The Effects of Moons on Saturn s Ring System

Transcription

1 The Effects of Moons on Satun s Ring System Kisten Lason Physics Depatment, The College of Wooste, Wooste, Ohio 44691, USA (Dated: May 10, 007) The ing system of Satun is a complex inteaction between numeous paticles, moons, and Satun. A pogam was ceated to simulate the diffeent aspects of the ing system and to study the effects of moons on the obiting paticles. Simply by consideing both the foce of gavity on each paticle and collisions between paticles, gaps fomed at eight diffeent esonant positions. I also studied the effects of collision elasticity and found that less elastic collisions cause the gaps to fom moe quickly. Futhemoe, the effects of shepheding wee studied by embedding an obiting moon in the ing paticles. By vaying the mass atio of Satun to the moon, it was found that the smalle the mass atio between Satun and the moon, the lage the gap cleaed in the ing and the lage the width of the inglet fomed in the cente of the gap. It was also obseved that esonances inteio to the moon contolled the inteio edge of gaps while esonances exteio to the moon contolled the exteio edge of the gaps. I. INTRODUCTION The ing system of Satun is a dynamically ich envionment with many phenomena to exploe. The complex inteactions between Satun s moons and the ings ae paticulaly inteesting. To bette undestand the obseved phenomena in ing systems, I ceated a compute simulation to model aspects of Satun s ing system like esonance and shepheding. Although all of the gas giants in ou sola system have ings, Satun has the lagest, most complex ing system. Also, thee is moe knowledge of Satun s ing system fom the wealth of infomation gatheed by the Voyage and Cassini satellite missions. The availability of infomation makes Satun s ings a a good choice fo modeling. Many of the obseved phenomena in ings ae ceated by inteactions between the ing paticles and obiting moons. Although thee wee only 18 known moons of Satun in 1997, that numbe has quickly gown to 57 and continues to ise as moe discoveies ae made. These moons wee not obseved befoe 1997 because 31 of these moons ae less then 10 km in diamete. Even these small moons though have visible effects on the ings. Figue 1: Cassini division and the obiting moon Mimas []. esonance means that T paticle = 1 T Moon, this elationship can be used to find the :1 esonance position 4π p GM 3 = 1 4π M 3 S GM S whee M s is the mass of Satun, M is the position of the moon, and p is the position of esonance. This equation can be massively simplified to A. Cassini Division The Cassini division in between Satun s A and B ings is a gap caused by a :1 esonance with the moon Mimas. The mass atio of Satun to Mimas to the ing paticles is about : : 1 [1]. Resonance happens when a ing paticle and an obiting moon have peiods that ae simple factions of one anothe. This causes the paticle and moon to always align at the same position in thei obit and the gavitational pull fom the moon will be geate at this point. Ove time, the small gavitational tugs fom the moon will add up causing the obit of the ing paticle to be petubed. When the ing paticle s obit is petubed, it will collide with othe paticles outside of the esonance aea and its obit will change causing a gap to fom in the ings. Satun s moon Mimas is in a :1 esonance with ing paticles in the Cassini division. Theefoe, evey one time that Mimas obits, the ing paticles will complete two obits. Since a :1 p = ( ) 1 3 M. Thus, the genealized esonance equation fo any esonance (m : n) whee m and n ae integes is given by p = ( n m) 3 M, which woks fo both esonances inside and outside the obit of the moon [3]. I was able to duplicate a :1 esonance in my model and tested how changes in the mass atio of Satun to moon to paticle effects the gap fomation. B. Shepheding Thee ae two obseved gaps in Satun s A ing: the Encke gap and the Keele gap. Many of the obseved gaps and tiny

2 obit. Theefoe, the velocity of the moon can be found by setting the centipetal foce equal to the gavitational foce v M M = F M = GM SM M, M M Figue : The attactive gavitational foce fom the moon esults in paticles being foced away fom the moon s obit. The black aows epesent the gavitational foce fom the moon while the ed aows indicate the net motion of the paticles. This pocess, called shepheding, is explained below. inglets ae caused by the foces fom tiny embedded moons in the ing paticles. The Encke gap is 35 km wide and ceated by the moon Pan [1]. Pan ceates a gap by focing paticles away fom its obit, a pocess known as shepheding. A tiny inglet is also obseved in the cente of the Encke gap. These paticles follow the obit of Pan and ae in a 1:1 esonance allowing them to stay in the cente of the gap. I also modeled shepheding by intoducing an embedded moon into the ing system and obseved the effects that changing the mass atio of the moon and planet had on the size of the gap [1]. Shepheding is the phenomenon of moons pushing ing paticles away fom thei obit. As seen in figue, an embedded moon will attact ing paticles as it obits. Howeve, the effective movement of the paticles is away fom the moon. Paticles obiting inteio to the moon will be moving at a faste velocity. Theefoe, the moon s gavitational foce will pull the paticle outwad and backwad. This will cause the paticle to lose some of its speed and it will fall close to the planet. As it falls close to the planet it will pick up speed again and its new obit will be futhe inwad fom the embedded moon. Paticles obiting exteio to the moon will be moving at a slowe velocity than the moon. Thus, the moon s foce will pull the paticle inwad and fowad. This will cause the paticle to gain speed which will make its obit bigge and it will move away fom the embedded moon. As it moves futhe outwad, the paticle will stat to slow down and its new obit will be futhe outwad elative to the moon. Theefoe, even though the moon is attacting the ing paticles, the combination of the planet s gavitational foce and the moon s gavitational foce actually causes the paticles to be pushed away fom the embedded moon [4]. C. Obit of Moon When ceating the pogam, all of the foces had to be taken into account. The obit of the moon could simply be found by using centipetal foce and Newton s law of gavitation since it was assumed that the moon taveled in a cicula whee M M is the mass of the moon, M S is the mass of Satun, M is the distance between Satun and the moon, v is the moon s velocity, and G is the gavitational constant [5]. By solving fo the velocity it can be used to obtain the moon s peiod 4π T M = M 3 GM, S which is just Keple s Thid Law. Then, by finding the angula fequency of the obit ω M = π T M, whee ω M is the angula fequency, the x and y positions of the moon can be found ove time by the equations [5] and x = M cos(ω M t), y = M sin(ω M t). D. Obit of Paticles Finding the obit of the ing paticles is moe complicated than the moon because foces fom both Satun and the moon must be consideed. To find the foce fom the moon and Satun, thei distances fom the paticle must fist be calculated. Then by setting Newton s second law equal to Newton s law of gavitation, the acceleation of the paticle can be found. m a = F = GMm ˆ. Theefoe, the net acceleation of the paticles fom both moon and Satun s foces is [5] a = GM M 3 + GM s s s 3. whee is the distance fom the paticle to the moon and S is the distance fom the paticle to Satun.

3 3 Figue 3: Both the moon and Satun s gavitational foce affect ing paticles. Figue 4: When paticles collide, they lose some of thei enegy and thei adial velocity deceases. II. IMPLEMENTATION OF PROGRAM The pogam simulated the ings by obiting appoximately 5000 paticles aound a cental planet. Enough paticles ae needed to fom a sufficient ing system; howeve if too many ing paticles ae used, it will take too long to un the pogam. Futhemoe, collisions had to be included into the simulation because as the paticles obits ae petubed by the obiting moons, they will collide with othe paticles in the ings and thei obits will be changed. The pogam begins by initializing the paametes to default settings. These paametes can then be changed by the use by using the gaphical inteface. Each paticle is then advanced by the time step, dt, and its new position is found using fouth ode Runge-Kutta integation. Afte the paticles and the moon ae advanced, the pogam checks to see if any paticles ae close enough to one anothe to collide. This is detemined by the collision adius paamete. The new velocities of the colliding paticles ae calculated and the gaphics ae efeshed with the new positions of the paticles. A. Collisions When the ing paticles get close enough, they will collide inelastically. I defined my collisions so that the azimuthal velocity is conseved but the adial velocity is not. The elasticity of the collision is detemined by the elasticity coefficient which is used to change the adial velocity of the paticles. Only collisions between paticles wee taken into consideation and paticles did not collide with Satun o the moon. To compute the new velocities of the colliding paticles, the x and y position of the paticles wee fist conveted into and phi coodinates by and v = v x ( x v ϕ = v x ( y ) ( y ) + v y ) ( x ) + v y, whee is the distance between Satun and the paticle, v is the adial velocity, and v ϕ is the azimuthal velocity. They ae then boosted to thei cente of mass efeence fame by finding thei aveage velocity and subtacting this aveage fom thei oiginal velocities. These cente of mass velocities ae then multiplied by the elasticity coefficient and thei diection is changed by multiplying them by 1. If the elasticity coefficient is less then 1, the collisions will be inelastic. If the elasticity coefficient is equal to 1, they will collide elastically. The velocities ae then taken out of the cente of mass efeence fame and conveted back into x and y coodinates. III. DATA AND ANALYSIS Thee ae many paametes and many diffeent possibilities of phenomenon to test with the pogam. I tested how vaying the elasticity coefficient changed the fomation of the Cassini division and how changing the mass atio of Satun to the moon effected gap size and fomation. A. Collision elasticity The effects of collision elasticity on the fomation of a :1 esonance gap wee tested by vaying the elasticity coefficient while keeping all othe paametes constant. The moon obited at a distance of 150 with a mass of 1,000 and Satun had a mass of 1,000,000. Ove time, a gap fomed in the ings at a distance if 94 to 100 fom Satun. Since the position of a :1 esonance is 94, the inside edge of the gap is contolled by a :1 esonance with the obiting moon. Howeve, as the elasticity coefficient was inceased, it took longe fo the gap to fom. To see how the coefficient effected the speed at which the gap fomed, the numbe of paticles at a distance of 94 to 98 fom Satun was aveaged and plotted to see how this aveage numbe of paticles in the gap aea changed ove time as seen in figue 5. The cuves wee then fit with exponentials to see how quickly the gap fomed ove time. Fo an elasticity of 0, the numbe of paticles in the gap decayed quickly at a ate of ± pe peiod of the moon while an elasticity of 0.9 caused the paticles to decay at a slowe ate of ± pe peiod of the moon, see table I. Theefoe, it will take longe fo a gap to fom when the collisions have a highe elasticity.

4 4 Figue 5: The numbe of paticles at a distance fom Satun of 94 to 98 wee aveaged and plotted vesus time fo elasticities of 0, 0.5, and 0.9. The data was fit by exponentials whee the equation fo the lines ae e T /τ. As the elasticity of the collisions inceased, the numbe of paticles in the gap decayed at a slowe ate. Figue 6: The effects of an embedded moon at a distance of 150 with a mass of 100 obiting a planet of mass 100,000. The colo scale coesponds to the numbe of paticles at a given distance fom Satun whee dak blue is no paticles pesent and white is appoximately 550 paticles pesent.as time pogesses, the fomation of the gaps can be obseved. Table I: Compaison of ate of gap fomation to collision elasticity. The units of τ ae in Peiods of the moon. Elasticity τ 1 Chi Squaed ± ± ± B. Embedded moons and esonances I also an simulations with a moon embedded in the ing paticles to obseve the effects of shepheding. The mass atio of Satun to the moon was vaied to see how the mass affected obseved esonances and the gap caused by shepheding. Collision elasticity was kept at 0.5 and the moon obited Satun at a distance of 150. The data was plotted using an image plot in Igo that gaphed the adial distibution of the paticles ove time. As can be seen in figues 6 and 7, the embedded moon successfully cleas paticles away fom its obit. Futhemoe, it was obseved that paticles at the same distance away fom Satun as the moon wee not pushed away and fomed a inglet in the cente of the gap. Table II shows the edge position of the gaps and thei coesponding esonance. When the mass of Satun was inceased fom 100,000 to 1,000,000 and the moon s mass was held at 100, the moon cleaed a smalle gap aound its obit and moe esonances can be seen, as shown in figue 7. The :1 and 1: esonances took longe to appea with the lage mass atio and wee not as lage. Since the embedded moon did not clea as lage a gap aound itself as it did when the mass atio was smalle, moe esonances could be seen in ing paticles close to the moon that did not have the oppotunity to fom befoe. Fo all mass atios, it was obseved that the esonances exteio to the moon fom the exteio edge of gaps while esonances inteio to the moon fom the inteio edge of the gaps. I then an the simulation again, this time keeping the mass of Satun at 1,000,000 and changing the mass of the moon to only 10. At this lage mass atio, the moon did not have enough Figue 7: The effects of an embedded moon at a distance of 150 with a mass of 100 obiting a planet of mass 1,000,000. A smalle inglet can be seen in the cente as well as gaps at 8 diffeent esonance positions. gavity to cause esonance gaps to fom ove the obseved time peiod. The effects of shepheding could still be seen and the moon did clea a small gap aound its obit. Duing all thee uns, when the moon shepheded paticles away fom its obit, paticles diectly in the moon s obit stayed foming a small inglet in the cente of the gap. The gap size fomed by the moon and the inglet width changed depending on the mass atio of Satun to the moon. These data ae Table II: Position of gaps fomed when the moon s mass is 100 and Satun s mass is 100,000 and 1,000,000 Resonance Gap edge : Satun s mass = 100,000 Gap edge : Satun s mass = 1,000,000 : : :3 13 5:4 19 4: :4 18 : : 38 36

5 5 Table III: Compaison of Mass atios to gap size and inglet width of embedded moons. Satun s Mass Moon s Mass Gap size Ringlet width 1 100, ,000, ,000, summaized in Table III. The smalle the mass atio between the moon and Satun, the lage the gap size and the lage the width of the inglet fomed in the gap. IV. CONCLUSION I was able to successfully ceate a compute pogam to simulate a planetay ing system. By vaying the collision elasticity I studied how collisions effected the speed at which a gap fomed in the paticles at a :1 esonance position. It was detemined that the moe enegy that is lost duing collisions, the quicke the gap will fom. By changing the mass atio of Satun to the moon, a total of eight diffeent esonances wee also obseved. Futhemoe, the moon was embedded in the ing paticles and the effects of shepheding wee succesfully simulated. The geate the mass atio between Satun and the moon, the smalle the gap fomed by shepheding. Also, when the moon shepheded paticles away fom its obit, some paticles emained in the cente of the gap, caught in the same obit as the moon and fomed a tiny inglet. This occuence is consistent with obsevations of the Encke Gap in Satun s A ing. The Encke gap is cleaed by the small moon Pan, howeve a small inglet emains in the cente of the gap [4]. When the effects of the :1 esonance cleaed out the gap in the ing paticles, paticles wee foced out of the gap towad the planet and the inteio edge of the gap was detemined by the esonance. This is consistent with obsevations of the Cassini division between Satun s A and B ing. The :1 esonance with Mimas contols the inteio edge of the Cassini division and thee ae a geate numbe of ing paticles at this edge [1]. All fou esonance positions inteio to the obit of the moon that wee obseved detemined the inteio edge of the gap causing paticles to collect at that edge. Howeve, esonant positions exteio to the obit of the moon detemined the exteio edge of gaps and caused paticles to collect on the exteio edge. I believe that the diffeence in the way the gaps ae fomed inteio to the moon as to those fomed exteio to the moon is caused by the shepheding effect. The shepheding effect of moons causes paticles that ae inteio to thei obit to be pushed futhe in and paticles exteio to thei obit to be pushed futhe out. Theefoe, when the gavitational tugs fom the moon build up ove time at the esonance positions, this causes paticles inteio to the moon to be slowed down and thei obit will become smalle causing them to fall into an obit close to the planet. Paticles exteio to the moon will be speed up by the gavitational foce fom the moon causing the size of thei obit to incease and they will move futhe away fom the planet. Theefoe, whethe it be esonance o shepheding, the attactive gavitational foce of the moon in effect pushes ing paticles away. Oveall, by simply including gavity fom Satun and the moon and collisions of paticles, the pogam was able to successfully simulate both the effects of esonance and shepheding in ing systems. Tests wee conducted on the effects of collision elasticity on the speed of gap fomation and it was found that the less elastic the collisions wee, the geate the ate at which the paticles evacuated the gap. Futhemoe, the effects of embedding moons into the ing paticles and changing the Satun to moon mass atio wee studied. I noticed that my simulation modeled obseved aspects of Satun s ing system like esonance gaps and shepheded gaps coectly. It was also found that the lage the mass atio, the smalle the gap fomed by shepheding and the smalle the inglet fomed in the gap. Finally, I was able to show that esonance detemined the exteio edge of gaps foming outside of the moon s obit while esonances detemined the inteio edge of gaps foming inside of the moon s obit Thus, the attactive foce of the moon actually pushes paticles away to fom gaps by esonance and shepheding. [1] E. D. Mine, R. R. Wessen, and G. N. Cuzzi, Planetay Ring Systems (Paxis Publishing, 007). [] E. Piazza Cassini-Huygens Mission to Satun and Titan Jet Populsion Laboatoy, Califonia Institute of Technology, , [3] C. D. Muay and S.F. Demott, Sola System Dynamics (Cambidge Univesity Pess,1999). [4] I. Pate and J. Lissaue, Planetay Sciences (Cambidge Univesity Pess,001). [5] A. M. Fidman and N. N. Gokavyi, Physics of Planetay Rings: Celestial Mechanics of Continuous Media (Spinge,1994).