PTYS 554 – Evolution of Planetary Surfaces Homework #3 – Assigned 10/29, due 11/12 1) If roughly 10 major basins (>900 km in diameter) formed on the Moon during late heavy bombardment. How many craters greater than 1km in size formed during this period? [Hint: Use the slopes of the power laws shown in class and solve for the constants, don’t forget the slope changes value at certain crater diameters.] Show that the gravitationally enhanced cross-section of the Moon is given by: ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛+2_21vvrescapeMoonMoonπ If all these objects approach the Earth/Moon at v=15 km-1, how many hit the Earth during the same late heavy bombardment period? The actual impact speeds are given by: € vi2= v2+ vesc2 What speeds do they hit each body at? How much extra impact energy did the Earth receive compared to the Moon? When the velocity is size-independent like this, does most of the delivered impact energy come from the rarer large impacts or the more numerous small ones (and does the same hold true for craters less than 1km in size)? Assume Lampson scaling to link energy and crater diameter. 2) Impacts on Mars and Venus: The dense atmosphere on Venus effectively screens out many impacting bodies. A projectile at velocity v experiences a ram-pressure from the atmosphere, if this pressure exceeds the strength of the material then the projectile fragments. The ram pressure is the product of the amount of atmospheric material scooped up by the projectile and the momentum of that material per unit area. Show that it equals: atmosphereramvPρ2≈ where ρatmosphere is the atmospheric density. The hydrostatic equation gives the pressure variation with height as: ( )ATMHzSgkTHwhereePzPµ=≈− where Ps is the surface pressure, k is Boltzmann's constant and µATM is the molecular weight of the atmospheric particles. Convert the atmospheric pressure equation to density. What is the atmospheric surface density and scale height for Venus, Earth and Mars. Use temperatures of 750, 270 and 200K and surface pressures of 100, 1, 0.01 bars respectively.If a projectile barely makes it to the surface without fragmentation on Mars, at what altitude will it break up if it had hit Venus. Assume the lowest possible impact speed in both cases. One way to recognize meteors is by their fusion crust i.e. the exterior if the rock is melted during its passage through the atmosphere. How hot do the gases at the leading edge of the meteor get, just before impact into the martian surface? (assume they’re adiabatically compressed). Is this hot enough to melt rock? How deep does this thermal disturbance penetrate into the meteorite? 3) Maxwell crater ejecta model. Maxwell’s Z-model describes the shape of ejecta streamlines in polar coordinates: Show that the maximum depth of excavation is: (where D is the transient crater diameter, ~ 80% of the final diameter) Show that the fraction of material from the transient crater cavity that was excavated is: (assume a hemispheric shape for the transient cavity). Show also that (z-2) is the tangent of the angle (above horizontal) at which the ejecta leaves the crater. Laboratory impacts and explosion testing show this angle to be close to 45 degrees. Given that, what value of z is appropriate for impacts? What then are the numeric values for the excavated fraction and maximum depth of excavation relative to the diameter? Meteor crater is 1200m across. From what range of depths did its ejecta come from? What is the volume of the ejecta blanket (ignoring the density change the impact caused)? Why is the number you calculate more than the actual ejecta blanket volume? The deepest stratigraphic unit that meteor crater excavated was the Coconino sandstone, where in the eject blanket did this material end up? In the previous homework we calculated the lunar crust to be ~50km thick. What sized craters on the lunar surface may have excavated mantle material? € r = ro1− cosθ( )1 z−2( )€ D2z − 2( )z −1( )1−zz−2€ z − 2z + 14) Crater shapes. Simple craters tend to be parabolas with h/D ~ 0.2. Ejecta blankets decrease in thickness according to the distance from the crater center cubed. If volume is conserved in the crater creation process then derive the height of the rim (hr) relative to the depth of the crater (h). hr/h is observed to be ~0.2 Was our assumption of volume conservation a good one? 5) Here, we’ll plot up some real crater counting data from the Lunae Planum region of Mars. The crater sizes and positions are available as a comma-separated-value file (Latitude, Longitude, Diameter) on the class website and come from the new catalog of Robbins and Hynek (JGR, 2012). The area represented is bounded by 290-300E and 0-20N. How many square kilometers is this? (Mars Radius = 3396.19km) Make an incremental plot of these data in the standard form that we discussed in class. Use the information below to date this surface as best as you can. Do the craters in different size ranges tell a consistent story? What’s a plausible explanation for any differences? Expected crater counts at different sizes for surfaces of different ages (isochrons) from Hartmann (Icarus, 2005). Diameter bins are spaced a factor of sqrt(2) apart. The diameter in the left column is the lower-diameter edge of that size bin i.e. first size bin is 1km to 1.41km, second bin is 1.41km to 2km etc… Anything younger than 1Gyr can be dated by linearly rescaling the 1Gyr column. The numbers in parenthesis are the exponents i.e. 2.0(-1) is 0.2