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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 95, NO. A1, PAGES 39-50, JANUARY 1, 1990

RADIAL DIFFUSION MODELS OF ENERGETIC ELECTRONS AND JUPITER'S SYNCHROTRON RADIATION

1. STEADY STATE SOLUTION

Imke de Pater

Astronomy Department, University of California, Berkeley

Christoph K. Goertz

Department of Physics and Astronomy, University of Iowa, Iowa City

The results of a computer code modeling the radial diffusion of equatorially confined energetic electrons in Jupiter's inner magnetosphere are compared with spacecraft as well as ground- based radio (synchrotron radiation) data. We find that the synchrotron radiation spectrum cannot be reproduced without a significant hardening of the electron spectrum between L = 3 and L = 1.5. This hardening may be due to energy degradation by Jupiter's ring particles. Our calculations also suggest that there may be larger-sized material outside Jupiter's ring up to L • 4 or Io's orbit.

1. Introduction

The physical processes that affect the distribution of energetic electrons in the inner regions of planetary magne- tospheres are of general interest. Whereas in situ observations are Possible in the Earth's magnetosphere, they are obviously much more difficult to make in Jupiter's inner magnetosphere. Because the radiation belts of Jupiter are very intense, spacecraft have not and will not spend a great deal of time in the inner Jovian radiation belts. Re- mote sensing observations of Jupiter's nonthermal or synchrotron radiation at decimetric wavelengths (DIM) have provided important information about the structure of the Jovian magnetosphere inside L~4 [e.g., Berge and Gulkis, 1976; de Pater, 1983, 1981b, c] on a continuous basis. In the following, L is defined as the ratio of the radial distance r in the equator to the planetary radius Rj (r= L Rj). For a dipole field it is equal to the Moilwain L shell parameter for the field line crossing the equator at r=L R.•. In 1973 and 1974 the spacecraft Pioneer 10 and 11 flew through Jupiter's inner magnetosphere. Pioneer 10 was near the equatorial plane and penetrated to L--3, whereas Pioneer 11 traversed this region at high latitudes and reached a minimum L value of 1.6. In contrast to these %napshots," DIM observations yield many sequences of global images of the inner magnetosphere. However, these images need to be deconvolved to give information on the distribution of the relativistic synchrotron radiating electrons. Several authors [Thorne, 1965; Chang and Davis, 1962; Mead and Ness, 1973; Birmingham et al., 1974; de Pater, 1981b, c] have constructed DIM brightness distributions from models of energetic electron distributions and magnetic field configurations. Spacecraft flyby data can now be used to provide boundary conditions for such models, and thus yield improved synchrotron radiation predictions. In ad- dition, spacecraft data allow us to remove one uncertainty,

Copyright 1990 by the American Geophysical Union.

Paper number 89JA01240. 0148-0227/90/89JA-01240505.00

39

namely, the source spectrum, and thus allow for a much better determination of the transport and loss processes active in the Jovian magnetosphere.

The energetic electron distribution is usually calculated by solving the so-called lossy radial diffusion equation [Go- ertz et al., 1979]. Particles diffuse radially inward from an external source (usually incorporated into the models as a boundary condition at large values of L). Diffusion is driven by fluctuating electric and/or magnetic fields such that the first (•) and second (J) adiabatic invariants are conserved. The diffusion coefficient is assumed to be of the form Dr.r.=DoL", where n is approximately equal to 3 [e.g. Goertz et al., 1979). Particle losses are either due to absorption by large solid objects (moons or rings) and/or processes violating these two adiabatic invariants. In the outer magnetosphere beyond Io's torus, only pitch angle scattering into the atmospheric loss cone is important. In the inner magnetosphere and especially in the synchrotron generation region, which this paper is con- cerned with, synchrotron radiation is another important loss mechanism. Near the atmosphere of Jupiter (at L= 1) electron-neutral collisions also become dominant. Since these collisions cause the electrons to be entirely lost from the magnetosphere very close to L--l, this loss is replaced by the boundary condition that the particle fluxes go to zero at L=I.

Synchrotron radiation losses were only properly considered by Birmingham et al., [1974]. However, these authors assumed a monoenergetic source spectrum of particles at large distances (L=15) from Jupiter. After the various spacecraft fiybys we know the spectrum of particles at different L-shells quite accurately; hence Birmingham et al.'s work can be improved. Since electron losses in the Io torus are not of interest here, we will use the observed particle fluxes at the inner edge of the torus (L=6) as our boundary condition. We solve the steady state equation for radial diffusion of particles, using the observed electron spectra near Io's orbit and include synchrotron radiation losses. Physical loss processes are replaced by a phenomenological loss term in sections 2 and 3. We will show as other authors have done before us [e.g., Goertz

40 DE PATER AND GOERTZ: RADIAL DIFFUSION

et al., 1979] that such losses are needed to reproduce the observed radial profile of the energetic electrons. Specific examples of losses due to absorption effects by Jupiter's moons, ring and extended atmosphere/ionosphere, which may be responsible for the local losses will then be considered in section 4. We will determine the diffusion and loss parameters which best fit the particle flux as a function of distance from Jupiter as measured by the Pioneer 10 and 11 spacecraft, as well as the synchrotron radiation characteristics (e.g., spatial distribution, spectrum} as measured on Earth. The most relevant observations which we try to reproduce are (1) the radial profile of the omnidirectional integral electron flux of Van Allen [1976], in particular the position of the sharp drop near L=2, (2) the evolution of the energy spectrum of the electrons between L=6 and L=3, (3} the radial profile of the DIM emission, in particular, the position of the peak intensity and the width of the peak, and finally, (4) the spectrum of the DIM emission.

In the future we hope to solve the time- dependent radial diffusion equations, to better understand the origin of time variations observed in Jupiter's synchrotron radiation [e.g., Hide and Brannard, 1976; Klein et al., 1972; Klein, 1976; Bolton et al., 1989]. We therefore will derive the time-dependent equations in this paper and use o• =0 to get the form appropriate for this paper.

To simplify the entire problem, we approximate Jupiter's magnetic field by a dipole field with a magnetic moment of 4.28 G R• and consider only equatorially confined particles. Although we know that this is not right (Jupiter's field shows higher-order moments which clearly show up

The synchrotron radiation losses are represented by the second term on the right-hand side of (1) [Birmingham et al., 1974]:

with:

dt =/c(•,) o--a; x (4a)

e4 (4b)

o•X=o•[Pa/'( 1+ ,•B ) f] (4c) We should note that when off-equatorially mirroring par-

ticles are considered, synchrotron emission changes not only a particle's energy, but also its pitch angle. The bounce averaged rates for these changes are given by Schulz [1977]. It will be essential to include these effects when constructing two-dimensional maps of Jupiter's synchrotron radiation.

Local losses, such as those due to satellites, ring particles, and pitch angle scattering, are represented by the last term in (1). This form is only an approximation but widely used (for its justification the reader is referred to Schulz and Lanzerotti [1974]). We will first treat r as a parameter without concerning ourselves with specific physical mechanisms which may be responsible for the losses. These will be discussed in section 4.

in the synchrotron radiation characteristics, and electrons The phase space density, f, is related to the observable are found at quite high latitudes [e.g., de Pater, 1983; de directional differential in energy (E=Tmoc •) electron flux Pater and Jaffe, 1984]), it will provide a good first approx- j• (E,a,L,t)' imation to the problem, since most of the synchrotron radiation is produced by electrons confined to the magnetic ja(E(p,J),a(p,J),L,t)----f(p,J,L,t)p • (5) equatorial plane.

The equatorial pitch angle is a. However, the most com- 2. The Model monly observed quantity is the integral flux:

Several authors [e.g., Birmingham et al., 1974] have shown that the evolution of the drift and bounce averaged phase space density f(t•,J,L,t) is described by the equation

Ja (E,a,L,t)= fE• ja (E',a,L,I)dE' The omnidirectional fluxes are:

(6)

•-•,' ø-- (ø• •' oø-/c)- • •- •t (1) -- OL L 2 dt j(E,L,t)=2•r fo r ja (E,a,L,t)sinc•da=

The first and second adiabatic invariants, • and J, respectively, are related to the particle's momentum p:

r' (2a) •---- 2mob m

J=jp,,a, (2b) In (2a), mo is the electron's rest mass and Bm is the

magnetic field at the particle's mirror point. The second adiabatic invariant is defined as the integral along the field line of the parallel momentum Pl] for a complete bounce path. We will only consider equatorially mirroring particles (J=0) in a dipole field geometry. Thus

so (3) --L 3

with Bo= 4.28 G.

=1g h(E,•,,,) (7a)

J( E,L,t)=2•r fo • Ja ( E,a,L,t)sinada= =Sx. H(E L,t) (7b)

Our model does not predict the variation of off- equatorially mirroring particles (a:• •), and thus we cannot predict the functions h(E,L,t) and H(E,L,t). Goertz et M. [1979] have argued that these functions do not vary significantly with E and L, because the pitch angle distribution between 4<L<12 is of the form sin ka, with k varying from 3.5 to 4. Thus H(L) varies only between 7.4 and 7.9. We will thus assume them to be constant.

The synchrotron volume emissivity at a frequency v of equatorially mirroring particles which contribute domi- nantly to the synchrotron emission from the equator is given by

DE PATER AND GOERTZ: RADIAL DIFFUSION 41

where

(s)

V/• Bo• 9.5x10 --29 CB= 4•'rno •oe3C L 3 L 3 (9a)

K•13 is the modified Bessel function of order 5/3 and •,c, the critical frequency, is given by

3 eB •2

The particle's energy, E=•/moC 2, is here and in the following expressed in MeV. The observed emission intensity $• is obtained by integrating (8) along the line of sight and multiplying it with the solid angle •B of the beam received by a detector at Earth:

S•=9• f l•(s)ds= :a•aa f t•(•')d•' (10a)

where Rj--7x10 ? m is the Jovian radius and s' is a normalized distance along the line of sight. The common radio-astronomical unit for $• is 1Jy: 10- 26W m- 2Hz- t. Thus $• expressed in janskies is

_ f 1 f S•,=2.2 xlO 3•B L3(,,) f j• (E,L,t)F( •-[)dEds' (lOb)

where we have made the reasonable assumption that for synchrotron radiating electrons --r--=c. '7too

Our aim is thus to solve equation (1) for the phase space density profile assuming various forms for Dr r and r and to calculate the resulting particle fluxes (6) and (7) as well as the synchrotron radiation (10). These will then be compared to observations in order to determine the "best" values for Dr r and r. Details of the numerical procedure for solving (1) are described in the appendix.

3. Results

3.1 Comparison With Pioneer Data The typical energy of the synchrotron radiating electrons

is several tens of MeV in the radiation belts, which corre- sponds to energies around 1 MeV at L= 6 if the particles are transported adiabatically. Unfortunately, the energy spectrum in this energy range is not well established. Van Allen [1976] used the data from five Pioneer 10 electron detectors and finds

j(E,L=6,t)=K(t)E-a(I+E/Eo) -b (11) with a-'l.5,b=3.5-4-1 (the variation comes from the difference between inbound and outbound data), Eo=100, and K(t) is 2x10•m -2 s-XMeV -•. The value of b affects the spectrum mostly at high energies (E • 10 MeV), while at low energies (E •< 1 MeV) the spectrum resembles a power law with index a. This spectrum indicates a power

law with exponent of 1.5 near Io for the energies of interest. Mcllwain and Fillius [1975] represent their three detector results by a spectrum with a=O,b=4, and Eo =15 MeV at L= 5.56. This spectrum would be fiat, i.e., hard, for the energies of interest. Divine and Garrett [1983] show a spectrum at L= 6.2 which for E • 1 MeV is very similar to the one of Van Allen. Using the lower-energy data from the Voyager spacecraft, they find that at L= 6 the phase space density has a power law in energy for E •> 0.1 MeV with an exponent of 3.72. This would yield a power law for j(E) with an exponent of 2.72 for E •< 0.5 MeV (where E,,, p2) and 1.72 for E> 5 MeV (where E,,, p). This indicates that the spectrum may be even softer than indicated by Van Allen's equation. In the following we will see that the synchrotron radiation data require a harder spectrum than that given by Van Allen and Divine and Garrett. We will argue that the hardening is due to physical processes occurring inside the Io torus and is not due simply to an incorrect spectrum used at L= 6. For that reason we will treat a as a free parameter and vary it from a=0 to a=2. A conservative (in terms of the spectral hardness at L= 6) and reasonable value is a=l.2. Most of our results will be based on this value.

Figure 1 shows the four different energy spectra for the integral flux J(E) as used in our calculations, with superimposed Van Allen's [1976] data: spectrum 1; a=0,b= 4,Eo=15; spectrum 2; a=l.2,b=3.5,Eo=100; spectrum 3; a=l.5,b=3.5,Eo=100; spectrum 4; a=2.0,b=3.5,Eo=100. We will use energy spectrum 2, unless otherwise noted.

Our calculations start off with a radial profile in phase space density, which is equal to zero everywhere, except at L=6, where the values are calculated according to (11). A first order solution of the time-dependent radial diffusion equation is obtained in one time step of 10 s s; then synchrotron radiation and local losses are included and the program continues with time steps equal to 2 x 104 s. The steady state profile is established after about 10 ?- 10 s s or 2-3 years.

In these calculations we assume the diffusion coefficient Drr=D L" with n=3 The lifetime of particles against O , ß

local losses is represented by r=roL m. Goertz et al. [1979]) show analytically that good fits to the data at L>3 can be obtained for various combinations of n,m and to. In this paper we only show results of calculations in which the local losses are independent of distance (results for calculations in which the losses change with distance (e.g., n=Z,m=-0.6) are very similar). Good fits to the particle observations for spectrum 2 were obtained with Do=l.5-2x10 -9 and ro=2--1.5x10 ? s; these are shown in the following figures. The parameters Do and ro which best fit Van Allen's data points when spectra 1 and 4 are used are slightly different: Do=2X10 -ø with ro=lX10 ? for spectrum 1, and Do=3X10 -9 and ro=SXlO 6 for spectrum 4. Since the local losses are assumed to be independent of energy, the slopes of the spectra with and without local losses are equal, and we only show spectra for the results including local losses.

Since Van Allen's particle detectors measure J•/2(E,L) at specific energies we show the profiles of these quantities as a function of L and energy in Figures 2 and 3, respectively. The increase of the flux with decreasing L in Fig- ure 2 is, of course, due to the betatron acceleration of the inward diffusing particles. Local losses lead to a slower


5 -1.5

spectrum 1' a=0, b=4, Eo=15

spectrum 2: a=l.2, b=3.5, Eo=100 -

spectrum 3: a=l.5, b=3.5, Eo=100 -

spectrum 4: a=2.O., b=3.5, Eo=100 - _

_

-1 -.5 0 .5 1 1.5 2 log energy

Fig. 1. Various energy spectra for the integrated quantity Y• superimposed on Va. Allen's [1976] data (squares). Note that the scale along the Y axis can be adapted by choosing slightly different values for K(t) in (11).

I0

'"' 9

I I I I I

Eo:•O --- to: 1.5 x IO?sec

'-• ....... t, ....... -a-•. ,' .100 Me V -

i i

i - i i i -

I • $ 4 5 6

distance in Jovian rodii

Fig. 2. Integral particle flux Jz: as a function of distance, L. The 2

integral in (6) is taken from Emi, to oo, with Emi, equal to 1, 21, and 100 MeV, respectively, as indicated. The solid lines are calculations without local losses, the dashed lines include local losses (see text). The squares represent data points from Van Alien [1976].

increase than adiabatic diffusion alone would predict, as shown by the difference between the solid (without local losses, i.e., r•) and dot-dash (with local losses) lines. The squares indicate measurements of Pioneer 10 and 11.

\ %, ß

\,\ /

.I I I0 I00

energy (Me¾) Fig. 3. Integral particle flux J x as a function of energy, at different L-shells (6, 3, and 1.5) as indicated. The squares represent data points from Van Allen [1976].

Those points were obtained by extrapolating between the contours of flux shown by Van Allen et al. [1975]. The error bars are due to uncertainties in the assignment of L shell parameter. The data points can only be matched by assuming local losses (f/r) to be present, as noted before by Goertz et al. [1979]. Figure 3 shows the dependence of


J• as a function of energy at L=6 (solid line), L= 3 (dot- 2

dash line) and L= 1.$ (dashed line). The squares are data points from Van Allen ([1976] at L= 6 and 3. As shown in Figure 1, the shape of the spectrum at L= 6 is fairly well represented by the choice of parameters a=1.2,5=3.5, and Eo=100 in (11). After diffusing inward over 3 the spectrum matches the observed spectrum at L= 3 also fairly well. Unfortunately, there are no spacecraft data at L= 1.6 yet. In this regime we have to rely on Jupiter's radio emission for information on the electron spectra. Note the softening of the spectrum at L< 3; this is due to synchrotron radiation losses.

Figure 4 shows different curves J• (E_>21 McV, L) as a function of distance to show how well the value of r for the particular choice of Do=Jx10 -9 and n=3 is determined: the dot-dash, solid and dashed lines are for ro=5X106, 1.5x10 ? and 3x10 ? , s, respectively. If both the diffusion coefficient, Do, and the lifetime against local losses, ro are changed such that Do ro= constant, the slope of at L>2.5 does not change, but the turnover point will shift inward if Do•2X10 -9 or outward if Do<2X10 -9. An example of this case is shown in Figure $. The dot-dash, solid and dashed lines are for Do=l x10 -9, 2x10 -9, and 3 x 10- 9s- •, respectively. In all three curves the product Do ro =3 .0 x l O- 2.

Fig. 5. Integral particle flux Jz• for 21-MeV electrons as a function 2

of L. The product Doro=3.0xlO -:• for all curves. The diffusion coefficient Do is 1 0x10 -9 2 0x10 -9 and 3.0x10 -9 for the dot-dash, solid, and dashed curves, respectively.

--- xIo

3 4 5 6

distance in Jovian radii

Fig. 4. Integral particle flux Jz• for 21-MeV electrons as a function 2

of L. The diffusion coefficient is Do=2x10 -9, and n=3. The lifetime, to, against local losses is 5 x 10 6 ,1.5 x 10 ?, and 3 x 10 ? s for the dot-dash, solid, and dashed curves, respectively.

So far the calculations were performed with the parameter n=3. If n is decreased, the slope of J• decreases and the turnover point shifts toward larger values of L. We find that a good fit to the data without local losses can only be obtained if n~ 0. For larger values of n, local losses must be included to obtain reasonable fits to the data. Several authors have argued convincingly that n should be at least as large as 3 [e.g., Schardt and Goertz, 1983] and we do not consider solutions with n~ 0 acceptable. We will also see below that loss free radial diffusion provides a distinctly inferior fit to the synchrotron data.

It is important to note that the observed spectra fiatten more inside L= 6 [Van Allen, [1976] with decreasing distance from the planet than the calculated spectra. This hints at either energy- dependent losses such that higher- energy electrons have a larger lifetime r, or an energy- dependent diffusion coefficient such that high-energy electrons diffuse in at a faster rate than less energetic particles. A third possibility is pronounced changes in the energy spectra due to satellite/ring interactions. A similar conclusion is reached below from a comparison of the DIM radiation properties with the calculated spectra.

3.2 Comparison With Synchrotron Radiation Characteris- tics

The synchrotron radiation parameters which are of rel- evance in this study are the observed radial profiles and the spectrum.

Radial profile. The radial profile of the synchrotron radiation can be obtained from a scan through a two- dimensional image along the magnetic equator. The image, however, has been convolved with a Gaussian beam, so the observed radial profile is influenced by radiation from electrons with pitch angles slightly different from 90 ø . Furthermore, particle losses by the moons and Jupiter's ring are important. De Pater [1981b] suggested that the %houlder" in the radial profiles is due to the satellite and ring absorption effects. Due to the multipole character of the magnetic field the profile also varies from one rotational aspect of the planet to the next. We will not attempt to obtain a precise fit to the data, but merely use them to constrain the position of the radiation peak and the width of the profile near the shoulder. To avoid problems with Jupiter's thermal radiation, we took a scan through an image of the polarized fiux density. The shape of the radiation peaks in these scans is equal to those of the nonthermal intensity.


Figure 6a shows the predicted synchrotron emission $• narrow near the base. These curves continue to show that at a wavelength of 20 cm, with (dot-dash line; r•=3, Do= r•> 0, and that local losses are important. Figure 6b shows 2x10-9,%=l.Sx107) and without local losses (solid line; the synchrotron radiation profile when different electron r•= 3, Do=2X10-9; dashed line; r•= 0.1, Do=7X10 -9) for spectra are used, and local losses are included such that calculations using spectrum 2. Remember that in a loss the electron flux profile of Van Allen is represented well: free medium acceptable fits to the electron flux profile of the dot-dash line is for spectrum 2 (same curve as the dot- Van Allen [1976] were only obtained for r•_• 0; for r•> 0 local losses had to be included. The curves in Figure 6a are scaled to the same peak intensity. The squares roughly outline the observed radial profile [from de Pater and Jaffa, 1984]. Both the solid and dashed lines are obviously too narrow. Only the dot-dashed line agrees fairly well with the data near the peak of the radiation belt but is still too

--- 8OO

'- 600

400

._= 200

I I I I I I

•n=5, r o = oo

1 .... n=O.I, •'o =03 ß

. '\

(o) ' ' I I I I I I 0 I 2 3 4 5

distance in radii

---- 8OO

'- 600

"" 400

.=_ 200

-- IkJ '•.• spectrum I: a:O, b=4, E(•15 -- •=;\ \ ---- speclrum;):o:l? b=35E:100

i='\ • --- spe ctrum 4:ø: 2,'b:3.5:Eo ø:100 _ _

\.\ \ -- I -• --

- \ ",,. \

(b) I --F-----:---'T -• .... I'-"-' -1 I I o I 2 3 4 5

distance in radii

Fig. 6. Comparison of the calculated radial profile of the synchrotron radiation, S, with observations. The squares roughly outline the shape of the radial profile of the synchrotron radiation as observed by de Pater and Jaffe [1084] at 21 cm. a: The various curves are calculated with the following parameters: dot-dash line, with local losses, n=3,Do=2X10-9; ro=1.5x107; solid line, no losses, n=3,Do=2X10-9; dashed line, no losses, n=O.1,Do=7X10 -9. b: Radiation profiles for different input spectra as indicated. All curves are scaled to the maximum flux density. The scale along the Y axis is for the dot-dash curve in Figure 6a, with a beam area equal to 0.1 R• at Jupiter.

dash line in Figure 6a), the solid line for spectrum 1, and the dashed line for spectrum 4. Apparently, the softer the spectrum, the narrower the profile. This can be understood by realizing that at larger distances from the planet one obtains the synchrotron radiation from electrons with higher energies. With a hard spectrum, more electrons will contribute to the radiation at large planetary radii, than with a soft spectrum. Spectrum 2 obviously gives the best fit to the radiation profile. We may further note here that electron losses inside L,-, 1.2 are dominated by the losses due to synchrotron radiation. This process is responsible for the shape of the radial profile inside L,-, 1.2.

Figure 7 shows the synchrotron radiation profile at 50, 20, 6, and 2 cm, for the calculation with local losses (dot- dash line in Figures 6a and 6b). Note that the peak of the radiation in the calculations moves slightly outward from Jupiter with decreasing wavelength, while the profile becomes broader. This is due to the fact that the shorter wavelength radiation is emitted by higher-energy electrons, which generate synchrotron radiation further away from the planet. The difference in peak position and extent of the radiation belt between the 6- and 20-cm data has never been observed, likely due to the finite resolution of the telescopes, absorption effects by the rings and inner moons of Jupiter which may change the shape of the profiles, and time variations in the position and shape of the radial profile.

Spectrum. De Pater [1981b] gives the ratio between the integrated non-thermal flux densities measured at 21, 50 cm, and 6 cm: s-2x=0 93+0.04; S---eL=0 63+0.04. Im- '-q50 ' ,.q21 ß proved estimates for the thermal emission (disk-averaged brightness temperature) as given by de Pater and Massie

1600 - I

o 1200

E 800

..=_ 4OO

- 50 cm - ........ 21cm .... 6cm

..... ---- 2cm

-- i /•""%,. -- ................. x x .•

.... ___/ • .. ...... I I I I ,,I ., I , 0 I 2 3 4 5

distonce in radii

Fig. ?. Synchrotron radiation profile at 50, 21, 6, and 2 cm indicated by the solid, dotted, short dashed and long dashed lines, respectively. The calculations are the same as those for the dot-dash line in Figure 5 (with local losses). The scale along the Y axis is in millijanskies per beam area, with the beam area equal to 0.1 R•r at Jupiter.


[1985], deviations in absolute calibration from the Baars et al., [1977] flux density scale, and possible changes in time in the synchrotron radiation characteristics change these values and their error bars slightly. We adopt the ratios 0.91 4- 0.06, and 0.63 4- 0.06 respectively. These values agree with the ratios in flux density as determined by Roberts and Komesaroff [1965] in the same frequency range (0.3-5 GHz). Although our calculations are performed for equatorially confined particles only, the ratios of the flux densities calculated at the different frequencies should be the same as those observed, if electrons with different energies have the same latitudinal distribution. This will be the case if the equatorial pitch angle distribution is the same for particles with different energies. This seems, indeed, to be the case [e.g., Mcllwain and Fillius, 1975]. Beaming curves (total flux density as a function of Jovian longitude) at frequencies between 0.6 and $ GHz [e.g., de Pater and Dames, 1979; de Pater, [1980, 1951a] also imply that this is the case.

From Figure 7 we obtain the following ratios in flux density between the different wavelengths for spectrum 2 at L= 6: s-•(calculated) -- 0.64; • (calculated) -- 0.50. $$0 The significant difference between the observed and calculated ratios indicate an electron spectrum which is flatter or harder than that calculated. This implies the existence of a loss mechanism which is more efficient for low-than high-energy electrons, or an energy-dependent diffusion mechanism such that high-energy electrons diffuse inward at a faster rate than less energetic particles, so that the electron spectrum does not soften as much as expected otherwise. Of course, a harder input spectrum at L=6 would yield a harder spectrum in the synchrotron frequency region as well, and thus be in better agreement with the synchrotron frequency spectrum. Spectrum 1 yields ratios so-•=0.82 and os--•=0.63; spectrum 3 yields 0.51 and •0

0.38, respectively, and spectrum 4 yields 0.33 and 0.23, respectively. However, only spectrum 2 gives an acceptable fit to Van Allen's energy spectrum and the synchrotron radiation profile. Spectrum 1 does yield the correct synchrotron ratios but yields a much too wide radiation profile as shown in Figure 6b.

A possible hardening of the spectrum with decreasing distance from Jupiter has been suggested a few times in the past years. It was first suggested in 1968 by Branson [1968], based upon a comparison of the extent of the radio emission region at 75 and 21 cm. His data suggest that the emission region at 75 cm is slightly broader than at 20 cm, which implies that the electron spectrum at larger distances from Jupiter is softer than closer to the planet. De Pater [1981b] noted from her model calculations to sim- ulate Jupiter's radio emission that the spectrum as measured by the Pioneer spacecraft at L= 3 was too soft to account for the synchrotron radiation spectrum. Finally, we noted in section 3.1 that the Pioneer spacecraft observed a hardening of the spectra between L=6 and L=3.

4. Discussion of Physical Loss Processes

spectrum 2, 3, or 4. Spectrum 1 is a hard source spectrum and does not require hardening, but it would yield a radiation profile which is significantly wider in L than the observed profile. A spectral hardening could be obtained either by an energy-dependent diffusion term, or by an energy-dependent loss term in the radial diffusion equation. The diffusion process is still poorly understood, and no clear theoretical predictions have yet been put for- ward for an energy dependence of the diffusion coefficient. Several physical loss mechanisms cause energy- dependent losses in the particle distribution. These processes are pitch angle scattering, Coulomb scattering, and absorption effects by satellites and dust. We will discuss each of these processes below, and compare the effects of such losses to the observations. Since we find that each specific loss process alone does not yield a good fit, we keep the generic loss term •/ro in the diffusion equation and vary the parameters for each process until an optimum fit to the data is obtained. We use spectrum 2 as the electron spectrum at L= 6; at the end of the section we will comment on the use of a different input spectrum.

4.1 Pitch Angle Scattering

The most widely discussed loss mechanism in Jupiter's magnetosphere at L•3 is pitch angle scattering by resonant wave-particle interactions [Sentman and Goertz, 1978]. The lifetime of particles against this scattering, r=rpa, can be approximated [Thorne, 1983]

where -r=(1-vJ/cJ) -•12 the cyclotron frequency, and 6B the the total wide band amplitude of the resonant fluctuating magnetic field. Sentman and Goertz [1978] found from considerations of particle loss rates and pitch angle distributions that 6B is fairly constant at 10 -s G between œ= 4 and œ< 6. This reduces (12) to

"'• (13) r10 a • L3

with rc• •,4.6X10 9, and E expressed in MeV. Note that this loss rate has m=--3, which is not in good agreement with the result from Goertz et al. [1979] that m_•0 provides the best fit to the electron flux profile (for n=3).

We performed several calculations with different values for rs. We found that no effect on the J(L) and other curves is seen, as long as rs • 109. When r•10 9 the pitch angle scattering is so large that it inhibits energetic electrons from reaching the inner radiation belts. This implies that the total wideband amplitude of the resonant fluctuating magnetic field cannot be much larger than 2•10 -s G•, in agreement with the value reported by Sent- man and Goertz [1978]. It further implies that pitch angle scattering alone cannot account for the hardening unless the wave intensity increases with decreasing L. We cannot rule out such an increase on the basis of these calculations.

It was shown in section 3 that a hardening of the electron spectrum between L= 6 and the radiation region is needed to explain the synchrotron radiation, if the synchrotron radiating particles have diffused inward from L •> 6, and had an electron spectrum at L=6 as represented by our

4.2 Coulomb Scattering

Inside L= 3 Coulomb losses may play an important role. From Wait [1964, and references therein], we find that the loss rate due to Coulomb scattering (for E> 0.2) is


_ • o If(E) r+l/2 - •o o-• •/•(•+•)•/• ] (14)

where the total loss time, rco, for collisions with other particles of equatorial density ne(L) cm -3 is

1.3XlO 12 Tco= ne(L ) (15) We implemented (14) as a separate explicit term in (1),

and approximated the equatorial density ne(L) with help of the density model of Sentman and Goertz [1978]:

ne(L)=noexp -?'6s(t-t/L ) (16) We varied the constant no from 10 4 cm -3 (as given by

Sentman and Goertz [1978]) up to l0 s cm -3. The density no has to be larger than 106 cm -a to cause a notice- able effect in the synchrotron radiation parameters. When no is as large as 3x10* cm -3, the calculated radio spectrum approaches that of the observed values' s-2a-=0.83 S50 • and s-z-=0.6? In these calculations a diffusion coefficient S21 ' of Do=3x10 -9, rather than Do=2x10 -9, yields the best fit to the 3./2 profile and the radial profile of the synchrotron radiation. Increasing no more hardens the spectrum slightly more but also broadens the synchrotron radiation profile too much. Similar results can be obtained with the density expressed as

-L

ne ( L)=3.0x 10 ?[exp o•'g] (17)

Hence Coulomb losses in the radiation belts may provide a natural explanation for the hardening of the electron spectra. However, it requires extremely large densities (neutral, ions, electrons combined), at relatively large distances from the planet: 2x106crn -3 at L= 1.5, and a few times l05 particles cm -a at L= 2. This seems rather unrealistic. The maximum electron density in Jupiter's ionosphere is of the order of 105 cm -a [e.g., Atreya, 1986]. The detection limit from the Voyager UVS experiment of Ly a emission from neutral hydrogen atoms implies a density less than a hundred particles per cubic centimeter in the vicinity of Jupiter's ring between L= 1.2 and L= 1.8 (S. K. Atreya, private communication, 1988). Thus we do not believe that Coulomb scattering can account for the required hardening of the electron spectrum.

4.3 Satellites

There are two satellites which orbit Jupiter at distances less than L=6: Thebe at 3.1 Rs and Amalthea at 2.5 Rs. Since the drift motion of the electrons around Jupiter is energy dependent and in the same direction as the moons' motions around the planet, the absorption of electrons by these satellites is energy dependent [e.g., Mogro-Campero, 1976]. Since we deal with equatorial electrons only, they can only be absorbed at the two places where the magnetic equatorial plane intercepts the rotational equator. Hence the absorption effect is small [e.g., de Pater, 1981a, b, c]. Moreover, calculations show that if there is an absorption effect at all, it is opposite to what we need: it will soften rather than harden the electron spectrum.

4.4 Dust

Dust particles in Jupiter's ring will also absorb electrons or degrade their energy as they pass through dust grains. Since the grains are small, energy loss is more important than absorption for electrons with E •> 1 MeV. The energy lost per passage through a grain is

•E--np<z>=3a[MeV] (18) where •= 2.3 (MeV cmJ/g) is a typical normalized energy loss rate [Drain and Salpeter, 1979], p= 1 g crn -a is the grain mass density and <z>=4/3a is the effective path length through a spherical grain of radius a measured in centimeters. Since a,-.10 -4 cm [Burns et al., 1984], the energy loss in one pass through the grain is small. To calculate the average loss rate to be inserted in (1}, we need to average over one drift period T. The number of collisions an electron will encounter during one drift orbit is equal to its path length, I, divided by the mean free path, a, between collisions. The path length, l, is

1=2•rr• l-ldri•t with r s the gyroradius, g• and gdrilt=2rr/T the angular gyrovelocity, and drift velocity, respectively, and f, the fraction of the electron's path inside the dust ring, which is inclined at an angle O with respect to the magnetic equator:

) (20) LtanO

with D the thickness of the ring. Equation (20) is only valid for values f, <1. The mean free path between collisions, a, is

• (21) 0"-- ndust 'a 2 With the optical depth, rn=nau,tDrra •, we can write

the energy lost per drift orbit (using n=2.3 MeV cmJ/g and p=l g cm-a):

(22) T -- LtanO

and the loss rate

o__t __ > ,, • o_z._ •,-•,• o_z. (23) Ot T OE -- LtanO OE

where we have taken the electron speed to be equal to c. We note that the loss rate is proportional to the product of optical depth {observed) and dust particle size {not well known). The optical depth profile for the Jovian ring can be approximated as [Burns et al., 1084]

7xl 0 -- 6 atL--1.3-- 1.72

n•= (24) 5x10-- 5atL=l.72-- 1.81

Our calculations show that a best fit to the data is obtained with the product arR=Sxlo -4 t•m at 1.73 <L< 1.82, and a rR_•5 x 10-5 t•m closer to the planet, and a radial diffusion coefficient, Do =3-4 x I o- 9. This would re-


quire an optical depth an order of magnitude larger than the values quoted by Burns et al. [1984], if there are only 1-•m-sized particles. If the particles had a typical size of 10 {100) •m, the optical depth required to match the synchrotron spectrum would be 5 x 10- 5 {5 x 10- 6 } in the thickest part of the rings. This is consistent with the num- bers quoted by Showalter et al. [1987]. The spectral ratios for the synchrotron radiation are • =0.81 and • =0.64 S50 • S21 (as compared to the observed values of 0.91+0.06 and 0.65+0.06). It is interesting to note that Goertz e• [1988] have also recently suggested on the basis of entirely different arguments that the typical 3ovian ring particle size is significantly larger than 1 •m.

4.5 Summary'

All the calculations discussed above in sections 4.1-4.4 were carried out by adding the term under discussion to the local loss term f/r in (1). Figure 8 shows the lifetime of electrons which diffuse adiabatically into Jupiter's magnetic field such that they have an energy of 2.8 MeV at 2 Rj. The dashed line in Figure 8 shows the lifetime of these electrons against pitch angle scattering, the dotted line against Coulomb losses and the solid line against absorption by Jupiter's ring. The lifetime against the generic local loss term is indicated by the long-dash-short-dashed line. From this graph it is clear that losses due to Coulomb scattering based on reasonable density profiles are negli- gibly small, anywhere in Jupiter's inner magnetosphere. Losses due to pitch angle scattering based upon a constant

magnetic field fluctuation level exceed the generic local loss only at L > 4-5; they are more important for lower-energy particles closer to Jupiter, while higher-energy particles are influenced by these losses further out (note that the curves shown are for electrons having 2.8 MeV at 2 Rj, and only 0.3 MeV at 6 R•). Losses due to dust are the most important loss mechanism inside 1.8 R•.

We have no good understanding of the loss mechanism between L= 1.8 and L~ 4. In this region the observed optical depth of the tenuous gossamer ring is always less than 10 -4 [Sfiowal•er e• aL, 1985], and should not cause much absorption if the particle size is the same as in the denser rings. However, if we omit the local losses between L= 1.8 and L~ 4 altogether, the calculated slope in J,/2 is too steep in this region, and the synchrotron radiation profile S• too wide. Losses due to the energy degradation process by dust does not improve the slope in J,/2, nor the shape in S•. Optical depths as large as a rR~10 -4 •m at L> 1.8 would improve the profile J,/2 slightly, but broaden S• considerably. An L-dependent power law in optical depth (~ 1 /L or 1/L a) with a= const does not significantly change the shape of the profiles either. We therefore suggest that the losses are largely energy independent in the region 1.8<L<4, and perhaps due to some- what larger sized material than that from which the rings are made up. Although absorption of energetic electrons by material orbiting Jupiter is also energy dependent (see section 4.3 above), its effect will be small if the material is spread out over several L- shells. Note that such material may form a source for the micron -sized dust particles in Jupiter's ring.

12

11

10

COULOMB LOSSES - _

_

x.,,,

.,..

_

DUST _

GENERIC LOSS TERM

--

1 2 3 4 5 6 distance in radii

Fig. 8. Lifetime of particles against pitch angle scattering (dashed line), Coulomb losses (dotted line), dust (solid line), and local losses (long-dashed- short-dashed line).


Calculations which use a different electron energy spectrum at L= 6, give results different from those shown above. Using spectrum 1 (a=0,b=4, and Eo= 15), we obtain good spectral ratios s-2x-=o 8½3 and s-•-=0.67. The $5O ' $21 synchrotron radiation profile, however, is similar in width to the profile shown in Figure 6b (solid line), i.e., much wider than the observed profile. We therefore suggest that the electron spectrum at L= 6 is softer than McIlwain and

[onl (parm propo.. Spearurn a.5,Eo=100) yields ratios •=0.60 and •/-• =0.43, with a radial profile in the synchrotron radiation which is similar to the dot-dash line (spectrum 2) in Figure 6b.

5. Conclusions

We developed a computer code for radial diffusion of equatorially confined energetic electrons in Jupiter's inner magnetosphere. We compared the results to spacecraft as well as ground-based radio (synchrotron radiation) data. Initially, phenomenological local losses and those due to synchrotron radiation were included; these losses were later discussed in terms of pitch angle, and Coulomb scattering, and losses due to satellites and dust particles in Jupiter's ring.

We noticed that the synchrotron radiation spectrum cannot be reproduced without a hardening of the electron spectrum between L= 3 and L= 1.5. Of all the physical processes considered we found that energy degradation due to the ring particles is the most realistic cause for this effect. The calculations further suggest the existence of larger sized material out to at least L,,, 4 or maybe up to Io's orbit. Such material or Io itself could be the source of the dust in the rings. On the other hand, we cannot rule out the possibility that the intensity of whistler waves which scatter electrons into the atmospheric loss cone increases with decreasing distance. The upcoming Galileo mission, in particular the atmospheric probe, will provide new observations which one may expect provide a means for distinguishing between these possibilities.

with

rnec 2 1/2 /-/(•,)=•,•/•(•+ •,,s )

/T• o

(A2)

We then need to explicitly solve the equation

o_t_•:(•) •/(•)• (A3) We can rewrite (A3):

Integration yields

•--•'(t,)•t(•)d• (A4)

f( t+ A t) = f(t) exp[K(L) H(p)At] (A5)

To solve (AS), we need to estimate • in H(•). Since the variation in energy of the phase space density is an exponential rather than a linear function, we approximate this variation by piecewise exponentials:

and

f=f(Pi )exp -(•' i-i-1 - •'i ) (A6) 61

o__t_ • (A7) o l• -- 6 i

•qith (A6) we find Y(•,i-•.)-- (AS) $(•,/)

Hence

a. (A9) 61---- ln[$ (•,i)]- ln[y(•,i+! )]

Appendix

The diffusion equation (1) is solved in two half-time steps: in the first step the effect of synchrotron radiation losses on the phase space density is determined via an explicit difference scheme; in the second half-time step the effects of diffusion and local losses are calculated via a set of implicit equations.

Synchrotron Radiation Losses The synchrotron radiation is calculated from electrons

which diffuse radially inwards from Io's orbit or beyond, conserving their first adiabatic invariant •. This means that we must deal with grid points in t• which cover a range of many orders of magnitude. We therefore use a so-called explosive grid in •, and work with the variable A=ln •t, rather than directly with

The partial derivativ e o•t X in (4a) can be written as

Radial Diffusion

The radial diffusion and loss term in equation (1) are solved implicitly [Ricbtmyer and Morton, 1976]. Let be the new value of f at grid point 1 after time step At, and fk be the old value (actually the value after the synchrotron radiation losses have been included). We can then write

with

6f•=f•+•12-f•_•12

6( 6 f)•= f•+ •--2 f•+ f•_ •

We rearrange equation (A10) such that

•X=H(•,).• (A1) =- - •*+• (All) •t


where

At =(L_t__•_)-- 2 DoL 2 (aL)•

Bt=[(L__.•_),•-2+(L+_•),-• l •ot, • (aL)• +

n L 2

The parameter n is a• specified in the introduction. The tridiagonal matrix (A11) can be solved with

with

ff = + o, (A12)

A l •1= Bi --Gi.Ei_ 1

Dl+GiGl_l Gi = Bi_GiEi_ 1

Once E• and Gt are determined from /=1 (Jupiter's surface) to i=L=6, fp can be determined from /=L-l= 6--AL to /=2=I+AL.

Acknowledgments. We very much appreciate discussions with R. I. Klein regarding our computer code. This research was supported by NASA grant NAGW-776 and NAGW-1805, NSF grant AST-8514896 and AST-8900156 to the University of California in Berkeley and the Alfred P. Sloan Foundation. I. de Pater is an Alfred P. Sloan fellow.

The Editor thanks two referees for their assistance in evaluating this paper.

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C. K. Goertz, Department of Physics andAstronomy, University of Iowa, Iowa City, IA 52242

(Received September 7, 1988; revised June 12, 1989;

accepted June 14, 1989.)

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