the cold 5-eV electrons. The energy E* of the

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 88, NO. A10, PAGES , OCTOBER 1, 1983 THE NON-MAXWELLIAN ENERGY DISTRIBUTION OF IONS IN THE WARM IO TORUS John D. Rihardson and George L. Sisoe Department of Atmospheri Sienes, University of California, Los Angeles Abstrat. Observations of Io's torus indiate at whih the bulk of the plasma resides, and a that the majority of ions have energies of old eletron omponent. By balaning the energy ev, with a high-energy tail extending up to the flow between these omponents and the amount of orotation energy. We have found that suh a dis- energy lost to radiation, they obtained temperatribution an be established via the Coulomb ool- tures of 50-eV for the ool ion omponent and ing of ions heated at the orotation energy onto the old 5-eV eletrons. The energy E* of the main body of the ions and the shape of the energy distribution are funtions.of the transport loss time. Mathing E * with the data (E* = ev) requires transport loss times in the range days. Introdution 5-eV for the eletrons, in good agreement with observations. This result requires that the hot omponent ontain 20% of the ions and that the entire torus be doubly ionized. It has been suggested that eletron-eletron heating may also be an important soure of energy for the torus emissions. Thorne [1981] has shown that a large flux of eletrons an be baksattered from the auroral zones. Shemansky and Sandel [1982] argue that the loal time asymmetry in the eletron temperature inferred from the UV The bulk of the plasma in Io's warm torus (L 6-7.5) was observed by the Voyager 1 plasma emissions requires eletron-eletron heating, experiment to have energies of ev [Bagenal sine ion-eletron interations are too slow to and Sullivan, 1981]. The probable origins of this maintain a loal time asymmetry. However, it plasma are neutral louds of sulfur and oxygen has been shown that the onvetion eletri whih have esaped from Io and are ionized by field set up by outward moving plasma in the harge exhange and eletron impat ionization. tail an reate this observed asymmetry [Barbosa The aeleration of th se newly reated ions by and Kivelson, 1983]. Thus neither eletronthe orotation eletri field imparts an initial eletron heating nor ion-eletron heating an be energy of 540 ev to sulfur ions and 270 ev to ruled out. oxygen ions. Whihever of these two mehanisms provides the The question therefore raised is why do the predominant amount of energy for powering torus bulk of the ions have a muh lower energy than emissions, the ions will be ooled by their intertheir reation energy? Goertz [1980] proposed ations with eletrons. It is expeted therefore that mass loading near Io loally dereases the that the average ion temperature should be a funorotation speed, enabling ions to form at a tion of the transport time in the torus. lower energy. However, the UV emissions whih If transport is fast, this implies a larger should result from a loally onentrated soure soure of hot ions to maintain the torus density region are not observed [Shemansky, 1980]. Sine along with shorter ooling times, and thus a it does not seem possible to reate the ions at plasma with a higher mean energy. The opposite the observed energy, some ooling mehanism must is true for slow transport. Many methods have be found whih an provide the observed ion energy been used to obtain estimates of the transport distribution. Loss of energy from ions to the time. They inlude alulating the ion reation old eletrons is an obvious possibility. However, rate needed to ause the observed orotation lag in order to assess the likelihood of the possibil- outside of Io's torus [Hill, 1980], modeling the ity, one must determine whether the time required to ool to the observed temperature is onsistent with estimated residene times of the ions. The exhange of energy between ions and eletrons has been studied in the ontext of trying to provide enough energy to the eletrons to power the UV torus emissions. Plasma waves are not of suffiient intensity to provide the ioneletron energy exhange neessary to power the torus [Thorne, 1981; Barbosa et al., 1982]. If ions provide the energy for the torus emissions, this leaves Coulomb interations as the probable energy transfer mehanism. Barbosa et al. [1983] have modeled the exhange of energy in the torus via Coulomb ollisions. Their model onsists of a plasma with three MaxwellJan omponents: a hot ion omponent at the orotation energy, a ooler ion omponent Copyright 1983 by the Amerian Geophysial Union. Paper number 3A /83/003A ion partitioning in the hot torus [Shemansky, 1980], and omputing the ionization rate of the neutral louds [Smyth and Shemansky, 1983]. These methods yield quite different results, giving a range for the torus transit time of days. Our objetive in this paper is to find tile ion energy distribution F(E) for the Io torus that results when a ontinuous soure of hot ions is ooled by old eletrons. We also inlude the effet of the transport time for the loss of the ions in our model and will obtain an estimate of this radial transit time based on the observed energy of the torus. The Model We assume a steady state situation in whih a onstant influx of ions is added to the torus at the orotation energy. For mathematial onveniene we will not inlude individual ion speies but will instead treat an ion having average torus properties: a mass of 24, a harge state of 8097

2 8098 Rihardson and Sisoe: Brief Report 1.5, a density of 2000 ions/m 3, and a orotation energy of 400 ev. It is also assumed that energy exhange in the torus plasma is a smooth and ontinuous proess. This should be the ase, sine in Coulomb ollisions most energy transfer is a result of small angle interations rather than diret ollisions. The governing equation for our model is the equation for partile flux onservation along the energy axis. d [m(e) de - d--e - ] = S - L (1) where m(e) is the number of ions between E and E + AE and S and L are the soures and losses of ions with energy E. There are two soures of ions, harge exhange and eletron impat ionization (the loss of ions by harge exhange enters into the loss term below), so that the soure term an be written S = (n øn+ x + nøn- i ) 6(E - E) (2) O + - where n, n, and n are the neutral, ion, and eletron number densities, e and e are the harge exhange and ionizatioc rates, and 6 is the Dira delta funtion indiating that all new ions are reated at the orotation energy E. Ions are lost from the system by harge exhange and transport out of the torus: L = n m(e) + m(e) o (3) where is the transport time. Using the onservation equation for ions, x T dn + o - n + - n n e. (4) dt i ß together with the assumption of steady state (dn+/dt.'; O) and integrating (1) over energy give where de re(e) - -- K i m(e) de K = (1 + x) 1_ o ZOO. 1 E T and z is the average ion harge state (n- = zn+). Note that this is the only plae where the harge exhange and ionization rates, and thus impliitly the roll of the neutrals, enter our model. The important quantity is the ratio of these two rates, ex/ei, whih varies throughout the torus. We use a value of 1.5 for this ratio, onsistent with the results of Johnson and Strobel [1982], whih should be aurate to within a fator of two in most of the torus. We now need expressions for (de/dt) e and (de/dt) i, the rates of interhange of energy between ions and eletrons and between ions of different energy. Spitzer [1953] has derived an expression for the interations of two MaxwellJan populations, i (5) (6) dt 2 T 1 - T 2 3xlO5A1A2 T 1 T 2 - ß z = 22 ( /2 dt eq eq nlz1z2 A 2) where A is the atomi weight and the Coulomb logarithm in the numerial fator in.eq was evaluated for Io torus onditions. Using the relation Te/A e >> Ti/A i and rewriting (7) in terms of energies rather than temperatures give E Z 4 ( - )i de = f C + m(e') E'-E 3/2 de' C + = --5 «o (E'-E) 2.5x10 A. (7) E - E Z2n-A «(8) E 3/2 C xlO5A ' de = C- e - e ( )e e These expressions are for interations between Maxwellian populations. The eletron distribution is probably MaxwellJan; the ions, however, are not expeted to have a MaxwellJan distribution. Nevertheless, we will use these quations now and return to the question of their appliability later in this paper. Protons have been suggested as a possible intermediary in the transfer of energy from ions to eletrons [Thorne, 1982]. Calulations based on whistler dispersion in the torus region indiate the proton number density is about 0.1 the heavy ion density [Tokar et al., 1982]. The proton temperature should result from an equilibrium between energy gained by the protons from heavy ions and that lost by the protons to eletrons. Using (8), taking -- T = 60 ev, and using the "average ion" parameters given earlier, we find T_ 17 ev. Again by use of (8) it an be shown t at in order for protons to provide as muh energy to eletrons as the heavy ions do, a proton density equal to one third the ion density would be required. A proton omponent this large is not supported by observations. We onlude that protons do not ontribute signifiantly to the transfer of energy in the torus. Substitution of (8) into (5) yields an integral equation for the ion distribution m(e). To find the form of the solution of this equation, the ion interation term (8), whih governs the ex- hange of energy between ions, an be initially negleted. For ions interating only with eletrons, the solution is where E - E e -Q m(e) = m(e ) (E - E ) Q--l- KE e 3/2 and m(e ) is the density of ions at the orotation energy. To give an example based on representative values, a transport time of 50 days and a harge exhange rate 1.5 times the ionization rate yield a value of about 0.8 for Q. Next onsider what happens when the ion interation term is inluded. We do not attempt to e (9)

3 Rihardson and Sisoe: Brief Report 8099 find an exat analyti solution to the governing overestimate the number of ions at the highest and integral equation in this ase. However, general lowest energies. onsiderations show that there will be some energy The other extreme is a distribution shown in E* at whih the energy an ion loses to eletrons Figure la, in whih there are no ions with eneris balaned by the energy an ion gains from higher gies greater than E or less than E*. This is energy ions. The ions, instead of ooling to the learly not a realisti distribution, sine the eletron energy, ool to this intermediate energy. lower energy ions will approah a Maxwellian dis- The solution of the equation should then have ap- tribution on a time sale of a few days, muh proximately the same general form for the distri- shorter than the transport time. The lower ener- bution funtion as (4) but with E replaed by E*: m(e) = m(e ) (E - E,) -¾ (10) E - E Here we regard E, the energy of the bulk of the plasma, and y; the exponent, as parameters to be determined to give (10) the best fit to the atual solution. Note for future referene that an expression for m(e ) is obtained by integrating (10) over energy, giving + m(e ) -- n (1 - y) (11) E - * E The neessity for a distribution funtion of the form given by (10) is seen by evaluating (1) at E*. If the density m(e*) does not equal infinity, (1) yields a restrition on the transport time z: gy ions probably have a distribution lose to that shown by Figure la, while the higher energy ions have a distribution similar to Figure lb. The orret solution should be braketed by these two extremes. We now need expressions for (de/dt) and (de/dt) e for the dis tribution shown in Figure la. Butler and Bukingham [1962] have derived an expression for the loss of energy by an ion of speed V to a MaxwellJan eletron population with thermal speed 2 4- A A 2 d-- de - 8/ z m e n n A (_. e + ( + 2 e) A. (V) ) e e 1 1 e where m is the eletron mass and n A is the (15) Coulomb elogarithms. The exhange of energy be- tween an ion with funtion F( ) is speed v and the distribution z < E 3/2(1 + z-. x )/C-n- 8 days 1 (12) de * - z e n n A F(, ¾) dt m i This ondition is learly not physially valid, as the transport and ooling mehanisms are in- dependent of eah other; thus m(e*) must equal infinity., Sine there are two unknowns, E and ¾, two equations are needed to solve for them. The first is the ondition of energy flow balane at E*, F( *, ¾) = [( 2_ '2 ) 1-¾ +--f 1 v * ( 2_ - (1 + ) (v2- '2)1-¾]. [ 2( 2_ '2)1-¾ v *2 1-¾ ) d de de ( )e -- ( )i (13) 1-¾ ] _ f (m2_m*2) dm -1 (16) The seond is obtained by integrating (5) over energy. The ion interation term drops out, as this governs the exhange of energy between ions and auses no net hange in the total ion energy. This leaves the equation E - - E E, Ee3/2 (Ee-E)dE-- f K i m(e')de' i m(e) C n E E E (14) where m i is the ion mass and m and m are the veloities orresponding to E and E*. Using these equations in (5), we again want to solve for E* and ¾. The integral over (5) an again be used as one of the equations E E E j. m(e) (.ae) de : f K f m(e,) de, de * e * * E E E (17) Using (13) and (14), it is possible to solve for E* and ¾. Before showing any results let us return to the question of the expressions used for (de/dt) i and (de/dt) e. The expressions (8) used above are appropriate for interations between Maxwelljan populations. The situation our equations govern is one in whih there is a superposition of many Maxwellians, as shown in Figure lb, one at eah temperature between the orotation energy and E*, with the total density of eah separate population given by m(e). This representation will The seond equation in this ase is obtained by evaluating (5) at E, m(e ) [( ) de + (de. ).]E e 1 Results = Kn + (18) Table 1 shows the solutions for E and ¾ for our two ases as a funtion of the transport time z. Observations indiate the bulk of the plasma has energies between 55 and 75 ev. For

4 8100 Rihardson and Sisoe: Brief Report TABLE 1. Solution for E and ¾ for Two Cases as a Funtion of ß days E e T ¾ Power, 10 TM Non-Maxwellian Case Maxwellian Case the Maxwellian ase (Figure lb) this orresponds to transport times of days, and for the non-maxwellian ase (Figure la) to transport times of about 100 days. Thus the range of values for the transport time braketed by our two solutions is 25 to just over 100 days, with the atual time probably lying loser to the shorter values, as the Maxwellian ase should be muh loser to the real distribution than the non-mawellian ase. These values of the transit time orrespond to values of ¾ ranging from Figure 2 shows the distribution funtion for the ase where E* = 61 ev and ¾ = 0.87, whih is appropriate for the non-maxwellian ase with a 100-day transport time. The distribution is heavil skewed in favor of ions with energies near E. Half of the ions have energies between 61 and 75 ev, 20% have energies greater than 145 ev, and 10% have energies greater than 225 ev. Thus ions lose most of their energy in a small fration of the tranport time. The distribution funtion shown in Figure 2 should be quite aurate in the upper two thirds of the energy range shown. At lower energies (50 ev) the self ollision time is short enough (4 4 days) ompared to the transport loss time that a more Maxwellian profile should result. Also shown in Table 1 is the amount of power transferred from the ions to the eletrons for eah value of the transport time. This alulation is model dependent in that a volume must be hosen for the torus. Here we have assumed a volume of 2 x 1031 m 3 for the emitting region. It also depends on our assumption of 5 ev for the eletron temperature. The results an be ompared with a UV output of 1.6 x 1012 watts based on the same values for the volume and eletron temperature (D. E. Shemansky, private om- muniation 1983). For the Maxwellian ase, transit times whih give good agreement with observations of E*, days, give power outputs,-_>' 100e V OOeV m _ ;500eV (a) (b) 0 I00 2_ Energy ( e V ) Figure 1. Shown here are the two types of energy distribution used in alulating energy transfer rates. The non-maxwellian ase (a) has no ions of energy levels below E* or above E. The Maxwellian ase (b) is the distribution arrived at by summing up the assumed Maxwellian at eah energy, four of whih are shown here. The density sales used in (a) and (b) are not the same.

5 Rihardson and Sisoe: Brief Report O 7O '> 60 E* = 61 ev X :.87 Summary We have obtained a distribution funtion for ions in the hot Io torus of the form re(e) = m(e ) (E-E,)-¾ E -E o 50 ' 40 3O 2O I0 I Energy (ev) where is between 0.8 and 0.9. This is a highly non-maxwellian distribution with ions bunhed at energies near E*, whih, based on observations, is in the range ev. An E* in this range was found to require a transit time through the torus in the range 25 to 100 days. This, in turn, implies an ion reation rate of 6 x x 1028 ions/s, with shorter transit times and higher soure strengths being preferred. Aknowledgments. We would like to thank D. D. Barbosa, R. M. Thorne, and D. E. Shemansky for valuable disussions. This work was sponsored by the National Aeronautis and Spae Adminis- Figure 2. This shows an example of the distribu- tration under ontrats NAGW-87 and NAGW-333. tion funtion we have derived. This ase is The Editor thanks R. A. Brown and D. E. appropriate for the Maxwellian ase with a 100 Shemansky for their assistane in evaluating this day torus transport time. paper. 1-2 times less than observed. The non-maxwellian ase yields even less power. We do not onsider the stated differenes be- tween the observed and alulated powers neessarily to be a major problem. Fators of 2 or 3 are not outside the unertainties of these parameters. Also, as mentioned earlier, it is not neessarily a requirement that ion-eletron energy transfer power the entire torus emission, as eletron-eletron heating may also be impor- tant. Finally, we note that the average energy of the torus plasma has been reported to be about 60 ev for SII and 90 ev for Sill [Brown, 1981, 1982]. These numbers are onsistent with the values we obtain for the average energy shown in Table 1. It should be noted that we have not determined that value of the eletron temperature that gives equal rates of energy transfer from ions to eletrons and energy output as UV radiation. Instead we have hosen a value of 5 ev as typifying the eletron temperature found by other means [e.g., Shemansky and Smith, 1981; Barbosa et al., 1983]. These results are quite sensitive to this parameter, as the exhange rate of energy between ions and eletrons is proportional to T 3/2. For example,.onsidering just the Maxwellian ase, raising the eletron temperature to 7.5 ev inreases the transport time needed to math the observations from days to days. It also, however, redues the power provided to the eletrons by a fqtor of 3 to 3 x 1011 watts, a fator of 5 less than observed. Reduing the eletron temperature in our model to 3 ev allows us to provide enough energy to the eletrons to power the torus. However, an eletron temperature below 4 ev would not be onsistent with the UV observations. Thus small variations in the eletron temperature allow us to either inrease the torus residene time but run into diffiulty in powering the torus, or to provide enough energy to power the torus but require small residene times. Referenes Bagenal, F., and J. D. Sullivan, Diret plasma measurements in the Io torus and inner magnetosphere of Jupiter, J. Geophys. Res., 86, , Barbosa, D. D., and M. G. Kivelson, Dawn dusk eletri field asymmetry of the Io plasma torus, Geophys. Res. Lett., 10, , Barbosa, D. D., F. V. Coroniti, W. S. Kurth, and F. L. Sarf, Detetion of lower hybrid emissions in the Io plasma torus: bounds on anomalous eletron heating rates, paper presented at the 1982 Fall Meeting of the Amerian Geophysial Union, San Franiso, Calif., De. 7-15, Barbosa, D. D., F. V. Coroniti, and A. Eviatar, Coulomb thermal properties and stability of the Io plasma torus, in press, Astrophys. J., Brown, R. A., The Jupiter hot plasma torus: Observed eletron temperature and energy flow, Astrophys. J., 244, , S+ Brown, R. A., The probability distribution of gyrospeeds in the Io torus, J. Geophys. Res., 87, , Butler, S. T., and M. J. Bukingham, Energy loss of a fast ion in a plasma, Phys. Rev., 126, 1-4, Goertz, C. K., Io's interation with the plasma torus, J. Geophys. Res., 85, , Hill, T. W., Corotation lag in Jupiter's magnetosphere: Comparison of observation and theory, Siene, 207, , Johnson, R. E., and D. F. Strobel, Charge exhange in the Io torus and exosphere, J. Geophys. Res., 87, , Shemansky, D. E., Mass loss and diffusive loss rate of the Io plasma torus, Astrophys. J., 242, , Shemansky, D. E., and G. R. Smith, The Voyager 1 EUV spetrum of the Io plasma torus, J. Geophys. Res., 86, , Shemansky, D. E., and B. R. Sandel, The injetion

6 8102 Rihardson and Sisoe: Brief Report of energy into the Io plasma torus, J. Geophys. Tokar, R. L., D. A. Gurnett, and F. Bagenal, Res., 87, , 1982, The proton onentration in the viinity of Smyth, W. H., and D. E. Shemansky, Esape and the Io plasma torus, J. Geo?hys. Res., 87, ionization of atomi oxygen from Io, Astrophys , J., in press, Spitzer, L., Jr., Physis of Fully Ionized Gases, Intersiene, New York, J. D. Rihardson and G. L. Sisoe, Department Thorne, R. M., Jovian auroral seondary eletrons of Atmospheri Sienes, University of California, and their influene on the Io plasma torus, Los Angles, CA Geophys. Res. Lett., 8, , Thorne, R. M., Injetion and loss mehanism for (Reeived February 7, 1983; energeti ions in the inner Jovian magneto- revised June 7, 1983; sphere, J. Geophys. Res., 87, , aepted June 10, 1983.)