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Assignment 1 (Part 1): The stress state of deformed lithospheres, the thermal evolution of planets, surface processes, and planetary geomorphology [50 points total; Due as hardcopy in class on Tues., Jan. 30, 2024] Please answer the following questions on a separate set of pages (not on these pages). Show all your work for full credit. Be sure to use the appropriate significant figures and to be careful with units. Neatly and legibly present your solutions in the order given, clearly mark your final answers.
1. First, we consider flexural loading of the lithosphere of Venus. [10 points total] Figure 1 shows a topographic profile across part of a corona (a circular feature with a trench surrounding it) on Venus. We are going to interpret the trench and rise as a flexural feature due to loading. Figure 1. Bold line shows the smoothed topographic profile to interpret. a) Mark on Figure 1 the approximate distance over which flexure is deforming the lithosphere. [1]
b) Assuming that this distance is related to the flexural parameter (a), use the
expression and methodology shown in class to determine the elastic thickness (Te)
of the lithosphere on Venus. You may assume that g = 9 m s-2, n = 0.3, the
density contrast is 500 kg m-3, and E =100 GPa. [3] c) How does the elastic thickness on Venus compare with that of the continents on Earth? Why might this be a surprising result? [2] d) The base of the elastic layer is determined by a temperature of about 1000 K and the surface temperature of Venus is 700 K. What is the thermal gradient on Venus? [1] e) Thermal gradients on Earth are about 25 K km-1. What does this result imply about the relative rates at which the Earth and Venus are cooling down? [2]
v01172024-jmh f) How might you explain this difference in cooling rates? [1]
2. Next, we consider volcanism on Io. [20 points total] The velocity u of magma traveling upwards through a dike of width w is given by:
w^2 g Dr
u = (1)
h
where g is gravity, Dr is the density contrast between magma and the surrounding
rock, and h is the viscosity.
a) Examine the effect of each variable in turn and explain why this equation makes physical sense. [4] b) If the total height of the dike is d , write down an expression for the time taken for a packet of magma to get from the bottom to the top of the dike. [1] c) Also write down an expression for how long it takes the material in the dike to cool by conduction. [1] d) By comparing the expressions for the cooling time and the transit time, derive an expression for the minimum width of a dike which will allow magma to ascend all the way to the surface. [3]
e) For Io, assume that d = 20 km, Dr = 100 kg m-3, g = 1.8 m s-2, k = 10-6^ m^2 s-1^ and
h = 10^3 Pa s. Using this information, what is the minimum dike width? [1]
f) If the total horizontal length of the dikes on Io is L , and they all have a constant width w , write down an expression for the magma discharge rate (in m^3 s-1) from these dikes in terms of u , L and w. [2] g) The magmatic resurfacing rate on Io is about 1 cm yr-1. If the radius of Io is 1800 km, what is the corresponding magma discharge rate (in m^3 s-1)? [2] h) If dikes on Io are 1-m wide, use the information given above to determine what the total length of dikes L has to be in order to produce the observed resurfacing rate. [4] i) How easy (or not) would it be to spot these dikes from a spacecraft? Explain. [2]
3. Next we consider compressive stresses on Mercury. [10 points total]
v01172024-jmh Figure 3. From Burr et al., 2013 (Fluvial features on Titan: Insights from morphology and modeling: GSA Bulletin, v. 125; no. 3/4; p. 299-321; doi: 10.1130/B30612.1). A PDF of this paper is available on Blackboard. (b) As shown at https://nssdc.gsfc.nasa.gov/planetary/titan_images.html, the largest of the clasts, likely pieces of rock or frozen water or hydrocarbons, in Figure 3D is ~15 cm in size, with others ~5 cm in size. What sediment transport mode (e.g., wash load, suspended load, or bed load) is most likely to carry clasts like this (with these sizes and shapes)? [2] (c) Figure 4 shows the relationship between sediment size and the minimum flow depth needed to transport that sediment. Note that on Titan the working liquid is not water, like in Earth rivers, but instead is probably liquid methane or
v01172024-jmh nitrogen! You can use this diagram to determine the flow depth ( h ) required to move sediment of a certain grain size ( d ) by a particular mode of sediment transport in the stream. Based on your answers to (a) and (b), use Figure 4 to determine the range of flow depths implied by the clasts seen in Figure 3D. [1] Figure 4. From Burr et al., 2006 (Sediment transport by liquid surficial flow: Application to Titan: Icarus, v. 181; p. 235-242; doi: doi:10.1016/j.icarus.2005.11.012). A PDF of this paper is available on Blackboard. (d) The Burr et al. (2016; 2013) papers discuss the technical details of fluvial transport for Earth and their application to Titan. The flow depth can be used to estimate the volumetric flux of fluid in the channel, a quantity that hydrologists call discharge. Discharge ( Q ) is given by the Manning equation: (3), where w is channel flow width, h is channel flow depth, is the gravitational acceleration, S is the slope of the channel, and Cf is a friction coefficient. (Note: the square-root portion of Equation 3 is the flow velocity, and the product of w and h is the cross-sectional area of the channel.) Cf can be approximated as: (4), where ks is approximately 2.5 times the grain size ( d ).
