα∂αeq∂α2dt=1τb∫Dαα∂αeq∂α2dsvcosα.
Using (ds)2=(dr)2+r2(dθ)2+r2sin2θ(dφ)2 and the conservation of first
adiabatic invariant (Eq. 1), we can write
<Dαα>=2vτb∫λGλJDαα(α)cosαtanαeqtanα2×r2+∂r∂λ2+r2cos2λ∂φ∂λ21/2dλ,
where r and θ, φ are the position on the field line connecting
Ganymede with Jupiter (θ=π/2-λ). λ is
latitude and λG and λJ are mirror latitudes near
Ganymede and Jupiter, respectively. τb is the bounce period which is
set equal to half the bounce period for Jupiter at L= 15 (Orlova and
Shprits, 2011). The variation in magnetic field and electron density along
field lines is taken into account (Divine and Garrett, 1983; Eviatar et al.,
2001a). Whistler mode waves are assumed to be confined within latitudes
(measured from magnetic equator) < ±13∘ (Menietti et
al., 2008). Pitch angle diffusion coefficients calculated in the current
work are presented in Fig. 2 corresponding to two electron energies (200 eV
and 2 keV) by using a wave magnetic field of 10 pT.
Bounce-averaged pitch angle diffusion coefficient versus pitch angle
for whistler mode waves at two electron energies as indicated.
Precipitation flux
Pitch angle diffusion by whistler mode waves drives the magnetospheric
electrons into the loss cone, thereby precipitating these electrons in the
atmosphere of Ganymede. The differential flux of precipitation electrons as a
function of energy and pitch angle is given by (Kennel and Petschek, 1966;
Ni et al., 2012)
J(Eo,α)=J(Eo,αLC)IoZoα/αLCIo(Zo),
where Zo=DSD/<Dαα>αLC is an energy-dependant parameter defining the diffusion
strength near the loss cone. Io is the modified Bessel function of the
first kind and J (Eo,αLC) is the electron flux near the
equatorial loss cone. < Dαα > LC is the bounce-averaged pitch angle diffusion coefficient at the edge of
the loss cone. DSD is the strong diffusion rate determined by (Kennel,
1969)
DSD=2(αLC)2τB.
Total precipitation flux Φ is determined by the differential flux of
precipitating electrons inside the equatorial loss cone (e.g., Chang, 1983):
Φ= 2πBABeq∫E1E2∫0αLCJ(Eo,α)cosαsinαdEodα.
Equatorial pitch angle α (0 to αLC) maps to pitch angle
θo (0 to π/2) at the top of the atmosphere. From the
conservation of the first adiabatic invariant,
sin2α/Beq=sin2θo/BA;
using Eq. (27), we can write Eq. (26) as
Φ= 2π∫E1E2∫0π/2J(Eo,α(θo))sinθocosθodEodθo,
where sinα=sinθosinαLC. E1 and
E2 are the lower and upper limit for integration over energy.
Variation in O2 number density.
Volume excitation rate
For calculating the energy deposition of precipitating electrons in the
atmosphere of Ganymede we have used the analytical yield approach (AYS)
(Green and Singhal, 1979; Singhal et al., 1980; Singhal and Green, 1981)
described in Appendix A. In this approach the volume excitation rate (VER)
for exciting the kth state in gas i may be written as (cf. Eq. 28)
VER= 2π∫E1E2dEo∫0π/2dθo∫WkiEodEJEo,α(θo)×sinθocosθoU(E,Z′,Eo)ρ(Z)pki(E).
Here, U (E,Z′, Eo) is the three-dimensional AYS. Z′=Z/cosθo, Z is the penetration depth in gram meters per square centimeters,
ρ is the atmospheric mass density and pki is the excitation
probability for exciting the kth state in gas i. We use the model
neutral O2 atmosphere computed by Eviatar et al. (2001b). The model
O2 profile is presented in Fig. 3. We have also included atomic oxygen
with a constant mixing ratio of 10 %. Vertical column density of the
O2 profile is 2.9 × 1014 cm-2.
In the present work we have calculated the OI λ 1356 Å diffuse auroral intensities. This wavelength arises from the atomic oxygen
transition (2s22p43p←2s22p33s5S∘). We have considered three processes for the emission of this
line. (1) Dissociative excitation of O2, (2) direct impact on atomic gas
exciting the O (5S∘) state and (3) direct impact on atomic O
exciting the O (5P) state which cascades to O (5S∘).
Probabilities of the excitation of the OI λ 1356 Å wavelength from
these processes are
(1)P1=σd0.1σ1+σ2,(2)P2=0.1σ(5s0)0.1σ1+σ2and(3)P3=0.1σ(5P)0.1σ1+σ2.
Here, σ1 and σ2 are the total inelastic electron
impact cross section for atomic O and molecular O2, respectively.
σd is the cross section for dissociative excitation of O2.
It is taken from the works of Wells et al. (1971), Erdman and Zipf (1987),
and Itikawa et al. (1989). Total inelastic electron impact cross sections
σ1 and σ2 and cross sections for the direct excitation of
atomic O to the states (5S∘) and (5P) are taken from the work
of Jackman et al. (1977). Total inelastic cross sections σ1 and
σ2 as a function of electron energy are presented in Fig. 4. The
excitation cross sections σd, σ (5S∘) and
σ (5P) are plotted in Fig. 5.
Total inelastic electron impact cross section versus electron energy
for atomic oxygen (σ1) and molecular oxygen (σ2).
For calculating the precipitation flux (Eq. 24), we require the flux J
(Eo,αLC) at the edge of the loss cone. We have considered
three flux profiles by assuming the pitch angle isotropy.
Flux of magnetospheric electrons observed near Ganymede (Frank et al.,
1997; Paranicas et al., 1999).J(Eo)=a/Eobinunitsof(cm2ssreV)-1,
where a= 1.1 × 1012, b= 3.52 for the thermal component (9–100 eV),
a= 1.6 × 107, b= 1.1 for suprathermal electrons (100 eV–3 keV). Flux below 1 keV has been extrapolated.
To simulate the heating of ambient electrons by electrostatic waves, we
have considered the flux due to a Maxwellian distributionfM=nhπ3/2vh3exp(-v2/vh2),wherevh=(2Th/m)1/2.
Finally, to simulate the acceleration by electrostatic waves we have
considered the flux due to a kappa distribution function:
fκ=nhm2πEc3/2Γ(κ+1)κ3/2Γ(κ-1/2)11+EκEcκ+1.
Ec is the characteristic energy. Two values of parameter
κ (κ= 1 and 2) and nh= 12.5 cm-3 are used. Electron flux is obtained usingJ= 2Eof/m2.
m is the mass of electrons. Integrating the VER (Eq. 29)
over altitude gives the intensity in Rayleigh (1 R= 106 photons cm-2 s-1). It is
assumed that each excitation gives rise to a photon.
Electron impact excitation cross sections versus electron energy for
dissociation (σd) and direct atomic O (5S∘) and direct
atomic O (5P) excitations. Atomic O (5P) cascades into O (5S∘).
Results and discussion
In Fig. 1, it is observed that the calculated whistler mode wave growth rate
is mainly confined within the normalized frequency range ω‾(=ωr/Ωc)≈ 0.1–0.45. This may be compared
with the observations of whistler mode waves in the magnetosphere of Jupiter
and in the vicinity of Ganymede. Observations of plasma waves made by
Voyager 1 and 2 have been described by several authors (Scarf et al., 1979;
Gurnett et al., 1979; Gurnett and Scarf 1983; Coroniti et al., 1984).
Voyager 1 detected whistler mode emissions inside of 10 RJ in the range
of ω‾≈ 0.3–0.42. The same emissions detected by
Voyager 2 extended out to 20 RJ and beyond. Peak amplitudes of
chorus waves received by Voyager were found to be about 0.26 mV m-1 (10 pT amplitudes assuming parallel propagation). Whistler mode waves were also
received near Io torus during the Ulysses–Jupiter encounter (Stone et al.,
1992) in the frequency range ω‾≈ 0.04–0.2.
Recently, similar wave observations detected from the Galileo probe have been
described by Menietti et al. (2008). The observations indicate that chorus
emissions are observed commonly in the Jovian magnetosphere near the
magnetic equator in the approximate radial range 9 < r < 13 RJ . These emissions exist in the frequency range from ω‾= 0.02 up to, but seldom exceeding, about 0.46. Wave magnetic
field amplitude is found about 7 pT. Whistler mode waves have also been
observed in the vicinity of Ganymede with a frequency of ω‾≤0.5 (Gurnett et al., 1996).
Figure 2 presents bounce-averaged pitch angle diffusion
coefficients due to whistler mode waves. It is noticeable that the diffusion coefficients do not change much between
two electron energies (200 eV and 2 keV) for pitch angle
less than about 25∘. At a higher pitch angle (> 50∘), the coefficients for an energy of 200 eV are
negligible, whereas for an energy of 2 keV the coefficients
extend up to about 80∘. However, it is found
that the diffusion coefficients are several orders of
magnitude smaller below about 100 eV and even become
0 for below 50 eV. The diffusion coefficients depend
upon electron densities along the field line. Electron
densities (and temperature) have been calculated using the
analytical expressions provided by Divine and Garrett (1983)
for the inner disc (7.9 <r< 20 RJ) in
Jupiter's magnetosphere. These expressions are:
Ne=Nexp-rλ-ZoH2,kT=Eo-E1exprλ-ZoH2.N is interpolated from Table 1 and
H=(1.82-0.041r)RJ,Zo=7r-1630RJcos(ℓ-ℓo),Eo= 100eV,E1= 85eV,ℓo= 21∘.
Here, r is the Jovicentric distance in RJ. λ
is latitude (radian), and ℓ is longitude (degree in System III;
1965). The model is based primarily on in situ data returned by experiments on the Pioneer
and Voyager spacecraft, supplemented by earth-based
observations and theoretical considerations. The model
represents the data typically to within a factor of
2±1 except where time variations, neglected in
the model, are known to be significant. Further, we have
made a high-density approximation ((ωpe/Ωe)2≫ω/Ωe) in the
calculation of diffusion coefficients. In this work the ratio ωpe2/Ωe2 is 2–400. This range is due to
variation in electron density and magnetic field along the field line
connecting Ganymede to Jupiter. Glauert and Horne (2005) have presented
a calculation of diffusion coefficients (PADIE code) for any ratio of β(=ωpe/Ωe). It is found that the high-density
approximation, at low electron energy (< 10 keV), agrees with the
PADIE results for β≥ 10 but underestimates diffusion
coefficients by 1 order of magnitude near the loss cone for β≈ 1.5. Resonant energy (EM=B2/8πne) at the
magnetic equator is found ∼ 600 eV and increases up to 30 keV along
the field line connecting Ganymede to Jupiter.
Equatorial parameter values for Jupiter's
thermal charged particle populations (from Table 7 of
Divine and Garrett, 1983).
Jovicentric distanceElectron densityr,RJlog N, cm-37.92.2510.01.4820.00.20
The O2 number density presented in Fig. 3 is based on the Bates (1959)
atmosphere model applied in the region r≤ 1.38 RG and a coronal
model in the exosphere region (Eviatar et al., 2001b). O2 surface
density is 1 × 108 cm-3, and vertical column density
is 2.9 × 1014 cm-2. The atmosphere model is similar to
that constructed by Feldman et al. (2000).
Electron impact total ionization and total
inelastic cross sections (in units of 10-16 cm2) for molecular oxygen and atomic oxygen gas
(Jackman et al., 1977).
O2OElectron energy (eV)Total ionizationTotal inelasticTotal ionizationTotal inelastic12.80.03.140.00.4120.320.221.232.233.132.327.729.124.525.451.427.629.218.419.081.823.825.313.514.013019.320.69.8610.220714.915.97.087.3533011.011.75.025.214687.928.540.810.965251.622.200.760.898361.181.600.550.6513300.831.140.390.4621200.580.800.270.3230000.440.600.210.24
Calculated intensities of OI 1356 Å emissions in Rayleigh due to magnetospheric
electrons (Eq. 31).
a Using total magnetospheric electron flux; O2 column density
2.9 × 1014 cm2.b Using precipitation flux; whistler mode wave amplitude 10 pT; O2
column density 2.9 × 1014 cm2.c Using precipitation flux; wave amplitude 20 pT; O2 column density 2.9 × 1014 cm2.d Using precipitation flux; wave amplitude 20 pT; O2 column density 1.0 × 1015 cm2.e Using total magnetospheric electron flux; O2 column density 1.0 × 1015 cm2.
In Fig. 4 we present the electron impact total inelastic cross sections for
atomic O and molecular O2. These cross sections have been calculated
using the analytical expressions given by Jackman et al. (1977). For
forbidden electron impact excitations, the cross section is written
as
σ=qoFW21-WEαβWEΩ.
The formula for allowed excitation is
σ=(qoF) 1-W/EαβEWln4ECW+e,
where qo= 6.513 × 10-14 eV2 cm-2. W is the
parameter for the low-energy shape of the cross section and, in most cases
is close to the energy loss. F is the optical oscillator strength, and e is the
base of natural logarithm. α, β, Ω and C are
adjustable parameters. The electron impact ionization cross sections are
calculated from
σ=σoA(E)Γ(E)tan-1Tm(E)-To(E)Γ(E)+tan-1To(E)Γ(E),
where
A(E)=K(E+kB)lnEJ+JB+JcE,Γ(E)=ΓSE(E+ΓB),To(E)=TS-TA(E+TB),Tm(E)=(E-Ii)2.σo= 10-16 cm2 and K, kB, J, JB, Jc,
ΓS, ΓB, TS, TA and TB are
adjustable parameters. Ii is the threshold energy for ionization. E is
electron energy. These are semiempirical cross sections based upon the
experimental data. Various energy-dependent and other parameters appearing
in Eqs. (37)–(39) are taken from Jackman et al. (1977). For atomic oxygen, 35 excited and Rydberg states and 3 ionizations have been included. For
molecular oxygen, 12 excited and Rydberg states and 7 ionization states have
been included in our calculations. For excitations of atomic oxygen
presented in Fig. 5, calculations have been made using Eq. (37). The electron
impact total ionization and total inelastic cross sections for molecular
oxygen and atomic oxygen are also presented in Table 2.
In Table 3 we have presented the diffuse auroral intensities of OI λ1356 Å emission resulting from the precipitation of magnetospheric
electrons observed near Ganymede (Eq. 31). It is seen from Table 3 that the
intensities due to precipitation of thermal electrons are negligibly small because diffusion coefficients are negligible for electron energies below
100 eV. Further, from the table it is also noticeable that total intensities
for both thermal and superthermal electron components exist in the range of 3–35 R. These intensities are too small to account for the observational
diffuse auroral intensities. The results of present calculations are in
agreement with the conclusions reached in earlier works (Eviatar et al.,
2001b). As discussed in the earlier works of Eviatar et al. (2001b),
Lavrukhin and Alexeev (2015), and McGrath et al. (2013), some acceleration
mechanism is required to energize the electrons. The processes that lead to
the acceleration of electrons can be different in nature. In this paper, we
consider the heating and acceleration by electrostatic waves observed in the
magnetosphere of Jupiter and in the vicinity of Ganymede. For electrostatic
waves, the wave normal is nearly perpendicular to the magnetic field lines.
The component of phase velocity parallel to the field lines is high. An
electron trapped in the wave may experience a substantial increase in
velocity as it accelerates to keep up with the parallel component of the
phase velocity. Particles could be accelerated by up to several times their
initial velocity by this mechanism (Swift, 1970). The energetic electrons
are subject to a frictional force in the ambient electron gas (Mantas,
1975):
OI 1356 Å intensity in Rayleigh (R) versus electron temperature
(Th) for the Maxwellian distribution (Eq. 20). Whistler wave amplitude is
indicated.
Intensity of OI 1356 Åemission in Rayleigh (R) versus
characteristic energy (Ec) for a kappa distribution with κ= 2. Whistler wave amplitude is indicated.
Same as in Fig. 7 for κ= 1.
Volume excitation rate versus altitude for dissociation O2,
direct O (5S) and cascading O (5P) excitations. A Maxwellian
distribution with Th= 150 eV is used, and whistler mode wave amplitude
is 10 pT.
dE/ds=-βNe/E,
where β= 2.59 × 10-12 (eV2 cm-2), Ne
is the ambient electron density, E is electron energy and ds is the element of
path length. The ambient electrons may be heated in this process.
Electrostatic ECH waves have been observed in Jupiter's middle
magnetosphere, with amplitudes of 1–5 mV m-1 (Scarf et al., 1979; Gurnett
and Kurth, 1979). These waves have also been detected near Ganymede, confined
to within a few degrees of the magnetic equator (Kurth et al., 1997; Gurnett et
al., 1996). The electrostatic ECH wave turbulence may heat and accelerate
the ambient. These electrons may precipitate into the atmosphere of Ganymede
via pitch angle diffusion by whistler mode waves and may produce diffuse
auroral emissions. We simulate the heating of ambient electrons using a
Maxwellian distribution (Eq. 32). Calculated diffuse auroral intensities of
OI λ 1356 Å emission are presented in Fig. 6 as a function of
electron temperature (Th). Calculations of this plot have been
performed at two amplitudes of whistler mode wave: 10 and 20 pT. It is
evident that intensities of about 70 R may be obtained from 10 pT for
a temperature of 150 eV. A higher temperature may yield intensities of up to 200 R.
Further, the wave amplitude of 20 pT may produce somewhat higher intensities as
compared to values for 10 pT at the same temperature. This is due to large pitch
angle coefficients for 20 pT since the diffusion coefficient scales as a square of the wave amplitude.
The acceleration of ambient electrons is simulated by a kappa distribution
function (Eq. 33). Estimated diffuse auroral intensities are depicted in
Figs. 7 and 8, corresponding to two values of κ= 2 and κ= 1, respectively. It is noted from Fig. 7 that the intensity ≈ 50–100 R
may be obtained for a characteristic energy Ec≈ 100–150 eV.
These values are higher in comparison to intensities obtained
from the Maxwellian distribution (Fig. 6). In Fig. 8, intensities for κ= 1 are somewhat higher (about 80 R for a characteristic energy 50 eV).
This is due to the fact that more higher-energy electrons exist in the tail
of a kappa distribution. It is also evident from Figs. 7 and 8 that
intensities are flattened out for a characteristic energy ≥ 200 eV.
Flattening is more in the case of κ= 1 as compared to κ= 2. This is probably due to the use of 3 keV as the upper limit of
integration over Eo in Eq. (29). So, this may not capture higher
energies in the tail of the distribution.
Finally, in Fig. 9 we show the altitude profile of the volume excitation rate
for the Maxwellian distribution by considering the whistler wave amplitude 10 pT
and Th= 150 eV. It is noticeable that the electrons deposit their
energy within about 200 km of the surface of Ganymede. Contribution
from direct excitation of atomic O (5S) 0.54 R is obtained, while
cascading contributes only 1.4 R. The maximum contribution of 72.9 R
appears to be from dissociative excitation of O2.
Conclusions
Diffuse auroral intensities of OI λ 1356 Å emission have been
calculated, resulting from pitch angle diffusion of magnetospheric electrons
by whistler mode waves. Three cases are considered for the estimation of
precipitation flux into the atmospheric loss cone. First, the intensities
are estimated due to the precipitation of magnetospheric electrons observed
near Ganymede. Next, the heating and acceleration of magnetospheric
electrons by electrostatic waves is considered. Maxwellian and kappa
distributions are used to simulate heating and acceleration, respectively.
The AYS approach is used to calculate the energy
deposition of electrons in the atmosphere of Ganymede. The following main
conclusions are reached.
Intensities of OI λ 1356 Å emission estimated from the
precipitation of magnetospheric electrons observed near Ganymede are
inadequate to account for the observational diffuse auroral intensities ≤ 100 R. This is in agreement with conclusions reached in earlier works
(Eviatar et al., 2001b). Some acceleration mechanism is suggested to
energize the magnetospheric electrons.
The use of a Maxwellian distribution to simulate the heating of electrons by
electrostatic waves can provide a diffuse auroral intensity of about 70 R corresponding to a temperature of 150 eV.
The use of a kappa distribution in the case of κ= 2 having a characteristic energy of 100 eV yields an auroral intensity of about 100 R.
However, for a distribution with κ= 1 intensities of similar
magnitude can be obtained for a characteristic energy of 50 eV.
The contribution from the direct excitation of atomic oxygen is about
1 %, and the cascading contribution from atomic oxygen is about 2 % of the
total estimated intensity.
The potential findings of the present study may be relevant for the
present JUNO and future JUICE missions to Jupiter. These missions will
provide new data on electron densities, electron temperature and whistler
mode amplitudes in the magnetosphere of Jupiter near Ganymede.
Data availability
We have taken data from various journal articles, whose references are given in the text of the
paper.
Analytical yield spectrum
Green and colleagues (Green and Singhal, 1979; Singhal et al., 1980; Singhal
and Green, 1981) studied the energy degradation of monoenergetic electrons
in planetary atmospheric gases using a Monte Carlo technique. A function
which they have called “yield spectra” is obtained from the Monte Carlo
simulation. Two-, three-, four-, and five-dimensional yield spectra are
defined as follows:
Two-dimensional yield spectra U (E, Eo) are defined as
U(E,Eo)=N(E)ΔE(eV)-1,
where N (E) is the number of electrons in the bin
centered at E which result after the incident electron
of energy Eo and all its secondaries, tertiaries, etc., have been completely degraded in energy. Similarly 3-D
yield spectra U (E, Z, Eo) are defined as
U(E,Z,Eo)=N(E,Z)ΔEΔZ(eV)-1(gmcm-2)-1.
Here, N (E, Z) is the total number of inelastic collisions that exist in the
spatial interval ΔZ around Z and in the energy interval ΔE
centered at E. Z is the longitudinal distance along the Z axis, scaled by an
effective range R (Eo). The numerical yield spectral function generated
by Monte Carlo simulation is represented analytically.
Two-dimensional YS is parameterized in the form
Ua(E,Ek)=Co+C1χ+C2χ2,
where Co, C1 and C2 are external parameters, and
χ=EkΩE+L,
where L=1 eV, Ω=0.585,Ek is incident
electron energy in kiloelectronvolt, and E is in electron volts. The
3-D YS is represented in the form
Ua(E,Z,Ek)=∑i=02Ai′R3χiGi(Z),
where each Gi is a micro-plume of the form
Gi,=exp-βi2Z2+γiZ.
The 3-D YS is constrained to reduce to 2-D YS upon
integration over Z. It is found that various parameters
appearing in Eqs. (A5) and (A6) are not too different from
gas to gas when distance is expressed in gram per centimeter square.
Thus, AYS has a universal character.
The authors declare that they have no conflict of interest.
Acknowledgements
This work was supported by the Planetary Sciences and
Exploration Programme (PLANEX), the Indian Space Research Organization
(ISRO) and PRL, Ahmedabad, under the sanctioned project scheme. Calculations reported in
the present work were carried out at the Computer Centre, Banaras Hindu
University.
The topical editor, C. Owen, thanks two anonymous referees for help in evaluating this paper.
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