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 satellite-planet combination and write up a short summary (min. 2 pages, max. 5) of the results. This will count 30% towards their final grade.
If you are unfamiliar with Linux/Unix 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 README.txt file.
Student Writeup
What I want you to do is to pick a satellite-planet pair and to examine the orbital evolution of the satellite. You should show at least one orbital evolution scenario which gives the correct present-day value of the satellite’s semi-major 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 influence 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 Linux/Unix
Linux uses a hierarchical directory structure e.g. /home/nimmo/src/tidal/
To move to this directory, type cd /home/nimmo/src/tidal
To move up one level, type cd ..
To find out which directory you are in, type pwd
To create a new directory called fish, type mkdir fish
To move a file called chips to the directory fish type mv chips fish (NB if the directory fish does not exist, this will rename fish to chips)
To list all the files in the directory you’re in, type ls or ls * ; ls –l gives more details on each file.
To list all the files starting with “f” type ls f*
If you’ve lost a file called fish, type find . –name fish -print
Linux files are of the form name.type; for instance, a fortran code is tidal.f, a text file is tidal.txt, a postscript file is tidal.ps and so on.
To delete a filed called chips, type rm chips; to delete a directory fish, type rmdir fish; to delete all files starting with “f” type rm f*
To edit the chips file using a text editor type dtpad chips
To print the chips file type lpr –Pprint_bill_cr chips, and to observe the printing queue type lpq –Pprint_bill_cr
To halt a program temporarily type ctrl+Z; 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 ctrl+C
To find 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 <process number>
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 semi-major 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
The three variables we are interested in are primary rotation rate, 1, in radians sec-1, the satellite’s semi-major axis a and eccentricity e.
The total gravitational effective mass is given by
(1)
where m1 and m2 are the mass of the primary and satellite, respectively, and G is the gravitational constant. Kepler’s 3rd law may be written a3n2= where n is the mean motion of the satellite. We are also going to assume that m1>m2 (this simplifies 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
(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
(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
(4)
From (2) we have
(5)
We can rewrite equations (4) and (5) in a more compact form as follows
(6)
Here Lo refers to the orbital angular momentum and we have dropped the subscript from the primary rotation rate . Note that d/dt depends only on dL0/dt and not dE/dt.
To complete the problem, we need the total rate of energy dissipation dE/dt due to tides on the primary and secondary, and the rate of change of orbital angular momentum dLo/dt 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
(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
(8)
Given initial values for a, eand, equations (7) and (8) are used to determine dLo/dt and dE/dt, which can be substituted into equation (6) to find da/dt, de/dt and d/dt and thus update a, e and .
Appendix 3 – Time-varying Q
Recall that the rate at which things happen depend on the Love number k2 and the dissipation factor Q. The satellite Love number can be specified, but usually we just calculate it using the expression given in the lectures:
Here is rigidity (note that previously denoted something different!), is satellite density, g is gravity and R is satellite radius. With the radius and mass specified, and g can be calculated directly.
The basic model for dissipation in a uniform viscoelastic body is given by the following equation:
(1)
where Q0 is a constant factor determined by the overall satellite properties (we will assume Q0=1), n is the mean motion of the satellite and is the Maxwell time, defined as , where is the rigidity and is the viscosity of the material (in this case, ice). As one might expect, Q is maximized where the forcing frequency (n) is comparable to , and is much larger when the forcing frequency is far away from . Remember that a large Q means a small dissipation!
Applying equation (1) is complicated by the fact that the viscosity of ice is strongly temperature dependent:
(2)
Here 0 is a reference viscosity at reference temperature T0 and is a constant telling us how strongly temperature-dependent the viscosity is.
There are two sources of heat: tidal dissipation (which depends on Q) and radiogenic heat production (which is independent of Q). Since Q is also temperature-dependent, it is obvious that interesting feedbacks can occur. We assume that the heat is removed from the satellite by convection in the ice shell. This convective heat flux is given by
(3)
where k is thermal conductivity, is thermal diffusivity, g is gravity, is density, is thermal expansivity. Note that the convective heat flux is also viscosity- (and thus temperature-) dependent – more feedbacks.
So the thermal evolution of the satellite is as follows:
(4)
where Tiis the internal temperature of the satellite of mass ms, radius Rs and specific heat capacity Cp, HRis the total radiogenic heat production and Ht is the total tidal heat production.
We use equation (4) to update the internal temperature, then equation (2) to update the internal viscosity, then equation (1) to update Q and thus calculate the tidal dissipation for the next timestep.