Assignment 1. Due Feb. 11, 2019

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1 Assignment 1 Due Feb. 11, 2019 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) Kepler s Laws Objective By reproducing ellipses via the "string-and-pencil method," you will draw ellipses and determine their eccentricities; by measuring the orbits of five of Jupiter's moons, you will test Kepler's third law; and by using characteristics of Pluto's orbit, you will confirm Kepler's second law. Procedure Kepler's three laws are simply a mathematical way of describing motions of objects that orbit a large central mass, such as the planets which orbit around the Sun or the moons which orbit around Jupiter. This lab explores each of Kepler's three laws. Please write down all work, calculations, and answers on separate sheets. Kepler's First Law: This law states that orbiting objects travel in elliptical paths with the central mass at one focus. Here you will get acquainted with ellipses by sketching one yourself. (a) Get two tacks and a piece of string. On your paper, place the two tacks a small distance apart, pinning down the ends of the string. Be sure to leave some slack in the string. Using the string as a guide (i.e., place the pencil inside the string loop and pull the loop taut), draw an ellipse. Use the Figure below as a guide. 1

2 (b) Now measure and write down the distance between the foci AND the length of the major axis of the ellipse. Make your measurements in cm. (c) Divide the distance between the foci (2r) by the length of the major axis (2a). This quantity is known as the eccentricity (e): e = 2r 2a = r a (1). Note that the quantity a is referred to as the semi-major axis length. What is the eccentricity of the ellipse you drew? (d) What familiar shape is an ellipse with e=0.0? Kepler's Third Law (yes, this is out of order!): This law states that the periods (P) and semi-major axes (a) of two bodies (1 and 2) orbiting a common object are related by: 2 P body1 a body1 2 a body 2 = Pbody 2 (2). 3 3 Here you will verify this law for the five largest moons orbiting Jupiter: Almathea, Io, Europa, Callisto, and Ganymede. This link has an animation that may be useful: 2

3 a) Create a table to hold the values of orbital period (P), semi-major axis (a), and P 2 /a 3 for all five moons. b) Determine the semi-major axes of the moons (in km) and fill in your table. c) Determine the orbital periods of the moons (in hours) and fill in your table. d) Calculate and examine the tabulated values for P 2 /a 3. Does Kepler s third law seem hold? Your numbers may not be exactly as you expected, so comment on any potential sources of error in your data/measurements or any other factors not being taken into account. Kepler's Second Law: This law states that 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). Here you will calculate the difference in this speed using Pluto as an example. You can assume that Pluto s orbit has an eccentricity e = 0.25 and a semi-major axis length a = 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 a and e. Give your answer in km. (b) Determine the distance (D perihelion ) between Pluto and the sun at perihelion. Again, you should be able to determine this using just a and e. Give your answer in km. (c) Determine the ratio between Pluto s orbital velocities at aphelion and perihelion, i.e. 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: Area = 1 Dvt (3), 2 where D is the distance from Pluto to the Sun, v is orbital velocity, and t is time. Remember that Kepler s Second Law states that planets sweep out equal areas in equal 3

4 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 2 D perihelionv perihelion Δt = 1 2 D aphelionv aphelion Δt (4). 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. Give your answers in km s -1. 4

5 (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 small test particle of mass m resting on the surface of a large, 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 m 3 kg -1 s -2 ). Note that for the large, 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. Assume density is constant. Show your work. Your resulting equation will express the incremental change in gravitational potential energy as a growing planet adds more material during accretion. HINT: Start with: du(r) = GM spherem shell r (3), and ask yourself what the expressions for M sphere and m shell would be. Some substitution and algebraic manipulation will be needed from there. Your answer should be in terms of radius and density. (b) Now we will let our planet grow. Integrate the equation you derive in part (a) over radius, r, from 0 to R to derive an expression for U undifferentiated. This is the gravitational potential energy after accretion of a uniform (undifferentiated) spherical planet of total radius R and (constant) average density ρ. Show your work. Remember that the result you seek is an equation for 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 (pre-accretion) temperature of the material, calculate the temperature after accretion of an undifferentiated (a) Mercury, (b) Mars, (c) Earth, and (d) Vesta. Look up their masses, radii, average densities, etc., and assume a 5

6 constant specific heat capacity (C p ) of 1,000 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 = M planet CpT (4), where M planet is the mass of the planet in question. (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 such a calculation, we would ignore 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 = Gπ 2 ρ 2 R 5 (5) This is the gravitational potential energy of a differentiated planet, where ρ is the (constant) average density. 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. Look up all the data you need, and assume C p is 1,000 J kg -1 K -1. Assuming these objects were solid after accretion, what effect would core formation (i.e. differentiation) have on each of these objects? 6

7 (3) Scale of the Solar System Distance: A Scale Model Assume the length of a long corridor is: meters. We will design a scale model of our solar system that we will place in this corridor. Let the Sun-Pluto distance ( 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 flew by the Kuiper belt object Ultima Thule on January 1, Ultima Thule was at a distance of 43.4 AU from the Sun when New Horizons made its closest approach. Ground controllers sent a command to New Horizons to take an image of Ultima Thule at closest approach. Assume the command was sent at 1200 UTC (i.e. at 12 pm Coordinated Universal Time, i.e. 12 pm GMT) on the day of closest approach to Ultima Thule. Also assume that, upon receiving the command, the spacecraft instantaneously acquired the image and transmitted it back to Earth. When (time and date) would you expect the transmitted image to begin to be received back on Earth? 7

8 (4) 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 planet, 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 uniformly 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 relative density. d. A huge collection of comets and other icy objects (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 objects in the Kuiper belt. 8

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