Astronomy 310 Assignment 4

The following assignment covers Chapters Eight to Ten. Please answer the following questions and submit your work using the assignment link. Show all your work in a clear and concise fashion, and include each step you take to reach the answer. The way in which you obtain your answer is as important as the final answer, itself, and marks will be given for showing all the relevant steps. Please refer to the appendix in the textbook (pages 392–395) to obtain any needed planetary values, and to Table 2-2 on page 41 for any needed constants. Submit your answers to your tutor for grading and feedback using the assignment link. You may submit your assignment (a) typed as a word-processed document, or (b) hand-written and scanned as a PDF. If you opt to handwrite your assignment, be sure to write legibly. Always keep a backup copy of your assignment. Note: Assignments must be submitted as .doc, .docx, or .pdf files. Each question has its point value marked in bold, for a total of 100 marks on this assignment. 1. [12 marks total] Problem 14 in Chapter Ten of the text (page 235): Make a diagram showing the pressure as a function of depth inside the outer 3000 km of the Moon, Mars, Earth and Jupiter. Contrast the depths at which the pressure reaches the maximum laboratory pressure of roughly 40 kbar. 2. [13 marks total] (a) [1] Suppose on the Moon, a meteoroid creates a crater in the shape of a hemisphere with a radius of 1 km. Calculate the mass of material excavated, assuming a density of 1920 kg=m3 for regolith. (b) [1] When the meteoroid makes the crater, the excavated material is raised vertically by a distance equal to the radius of the crater, 1 km. Calculate the potential energy required to lift the material. (c) [3] The meteoroid that created the crater was spherical and made of rock, and it had an impact velocity of 5 km=s. Assuming that energy was conserved in the collision, what was the size of the meteoroid? (d) [5] Comets have a density of one third the value of rocky meteoroids, but can travel three times as fast. If a comet the same size of the meteoroid had impacted and caused the crater, with a density of 1000 kg=m3 and an impact velocity of 15 km=s, what size crater would have been made? Would it be larger or smaller than that made by a rocky meteoroid? (e) [3] Consider the material excavated by the impactor, the ejecta. Ejecta blankets are found on both Mercury and the Moon, but on Mercury, the ejecta is generally much closer to the crater compared to the Moon. Explain why this is. 1 3. [9 marks total] Both the continents and the ocean have plates, made of Earth’s crustal material, that float on the Earth’s mantle. Oceanic plates are covered by the ocean to its full depth and float on the Earth’s mantle. The continental plates float on the mantle as well, as seen in the figure, which shows a depth profile below sea level for each type of plate. Determine the depth of the continental plate (dcp) using the data given, assuming the area under consideration is the same in both cases and the mass of each cross-section is also the same. cp = 2700 kg=m3 do = 4 km m = 3300 kg=m3 dop = 7:5 km op = 2900 kg=m3 o = 1000 kg=m3 4. [13 marks total] (a) [7] Problem 11 in Chapter Ten of the text (page 315): Smith and others (1979b) discuss a 20-km-thick crust of sulfur and silicates on Io, with perhaps a 1-km-deep layer of liquid sulfur involved in the eruptions. Johnson and others (1979) used the geometry of the erupting plumes to calculate an average resurfacing rate all over Io of 3 10??4 to 0:1 cm=y. The radius of Io is 1820 km. If a layer roughly 10 km deep is assumed to cycle through the eruptions, how long does it take for all this material to go through the cycle of eruption, burial, remelting and reeruption? Has most of the material been through the cycle less than once, once, or more than once? (b) [6] Problem 12 in Chapter Ten of the text (page 315): Volcanologist A. Rittmann (1962) discussed nine terrestrial eruptions with eruptive rates of about 0:03 to 3; 000 km3=y (for individual volcanoes). Taking these as lower and upper limits on the eruption rates of Io volcanoes and assuming that there are eight eruptions on the average at all times, calculate the cycle time for the upper 10 km of Io’s crust. Compare this with the result in [part (a)]. 2 5. [8 marks total] (a) [4] If we want to greatly simplify the composition of Earth, we could suppose that Earth was only composed of uncompressed iron, with a density of 8000 kg=m3 and uncompressed rock, with a density of 3000 kg=m3. Calculate the percentage of Earth’s volume made of rock and iron. (b) [4] Natural diamonds can form under high pressure conditions in the Earth’s mantle at depths around 165 km on average. Diamonds require a minimum pressure to form and cannot form at depths that are shallower. What is the diameter of the smallest body where the pressure at its center could allow the formation of diamonds? Assume a density of 3000 kg=m3 for both this body and the density at 165 km inside Earth. 6. [12 marks total] Problem 13 in Chapter Nine of the text (page 269): Consider a typical Apollo asteroid of diameter 2 km approaching the Moon on an orbit with aphelion in the outer asteroid belt and perihelion at Earth’s orbit. (a) [3] [With] what velocity does it approach the Earth-Moon system? (b) [3] [With] what velocity does it hit the Moon? (c) [4] What energy, in joules, is expended during the collision? (d) [2] What diameter crater does it make? 7. [16 marks total] Instead of using a constant density as representative of a planet’s interior, we are going to consider a variable density profile. Assume that the density (r) at some distance inside the planet is given by (r) = 0 R2 r2 ; where R is the radius of the planet, r is the distance from the center outward, and 0 is the density at the surface (where r = R). (a) [2] The mass of the planet can be found by M(r) = R r 0 4r2(r)dr: Find the mass M(r) contained inside a radius r, using integration. (b) [2] Find the mean density of the planet in terms of 0. (c) [2] Find the acceleration due to gravity, g, at a radius r. (d) [4] Using the hydrostatic equation, find the general solution for the pressure P(r) inside the planet. Note that you can assume that the pressure is zero at the surface. (e) [3] Assume the planet contains a small solid core of radius r = 0:08R. Write the expression for the pressure at the top of this solid core. (f) [3] At this same radius (r = 0:08R), write the expression for the pressure of a planet with an interior that has a constant density, like that worked out in the textbook, and the same mean density found in (b). Are pressures higher in the constant density or variable density planet? 3 8. [8 marks total] We know the terrestrial planets formed by aggregation of debris from the solar nebula. We want to calculate the maximum size of object that can form by aggregation before self-gravity causes it to pull itself into a round shape. Our analysis is assisted by considering Earth’s mountains. It is clear from Earth’s geology that of its many mountains, none are taller than 10 km high. This indicates that the strength of rock at the center of the base of a mountain cannot support a structure higher than 10 km. The pressure at the middle of the base of a mountain is equal to the weight of a column of rock (with unit cross sectional area) as high as the mountain. If a mountain were taller than 10 km, it would slump or begin to be overtaken by self-gravity. We want to use this fact to determine the maximum size of an object that can form without being overtaken by self-gravity. Assume you have two cubes of rock (density of 3000 kg=m3) of equal mass (M) and size (length d) that are face to face. They are so large that the only force holding them together is pressure caused by gravitational attraction, but not so large that they flow into a spherical shape due to self-gravity. The force between the two cubes is F = GM2 d2 : Determine the maximum size of rectangular body that could be made from the two cubes. 9. [9 marks total] Problem 13 in Chapter Ten of the text (page 315). Note: Section (a) should read “Mars is at perihelion,” not “prehelion.” (a) [5] Mars is at perihelion. A polar ice field with dirty ice having an albedo of 40% is located so that the sun is 10 above the horizon. The emissivity of the ice-soil mixture is 0.5, and half the sunlight gets through the Martian atmosphere at the slant angle mentioned. Calculate the equilibrium temperature of the Martian surface. (b) [3] A landslide occurs, exposing a cliff face with an 80 slope, which is fully exposed to the Sun. Calculate the equilibrium temperature of the ice-soil mixture of the cliff face. (c) [1] Comment on the future erosion of the cliff area. 4