Estimation of Magnetospheric Plasma Parameters from Whistlers Observed at Low Latitudes

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1 Iteratioal Joural o Astroomy ad Astrophysics, 13, 3, Published Olie December 13 ( Estimatio o Magetospheric Plasma Parameters rom Whistlers Observed at Low Latitudes Mohd Alta 1, Mohammad Mushtaq Ahmad 1, Ja Mohammad Baday 1 Departmet o Physics, Natioal Istitute o Techology, Sriagar, Idia Departmet o Civil egieerig, Natioal Istitute o Techology, Sriagar, Idia altaig@redi.com Received March, 13; revised April, 13; accepted April 8, 13 Copyright 13 Mohd Alta et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. ABSTRACT Whistler observatios durig ighttimes made at low latitude Idia groud statios Jammu (geomag. lat., 9 6'N; L = 1.17), Naiital (geomag. lat., 19 1'N; L = 1.16) ad Varaasi (geomag. lat., 1 55'N; L = 1.11) are used to deduce electro temperatures ad electric ield i the viciity o the magetospheric equator. The accurate curve ittig ad parameter estimatio techique are used to compute ose requecy ad equatorial electro desities rom the dispersio measuremets o short whistlers recorded at Jammu, Naiital ad Varaasi. I this paper, our aim is to estimate the Magetospheric electro temperatures ad electric ield rom the dispersio aalysis o short whistlers observed at low latitudes by usig dieret methods. The results obtaied are i good agreemet with the results reported by other workers. Keywords: Magetosphere; Electro Temperature; Electric Field; Electro Desity 1. Itroductio It is well kow that lightig discharges are accompaied by the geeratio o electromagetic waves i a wide requecy rage [1,]. Wave eergy ca peetrate ito the magetosphere ad propagate almost alog geomagetic ield lies to the opposite hemisphere where it is recorded by a radio receiver called whistler. The dyamic spectrum o the recorded sigal is typically dispersed i the spectrogram. These sigals sometimes proceeded by a associated sigal with a udispersed dyamic spectrum ad are geerated durig the same lightig discharges but propagate i the Earth-ioosphere waveguide [,3]. Whe this sigal is recorded ad coicides approximately with the momet o lightig discharge, its time delay does ot usually exceed. s [] as the velocity o wave propagatio i the Earth-ioosphere waveguide is close to velocity o light, this sigal is called a atmospheric or seric. The whistler sigal itesity is ormally greatest at ew khz withi the ELF/ VLF bad; 1 - khz (Carpeter, 196). Whistlers represet a iexpesive ad eective method or obtaiig various plasmaspheric parameters like electro desity, electro temperature, electric ield etc. i the magetosphere, but the experimetal results published up to ow reer maily to higher latitudes [5-7], ad a systematic descriptio o the mai eatures o the plasmaspheric electro desity based o large quatities o whistler data is still lackig at high latitudes, with the exceptio o work by Park et al. [8]. Recetly Tracsai et al. [9] have processed whistlers recorded at Tihay, Hugary (L = 1.9) betwee December 197 ad May 1975 i order to study the distributio o equatorial electro desity ad total electro cotet i lux tubes havig L- value, lyig i the rage L = At low latitudes, the exploratio o whistlers or electro desity determiatio has oly bee carried out by Lalmai et al. [1]. I this paper the equatorial electro desity, equatorial electro temperature ad east-west compoet o electric ield at low latitudes usig the whistler data observed at our groud statios Jammu, Naital ad Varaasi have bee estimated. At middle ad high latitudes, both satellite ad groudbased whistler data were exploited ully to reveal ew acts about the structure ad dyamics o the ioosphere ad magetosphere. These achievemets icluded the discovery o the plasmasphere, plasmapause, ad bulge [11], idetiicatio o the mechaism o ioosphere-protoosphere couplig [5,1] ad the measuremet o the

2 magetospheric electric ield [13]. Although the applicatio o whistlers to diagostic o electro temperature o high latitudes has bee discussed sice the early 196s [6,1,15], this problem still seems to be at a early stage o its developmet at low latitudes. At low latitudes, whistler data have bee used or determiig electro temperatures, electric ield etc. or uderstadig the magetospheric pheomea. We cosider the methods o traditioal diagostics o magetospheric parameters, such as electro plasma desity, the large scale electric ield ad possible temporal variatios o the magetic ield at the magetospheric equator, whe both ad t are kow. Whe oe or both o these parameters are ot kow the dyamic spectra o whistlers ad/or serics eed to be extrapolated. Method o this extrapolatio is subsequetly cosidered. The we estimate the equatorial electro desity, electro temperature, ad electric ield i the equatorial magetosphere based o the aalysis o the dyamic spectra o whistlers. Whistler studies i Idia, which have bee i progress sice 1963, have made sigiicat cotributio to the propagatio o low latitude whistlers ad uderstadig o the structure ad dyamics o the low latitude ioosphere [16-18]. For the aalysis o o-ose whistlers, a umber o methods have bee proposed [19]. The ose requecy o the whistler data used i estimatig electro desity, electro temperature ad electric ield has bee computed by meas o accurate curve ittig method developed by Tarcsai [19] based o least squares estimatio o the two parameters, zero requecy dispersio D o, equatorial electro gyrorequecy He i Berard s approximatio. This matched ilterig techique developed or the aalysis o whistler waves icreases the accuracy o aalysis ad speed o data processig [17,18,]. The techique employs dispersive digital ilters whose requecy-time respose is matched to the requecy-time respose o the sigal to be aalyzed. Due to high resolutio ad time domai, may ie structure compoets with amplitudes dierig i requecy ad time are see i dyamic spectra []. The accuracy ad eectiveess o the techique have bee discussed at legth by aalyzig a large umber o whistlers both o the groud statios (rom the low to the high latitudes) ad oboard rockets/satellites [18,-]. Electric ields are closely related to ad cotrol most o observed gyophysical pheomea such as the bulk motio o the magetospheric plasma, the curret systems i the magetosphere ad the ioosphere, ad to the acceleratio o plasma particles i the Earth s magetosphere. The role o the electric ield i cotrollig the bulk motio o the plasma has bee recogized i all the theoretical studies o the various dyamic processes takig place i the Earth s magetosphere although adequate experimetal techiques or the precise measuremets o such ields i the ioosphere ad magetosphere were ot available or quite some time. The observed cross-l motios o the whistler ducts are beig used curretly or obtaiig the east-west compoet o the electric ields i the plasmasphere durig substorm periods as well as quiet times [,5]. The tidal orces i the Earth s atmosphere cause motio o the plasma across the magetic ield lies ad give rise to electromotive orces. The geeratio o electric ield by the motio o coductig plasma across the magetic ield is aalogous to dyamo actio ad the theory dealig with the electric ield geeratio by this mechaism is kow as dyamo theory. The electric ield geeratio mechaism i the ioosphere has bee developed by various workers [3,]. Electric ield measuremets have bee carried out i the equatorial E- regio o the ioosphere by may workers. These measuremets reveal the existece o east-west electrostatic ield ragig rom 1 to mv/m. The whistler method o obtaiig the east-west compoet o the electric ield has the advatage o exteded time coverage ad remarkable property o beig directly ivolved i the motio o magetospheric tubes or ducts o ioizatio. Further, the groud-based whistler determiatios o electric ields are comparatively easier ad the equipmet used ca be moitored with relative ease o a routie basis. It is precisely or this reaso that the groud-based whistler studies o electric ields are still cotiued at a umber o statios spread all over the world. I this paper we irst preset the whistler data used or the aalysis recorded at Jammu, Naiital ad Varaasi. This is ollowed by a presetatio o a outlie o the method developed by Tarcsai [19] rom which electro desity, electro temperatures, ad electric ield i the viciity o magetospheric equator are evaluated. Fially the results are discussed ad compared with those reported by other workers.. Data Selectio ad Method o Aalysis At low latitudes, the whistler occurrece rate is low ad sporadic. But oce it occurs, its occurrece rate becomes comparable to that o mid-latitudes (Hayakawa et al., 1988). Similar behavior has also bee observed at our low latitude Idia statios. All the Idia statios are well equipped or measuremets o VLF waves rom atural sources. For the preset study, the whistler data chose correspods to Jue 5, 1997 or Jammu, 5 March 1971 or Naiital ad 19 February 1997 or Varaasi. O 5 Jue 1997 at Jammu statio whistler activity started aroud 1 h IST (Idia Stadard Time) ad lasted up to 5 h IST. Durig this period about 1 whistlers have bee recorded [5]. O 5 March 1971 at Naiital statio whistler activity commeced aroud

3 1 IST ad lasted up to 5 IST. Altogether more tha hudred whistlers were recorded ad the occurrece rate showed a eeble but discerible periodicity [6]. O 19 February 1997 at Varaasi statio whistler activity started aroud 3 IST ad lasted or about oe hour up to 3 IST. Durig this period several whistlers were recorded [7]. Figure 1(a) presets dyamic spectrum o short whistlers (marked A, B, C, D, E, F ad G, selected or the aalysis) i the requecy bad KHz recorded at Jammu at 1 IST o Jue 5, I the requecy bad KHz large umber o requecy compoets are missig ad sigals resemble more like emissios rather tha whistlers. Further, VLF waves i both the requecy bads do ot appear simultaeously, rather they appear alterately. Figure 1(b) shows dyamic spectrums o short whistlers (marked 1,, 3 ad, selected or the aalysis) ad VLF emissios recorded at Jammu at 17 IST. Whistlers are baded ad diused i the requecy rage KHz ad are repeated i time. The time iterval betwee the evets is ot cojammu LT 5 (khz) 3 Figure. Dyamic spectrum o whistlers recorded at Naiital March 5, Whistlers are marked by W1, W, W3, W ad W t (sec) (a) Jammu LT (khz) t (sec) (b) Figure 1. (a) Dyamic spectrum o whistlers recorded at Jammu Jue 5, Whistlers are marked by A, B, C, D, E, F ad G. (b) Dyamic spectrum o whistlers recorded at Jammu Jue 5, Whistlers are marked by 1,, 3 ad. stat. Uusual VLF oises are also see i the spectrum. Figure shows dyamic spectrum o short whistlers selected or the aalysis recorded at Naiital o March 5, The soograms o sample whistlers (marked W1, W, W3, W ad W5) are arraged i a sequece or dieret time o arrival. Figures 3(a) ad (b) shows the dyamic spectra o short whistlers (selected or aalysis) recorded at Varaasi o 19 February 1997 at 17 IST ad 338 IST respectively. Tarcsai [19] has developed a curve ittig techique or the aalysis o middle ad high latitude whistlers. This techique has also bee applied successully to those low latitude whistlers whose propagatio path are low below L = 1. [8-31]. Further techique is oud suitable ot oly or log ad good quality whistlers but also or short ad ait whistlers. The computer programme writte or the purpose requires iput data such as requecy time (, t) values scaled at several poits alog whistler trace appropriate or F, zero requecy dispersio (Do), ad a suitable ioospheric model etc. The output results iclude the L-value o propagatio, equatorial electro desity, total tube cotet etc. we

4 6. (khz) (khz) Varaasi a :17 W T (sec) Varaasi e 19::97 3:38 6. W T (sec) Figure 3. Dyamic spectra o whistlers recorded at Varaasi February 19, Whistlers are marked by W1 ad W. have adopted this programme or the aalysis o ighttime whistlers recorded at our statio Naital, Varaasi ad Jammu durig quiet days. At low L-values, the curve ittig method o Tarcsai [19] would ot chage too much the equatorial electro desity ad total electro cotet values compared to the systematic errors which are iheret i all o the existig ose extesio methods. These systematic errors origiate rom the approximatios used or the reractive idex ad or the ray path i the derivatio o the aalytic expressios or the dispersio ad rom the dierece betwee the theoretical ad actual distributio plasma alog the ield lies [3]. To examie its validity we aalyzed ew whistlers recorded at Jammu usig this method as Dowde Allcock [31] Q-techique. Both methods yielded results withi ±1%. Further, it is to be oted that the Tarcsai s method has successully bee used i the aalysis o low latitude whistler). For the determiatio o D o, ad t approximate uctio or the dispersio o whistlers is give by [19] D t D 1 Heq o Heq A where D o = zero requecy dispersio Heq = equatorial electro gyrorequecy = Wave requecy t = travel time at requecy, ad Here (1) 3 1 A () 1 where is the ose requecy or which travel time t is writte as t Heq D o 1 1 (3) I the causative seric is ukow, ad the travel times at dieret requecies o the whistler traces are measured with respect to a arbitrary time origi, the it is ecessary to itroduce a ew parameter T, which gives the dierece i time betwee the chose origi ad the actual causative seric. Usig T ad Equatio (1) the * t ca be writte as measured travel time D Heq A o Heq 3 Heq Heq * t t T T Do Heq Heq Heq T I this equatio there are our ukow parameters D o, He, T ad. Tarcsai [19] has developed a computer program to solve Equatio () or the ukow usig successive iteratio method. I this method those values o D o, He, T ad are searched which give best it to the measured parameters. Ater Park (197) ad usig Equatio (3) or t. 5 / 5 3 eq Ke t L Ke Do Heq 1 / 1 3 NT KT tl KTDo (5) Heq 5 / 5 3 N K1tL K1DoHeq Where the costats K / e ad K / T are weakly depedet o ad Λ Tracsai [19]. Usig Equatio (5) ad aalyzig whistlers show i Figures 1-3 recorded at Jammu, Naiital ad Varaasi, ose requecy, equatorial electro desity eq ad total electro cotet N T i a lux tube o uit cross sectio ()

5 3 has bee evaluated. The the equatorial electro temperature T eq was estimated rom ose requecy computed rom Tarcsai [19] method or a give model o electro distributio or our aalysis. A diusive equilibrium model similar to that adopted by [5,18,19,7], was employed which was represeted at the height 1 km by a electro desity 1 3 electro/cm 3, O + = 9%, H + = 8%, ad He + = % at the temperature re T o 1 K. The electro temperature i the magetosphere (T e ) is related to the electro temperature at the reerece level T by the equatio. re e re R Rre T T (6) where R ad R re are the correspodig geocetric distaces. We took two values o, ( = 1 ad ). For the case o =, T e remais almost costat ad or the case o =, T e icreases rapidly with height, oe expect the actual value o to lie betwee these two extremes. The results o the calculatio o, He, D o, L, eq ad T eq or the whistlers uder cosideratio are show i Table 1 (or = 1 ad ). The whistler ose requecy is related to its path L by the approximate relatio L (7) Where is i khz ad a cetral dipole magetic ield is used to represet the geomagetic ield. The ose requecy ad the miimum equatorial gyrorequecy alog the path o propagatio are related as [33] / / Heq Ho o K K R R (8) Where K / =.38 or a diusive equilibrium model o the ield-lie distributio o ioizatio, Heq ad Ho are the equatorial gyrorequecies at geocetric distaces o R ad R o (Earth s surace) respectively. I the equatorial plae the covectio electric ield, deies as positive i the eastward directio, is give by E V B B (9) I the case o magetic equator we obtai rom Equatio (8) 3 dr dt Ew Bo Ro R (1) where B o represets the geomagetic ield stregth at the 3 Table 1. Parameters o whistlers observed at Jammu, Naitial ad Varaasi groud statios estimated rom the whistler dispersio aalysis usig accurate curve ittig techique. W is the whistler umber, IST is the Idia Stadard Time, D o is the dispersio o whistler, is the whistler ose requecy, Heq is equatorial gyro requecy, L-value is i earths radii, eq is the equatorial electro desity ad T eq is equatorial electro temperature. W Statio Dates & Year IST D o (sec 1/ ) (KHz) Heq (KHz) L Value eq (cm 3 ) = 1 T eq (ev) = T eq (ev) 1 Jammu 5 Jue :: ± 1.. ± ±.7.5 ± ± Jammu 5 Jue :7: 81.9 ± ± ±.3.35 ±.5 ± Jammu 5 Jue 1997 :7: ± ±. 1.7 ±.5.39 ±.7 7 ± Jammu 5 Jue 1997 :7: ± ± ±.3.38 ±. ± Jammu 5 Jue 1997 :1: 8.8 ± ± ± ±.1 93 ± Jammu 5 Jue 1997 :1: ± ± ± ±. 61 ± Jammu 5 Jue 1997 :13: 35.5 ± ± ± ±.5 88 ±..6 8 Jammu 5 Jue 1997 :13: ± ±.1 1. ±.8.1 ±.3 63 ± Jammu 5 Jue 1997 :1: 6.1 ± ± ± ±. 3 ± Jammu 5 Jue 1997 :1:55.8 ± ± ± ±.8 35 ± Jammu 5 Jue 1997 :15: ± ± ±.. ±. 76 ± Naiital 5 March 1971 :3:. ± ±..35 ± ±. 98 ± Naiital 5 March 1971 :9: 18.6 ± ± ±.8.67 ±. 136 ± Naiital 5 March 1971 :56: 18.5 ± ± ± ±. 5 ± Naital 5 March 1971 :3: 13. ± ± ±.7.89 ±. 5 ± Naiital 5 March 1971 :3: 15. ± ± ± ±. 9 ± Varaasi 19 Feb :17: 11.9 ± ± 13.5 ± 6.1 ±. 7 ± Varaasi 19 Feb :38: 13.5 ± ± ± 31.9 ±.1 5 ± 5.1.

6 Earth s surace ad E w is the westward compoet o the magetospheric electric ield. With the help o Equatios (7) ad (8) the covectio electric ield i a dipole model i the equatorial plae ca be obtaied as [3] w E t (11) d d V m With the ose requecy expressed i hartz oe ca directly estimate the covectio electric ield rom the 3 slope o usig Equatio (1). The variatio o 3 with tim e or the whistlers observed at Jammu, Naital ad Varaasi are give i Figures -6 respectively. 3. Results ad Discussio Several attempts have bee made to use whistlers as a diagostic o the electro temperature o the magetosphere besides the traditioal methods o diagostics o electro temperature i the magetosphere. The irst such attempt was probably made by Scar [35] who estimated this temperature rom the thermal atteuatio o ose whistlers at the upper cut-o requecy. This method was developed by Liemho ad Scar [36,37], but, to / : 1:7 :1 :13 :1 :15 :7 t (IST) 3 Figure. Plot betwee ad t or the case observed at Jammu o 5 Jue / : :3 :9 :56 :3 :3 5: t (IST) 3 Figure 5. Plot betwee ad t or the case observed at Naital o 5 March / : :17 :38 t (IST) 3 Figure 6. Plot betwee ad t or the case observed at Varaasi o 19 Feb our kowledge, was ever used i practice, perhaps or two reasos. Firstly, it is diicult to decide whether the whistler upper cut-o requecy is determied by wave atteuatio or by propagatio eects. Secodly, the i- rom i situ ob- terpretatio o the whistler cut-o requecy i Scar s method is very sesitive to the aisotropy o the electro distributio uctio, which ca i geeral, be determied oly by i-situ measuremets. I a alterative approach to this problem, Mc Chesey ad Hughes [3] measured the electro desity at the magetospheric equator ( eq ) by whistler dispersio aalysis, ad i the topside ioosphere servatios o LHR oise. The ratio o these desities was itted to a diusive equilibrium model o electro desity distributio with temperature as a parameter. The mai assumptio was that the electro temperature did ot chage alog the magetospheric magetic ield lie. However, this assumptio seems to be icompatible with satellite measuremets o electro temperature, equatorial temperatures ca be up to a actor o 1 larger tha those at ioospheric altitudes [38,39] ad eeds to be abadoed i urther modiicatios o this method. A dieret approach was take by Guthart [] who attempted to estimate magetospheric electro temperature rom its eect o whistler group velocity, assumig a gyrorequecy model electro distributio. He predicted that the thermal eect o whistler spectra should be largest at requecies ear the upper cut-o requecy o ose whistlers. However, the size o the eect was less tha the experimetal error associated with whistler spectral aalysis. This coclusio eabled Guthart to estimate a upper boud o the magetospheric electro temperature o 1 K 1.7 ev. By cotrast, Kobelev ad Sazhi [1] have argued that thermal eects i the viciity o the plasmapause ca be estimated by compariso o observed ad theoretical whistler dispersio curves. Assumig a electro desity distributio alog ield lies, they obtaied values o electro temperature

7 5 i the rage 7-19 ev, depedig o the value o the parameter. This temperature correspods to a average temperature o all electros cold oes with eergies 1 ev plus small hot compoets with eergies o the order 1 kev. I both these papers the eect o variatio o the electro temperature alog the magetospheric magetic ield lie was eglected as was doe by Mc Chesey ad Hughes [3]. More accurate aalysis by Sazhi et al. [], based o the DE-1,, 3, models described earlier, let to a result rather close to that o Guthart [], amely, the magetospheric electro temperature was estimated to be blow ev ad depeds o the choice o electro distributio model. Sazhi et al. [3] discussed dieret approaches to this type o diagostic techique ad have cocluded that the most eective way to estimate the electro temperature with the help o groud observed whistlers would be to use ose whistlers with the well-deied upper brach ad compare whistler group delay times at the ose ad at the upper cut-o requecy. Recetly, Sazhi et al. [] have exteded this approach o aalysis to a larger umber o whistlers i order to get statistically more sigiicat results. They have show that the estimated magetospheric electro temperature strogly depeds o the choice o model o electro distributio alog the magetospheric magetic ield lie. The whistler data recorded at Jammu, Naiital ad Varaasi at dieret times ad or dieret magetic activities have bee aalysised to estimate the magetospheric electro temperature i the viciity o magetospheric equator at low latitudes. We have applied the curve-ittig techique o Tarcsai [19] or our o-ose whistlers at these statios to derive the magetospheric electro temperatures. The estimated temperature o magetospheric electros ierred rom the whistler data show i Table 1 is about.8 ev or the value o = ad is about.5 ev or the value o = 1. Our mea value o T eq obtaied usig the diusive equilibrium model estimated by the method o curve-ittig techique o Tarcsai [19] is ~.5 ev, slightly smaller with other estimates o electro temperature i the equatorial plasmasphere (see Guthart, 1965, 1973; Sazhi et al., 1993; Sazhi et al., 199, 199). Magetospheric temperatures are quite variable ierred temperatures betwee K ad 3 1 K; similar electro temperatures were deduced i a more detailed study by Decreau et al. [5] ad it would be uwise to attempt to geeralize our results. We have also estimated the above electric ield o the basis o dipole geomagetic ield. However, both groud-based ad satellite broe magetometer data shows that ast chages i the magetic ield takes place durig substorm commecemet ad substorm expasio ad recovery phase. Wag ad Kim (197) have discussed the decayig rig curret ad the electric ield that may be associated with that decay. Thus, the departure rom the dipole ield model gives a iduced electric ield due to temporal chages i the geomagetic ield. I the preset study the magetospheric electric ield i the plasmasphere at dieret L-values is oud to be eastward i the pre-midight sector ad westward i the post-midight sector, i agreemet with the published results [1,13,6]. The magitude o the eastward elec- tric ield is about.35 mv/m i the equatorial plae o Jammu. The westward electric ield comes out to be about.7 mv/m or Naital ad.1 mv/m or Varaasi. Thus the preset results o electric ield agrees with the results o Ioosode observatios o the ight-time F-layer durig substorm, ad are i good agreemet with the results reported by Park [6]. I the Ioosode studies, it was oud that durig witers, the F-layer settles dow to a quasi steady state. Its respose to the magetospheric substorm activity cosists o a large scale distortio with F-layer lited upward i the pre-midight sector ad pushed dowward i the post-midight sector. This distortio was iterpreted as the result o E B drit by a eastward electric ield beore midight ad by a westward ield ater midight. This reversal i the electric ield directio is coirmed by the low latitude whistler results reported here. The reversal o the electric ield directio ear midight as show i Figures -6 with the westward compoet durig post-midight hours ad a eastward compoet durig pre-midight hours was also observed by the barium cloud techique ad by electrostatic probes o balloos.. Coclusio The o-ose whistlers recorded at Jammu, Naiital ad Varaasi have bee aalyzed to estimate the equatorial electro desity, electro temperature ad electric ield i the equatorial magetosphere. The estimated tempera- are slightly smaller compared to ture ad electric ield the estimated value o other workers. This prelimiary test o our method o temperature diagostic is rather ecouragig. However, beore this method ca be recommeded or practical applicatios, we eed to speciy the model o electro desity, temperature distributio, ad electric ield i the magetosphere more accurately, so as to have a better estimate or the eect ducted ray paths ad icrease the precisio o determiig whistler parameters. Actually it is the irst attempt to estimate the above metioed parameters by usig o-ose whistler data recorded at Jammu, Naiital ad Varaasi. 5. Ackowledgemets M. Alta, M. M. Ahmad ad J M Baday are thakul to Director, NIT Sriagar, Kashmir, Idia or his costat

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