Assignment 1 Due Jan. 31, 2017 Show all work and turn in answers on separate pages, not on these pages. Circle your final answers for clarity. Be sure to show/explain all of your reasoning and that your work is neat. If I cannot read or understand what you did, you will lose points (potentially all of them). (1) Planetary Astronomy I showed in class using visualization software what the planets look like in the night sky. Over several nights nights (and, optionally if you are so inclined, multiple times on each night), try to find some of the following objects in the night sky and make some observations (record the time and map the location in the sky for each observation you make using the attached sky map). The objects you can look for after sunset/evening are: Moon, Mars, Venus, Uranus, and Neptune. The objects you can see late at night or very early in the morning/before sunrise are: Mercury, Jupiter, Saturn. How do to the observations you make relate to the ideas of Copernicus and Kepler? 1
Winter Night Sky North The BIG DIPPER www.liebacklookup.com East pollux castor GEMINI, the Twins CASSIOPEIA West CANIS MAJOR sirius betelgeuse ORION, the Hunter rigel South How to use this Sky Map Hold the map so that the direction you are facing is at the bottom of the map Hold the map up toward the sky Constellation names are in ALL CAPS, star names are in lower-case
(2) Planetary Formation: Accretion and Differentiation In this problem, we will calculate the temperature change caused by core formation during planetary accretion and differentiation for Mercury, Mars, Earth/Venus, and Vesta. (a) The gravitational potential energy, U(r), felt by a test particle of mass m on the surface of a spherically symmetric mass M(r) of radius r is given by: U(r) = GM(r)m r (1), where G is the universal gravitational constant (6.673 x 10-11 m 3 kg -1 s -2 ). Note that for the spherical mass: M(r) = (volume)(density) M(r) = 4 3 πr3 ρ (2). Derive an equation for the change in gravitational potential energy, du(r), caused by adding a thin shell of mass m shell to an existing sphere of radius r and mass M sphere. Show your work. Your result will express the incremental change in gravitational potential energy as a growing planet adds more material during accretion. HINT: Start with the formula: du(r) = GM sphere m shell r (3), and ask yourself what the expressions for M sphere and m shell would be. (b) Now we will let our planet grow. Integrate the expression you derive in part (a) from 0 to radius R to derive an expression for the gravitational potential energy after accretion of a uniform (undifferentiated) spherical planet of total radius R and density ρ. Show your work. The result is the gravitational potential energy of an undifferentiated planet. (c) Using the result from part (b), and assuming all of the gravitational potential energy is converted into heat, and ignoring the initial temperature of the unaccreted material, calculate the temperature after accretion of an undifferentiated (a) Mercury, (b) Mars, (c) Earth, and (d) Vesta. Look up their masses, radii, etc., and assume a constant specific heat capacity (C p ) of 1000 J kg -1 K -1. Assume that the heat or thermal energy (E T ) of an object of mass m at temperature T can be expressed as: E T = mcpt (4), 2
where m is the mass of the planet in question (note this is not necessarily the same m as in Equations 1, 2, and 3). (d) One could now do calculations similar to the ones you did in parts (a) and (b) to find the gravitational potential energy after accretion of a differentiated planet of mass M and radius R with a core of radius 0.5R that has twice the density of its mantle (i.e., ρ core = 2ρ mantle ). Note that in this calculation, we are ignoring the crust, i.e. this planet is simply a core with a mantle around it. After doing that work, you would get an expression that looks like this: U differentiated = 272 243 Gπ 2 ρ 2 R 5 (5) This is the gravitational potential energy of a differentiated planet. EXTRA CREDIT: Derive Equation (5). Subtract the equation you derived for the gravitational potential energy of the undifferentiated planet, i.e. your answer for part (b), from that of the differentiated planet, i.e. Equation (5), to obtain an expression for the change in gravitational potential energy due to differentiation. (e) Using your answer to part (d) and again applying Equation (4), estimate the increase in temperature caused by core formation in (a) Mercury, (b) Mars, (c) Earth, and (d) Vesta, assuming that the gravitational potential energy release is spread evenly throughout the planet. Again, look up the data you need, and assume C p is 1000 J kg -1 K - 1. Assuming these objects were solid after accretion, what effect would core formation have on each of these objects? 3
(3) Kepler s Laws Objects in elliptical orbits sweep out equal areas in equal times. This implies that the orbital speed of a planet around the sun is not uniform. It moves fastest at the point closest to the sun (known as the perihelion) and slowest at the point farthest away (known as aphelion). In this problem, we will calculate the difference in this speed using Pluto as an example. Pluto's orbit has an eccentricity e = 0.25. Its semi-major axis is 5.9 x 10 9 km. (a) Determine the distance (D aphelion ) between Pluto and the sun at aphelion. You should be able to determine this using just the semi-major axis and the eccentricity. (b) Determine the distance (D perihelion ) between Pluto and the sun at perihelion. Again, you should be able to determine this using just the semi-major axis and the eccentricity. (c) We now want to determine the ratio between Pluto's velocity at aphelion and perihelion: v aphelion /v perihelion. To do this you need to find the area swept out by Pluto's orbit. This can be approximately described as a triangle with: where D is the distance from the Sun, v is velocity, and t is time. Remember that Kepler s Second Law essentially states that planets sweep out equal areas in equal times. This means that the area swept out in some fixed time interval (Δt) is the same at perihelion as it is at aphelion. Therefore we can say: (1), 1 2 D v Δt = 1 perihelion perihelion 2 D v aphelion aphelionδt (2). Using Equation (2), derive an expression for v aphelion /v perihelion. (d) Given that Pluto's minimum orbital velocity is 3.7 km s -1, determine values for v aphelion and v perihelion. 4
(4) Scale of the Solar System Distance: A Scale Model Assume the length of a long corridor is: 202.4 meters. We will design a scale model of our solar system that we will place in this corridor. Let the Sun-Pluto distance (39.5294 AU) be equal to the length of the corridor. On this scale: a. What would be the diameter of the Sun in our model? b. What would be the diameters for each of the major planets (and Pluto) in our model? c. What would be the diameter of the Moon and its distance from Earth in our model? d. How far from the Sun would we place each of the major planets (and Pluto) in our model? Use the mean orbital distances from the Sun in your scale calculations. e. How far from the Sun would we place the nearest star (Alpha Centuri, which is located 4.3 light-years away) in our model? Time: Communications f. How many times can a photon travel the Earth s circumference in one second? g. The New Horizons spacecraft recently flew by Pluto. At 1200 UTC on the day of closest approach to Pluto, assume that ground controllers sent a command to New Horizons to take an image. Upon receiving the command, assume the spacecraft instantaneously acquires the image and transmits it back to Earth. Assuming Pluto was at its mean heliocentric distance during the New Horizons encounter, when would the transmitted image begin to be received at Earth? 5
(5) Properties of Objects in the Solar System On the attached pages, you are given tabular data (size, density) about solar system objects (planets, major moons, asteroids). a. Make a plot of object radius as a function of density. Label each of your data points so we know what object is which. [Hint: you may want to use a log-log plot rather than a linear plot. Extra credit: Explain why a linear plot not be such a good idea.]. On your graph, can you identify groupings of objects? Draw boundaries around these clusters of data points and explain what they might represent (e.g. in what way(s) are the objects included in each group similar; in what way(s) are the groups distinct from each other in terms of physical properties). b. Assume that the total mass of the major planets, plus Pluto and the asteroid Ceres, is distributed in a disk around the Sun (one that is 1 cm thick between 0.2 AU and 50 AU). Divide this disk solar system into 10 zones (one zone for each major planets, plus Pluto and Ceres), and compute the density within each zone. [Hint: You should draw a sketch of the solar system to figure out the size of each zone, e.g. the outer and inner radii of each part of the disk.] For example, the Earth zone would extend from the mean orbital distance of Earth halfway to Venus and halfway to Mars (i.e., from ~0.86 to ~1.25 AU). Distributing the Earth s mass into this zone yields a density of ~10 g/cm 3 ). [Hint: You will need to determine the volume of each of your zones in order to compute the density.]. c. Based on your calculations in (b), make a plot of density as a function of distance from the Sun. [Does this need to be a linear or a log plot?] Discuss trends in the density, noting especially zone(s) of substantially lower or higher density. d. A huge collection of comets (the Kuiper belt) is believed to extend from 35 to 50 AU from the Sun. The total population of these objects is estimated to be as high as 30,000 objects with diameters between 100 and 400 km. If these objects are taken into consideration in the density calculations, do they noticeably affect the calculated density distribution in (c)? Assume an average diameter of 200 km and an average density of 1 g/cm 3 for the comets in the Kuiper belt. 6