PTYS 554 – Evolution of Planetary Surfaces Homework #1 – Assigned 9/10, due 9/24 1) Geotherms and satellite heating. If Io’s surface heat flux is 3 W m-2 then how much heat is produced per kilogram in the interior? Compare this with what a typical piece of solar system rock (chondrite) produces via radioactive decay i.e. 4x10-12 W kg-1. If Io’s bulk composition is chondritic then what fraction of Io’s heat comes from radioactivity rather than tides? If Europa has liquid water 4km below the surface and the average surface temperature is 110K then what is Europa’s heat flux? How much radiogenic heat is produced in the rocky portion of Europa via radioactive decay? What fraction of Europa’s heat comes from radioactivity rather than tides? If there were no tidal heating on Europa the how thick would the ice-shell be? How thick would the ice shell be on Ganymede if the rocky portion of that body produces radiogenic heat at the chondritic rate and the surface temperature is similar to Europa? (Europa’s H2O layer is ~150km thick and the radius of Ganymede’s rocky interior is about 68% of the body.) 2) Moments of inertia. A lot can be discovered from a planet from its moment of inertia. Moment of inertia depends on the geometry of the object sphere vs empty shell vs point etc… but in general is given by k M R2, where M is the mass, R is the radius and k is a constant. In a differentiated planet (radius Rp with density ρc in a core of radius Rc and density ρm in the mantle surrounding the core) the moment of inertia is: € Idifferentiated=251+ c x51+ c x3" # $ % & ' MPRP2 ( )PcmmcRRxandcwhere =−=ρρρ. Titan’s mean density is 1880 kg m-3, assume that it’s made up only of different phases of water (~1000 kg m-3) and rock (~3300 kg m-3) and that it’s fully differentiated. What moment of inertia factor (k) do you expect Titan to have? Cassini tracking data published recently has shown this value to be 0.34. Compare this number to what you expected from the above calculation. What do we learn about Titan from this comparison?3) Isostasy and gravity anomalies. On Venus plate tectonics is absent. Down-welling flows in Earth’s mantle are usually associated with subduction whereas on Venus it’s thought to cause shortening of the crust. Think of a linear strip of the lithosphere that has a width wo. It gets compressed and reduced in width to w. i.e. the compression factor is Cf = wo/w. This compression builds mountains that are supported by Airy isostasy i.e. they float like icebergs in the Venusian mantle. Show that the mountain height is given by ( )1−−=fmcmLCThρρρ where the mantle and crust densities are ρm (~3300 kg m-3) ρc (~2750 kg m-3) respectively and TL is the thickness of the crust. How tall do these mountains get when crustal rocks get compressed by a factor of two? Assume a representative venusian crustal thickness of 70km The very large lunar south pole Aitken basin is currently about 8km deep and has no major free-air gravity anomaly associated with it. What does that tell us about how the basin is supported? If this impact originally excavated all the way through the crustal material (density 2800 kg m-3) to the mantle (3300 kg m-3) then how thick is the lunar crust? Why does doing a Bouguer correction over a compensated feature result in a large anomaly? Northern Europe is still rebounding from the last ice age. What free-air and Bouguer anomalies would you expect to see over this region. Which would be larger? If the signs of these anomalies were both reversed then what plausible geological scenario could you be looking at?4) Practicum… Pull the latest gravity map for Mars (produced by Mars Reconnaissance orbiter) from the PDS website. Go to: http://geo.pds.nasa.gov/missions/mro/gravity.htm Click on ‘RSDMAP - Radio Science Digital Maps’ and then ‘data’ and then ‘rsdmap’. This latest version is complete up until degree and order 95 (the data file is JGMRO_110B2_ANOM_095.IMG with label file JGMRO_110B2_ANOM_095.LBL). This is the free-air anomaly in milli-gals i.e. the gravity has been corrected down to the martian geoid. This file is evenly spaced in longitude and latitude with 360 samples by 180 lines (i.e. 1x1 degree). The top left corner of the image is at -180E 90N. I’ve posted a equivalently spaced file of martian topography on the class-website for your computational convenience. What is the real resolution of these gravity data? Do a Bouguer correction to the free-air gravity map (careful of your units!). Generate a nice figure of the two anomaly maps and topography. What major (planetary-scale) feature shows up in the martian Bouguer map that doesn’t appear in the free-air map? What do you infer about this feature? We’ll check out some specific features. Take a free-air profile over Olympus Mons (227E, 18N) and compare to an equivalent Bouguer profile. How does the magnitude of the anomaly compare? What does that mean for the support of Olympus Mons. Compare equivalent profiles over the Hellas impact basin (70E, 40S) and Utopia basin (110E, 45N). How do these features differ in their compensation? What explains the different gravity anomalies in each case? As we discussed in class, Utopia has a great deal of fill (let’s say it’s all basalt). Estimate the thickness of the fill from the free-air anomaly. Is the giant canyon, Valles Marinaris (285E, 8S), in isostatic equilibrium? Why should we be cautious of these results? Optional extra for the motivated…. Convert the Bouguer anomaly map to crustal thickness by assuming isostatic equilibrium and constant density everywhere. Compare profiles and profiles with figures 1 & 2 in Zuber et al. (Science, 287, 1788, 2000). [Topography data are available on the class website, for this question you can ignore geoid undulations if you like and just assume topography is instead relative to the