Whistler-mode instability in magnetospheres of Uranus and Neptune

ARTICLE IN PRESS

Planetary and Space Science 56 (2008) 310–319 www.elsevier.com/locate/pss

Whistler-mode instability in magnetospheres of Uranus and Neptune A.K. Tripathi, R.P. Singhal Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi 221005, India Received 19 September 2006; received in revised form 17 September 2007; accepted 18 September 2007 Available online 22 September 2007

Abstract Whistler-mode instability in the magnetospheres of the outer planets Uranus and Neptune is investigated using an anisotropic kappa loss-cone distribution and comparisons have been made with the observations made by Voyager 2. Normalized temporal growth rates have been evaluated numerically at two representative radial distances at each planet. Parametric studies have been performed by changing plasma parameters: cold and hot electron densities, hot electron temperature and temperature anisotropy. It is found that whistler-mode emissions observed at lower radial distances cannot be reproduced in our calculations. Electron pitch-angle diffusion and energy diffusion coefficients have been obtained using the calculated growth rates. The present calculations show that electrons of energy above about 20 keV may be able to precipitate into the planetary atmospheres of both planets. Present studies should be helpful in making estimates on scattering properties of whistler-mode waves and thus contribute to a better understanding of the auroral activity in the planetary atmospheres. r 2007 Elsevier Ltd. All rights reserved. Keywords: Whistler-mode instability; Uranus magnetosphere; Neptune magnetosphere; Diffusion coefficients; Strong pitch-angle diffusion

1. Introduction The Voyager 2 encounter of outer planets Uranus and Neptune provided the first opportunity of the observations of plasma waves in and near the magnetospheres of these planets (Gurnett et al., 1986, 1989). The observations include electrostatic waves upstream of the bow shock, turbulence in the shock, electrostatic cyclotron harmonics (Bernstein wave) associated with harmonics of the electron– cyclotron frequency and whistler-mode waves in the magnetospheres, electrostatic and electromagnetic ion– cyclotron waves, broadband electrostatic noise in magnetotail, and many other types of plasma waves. In the recent past Kurth (1991, 1992) and Kurth and Gurnett (1991) have provided for the first time, overviews of comparative study of plasma wave spectra at the outer planetary magnetospheres. They have also discussed the various wave modes present in each case. Recently up-to-date views of the plasma waves at each outer planet have been Corresponding author. Fax: +91 542 2368428.

E-mail addresses: [email protected] (A.K. Tripathi), [email protected] (R.P. Singhal). 0032-0633/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2007.09.003

reviewed by Zarka (2004). Whistler-mode waves exist in a plasma environment which both supports the growth of the waves and is modified by interaction with waves. Wave– particle interactions provide the channels, through which the waves can accelerate, scatter or thermalize the plasma. The whistler-mode emissions (chorus and hiss) play an important role in the dynamics of magnetospheres of Uranus (Kurth et al., 1991) and Neptune (Gurnett and Kurth, 1995) by controlling pitch-angle scattering and loss of magnetically trapped radiation belt particles. Whistlermode waves in the inner magnetospheres of Uranus (Coroniti et al., 1987) and of Neptune may be the source of pitch-angle diffusion and contribute to precipitated fluxes of energetic electrons (Kurth and Gurnett, 1991). Whistler-mode emissions are also triggered by lightning generated whistler. Plasma waves such as lightning generated whistler provide also important diagnostics tools, from which fundamental plasma parameters such as the electron density can be computed. As is well known, whistler-mode waves are electromagnetic waves in the magnetized plasma that propagate through the magnetosphere at frequencies below the electron–cyclotron frequency (fc) and electron plasma frequency (fp). Because the

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

magnetic field introduces anisotropy in the plasma, whistler waves tend to be guided along the ambient magnetic field (Gurnett et al., 1990; Stenzel, 1999). Anisotropic electron distributions occur naturally in outer planetary magnetospheres. The temperature anisotropy is caused by the different mobility parallel and perpendicular to the ambient magnetic field and by the loss-cone property of open-ended magnetic field. This temperature anisotropy can, under certain conditions, lead to the instabilities which transfer plasma kinetic energy into wave energy (Kennel and Petschek, 1966; Tripathi and Misra, 2004). Thus the presence of temperature anisotropy is the prominent source of providing free energy and of generating whistler-mode instability in magnetized plasma. In situ measurements by Voyager 1 and 2 in the inner magnetospheres of the outer planets show a well developed suprathermal tail of energetic electrons (Belcher, 1983; Sittler et al., 1983, 1987; Belcher et al., 1989). The electron distributions are composed of a Maxwellian thermal (cold) component and non-Maxwellian suprathermal (hot) component. The hot plasma component can be modeled by loss-cone bi-Lorentizan (or kappa) distribution function (Summers and Thorne, 1991). In the present work, we have studied the whistler-mode instability at Uranus and Neptune due to electron temperature anisotropy in suprathermal component and compare our results with whistler-mode observations given by Voyager spacecraft. We have also calculated the pitch-angle and energy diffusion coefficients from resonant interactions with whistler-mode waves using the recent work of Singhal and Tripathi (2006). The temporal growth rates of whistler-mode waves in the magnetospheres of both planets have been calculated in terms of a linear plasma wave instability analysis without making any further approximations. The temporal growth rates are calculated at two radial distances for each outer planet. The normalized growth rate is also calculated by varying the plasma parameters: cold electron density (nc), hot electron density (nh), hot electron temperature (Th), temperature anisotropy (Ah) and spectral index (k). The organization of paper is as follows: in Section 2 detailed observations of whistler-mode waves for both planets are presented. In Section 3 we present the dispersion relation of whistler-mode waves propagating parallel to the ambient magnetic field. In Section 4 we describe and discuss plasma parameters in the magnetospheres of both planets which are used in the present study. In Section 5 we present the calculated normalized pitch-angle and energy diffusion coefficients. We summarize our results of temporal growth rates in Section 6. A detailed discussion and brief conclusions of the present study are given in Section 7. 2. Observations of whistler-mode waves The observations of the plasma wave spectrum observed at Uranus by Voyager 2 were first reported by Gurnett et al. (1986). Voyager 2 traversed the radial distances from

311

6.3 to 9 RU and magnetic latitudes (l) from 131 to 19.51 during the interval SCET from 2000 to 2100. An important result of the Voyager findings is that the most intense wave activity in the outer planet magnetospheres (generally within 15 planetary radii) takes place close to the planet where plasma density is high and the flux of resonant particles is sufficient to drive the various instabilities. Strong whistler-mode emissions (chorus and hiss) are observed at 46 RU in the inner magnetosphere of Uranus. On the outbound, nightside leg of the trajectory, the whistler-mode emission is very intense. The whistler-mode displays a high degree of asymmetry from inbound to outbound. A large part of this is most likely due to the high magnetic latitude of the inbound pass and nearly equatorial passage on the outbound leg. As Voyager passed through the ring plane, the plasma wave investigation recorded many strong electromagnetic (whistler) plasma waves, with intensity peaks in the inner region within 12 RU. Strong whistler-mode emissions were detected during the outbound pass (nightside) of Uranus (Scarf et al., 1987). These very intense emissions have average f/fc values approximately equal to 0.2 when the spacecraft was within from 151 to 201 of the magnetic equator. The maximum electric field intensity occurs at 1800 SCET (4 RU) in the range of 100 mV/m. The field intensity of wave decreases to lower value 10 mV/m as the radial distance increase up to 15 RU. Low intensity plasma waves were also observed at o8 RU (Gurnett et al., 1986). Coroniti et al. (1987) recently analyzed the pitch-angle diffusion associated with these strong whistler-mode emissions on the outbound leg (6–9 RU) of the Uranus encounter. They have concluded that waves are strong enough to put from 5 to 40 keV electrons on strong diffusion. Further, they conclude that the precipitated energy can account for the reported UVS aurora (Gurnett et al., 1986; Coroniti et al., 1987; Scarf et al., 1987; Kurth and Gurnett, 1991). Gurnett et al. (1989) have reported the first observations of plasma-wave spectrum by Voyager 2 during the encounter with Neptune. These plasma waves in the inner magnetosphere occurred at the magnetic equator (l ¼ 01) crossings on the inbound leg at about 0023 SCET and near 0800 SCET on the outbound leg. Gurnett et al. (1992), Kurth (1992) has reported that significant whistler-mode emissions could exist near the magnetic equator in the intermediate radial distance ranges say 3oRo8 RN as in the other planetary magnetospheres. The intensity of these emissions is quite weak from 10 to 30 mV/m. The variation of field intensity, magnetic field strength and electron– cyclotron frequency with radial distance have the similar characteristics as occur in Uranus. The absence of intense whistler-mode emissions is probably due to the low trapped radiation intensities in Neptune’s magnetosphere, compared to other planetary magnetospheres (Krimigis et al., 1989). Kurth and Gurnett (1991) reported the pitch-angle scattering of these weak whistler-mode emissions on the outbound leg of the Neptune encounter. They have

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

312

concluded that waves are weak to put electrons on strong pitch-angle diffusion (Kurth et al., 1991; Kurth and Gurnett, 1991; Gurnett and Kurth, 1995). 3. Dispersion relation We consider a plasma in which the entire electron and proton distribution is the superposition of a two component bi-Lorentzian (kappa) distribution with loss-cone feature fk, introduced by Summers and Thorne (1991, 1995) and a bi-Maxwellian distribution. The cold (thermal) electron and proton components are represented by biMaxwellian and hot (suprathermal) electron and proton components by kappa loss-cone distribution. The biMaxwellian distribution is given by ! v2k 1 v2? M f s ðv? ; vk Þ ¼ 3=2 2 exp  2  2 , (1) aks a?s p a?s aks with associated perpendicular and parallel thermal speeds a?s ¼ ð2T ?s =ms Þ1=2 ; and

AM s ,

AM s ¼

aks ¼ ð2T ks =ms Þ1=2

(2)

the temperature anisotropy, is given by

a2?s  1. a2ks

(3)

The kappa loss-cone distribution is given by f ks ðv? ; vjj Þ ¼

1 Gðk þ ‘ þ 1Þ p3=2 y2? yjj k‘þ3=2 Gð‘ þ 1ÞGðk  1=2Þ !ðkþ‘þ1Þ  2‘ v2jj v? v2?  1þ 2 þ 2 . y? kyjj ky?

ð4Þ

The parameter y?s and yks are effective thermal speeds given by     2k  3 1=2 1 T ?s 1=2 ; (5) y?s ¼ k ð‘ þ 1Þ1=2 ms yjjs

    2k  3 1=2 T ks 1=2 ¼ : k ms

(6)

The distribution function has been normalized to 1. Again, the parameters T? and TJ are temperature components and Aks , the temperature anisotropy, given by Z 1 T ?s ¼ ms v2? f s d3 v, (7) 2 Z (8) T jjs ¼ ms v2jj f s d3 v, Aks ¼

T ?s  1 ¼ ð‘ þ T jjs

y2 1Þ ?2 s yjjs

relative number of particles in the high-energy tail of the distribution. Under the limit ‘-0 and k-N the distribution reduces to the well-known bi-Lorentzian and Dorry– Guest–Harris loss-cone bi-Maxwellian distribution, respectively. Since for parallel wave propagation the loss-cone index plays no role, therefore we set it zero in the present calculations. Employing distributions (1) and (4), the dispersion relation for the parallel propagating (i.e., k?=0, k=kJ) whistler-mode waves in R-mode can be written (Summers and Thorne, 1995; Mace, 1998)     X o2ps  c2 k 2 o k k o þ 2s O s ¼1þ As þ As þ kyjjs o2 o2 kyjjs s " #) 1=2 3=2 ðk  1Þ k1 o þ 2s O s   1=2 . ð10Þ Z k kyjjs k ðk  3=2Þ k1 Summation over species s involved in (10) is carried on cold and hot components of electrons and protons. For cold component k-N, Aks ! AM s , expression (10) reduces into purely cold components:    X o2ps  c2 k 2 M M o þ 2s Os ¼ 1 þ  A þ A s s o2 o2 kajjs s    o o þ 2s O s þ Z , ð11Þ kajjs kajjs where o is the complex wave frequency (o ¼ or+ig), k ¼ |kJ|, and for particles species s the plasma frequency is given by o2ps ¼ 4pn0s q2s =ms , the cyclotron frequency Os is given by Os ¼ jqs jB0 =ms c, charge sign is 2s ¼ qs =jqs j  qs, ms and ns are, respectively, particle charge, mass and number density of species s. c is the speed of light. B0 is the ambient magnetic field. The functions Z k1 and Z appearing in (10) and (11) are the modified plasma dispersion function of Summers and Thorne (1991) and the plasma dispersion function (Fried and Conte, 1961), respectively. The modified plasma dispersion function is defined by the expression Zk ðxÞ ¼

(9)

G is gamma function. The parameter ‘(40) is the ‘losscone index’ and the parameter k (X2) is the ‘spectral index’; ‘ is a measure of the angular size of the effective loss-cone region (qfk/qv?40), and k is a measure of the

Gðk þ 1Þ  1=2Þ Z þ1 ds  ; kþ1 2 1 ðs  xÞð1 þ s =kÞ k3=2 Gðk

ImðxÞ40

ð12Þ

with a suitable extension to Im(x)p0 by analytic continuation, where x ¼ x+iy and xns ¼ o  nOcs =kk yks , in the limit as k-N, the above expression (12) reduces to wellknown Maxwellian plasma dispersion function ZðxÞ ¼

 1:

1

p1=2

1 p1=2

Z

1

2

es ds; 1 s  x

ImðxÞ40

(13)

with Zns ¼ o  nOcs =kk aks . Expression (10) is broken into real and imaginary parts and a numerical technique is used to find the real and imaginary components of o (Abramowitz and Stegun, 1970; Summers et al., 1994).

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

4. Plasma parameters In this section we describe the basic plasma parameters measured in the inner magnetospheres of outer planets Uranus and Neptune and used in the present studies. Normalized temporal growth rate calculations have been performed at two radial distances with spectral-index k ¼ 2 and loss-cone index ‘ ¼ 0 for each planet using the plasma parameters given in Table 1. Subscripts c and h are used for referring to cold (thermal) and hot (suprathermal) electrons. Thermal electrons are assumed purely isotropic and for hot the ‘best fit’ electrons temperature anisotropy Ah ¼ T?h/TJh ¼ 1.1 is used. A representative value of k ¼ 2 is used in calculation. Protons with a constant hot component temperature 104 eV have also been included and remaining proton parameters are the same as for electrons.

313

( ¼ nc+nh) ranged between 0.02 and 1.0 cm3, and the total temperature (neTe ¼ ncTc+nhTh) ranged from 10 eV near the terminator to 100 eV within nightside hemisphere. Tc was 7 eV near the terminator and from 10 to 30 eV within the nightside hemisphere. The nh ranged between 0.01 and 0.3 cm3 with Th ranging from 20 to 200 eV near the terminator and from 500 eV to 2 keV in the nightside region. The data at two radial distances 5.5 and 12 RU are summarized as follows: (i) at R ¼ 5.5 RU, nc were from 0.5 to 1.0 cm3, Tc between 17 and 25 eV, nh ranged between 0.20 and 0.30 cm3 and Th were from 300 to 500 eV. (ii) At R ¼ 12 RU, nc ranged between 0.03 and 0.055 cm3, Tc were from 70 to 80 eV, nh between 0.02 and 0.03 cm3 and Th were from 900 eV to 1 keV. These observational data of plasma parameters for Uranus are taken from the work of Bridge et al. (1986) and Sittler et al. (1987). 4.2. Neptune

4.1. Uranus A survey of the low-energy and high-energy plasma electron environment during the Voyager 2 encounter with Uranus has been given by Bridge et al. (1986). Sittler et al. (1987) have also presented the results of an analysis of the Voyager 2 plasma science experiment (PLS) electron measurements made during the Uranus encounter. The inner magnetosphere appears in the PLS measurements as the region of high particle intensities measured between 16 h (R ¼ 7 RU inbound) and 23 h (R ¼ 18 RU outbound). The plasma energy density sampled by the PLS instrument is negligible compared to the energy density of the magnetic field (bo0.01) throughout the inner magnetosphere. The cold (nc) and hot (nh) electron density components were roughly symmetric relative to closest approach for p6 RU, whereas the electron temperatures cold (Tc) and hot (Th) components were significantly greater during the outbound pass (nightside hemisphere) than those observed inbound (near the terminator). For Ro6.3 RU, Tc were from 5 to 10 eV during the inbound pass and from 10 to 30 eV outbound. Th were p300 eV inbound and p1 keV during the outbound. The large drop in nh and rise in Th after 1939 SCET occurred when the spacecraft possess more negative potential. The observations through PLS reveal that electron fluxes are much larger in the nightside hemisphere whereas during dayside magnetosphere no significant electron fluxes (E410 eV) were observed. For R45 RU, the total electron density ne

An initial survey of the low-energy electron environment through PLS experiment during Voyager 2 encounter at Neptune has been presented by Belcher et al. (1989). Electron cold (hot) density and temperature component profiles data obtained during the Neptune encounter by the Voyager 2 PLS instrument have been described by Zhang et al. (1991). The electron densities in the inner magnetosphere are estimated about 0.1 cm3 at radial distance 4 RN and decreases to values around 0.01 cm3 at distance 12 RN and electron temperature increases from 7 to 25 eV. The observational data show large variability with coordinated Universal Time (UT) at Voyager, radial distances and magnetic latitude. The PLS instrument covered the radial distance less than 16 RN and magnetic latitudes from 01 to 451. The instrument near 237/0420 UT, close to the planet and near the magnetic equator detected the maximum electron density and temperature 2 cm3 and 90 eV, respectively. After 237/0435 UT when electron becomes cold, the variation of electron parameters outbound is very similar to that observed during the inbound leg. Like the total electron temperature, Tc, decreases toward closest approach. The ratio of the hot electron density to the total electron density ne is 1 in the hot plasma region. At the time when the cold electrons are present, hot electrons are a small fraction of the total population; the ratio nh/ne is usually a few percent. For most of this time period the hot electron temperature is from 100 to 200 eV. The electron temperature remains roughly constant between radial

Table 1 Observed and used plasma parameter Planet

Distance

nc (cm3)

nh (cm3)

Tc (eV)

Th (eV)

B0 (nT)

Uranus

R ¼ 5.5–12 RU

Observed 1–0.03 Used 60–0.06

Observed 0.30–0.02 Used 0.60–0.022

Observed 17–80 Used 17–80

Observed 300–1000 Used 300–2000

111–7.5

Neptune

R ¼ 4.5–11 RN

Observed 0.50–0.03 Used 5–0.11

Observed 0.03–0.002 Used 2–0.012

Observed 6–30 Used 6–30

Observed 70–290 Used 200–580

71–6.8

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

314

distances 2 and 8.5 RN; beyond 8.5 RN the temperature increases rapidly until about 14 RN. The observational data at two radial distances 4.5 and 11 RN are given as: (i) at R ¼ 4.5 RN, nc were from 0.30 to 0.50 cm3, Tc between 6 and 8 eV, nh ranged between 0.01 and 0.03 cm3 and Th were from 70 to 80 eV. (ii) At R ¼ 11 RN, nc ranged between 0.03 and 0.06 cm3, Tc were from 20 to 30 eV, nh between 0.002 and 0.003 cm3 and Th were from 260 to 290 eV. The data of plasma parameters at Neptune used in the present calculation are taken from Zhang et al. (1991).

5. Electron diffusion coefficients Plasma waves can transfer energy and momentum to particles in a plasma, plasma waves are therefore of fundamental importance for understanding the auroral energetics of the planets. It has been known for many years (Kennel and Petschek, 1966) that the wave–particle interactions play a fundamental role in controlling the loss of radiation belt particles in planetary magnetospheres. Whistler-mode waves are ubiquitous in inner magnetospheric regions. Their generation is attributed to nonrelativistic cyclotron resonance o  kk vk  nOce ¼ 0

(14)

with O ¼ 2pf, n ¼ 1, 2, 3,y,kJ the parallel wave vector, and vJ| the particles parallel velocity. Wave growth occurs when a velocity inversion (qfk/qv?40) exists in the particle population. Whistler-mode waves can resonate with the electrons of 100–1000 keV (hiss) or 1–10 keV (chorus), causing their pitch-angle scattering in the magnetic loss-cone. Whistler-mode emissions can cause both pitch-angle and energy diffusion of magnetospheric electrons and scatter them into the planet’s atmosphere producing auroral emissions. The scattering properties of waves depend on the intensity of plasma waves. At Uranus and Neptune, aurora has been observed at high magnetic latitudes (Broadfoot et al., 1986; Coroniti et al., 1987; Mauk et al., 1994) suggesting a source of particles in the outer magnetospheres. These authors propose that keV electrons are most likely the primary precipitation energy sources for the aurora. We have therefore calculated the pitch-angle and energy diffusion coefficients using representative temporal growth rate profile of each planet: (i) for Uranus at radial distance 12 RU and (ii) for Neptune at radial distance 11 RN. Calculations have been performed using the diffusion coefficient expressions given by Lyons (1974) and also Singhal and Tripathi (2006).The expressions for the normalized electron energy ðK¯ e Þ, normalized pitch-angle diffusion coefficient (Dnaa ) and time constant for pitch-angle diffusion (taa) are defined as K¯ e ¼

Ke 2 B0 =8pne

,

(15)

"

Dnaa

B2 ¼ hDaa i Oe wave v2 B20

#1 ,

(16)

and 1=taa ¼ Daa =v2 ,

(17)

where Ke is parallel electron energy, v is electron speed, Daa is pitch-angle diffusion coefficient, Bwave is wave magnetic field intensity and B0 is ambient magnetic field intensity of the planet. Other details of pitch-angle and energy diffusion coefficients are given in Lyons (1974). For Uranus at 12 RU, the electron energies for K¯ e ¼ 8 and 20 are 18.64 and 46.6 keV, respectively. The average pitchangle diffusion coefficients /DaaS at these energies are 8.87  103 and 0.55, respectively. For Neptune at 11 RN, the electron energies for K¯ e ¼ 12 and 20 are 13.2 and 22 keV, respectively, and /DaaS values are 0.041 and 0.99, respectively. We have taken wave magnetic field intensity Bwave ¼ 102 nT for both planets. For Uranus the values of taa at 12 RU are 4.76  104 and 0.78  104 s for electron energies 18.64 and 46.6 keV, respectively. In case of Neptune at 11 RN these are 9.1  103 and 0.39  103 s for electron energies 13.2 and 22 keV, respectively. It may be useful to compare taa with the time constant tSD for ‘strong’ pitch-angle diffusion at both planets. In the strong diffusion regime, particles diffuse across the loss-cone in less than a quarter-bounce period, with the result that the precipitation mechanism saturates and the particle flux is driven isotropic. The particle precipitation rate is then independent of the amplitude of the scattering waves, and depends only on the particle bounce time and the geometric size of the loss-cone. The parameter tSD is given approximately by (Thorne, 1983) tSD ¼ 3:6RL4 =V ,

(18)

where L is the magnetic shell parameter, R is the radial distance from the planet and V is the electron thermal velocity. For Uranus the calculated values of tSD at L ¼ 12 are 2.29  104 and 1.45  104 s for 18.64 and 46.6 keV, respectively. In case of Neptune at L ¼ 11 these are 1.84  104 and 1.43  104 s for 13.2 and 22 keV, respectively. Thus to put tens of keV electron on strong diffusion (taaEtSD) the required wave amplitudes for Uranus and Neptune must be about (2–14)  103 nT and about (1–7)  103 nT, respectively. 6. Results In Fig. (1) we compare the observed distribution functions using Eqs. (1) and (4) for k=2 and ‘=0. The fit has been shown for planet Uranus using the representative physical plasma parameters. From this figure we note that the value of k=2 gives a reasonably good fit with the observed electron distribution function. No observations of electron distribution function are available for the planet Neptune.

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

Uranus x 1/2

Observed Calculated

f (sec3/cm6)

10-27

1x10-29

10-31

10-33 0

10

20

30

(103

V

40

50

Km/sec)

Fig. 1. Distribution function versus velocity of electron. Calculated value is obtained using Eqs. (1) and (4) of text (k=2, ‘=0). For planet Uranus: nc=0.63 cm3, nh=0.25 cm3, T c? ¼ T ck ¼ 17 eV; T h? ¼ T hk ¼ 300 eV.

125 κ=2

Uranus R = 12 RU nc = 0.06 cm-3 nh = 0.044 cm-3 Tc = 80 eV Th = 1000 eV

100 3

γ- x 106

75

50

25

0 0.00

0.02

0.04

0.06

0.08

0.10

ω- r Fig. 2. Normalized temporal growth rate g¯ ð¼ g=Oe Þ versus normalized real frequency o ¯ r ð¼ or =Oe Þ for the whistler-mode instability with variation of spectral index k at Uranus R=12 RU.

We first consider the sensitivity of our results to the spectral index k. Fig. 2 shows the variation of normalized temporal growth rate g¯ ð¼ g=Oe Þ with real frequency o ¯ r ð¼ or =Oe Þ for planet Uranus at radial distance 12 RU for two values of k. There is decrease of maximum value of temporal growth rate with increase of k values. Since for higher k, there are lower numbers of resonant electrons in the high-energy tail of the distribution. The

315

variation of growth rate with cold electron density is shown in Figs. 3(a) and (b) for both planets. In Fig. 3(a) the value of growth rate increases as nc is increased. Although increasing the density nc does not show systematic behavior of the peak value of o ¯ r . Also it may be noted that the real frequency o ¯ r shifts to a somewhat lower value as nc is increased. This behavior follows from the approximate dispersion relation given in Eq. (6) of Mace (1998), i.e., o ¯ r ¼ ðk2 c2 =o2pe Þ=ð1 þ k2 c2 =o2pe Þ. However, in Fig. (3b), o ¯r first increases as nc is increased from 3 to 4 cm3 and then decreases as nc is increased to 5 cm3. This behavior cannot be explained by the simple approximate relation of Mace. The effect of changing hot electron number density is shown in Figs. 4(a) and (b). It is found that the value of growth rate increases as nh is increased. Also the peak value of normalized real frequency o ¯ r shifts to a higher value as nh is increased. This effect is more pronounced at Uranus than at Neptune. The enhancement is due to increase in fraction of suprathermal particles which are the source of free energy for whistler-mode emissions. In Figs. 5(a) and (b) we plot the variation of growth rate with Th. It is found that there is proportional increase of maximum value of growth rate with Th values. In Figs. 6(a) and (b) the effect of hot electron anisotropy (Ah) is shown. Increase in the hot electron anisotropy increases the growth rate as well as the bandwidth for both the planets. Also, with increase in Ah the peak shifts towards higher frequencies. This result follows from an approximate expression for parallel whistler wave growth rate given in Eq. (8) of Mace (1998), g¯ / ððo ¯ r =ð1  o ¯ r ÞÞ  Ah Þ. As Ah is increased the range of o ¯ r for which the growth is possible also increases, giving a larger bandwidth. Finally, the calculations show that at lower radial distances (at 5.5 RU and 4.5 RN) the observed values of nc and nh do not produce whistler-mode emissions. The reasons may be firstly, the present calculations have been done using a linear analysis of waves which may not be applicable. Second and more likely may be the fact that the whistler-mode emissions observed in these cases may have been produced at more favorable (lower resonant energies) locations and then propagated to the location where these have been observed. Because the most intense whistler-mode emissions usually occur near the magnetic equator, it is possible that the spacecraft did not sample the proper region of the magnetosphere to detect these emissions. In Figs. 7(a) and (b) we show the normalized pitch-angle and energy diffusion coefficients for electron interactions with whistler-mode waves. From both figures it is noted that energy diffusion coefficients Dav and Dvv are several order of magnitude smaller than pitchangle diffusion coefficient Daa. Observed electric fields of the whistler-mode waves measured by Voyager 2 for planet Uranus and Neptune (Kurth and Gurnett, 1991) are of the order of 105 V/m. For Uranus and Neptune the calculated group velocities vg ¼ ðdo=dkÞ are 8.22  106 and 6.4  106 m/s, respectively. These give magnetic field intensity of the waves ðB ¼ E=vg Þ for respective planet as 12.2  103 and 1.56  103 nT, respectively. The time constant for

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

316

5

2.0 Uranus R = 5.5 RU nh = 0.60 cm-3 Tc = 25 eV Th = 350 eV

nc = 60 cm-3 40 20

4

γ x 106

γ x 106

1.5

1.0

0.5

3

Neptune R = 4.5 RN nh = 2.0 cm-3 Tc = 8 eV Th = 200 eV

nc = 5.0 cm-3

4.0

3.0

2 1 0

0.0 0.02

0.04

0.06 ω- r

0.08

0.10

0.04

0.06 ω- r

0.08

0.10

Fig. 3. Normalized temporal growth rates (¯g) versus normalized real frequency (o ¯ r ) with the variation of cold electron density (nc) at two planets as marked in (a) Uranus R=5.5 RU and (b) Neptune R=4.5 RN.

125 nh = 0.066 cm-3

75

15 γ- x 106

γ x 106

100

20

Uranus R = 12 RU nc = 0.06 cm-3 Tc = 80 eV Th = 1000 eV

0.044

50

nh = 0.036 cm-3

Neptune R = 11 RN nc = 0.11 cm-3 Tc = 30 eV Th = 290 eV

0.024

10

0.022

0.012

5 25 0

0 0.02

0.04

0.06 ω

0.08

0.10

0.02

0.04

r

0.06 ω r

0.08

0.10

Fig. 4. Normalized temporal growth rates (¯g) versus normalized real frequency (o ¯ r ) with the variation of hot electron density (nh) at two planets as marked in (a) Uranus R=12 RU and (b) Neptune R=11 RN.

γ x 106

200

Uranus R = 12 RU nc = 0.06 cm-3 nh = 0.044 cm-3 Tc = 80 eV

50 Th = 580 eV

Th = 2000 eV

40

γ x 106

250

150 100

1000

Neptune R = 11RN nc = 0.11 cm-3 nh = 0.024 cm-3 Tc = 30 eV

30 20 290

10

50 500

145

0

0 0.02

0.04

0.06 ω r

0.08

0.10

0.02

0.04

0.06 ω

0.08

0.10

r

Fig. 5. Same as in Fig. 4 but for the variation of hot electron temperature (Th).

diffusion, observed (calculated from observed electric field) and the required magnetic fields are given in Table 2. It may be noted from Table 2 that at Uranus and Neptune

electrons of energy above 20 keV may be able to precipitate into the atmospheres. We also find in our calculations that at Uranus and Neptune taa is very large for electron

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

600

250

0.20

300

Neptune R = 11 RN nc = 0.11 cm-3 nh = 0.024 cm-3 Tc = 30 eV Th = 290 eV

300

γ- x 106

400 γ x 106

350

Uranus R = 12 RU Ah = 0.30 nc = 0.06 cm-3 -3 nh = 0.044 cm Tc = 80 eV Th = 1000 eV

500

200

200

Ah = 0.30

150 100

100

317

0.10

0.20

50

0.10

0

0 0.05

0.10 ω- r

0.15

0.20

0.05

0.10

0.15 ω- r

0.20

0.25

Fig. 6. Same as in Fig. 4 but for the variation of temperature anisotropy (Ah).

101

101

Dαα

10-1

Neptune

R = 12 RU nc = 0.06 cm-3 nh = 0.044 cm-3 Tc = 80 eV Th = 1000 eV

10-2

Normalized electron diffusion coefficients

Normalized electron diffusion coefficients

Uranus 100

Dαv

10-3 1x10-4

Ke= 20 Ke= 8

Dvv 1x10-5 10-6 10-7

100

R = 11 RN nc = 0.11 cm-3 nh = 0.024 cm-3 Tc = 30 e V Th = 290 e V

Dαα

10-1

10-2 Dαv 10-3

Ke= 20 Ke= 12

1x10-4 Dvv 1x10-5

10-6 0

10

20

30 40 50 Pitch angle (degrees)

60

70

0

80

10

20

30 40 50 Pitch angle (degrees)

60

70

80

Fig. 7. Normalized electron diffusion coefficients versus pitch-angle using temporal growth rate profile of Fig. 4.

Table 2 Pitch-angle diffusion parameters Planet

Distance

K¯ e

Ke (keV)

taa (s)

Eobs (V/m)

tSD (s)

Bobs (nT)

Breq (nT)

4

4

5

2

Uranus

R ¼ 12 RU

8 20

18.64 46.60

4.76  10 0.78  104

2.29  10 1.45  104

(1–3)  10 (1–3)  105

(1.22–3.66)  10 (1.22–3.66)  102

1.41  102 2.24  103

Neptune

R ¼ 11 RN

12 20

13.20 22.00

9.10  103 0.39  103

1.84  104 1.43  104

(1–3)  105 (1–3)  105

(1.56–4.68)  103 (1.56–4.68)  103

7.09  103 1.67  103

energies below 19 keV (K¯ e ¼ 8) and 13 keV (K¯ e ¼ 12), respectively. Therefore electrons of energies lower than these values cannot precipitate since the required electric field will be extremely large to satisfy the conditiontaa  tSD .

7. Discussion and conclusions We have for the first time made a comprehensive study of whistler-mode emissions using kappa-loss-cone distribution function in the magnetospheres of Uranus and

ARTICLE IN PRESS 318

A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319

Neptune. We have applied the dispersion relation for parallel propagating whistler-mode waves to the magnetospheres of both planets and have made comparisons with the observations made by Voyager 2. Whistler-mode emissions observed by Voyager 2 at both planets near ring plane crossing can be accounted for by our calculations. It may be noted from Table 1 that the plasma parameters used in present calculations (in particular nc and nh) are different at lower radial distance than the observed values since for observed parameters given in Table 1, whistlermode instability could not be obtained. In case of Neptune the calculations show that at very low values of cold electron density (o3 cm3) at radial distance o5 RN no whistler-mode emissions are possible. At radial distance o5 RN a somewhat larger cold electron density is needed than observed density. Comparing the present results with earlier calculation for Saturn (Singhal and Tripathi, 2006) we find that increasing Th results in somewhat larger bandwidth towards lower frequencies on all the three planets. The effect is more pronounced at Neptune and Saturn and less at Uranus. An increase in Ah produces larger bandwidth towards higher frequencies at all the three planets. Increasing nh gives a larger bandwidth towards higher frequencies at Uranus. However, at Neptune and Saturn the peak frequency shifts only slightly towards higher frequencies without a systematic change in bandwidth. We have also calculated the pitch-angle diffusion and energy diffusion coefficients for planets Uranus and Neptune at radial distances 12 RU and 11 RN, respectively. Comparing the present pitch-angle diffusion studies with our earlier study at Saturn (Singhal and Tripathi, 2006) we find that at Saturn the energy of electrons which can precipitate into the atmosphere is lowest at about 10 keV. For both Uranus and Neptune electrons above 20 keV will be able to precipitate into the atmospheres of planets. Coroniti et al. (1987) have concluded from their calculations for Uranus that the whistler-mode emissions are strong enough to put electrons in the range of 5–40 keV on strong pitch-angle diffusion. However, our calculations show that electrons above only 20 keV may be put on ‘strong’ diffusion. Further, Kurth and Gurnett (1991) have concluded that there is little or no pitch-angle scattering due to whistler-mode emissions at Neptune, while our results show that electrons of energies above 20 keV may be put on ‘strong’ pitch-angle diffusion. Present studies should be helpful in making estimates on scattering properties of whistler-mode waves and thus contribute to a better understanding of the auroral activity in the planetary atmospheres. Thus the whistler-mode waves can provide a source of energetic electrons for auroral precipitation. Acknowledgments This work was supported by a research associateship award to one of the authors (A.K.T.) by the Council of Scientific and Industrial Research, Government of India.

Calculations reported in the present work were carried out at the Computer Centre, Banaras Hindu University. References Abramowitz, M., Stegun, I.A., 1970. Handbook of Mathematical Functions. U.S. Government Printing Office, Washington, DC. Belcher, J.W., 1983. The low-energy plasma in the Jovian magnetosphere. In: Physics of the Jovian Magnetosphere. Cambridge University Press, New York, pp. 68–105. Belcher, J.W., Bridge, H.S., Bagenal, F., et al., 1989. Plasma observations near Neptune: initial results from Voyager. Science 246, 1478–1483. Bridge, H.S., Belcher, J.W., Coppi, B., et al., 1986. Plasma observations near Uranus: initial results from Voyager 2. Science 233, 89–93. Broadfoot, A.L., et al., 1986. Ultraviolet spectrometer observations of Uranus. Science 233, 74. Coroniti, F.V., Kurth, W.S., Scarf, F.L., Krimigis, S.M., Kennel, C.F., Gurnett, D.A., 1987. Whistler-mode emissions in the Uranian radiation belts. J. Geophys. Res. 92, 15234–15248. Fried, B.D., Conte, S.D., 1961. The Plasma Dispersion Function. Academic Press, New York. Gurnett, D.A., Kurth, W.S., 1995. Plasma waves and related phenomena in the magnetosphere of Neptune. In: Cruikshank, D.P. (Ed.), Neptune and Triton. University of Arizona Press, Tucson, pp. 389–423. Gurnett, D.A., Kurth, W.S., Scarf, F.L., Poynter, R.L., 1986. First plasma wave observations at Uranus. Science 233, 106–109. Gurnett, D.A., Kurth, W.S., Poynter, R.L., et al., 1989. First plasma observations at Neptune. Science 246, 1494–1498. Gurnett, D.A., Kurth, W.S., Cairns, I.H., Granroth, L.J., 1990. Whistlers in Neptune’s magnetosphere: evidence of atmospheric lightening. J. Geophys. Res. 95, 20967–20975. Gurnett, D.A., Kurth, W.S., Granroth, L.J., 1992. Plasma wave observations at Neptune. Adv. Space Res. 12, 47–54. Kennel, C.F., Petschek, H.E., 1966. Limits on stably trapped particle fluxes. J. Geophys. Res. 71, 1–28. Krimigis, S.M., Armstrong, T.P., Axford, W.I., et al., 1989. Hot plasma and energetic particles in Neptune’s magnetosphere. Science 246, 1483–1489. Kurth, W.S., 1991. Voyager plasma wave observations near the outer planets. Adv. Space Res. 11, 59–68. Kurth, W.S., 1992. Comparative observations at plasma waves at the outer planets. Adv. Space Res. 12, 83–90. Kurth, W.S., Gurnett, D.A., 1991. Plasma waves in planetary magnetospheres. J. Geophys. Res. 96, 18977–18991. Kurth, W.S., Gurnett, D.A., Scarf, F.L., Coroniti, F.V., 1991. Wave– particle interactions in the magnetosphere of Uranus. In: Bergstrahl, J., Miner, E.D., Mathews, M.S. (Eds.), Uranus. University of Arizona Press, Tucson, pp. 926–958. Lyons, L.R., 1974. Pitch angle and energy diffusion coefficients from resonant interactions with ion–cyclotron and whistler waves. J. Plasma Phys. 12, 417–432. Mace, R.L., 1998. Whistler instability enhanced by suprathermal electrons within the Earth’s foreshock. J. Geophys. Res. 103, 14643–14654. Mauk, B.H., Krimigis, S.M., Acuna, M.H., 1994. Neptune’s inner magnetosphere and aurora: energetic particle constraints. J. Geophys. Res. 99, 14781–14788. Scarf, F.L., Gurnett, D.A., Kurth, W.S., Poynter, R.L., 1987. Voyager 2 plasma wave observations at Uranus. Adv. Space Res. 7, 253–258. Singhal, R.P., Tripathi, A.K., 2006. Study of whistler-mode instability in Saturn’s magnetosphere. Ann. Geophys. 24, 1705–1712. Sittler Jr., E.C., Ogilvie, K.W., Scudder, J.D., 1983. Survey of low-energy plasma electrons in Saturn’s magnetosphere: Voyager 1 and 2. J. Geophys. Res. 88, 8847–8870.

ARTICLE IN PRESS A.K. Tripathi, R.P. Singhal / Planetary and Space Science 56 (2008) 310–319 Sittler Jr., E.C., Ogilvie, K.W., Selesnick, R., 1987. Survey of electrons in the Uranian magnetosphere: Voyager 2 observations. J. Geophys. Res. 92, 15263–15281. Stenzel, R.L., 1999. Whistler waves in space and laboratory plasmas. J. Geophys. Res. 104, 14379–14395. Summers, D., Thorne, R.M., 1991. The modified plasma dispersions function. Phys. Fluids B, 1835–1847. Summers, D., Thorne, R.M., 1995. Plasma microinstabilities driven by loss-cone distributions. J. Plasma Phys. 53, 293–315. Summers, D., Xue, S., Thorne, R.M., 1994. Calculation of the dielectric tensor for a generalized Lorentzian (kappa) distribution function. Phys. Plasmas 1, 2012–2025.

319

Thorne, R.M., 1983. Microscopic plasma processes in the Jovian magnetosphere. In: Dessler, A.J. (Ed.), Physics of the Jovian Magnetosphere. Cambridge University Press, New York, pp. 454–488. Tripathi, A.K., Misra, K.D., 2004. Effect of loss-cone distribution on the generation of whistler-waves in the magnetosphere. J. Atmos. SolarTerrestrial Phys. 66, 987–995. Zarka, P., 2004. Radio and plasma waves at the outer planets. Adv. Space Res. 33, 2045–2060. Zhang, M., Richardson, J.D., Sittler Jr., E.C., 1991. Voyager 2 electron observations in the magnetosphere of Neptune. J. Geophys. Res. 96, 19085–19100.