Volcaic activity ad tidal heatig of Satu s moo celadus P. Vaga Geodetic ad Geophysical eseach Istitute, Seismological Obsevatoy, Budapest, Meedek u. 18-111, ugay (vaga@seismology.hu 1. Pelimiay statemets coceig dyamics of celadus he medium size, spheical satellites of Satu tued i sychoized otatio i the ealy stage of thei histoy, what meas thei obital ad otatio peiods ae the same. Cosequetly thei tidal bulge has o motio i a coodiate system fixed to the body of the moo, theefoe thee is o tidal fictio ifluecig the axial otatio spi ad dyamical pocesses o ad withi the celestial object. his passivity disappeas howeve if the obit is locked i the esoace betwee the obital peiods of membes of the satellite system. his esoace geeates a foced ecceticity which leads to tempoal vaiatio of the tidal bulge due to cetal body of the system. celadus is oe of the iemost moos of Satu. he most emakable types of tectoic featues foud o celadus ae faults that ca u up to two huded kilometes log, which coss olde, tectoically defomed fomatios. he oigi of these global liea stuctues should be elated to tidal activity geeated by the celadus-dioe :1 esoace. idal heatig is esultig pehaps also fom this obital esoace with Dioe, which powes the wate volcaism imaged by Cassii spacecaft i 005. ig.1. Distibutio of liea stuctues o the suface of the ath ig. 1 shows lieamets detected o the ath ove the fome USS with the use of satellite telemetic ivestigatios duig eighties of XXth cetuy. (Bjukaov et al., 117
118 1984. hese stuctues, which wee foud i othe pats of the ath too, have egula distibutio ove the plaetay suface ad possibly ae elated to despiig of axial spi. Simila fault-lies ae visible o the suface of celadus (ig. what ca be i piciple also associated to esoace suppoted tides of celadus. ig.. Distibutio of liea stuctues o the suface of the ath he aim of ivestigatio caied out by the autho is to model cotibutio of the tidal pheomeo i dyamics of celadus. he mathematical modellig of the tide geeated omal (adial ad tagetial (tagetial distibutio i the icy matle of celadus was caied out. With the use of these esults the magitude of tidal heatig was estimated..desciptio of elastic defomatios of spheical body he stesses poduced by tidal foces wee studied by ihomogeeous diffeetial equatios of elasticity fo a gavitatig globe, which ae geeally used i eath tidal eseach. lastic defomatios i this system ca be chaacteised by dimesioless fuctios of Love ad Shida: ( descibes the vetical, ( the hoizotal displacemets ad ( gavitatioal potetial associated with elastic defomatio. μ=μ( ad λ=λ( ae the Lamé paametes, ρ=ρ( is the desity at a distace fom the cete. W=W( descibes the gavity potetial, while is the degee of spheical hamoics ad G is the gavitatioal costat. If the otatio system itoduced by Molodesky (1953 is i use the coespodig diffeetial equatios ae: 1 ( G 4 1 ( 4 M 1 ( 4 W W ( ( 1 W (All diffeetiatios ( ae caied out with espect to the adius
om amog the six bouday coditios thee efe to the suface (=a. By the defiitio of elastic theoy of tidal defomatios N(a=M(a=0. At the coe bouday =b N(b is equal to the hydodyamic pessue ad the value of M(b depeds o the stuctue of the coe. o detemie the omal (adial ad tagetial (hoizotal stesses the followig auxiliay elatios ae valid ( 1 N ( M 3. Model of celadus used fo aims of peset ivestigatio he data listed i able 1 show chaacteistics of Satu s moo celadus. able 1. Chaacteistics of celadus (Poco et al, 006 Obital chaacteistics Physical chaacteistics Semimajo axis ( :.37948 10 8 m Mea adius (a :.51 10 5 m cceticity(e : 4.5 10-3 Mass (M : 1.08 10 0 kg Obital peiod ( : 1.370 day Volume: 6.71 10 16 m 3 Mea suface tempeatue: 75 K Mea desity(ρ : 1610 kg m -3 ( Mass of Satu (M s : 5.68 10 6 kg Suface gavity: 0.078 m s - If the physical paametes of celadus ad aothe sigificat Satu moos ( Mimas, hetys, Dioe, hea, ita, ypeio ad Iapetus ae compaed it tus out that the celadus has a elatively high aveage desity (see able 1. It meas that due to the fact that his suface is composed by wate ice celadus should have a ihomogeeous ie stuctue with a deth coe. o the pupose of peset study a two layeed somewhat abitay selected model was used: - icy matle 1 /a>0.55 - ocky coe 0.55 /a 0 he physical paametes of such a atificial model of celadus ae descibed i able. able. Physical popeties of the model Mass (kg Volume (m 3 Desity (kg m -3 μ GPa λ GPa icy matle 6.15 10 19 5.59 10 16 1100 5 5 ocky coe 4.64 10 19 1.1 10 16 4150 0 0 119
I the above model the Lame paametes ae take equal (μ= λ ad the effect of iceasig hydostatic pessue alog the adius was ot take i cosideatio. he umeical solutio of the sixth ode diffeetial equatio i case of = fo the auxiliay fuctios ( expessed i elative uits withi the icy matle gives auxiliay fuctios M ( ad N ( i the fom show i ig. 3. he expeiece of the study of vey diffeet models of the ath shows, that the geeal ted emais simila fo essetially diffeet models of the matle i case whe the depth of the coe-matle bouday emais the same (Vaga, 1985, 1988, 199. If the coe adius deceases ( i.e. i case of educed ihomogeity of the plaetay body the elative maximums of N ( ad M ( ae displaced to the deepe pats of the matle. 4. idal heatig of celadus Due to sychoous otatio of celadus the tidal bulge vaies oly due to foced ecceticity caused by :1 obital tidal esoace with Dioe. he coespodig vaiatio of secod degee tidal potetial ca be give with equatio G M S a 3 1 e W cos a (1 e he omal ad tagetial secod degee tidal stesses ca be expessed with the use of auxiliay fuctios N ( ad M ( (ig. 3 W ( ( N N g a ( W ( 1130 ( ( ( M M g a he stess geeated eegies N o M i a volume V( ca be obtaied with 1
( ( V ( esults of calculatios caied out ae plotted o ig. 4. It ca be cocluded that most of tidal eegies ae cocetated i the middle of the somewhat abitay chose matle. It is so - idepedetly fom the selected model - because of featues of N ( ad M ( (ig. 3 ad due to i expessio a fo the tidal potetial. his eegy cocetatio i the middle of the matle ca be illustated by the distibutio of focal depths of quakes of the Moo (Nakamua, 1983. o eegy dissipatio withi the celadus (M =1.08 10 0 kg oe gets d J K M dt ee J is the mechaical equivalet of heat (=4.18 Joule, expesses the specific heat ( 0.. o calculate the wam up K (i degee/s we got fo dissipatio fom Δ 10 17 Joule ad Δ=1/ = 0.6851 day d 11 8.4 10 Joule/ s dt. his way fo the wam up we got K=1.4 10-8 degee/s. his meas that duig 300-350 yea the tempeatue elagemet i the middle matle ca be 00Kelvi. (o a compaiso fo the ath K=1.4 10-16 degee/s i.e. 3Kelvi duig 10 9 yea 5. idal fictio due to iegula axial otatio alog eccetic obit. Applyig Keple s thid law to the Satu_celadus system obital velocity of celadus i case of cicula motio is 1131
G 3 ( M S M a I case whe the axial otatio is sycoized ad due to the foced ecceticity caused by Dio the magitude of obital speed vaiatio is 1 (1 e 3/ 4 1/ 1/ 3/ G ( M S M a ~10-3 /s his meas that the libatioal motio of celadus o ist obit aoud Satu (ig. 5 is 8.6. ig.5. Libatioal motio of celadus his motio poduces a additioal heatig idepedetly fom the ie stuctue of celadus. he amout of heatig geeated i this way will be discussed i a futue cotibutio. 5. Coclusios. he tidal stesses ad eegies as it is show by ealie model calculatios fo a spheical o-otatig, elastic ad isotopic ath ae geeated chiefly i the middle pat of the s matle. his coclusio holds i case of aothe bodies of the sola system too. his cicumstace allows exted the taditioal field of eath tide studies to the poblems of plaetology. By defiitio tidal stesses ae equal to zeo both at the suface ad the cete of celestial bodies: at the plaetay suface N(=M(=0 while at the cete the tidal potetial W=0. he tidal heatig ca wam up ad melt the ie pat of the celadus at the depth iteval 15-160 km. his way, the foutai-like plumes detected by the Cassii missio ca be explaied. I case of educed of educed ihomogeety (expessed i lowe mea desity values, the tide geeated heat moves to the deepe pats of 113
the moo. Possibly that is the easo why i case of othe moos ofsatu thee is o simila volcaic activity obseved. idal fictio due to iegula axial otatio alog eccetic obit ca be a aothe souce of tidal heatig, which eeds futhe ivestigatios. Ackowledgemets he eseach descibed i this cotibutio was ealized i the fame of Gema- ugaia bilateal poject Ivestigatio of geodyamical pocesses by meas of Liouville equatio suppoted by the Deutsche oschugsgemeischaft ad by the ugaia Academy of Scieces. he ugaia poject paticipats appeciate suppot of the ugaia Sciece oudatio (Poject: OKA K60394. efeeces Bjukaov et al., 1984. Cosmogeological map of the USS of the scale 1 : 500 000. Pape peseted at the 7 th Iteatioal Geological Cogess, Sectio C 18, Nauka, Moscow. Molodesky M.S., 1953. lastic tides, fee utatios ad some poblems of the ath s stuctue. udi Geophys. Ist., 19 (146, 3-5. Nakamua Y., 1983. Seismic velocity stuctue of the lua matle, J. Geophys. es., 88, 677-686. Poco C.C. et al., 006. Cassii obseves the active south pole of celadus. Sciece, 311, 10 Mach, 1393-1401. Vaga P., 1985. Ifluece of exteal foces o the tiggeig of eathquakes, athq. Pedict es., 3, 191-01. Vaga P. ad Deis C., 1988. A study of the vaiatio of tidal Love umbes with ath stuctue., Geophys. asact., 34, 4, 63-8. Vaga P., 199. Stesses of luisola oigi actig at the coe matle bouday. Mauscipta Geodetica, 17, 351-355. 1133