Astronomy 310 Assignment 5

The following assignment covers Chapter Eleven. 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 85 marks on this assignment. 1. [10 marks total] Problem 13 in Chapter Eleven of the text (page 347): (a) [8] Calculate thermal escape times for H, He, and H2O on Mars with exosphere temperatures of 210 K and 1000 K. Comment on the volatile history on Mars if transient events ever heated its exosphere to 1,000 K for 1 My. (b) [2] Why is the calculation inadequate to fully explain the escape of Martian volatiles? 2. [10 marks total] (a) [4] Assuming a pure N2 atmosphere and a temperature of 300 K, calculate the ratio of the pressure at the top of the Yukon’s Mount Logan, at a height of 6 km, relative to the pressure at sea level. (b) [2] What would the pressure ratio have been on ancient Earth when it had a pure CO2 atmosphere? (c) [4] Is the pressure at the top of Mount Logan lower or higher on a hot day versus a cold day? Explain your reasoning. Assume the sea-level pressure is constant, no matter what the temperature is. 3. [8 marks total] Assume that a planet can have an atmosphere if the escape speed of the planet is 6 times larger than the thermal speed of the molecules in the atmosphere (also known as the root-mean-square molecular velocity). Suppose that a hypothetical object having the same mass and radius as Mercury, and an albedo of 0.1, orbits the Sun at just the right location for this condition to be met. Assume that its atmosphere is made of carbon dioxide (CO2). What is the radius of this object’s orbit around the Sun? 4. [10 marks total] Here we’re going to investigate the effect of a thick atmosphere on impactors by considering Jupiter, a planet with a thick atmosphere. (a) [2] The total mass of the atmosphere per unit area in kg/m2 can be calculated from the atmospheric pressure and surface gravity. This is called the column density of the atmosphere and is represented by the symbol . It is the mass of a column of atmosphere with an area of 1. Find the total mass of the atmosphere per unit area for Jupiter, assuming Jupiter has an atmospheric pressure of about 100 kPa and a surface gravity of 25 m/s2. (b) [1] A meteoroid with a radius of 1 km enters Jupiter’s atmosphere and travels downwards, perpendicular to the surface of the planet. What is the total mass of the atmosphere the meteoroid encounters if it hits the surface of Jupiter? 1 (c) [3] For atmospheres that are thick, meteoroids can break up if the mass of the atmosphere is the same as the mass of the meteoroid, itself. Write a general expression for the radius at which the meteoroid will break up. For the case of Jupiter, calculate the meteoroid radius at which break-up occurs if the density of the meteoroid is 2500 kg/m3. (d) [2] If the meteoroid came at it at an angle, rather than perpendicular to the surface, what would be the effect on the break-up radius? (e) [2] If the meteoroid was made of ice rather than rock, would the break-up radius be bigger or smaller? Why? 5. [7 marks total] Consider a parcel of dry air in the Earth’s atmosphere. Show that if you replace some portion of the air molecules (80 % N2, 20 % O2) by an equivalent number of water molecules, the parcel of air becomes lighter and rises. In light of this, explain how clouds form.1 6. [14 marks total] In “Mathematical Notes on Atmospheric Structure” on page 322 of the textbook, the author derives a pressure equation for the atmosphere. Here we will explore that equation. (a) [2] Derive the equation dP = ??MHPg kT dz by combining the hydrostatic equation with the ideal gas law. State any assumptions made. (b) [5] Rearranging the equation from part (a), we can write dP P = ??MHg kT dz Integrate both sides, assuming a constant gravity. Note that you should integrate from the surface to an arbitrary height above the surface. Take the surface pressure to be P0. (c) [2] Your answer in (b) should contain the quantity MHg kT . What is this quantity and what does it represent, physically? (d) [4] Venus and Mars both have atmospheric scale heights larger than that of Earth. For each planet, explain why. Include any relevant calculations to support your answer. (e) [1] Why might Mars have had a bigger atmospheric scale height 4 billion years ago? 7. [16 marks total] In reality, when we calculate a pressure profile for the atmosphere, the acceleration due to gravity is not a constant. Gravitational acceleration also changes depending on the distance from the planet’s surface. (a) [1] Write a physical equation showing how gravitational acceleration changes with respect to the distance z from a planet’s surface. (b) [5] Repeat the integration of both sides of the equation given in Problem 6 (b). This time do not assume constant gravity. Be sure to integrate from z = R at the surface this time to satisfy the boundary conditions for g in the system. (c) [2] Verify that the equation found in part (b) gives both the expected z value at the surface and the expected surface pressure. (d) [1] What is the pressure for very large values of z? (e) [2] Calculate the percentage change in g from the Earth’s surface to an altitude of 100 km. 1 Source: Adapted from de Pater, I., & Lissauer, J. J. (2001). Question 4.12E (p. 134). In Planetary Sciences. Cambridge: The Press Syndicate of the University of Cambridge 2 (f) [4] Estimate the error in Earth’s atmospheric pressure at 100 km calculated by the equation for atmospheric structure if a constant value for g is assumed. Assume a surface temperature of 280 K and a surface pressure of 1 105 Pa. Assume we are dealing with air molecules (80 % N2, 20 % O2). (g) [1] Does it seem reasonable to assume a constant value for g when calculating a pressure profile for the atmosphere? Explain. 8. [10 marks total] (a) [1] Write an expression for the density of the atmosphere in terms of the scale height H and the distance from the surface z. (b) [4] Use your answer in part (a) to derive an expression for the mass of a column of atmosphere of surface area 1m2, extending from the ground to a great height above the surface. (c) [3] The heat capacity Cp tells us how much energy it takes to heat up 1 kg of the atmosphere by 1 K. [This means it has units of J kg??1 K??1.] Considering only solar heating effects, write an expression for the time it takes to heat up a column of Earth’s atmosphere by 1 K. (d) [2] Calculate the time it takes for a column of Earth’s atmosphere to heat up by 15 degrees during the day. Assume the heat capacity for Earth is Cp = 300 J kg??1 K??1 and the surface pressure is 1 105 Pa. 3