Astronomy 310 Assignment 2

The following assignment covers Chapters Four to Five. 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 65 marks on this assignment. 1. [6 marks total] Problem 18 in Chapter Four of the text (page 97): Two optically thick infrared stars are at the same distance. Star A has peak radiation at 2:0 m [2.0 micro meters], and B at 4:0 m. Star A is 16 times as bright as B. What physical conclusions can you draw about these stars if they both radiate as blackbodies? Note: If a material is optically thick, very little light will pass through because the material is opaque. [See pages 323 to 325.] All blackbodies are optically thick. 2. [5 marks total] One statement of the virial theorem is as follows: For a gravitational bound system in equilibrium, twice the kinetic energy will equal the absolute value of the potential energy. Show that this is true for a planet in a stable orbit about a star. 3. [13 marks total] (a) [9] A typical diffuse cloud of atomic hydrogen has a temperature of 50K, and approximately 5108 hydrogen atoms per cubic meter. Assuming the cloud is completely composed of hydrogen, calculate the minimum mass (in solar masses) necessary to cause the cloud to spontaneously collapse. This is called the Jeans mass. Compare this to the estimated mass of diffuse hydrogen clouds of 1 to 100 solar masses. Are diffuse hydrogen clouds stable against collapse? (b) [4] Repeat the calculation of the Jeans mass for the dense core of a giant molecular cloud. Typical dense cores have approximately 1014 hydrogen atoms per cubic meter and temperatures of 150K. Again, assume the cloud is entirely composed of hydrogen atoms. Compare this to the range in masses for these objects of 10 to 1000 solar masses. Are these objects stable against collapse? 1 4. [8 marks total] (a) [5] For an object starting from rest, falling with a uniform acceleration a, the distance travelled d, in a time t, is given by d = 1 2 a t2: Consider a cloud of radius R that starts to collapse. Using the expression above, show that the time for a point on the surface to travel a distance R is given by r 3 2G ; where is the density of the cloud and G is the universal gravitational constant. You may assume the acceleration is constant. (b) [3] What you have derived is called the free-fall time. It is the time it takes for the cloud to collapse to a single point, assuming there are no outward forces. Interestingly, this expression does not depend on the size of the cloud. A cloud with a larger radius will also have more mass, which leads to a larger acceleration. Calculate the free-fall time for the dense core of a giant molecular cloud, and express your answer in years. As in question 3(b), assume the core of a giant molecular cloud has 1014 hydrogen atoms per cubic meter. 5. [6 marks total] Problem 22 in Chapter Four of the text (page 97): The condensation in Problem 21 [a condensation of gas and dust with Jupiter’s mass] is acted on by the central star, which has 1 MJ and is 5.2 AU away. Estimate the density that the condensation must have to keep it from being torn apart by tidal forces from the Sun. (Hint: the condensation would have to be outside Roche’s limit). It may be helpful to look at the solution for Problem 21, which has been worked out in the Study Guide. 6. [7 marks total] Uranium 238 decays into lead with a half-life of approximately 4.5 billion years. The equation for the number of lead atoms after a given time t is given by N(t) = N0 e??kt; where N(t) is the number of Uranium atoms at time t, N0 is the original number of Uranium atoms, and k is the time constant. See the Additional Notes from Unit 2, Section 5 for further discussion of the equation. (a) [3] Determine the time constant for Uranium turning into lead. (b) [3] A rock from the lunar highlands contains 55% of its original amount of Uranium 238, with the remaining 45% having decayed into lead. How old is the rock? (c) [1] A rock from the lunar maria contains 60% of its original Uranium 238 (with the remaining 40% having decayed into lead). Is this rock younger or older than the rock from part (b)? 7. [6 marks total] Problem 7 in Chapter Five of the text (page 125): (a) [3] You are traveling through space and come to a star of normal solarlike composition but with planets composed of refractory silicate minerals rich in aluminum, titanium, and calcium, and containing no water or ice. What can you conclude about the formation conditions? (b) [3] Compare these planets with our own Moon and comment on the formation conditions of the Moon. 8. [7 marks total] (a) [4] Show that the Safronov “rule of thumb” is exact for a planetary body with a density of 1790 kg=m3. Another way to state this is to show that vesc = 1 10??3 R for a planetesimal with this density, where vesc is the escape velocity in m=s, and R is the radius of the planet in m. (b) [3] What is the escape velocity of a 1000-km-diameter planetesimal if its density is 3000 kg=m3? 2 9. [7 marks total] If a planetesimal with velocity v is sweeping through a nebular cloud of smaller planetesimals of fixed size, it can be shown that the time t to grow to a radius R is given by t = 4R v p n ; where n is the density of the nebular cloud and p is the density of the planetesimal, itself. [See Problems 14 and 15 on page 125.] (a) [4] If the density of the nebular cloud is 10??7 kg=m3, estimate the timescale to accrete a body 1000 km across in the solar nebula at a distance of 1 AU from the centre of the nebula. Assume a reasonable p. (b) [3] If the accretion process is only 1 % efficient (only 1 in 100 collisions results in a mass gain), estimate the new timescale.