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# Topics in Glaciology EART 290

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This 4 page Class Notes was uploaded by Jillian Moore on Monday September 7, 2015. The Class Notes belongs to EART 290 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 46 views. For similar materials see /class/182199/eart-290-university-of-california-santa-cruz in Earth Sciences at University of California - Santa Cruz.

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Date Created: 09/07/15

ESS 298 Outer Solar System Computer Exercise Overview This exercise allows the students to follow the tidal evolution of an isolated satellite forwards or backwards in time and to investigate the circumstances under which tidal heating is likely to be important Each student will select a different satelliteplanet combination and write up a short summary min 2 pages max 5 of the results This will count 30 towards their final grade Ifyou are unfamiliar with LinuxUnix operating systems read Appendix 1 before continuing It will also help to have a copy of Week 1 s notes if you want to understand the theory underpinning this exercise Program This way to operate the program will be demonstrated in class Also take a look at the READMEtxt le Student Writeup What I want you to do is to pick a satelliteplanet pair and to examine the orbital evolution of the satellite You should show at least one orbital evolution scenario which gives the correct presentday value of the satellite s semimajor axis and eccentricity You should also analyze the results you obtain relating them to the thermal evolution of the satellite other observables and other nearby satellites Questions you may need to address in your writeup include but are not limited to the following 1 How large could the eccentricity have grown before the satellite encountered its neighbours and presumably smashed into them 2 As the orbit evolved is the satellite likely to have passed through any resonances with neighbouring satellites NB these satellites are also presumably moving outwards under the in uence of tides Remember that resonant encounters tend to pump up eccentricities and lead to additional tidal heating and that capture into resonance is only possible though not assured if the satellites are on converging trajectories Unfortunately neither of these effects can be easily modelled in our calculations 3 What is the effect of the initial temperature on the evolution of the satellite towards the present day 4 What evidence is there based on geological or impact cratering observations for the thermal evolution of the body Can you reconcile these observations with your results 5 What are the most dangerous simplifications in this model and how likely are they to affect the results In presenting your findings you should clearly state the parameters assumed and show the results obtained as well as addressing the above questions Appendix 1 Using LinuxUnix Linux uses a hierarchical directory structure e g h0menimm0srctidal To move to this directory type cd lhomenimmosrctidal To move up one level type cd To nd out which directory you are in type pwd To create a new directory called sh type mkdir sh To move a le called chips to the directory sh type mv chips sh NB if the directory sh does not exist this will rename sh to chips To list all the les in the directory you re in type ls or ls ls 1 gives more details on each le To list all the les starting with f type ls f Ifyou ve lost a le called sh type nd name sh p1int Linux les are of the form nametype for instance a fortran code is tidalf a text le is tidaltxt a postscript le is tidalps and so on To delete a led called chips type I39m chips to delete a directory sh type nndir sh to delete all les starting with f type rm f To edit the chips le using a text editor type dtpad chips To print the chips le type lpr Pp1intbillcr chips and to observe the printing queue type lpq Pp1intbilcr To halt a program temporarily type ctrlZ to put it in the background type bg and to bring it into the foreground type fg To halt a program permanently e g if it hangs type ctrlC To nd out which processes are operating type ps ef grep more hit the space bar to scroll down to kill a particular process type kill 9 ltpr0cess numbergt Appendix 2 Outline of theory We are going to consider a simple two body system consisting of a rotating primary and a satellite which is in synchronous rotation The two bodies will exert torques on each other which will change the primary s rotation rate and the satellite s eccentricity and semimaj or axis Angular momentum is conserved within the system but energy is not owing to dissipation Most of the theory outlined below is contained in Week 1 of the lectures Here we are just combining various aspects of it together Note that the theory below assumes bodies which are uniform which is obviously an approximation Notation 1 The three var1ables we are mterested 1n are pr1mary rotatlon rate 21 1n radlans sec the satelllte s semimajor axis a and eccentricity e The total gravitational effective mass is given by u Gm1m2 1 where m and Mg are the mass of the primary and satellite respectively and G is the gravitational constant Kepler s 3rd law may be written 613712 u where n is the mean motion of the satellite We are also going to assume that m1gtgtM2 this simpli es the algebra and keeps the results consistent with those in the lectures Energy Neglecting the rotational energy of the satellite the total energy of the system E is due to the rotational energy of the primary and the orbital energy of the satellite an is Evil CQf 2 where C1 is the polar moment of inertia of the primary Note that energy does not care about the eccentricity of the satellite Angular Momentum Again neglecting the rotational angular momentum of the satellite the total angular momentum of the system L which is conserved is Lm2y12a121 ez C1 21 3 Orbital Evolution To look at the orbital evolution of a system we need to consider the rates of change of energy and angular momentum From 3 we have d9 1 1 2 d9 6i ai ai Izmz lZ e e 612 C1 10 4 dt 6a dt 6e ch 691 ch 2 a dt 1 e dt dt From 2 we have d9 d9 diEaiE 6E 1 ml clgl 1 5 dt 6a dt 691 dt 2a2 dt dt We can rewrite equations 4 and 5 in a more compact form as follows d a d E 2612 261269 7 e 2H7 H 1 al ez 4 172 al eZQ 6 dt dt Kng mzey Q 0 mzeyC1 Here L0 refers to the orbital angular momentum and we have dropped the subscript from the primary rotation rate Q Note that dQdt depends only on dLgdt and not dEdt To complete the problem we need the total rate of energy dissipation dEdt due to tides on the primary and secondary and the rate of change of orbital angular momentum dLodt due to the tides on the primary We derived the tidal dissipation in the lectures the tidal torque takes a similar form The tidal torque on the primary gives rise to a rate of change of angular momentum as follows dL 3 k1 JszzRf dt EQI a6 215e2 7 Here k1 is the Love number for the primary k2 is the Love number for the satellite The total rate of energy dissipation due to tides in the primary and secondary is given by 2 5 12 2 5 12 di Ll M Q 215e2 k4 M u 7e2 8 dt 2 Q1 a6 a 2 Q2 16 6 32 Given initial values for a e and 9 equations 7 and 8 are used to determine dLodt and dEdt which can be substituted into equation 6 to nd dadt alealt and dQdt and thus update a e and 9 Appendix 3 Time varying Q Recall that the rate at which things happen depend on the Love number kg and the dissipation factor Q The satellite Love number can be speci ed but usually we just calculate it using the expression given in the lectures k 2L2 194 1 u 2 pgR Here u is rigidity note that upreviously denoted something different p is satellite density g is gravity and R is satellite radius With the radius and mass speci ed p and g can be calculated directly 2 The basic model for dissipation in a uniform viscoelastic body is given by the following equation 2 Q Q0 M 1 n r where Q0 is a constant factor determined by the overall satellite properties we will assume Q0l n is the mean motion of the satellite and ris the Maxwell time de ned as r77u where u is the rigidity and 77 is the viscosity of the material in this case ice As one might expect Q is maximized where the forcing frequency 71 is comparable to 11quot and is much larger when the forcing frequency is far away from 11quot Remember that a large Q means a small dissipation Applying equation 1 is complicated by the fact that the viscosity of ice 7 is strongly temperature dependent 77T 770 eXP7T To 2

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