Describing the Asteroids Motion - Report Example

Asteroids I.D. of the Asteroids Part 1. What concepts would you use to describe the asteroids motion? (10 points) Answer: Certain concepts of physics are critical in describing an asteroid’s motion. Kepler’s Laws are perhaps most important among them. According to Smith (2009), “We now know that Kepler’s laws are the consequence of bodies moving under the influence of gravity. This means that Kepler’s laws are universal. They not only describe the motion of the planets, but of satellites, comets, asteroids, stars and even galaxies.” Furthermore, Newton’s Laws of Motion and Newton’s Law of Gravitation help us in analyzing not only the asteroid’s motion around the Sun but also perturbations acting upon it (Bombelli 2013). For example, the massive planet Jupiter causes certain irregularities in the ideal elliptical orbit of asteroids due to its immense gravitational pull. 2. If the asteroid is moving at a constant speed in its orbit, does this also mean it is moving at a constant velocity? Explain your answer. (10 points) Answer: The asteroid has a constant speed but this may not imply that it has constant velocity as well. First of all, a body has velocity when it changes its position with respect to time in a certain direction. However, the asteroid revolves in a roughly elliptical path, not in a straight line. However, at any given instant, the asteroid does have a velocity component acting along the tangent to its elliptical orbit for that instant. This velocity component can be used for the purpose of calculations in theoretical physics. But since the asteroid is changing its direction constantly, this instantaneous velocity is also changing and the body is undergoing acceleration. So no constant velocity is possible. 3. Based on your answer above, is the asteroid accelerating? Explain. (10 points) Answer: It should be mentioned once again that even though the asteroid moves with a constant speed, it is moving along an elliptical path and it is never moving along a straight line n practical sense. This happens because the Sun pulls the asteroid toward itself by gravitational force. However, due to the component of its own inertia the asteroid tends to fly off along a tangential straight line to its orbit. But this is made impossible by the gravitational force constantly acting upon it. Now of force is acting on a body, it produces acceleration. In the asteroid’s case, this acceleration manifests as change of direction of the asteroid’s motion. In this way, speed remains constant and velocity remains changing. However, instantaneous velocity (that is, velocity over an infinitesimally small interval of time) can be assumed as constant for calculation purposes. 4. What force acts to hold this massive object in orbit around the Sun? What direction does this force act? (10 points) Answer: The force of gravitational pull exerted by the Sun keeps the asteroid moving along its orbit. At any arbitrary point on the elliptical orbit of the asteroid, the inertial motion of this asteroid develops a tendency of the body to fly off along a straight line into the space. However, Sun’s gravitational pull acts perpendicular to this straight line. This force, which drags the asteroid toward the center of the Sun, does not let the asteroid to fly off. Hence, the gravitational force of Sun acts in a direction toward the sun along a straight line. 5. This asteroid’s motion is affected by many different massive objects. Rank the following in order of increasing influence: the Sun, Mars, Jupiter, and other asteroids. Explain your answer clearly and specify the main determining factor for each body. (10 points) Answer: The most prominent effect is the gravitational pull of the Sun (1). Next, gravitational forces exerted by the Jupiter (2) cause perturbations in the asteroid’s motion. After Jupiter, Mars (3) exerts gravitational force on the asteroid. But other asteroids (4) may also exert significant gravitational force on our asteroid. However, if another asteroid (or group of asteroids) comes sufficiently near to our asteroid, then its gravitational force may become larger enough than that of Mars. 6. Assume this asteroid has the shape of a sphere. A rock sitting on the surface of the asteroid has a weight of 100 Newtons. What would the weight of this rock be if it were moved to a distance above the surface equal to the radius of the asteroid? It may be helpful to draw a picture to better visualize what is going on. How is the mass of the rock affected by its change in position? (15 points) Answer: Newton’s law of gravitation can be used to solve this numerical. Accordingly, the following mathematical expression can be used: GMm/r^2 = mg … (1) Let’s put G = Gravitational Constant M = mass of the asteroid m = mass of the rock in question r = distance between the rock and the asteroid (in this case, the radius of the asteroid) g = acceleration due to gravity Accordingly, mg = 100 N when the rock is sitting on the surface of the asteroid. Now, let gravitational acceleration in the new position (where the rock is raised at the distance of r from the asteroid’s surface) be a. Then, we have GMm/(2r)^2 = ma … (2) Using equations (1) and (2), we get mg/ma = (2r)^2/ r^2 Or 100/ma = 4 Or ma = 25 So the answer is 25 Newtons. 7. At some point in its orbit, our asteroid is moving with a speed of 20 kilometers per second. Determine the kinetic energy of the asteroid assuming it has a mass of 1000 kilograms. Be sure your answer is in the correct units of energy. (10 points) Answer: Although the asteroid does not have velocity but speed, at a given instant the asteroid has a certain velocity along the direction of the tangent to its orbit at that position. Now kinetic energy is given by the general mathematical expression (1/2) mv^2. In order to solve our problem, let’s put m = mass of the asteroid; and v = velocity of the asteroid. Therefore, kinetic energy or K.E. will be given by: K.E. = (1/2) 1000 X 20000^2 = 200000000000 So the answer is 2 X 1011 Joules or 2 X 108 Kilojoules. 8. If the distance from the Sun varied during our asteroids orbit, would its kinetic energy change? Would its potential energy change? Would the total energy (sum of kinetic and potential energy) of the asteroid change? (15 points) Answer: The total mechanical energy of the asteroid at any point of time is given by the equation: M.E. = P.E. + K.E. Where M.E. is the mechanical energy, P.E. is the potential energy, and K.E. is the kinetic energy. Now, according to Kepler’s Second Law, “A line joining a planet and the Sun sweeps out equal areas in equal intervals of time” (see Smith 2009). The same law can be applied to the case of our asteroid as well. So when the asteroid is near the Sun, it travels larger distance in a given time interval, say dt. On the other hand, when the asteroid is far away, it travels smaller distance in the same time dt. Consequently, the asteroid’s velocity and kinetic energy increase when it when it moves near the Sun. Its velocity and kinetic energy decrease when it moves away from the Sun. (For more details refer to Broholm 1997) According to the law of conservation of energy (COE), total mechanical energy must remain constant. So when the asteroid is far away from the Sun, its potential energy increases. And when it approaches closer to the Sun, its potential energy decreases. The total energy of the asteroid will remain constant. Part 2 1. As mentioned before, our asteroid is in the shape of a sphere and has a mass of 1000 kilograms. Determine the density (in grams per cubic centimeter) of this asteroid if its diameter is known to be 1.2 meters. Useful information: 1 kg = 1000 g, 1 m = 100 cm, volume of sphere = 4/3 πr3. Remember that the radius of a sphere is equal to half its diameter! Show all of your work. (20 points) Answer: According to the question, mass of the asteroid is 1000 kg, while its radius is 0.6 m. In C.G.S. units, the mass m = 10^6 gm and radius r = 60 cm. Now, density of a substance is given by density = mass / volume Volume of the asteroid is 4/3 πr3 Putting r = 60, and π = 3.141 (approx), we have volume = 904778.684 cc (approx) Therefore, density is given by (10^6)/ 904778.684 = 1.105 gm/cc (approx) 2. How does your calculated density (in grams per cubic centimeter) compare to the density of water? Would you expect this asteroid to float or sink in water based on your calculations? (15 points) Answer: In normal temperature and pressure, density of water is generally assumed to be around 1 gm/cc. The density of our asteroid is much higher, about 1.105 gm/cc. Therefore, the asteroid will sink in water. 3. One side of our asteroid is constantly illuminated by the Sun while the other side remains in the dark. Do you expect there to be a temperature difference between the light and dark sides? Explain why or why not. If the two sides are at different temperatures, how might heat transfer from one side to the other? Note that our asteroid does not have enough gravity to hold an atmosphere. (15 points) Answer: In almost every planetary climatic system, heat transmission occurs through fluids like gases inside the atmosphere of the planet. For example, in Earth transmission of heat can take place by convection currents, thanks to the atmosphere. However, if a planet or an asteroid has such an orientation that one of its two sides always faces the sun, then the side facing the sun will become extremely hot as we can see in the case of Mercury. On the other side, there will be a perpetual night and perpetual winter. In the case of our asteroid, somewhat similar situation will exist. The side facing the sun will have a very high temperature, and the side away from the sun will have a very low temperature. In the absence of an atmosphere, the only method of transmission of heat is conduction through the material of which the asteroid is built. Generally, asteroids have a rocky geology and metals like iron are less frequent in them. So, heat transmission through conduction in the ground of the asteroid will also be very limited. (Bertolli, Farinella, and Vokroutilicky 2003) 4. Occasionally an asteroid will break into fragments due to a collision. These fragments can leave the asteroid belt and some even make their way to Earth. Upon entering the Earths atmosphere, a fragment is heated to high temperature by frictional forces. What would happen to any water-ice contained within a fragment? Is this type of change considered a chemical change or a physical change? Explain. (10 points) Answer: The water-ice will rapidly vaporize into steam because the fragment will be subjected to immense heat as it will go through a vigorous and rapid frictional motion against the gases present in earth’s atmosphere. This type of change should be considered as a physical change. Although the water-ice will change from solid state to gaseous state very rapidly, it will undergo a liquid state too for at least a negligible fraction of second. Therefore, it will be a change only in the physical state of the water-ice and no chemical change will take place. 5. Asteroids are mainly composed of metals like iron and nonmetals like carbon. Briefly explain the differences between metals and nonmetals based on your knowledge of the periodic table. (10 points) Answer: Differences between metals and nonmetals (see The Periodic Table 2008) A metal generally possesses 1-3 electrons in the outer shell of its atom Prone to be oxidized Low electro-negativity Metallic oxides have basic characteristics A nonmetal generally possesses 4-8 electrons in the outer shell of its atom Good oxidizers Highly electronegative Nonmetallic oxides are generally 6. If the asteroid fragment contains carbon, it may burn when entering the Earths atmosphere. What is the most likely compound to result from this process? Which type of chemical bond would result from this process? Of the two broad classes of chemical reactions mentioned in this course, which type would this be? Be sure to fully explain all of your answers. (20 points) Answer: On rapid burning in sufficient amounts of oxygen as the asteroid fragment will enter the thicker layers of Earth’s atmosphere, carbon will undergo oxidation producing carbon dioxide gas. The carbon atom will form two covalent bonds. Each bond will hold one oxygen atom. Figure 1 In Figure 1, we can see a rough pictorial representation of covalent bonding among carbon and oxygen atoms giving rise to a CO2 molecule. The darker circle represents carbon atom while the light blue circles represent oxygen atoms. The small black circles are valence electrons of C-atom while the small white circles are the valence electrons of O-atoms. Note that the C-atom reaches stable configuration of 8 valence electrons by taking 2 atoms from each O-atom. Similarly, both O-atoms reach stable configuration of 8 valence electrons by sharing two electrons of the C-atom in their respective valence shells. The chemical reaction will be an oxidative process of carbon. Carbon will undergo oxidation since it will attach with two oxygen atoms. The whole process involves combustion of carbon; therefore it is an exothermic chemical reaction. (The Periodic Table 2008) Part 3 1. Due to friction with the Earths atmosphere, a large static electric charge could build up on our plummeting asteroid fragment. Would you expect our fragment to generate a magnetic field? Explain why or why not. (15 points) Answer: Theoretically at least, every moving electric charge generates magnetic field (College Physics 2013). So if static electric charge builds up on our plummeting asteroid fragment, then it will act as a charged body in motion inside the earth’s atmosphere and magnetic field. So the asteroid will give rise to its own magnetic field. However, this electromagnetic process will be very short lived since only within seconds the asteroid will burn out and completely disintegrate. 2. As the fragment falls through the atmosphere, it is heated and some of the material is vaporized. Explain how you could determine the composition of this hot vaporized material from the light it emits. (15 points) Answer: When a substance burns, it emits light. The emission spectrum of this light can help us in understanding the chemical composition of the burning substance. The chromatic distributions found the spectral analysis can detect the chemicals present. For example, if the burning part of the asteroid fragment predominantly emits blue light, then it can be deduced that it has a high percentage of metallic copper in it. On the other hand, if the emitted light rapidly changes from red to yellow to white, then one may deduce that the fragment contained iron. (College Physics 2013) 3. Some asteroid fragments are large enough to not completely burn up in the atmosphere and they end up on the surface of the Earth. It is possible for such a fragment to be radioactive. What is the chief cause of radioactivity? If you had a radiation detector that could measure the amount of radiation—but not the type of radiation—how could you determine which type of radiation was being emitted? (20 points) Answer: One of the chief reasons of radioactivity is that atoms of certain elements have unstable nuclei which attempt to attain stability by releasing some energy in the form of radiation. In the elements whose atoms have a large number of neutrons as compared to their atomic numbers (or numbers of electrons), radioactivity is very likely to take place. This is a common feature of elements that have very high neutron to proton ratios. In fact atoms with more than 83 protons in their large nuclei are unstable and show radioactivity. (The Periodic Table 2008) Determining the type of radiation depends on the source of radiation. If radiation is coming from an object such as smoke detector, it likely to be composed of alpha particles. However, cosmic rays might contain alpha and beta particles as well as gamma rays. So in order to determine the type of radiation, the source of radiation should be found out and studied. Furthermore, if very thick lead shields are the only way to block the radiation, then possibly gamma rays are being emitted. If maximum damage is caused in a living tissue due to exposure to radiation, then alpha particles have been emitted. 4. Conservation of energy tells us that energy cannot be created or destroyed. Clearly, energy is required for radioactivity to occur. Where is this energy coming from? (10 points) Answer: Nuclear binding energy keeps the constituents of the nucleus of an atom together in a coherent manner. But the protons are positively charged and they repel each other. When the number of protons in the nucleus is large enough, the repulsive electrostatic forces increase among them. Consequently, electric potential energy of the protons may increase so much so that the nuclear binding energy is no more able to keep them together. As a result, radioactivity takes place where the excess energy of the nucleus is radiated through the processes like emissions of alpha particles, beta particles, gamma rays, positron and high-energy protons. (College Physics 2013; Physics: Principles and Problems 2004) 5. How could the age of this fragment be determined? (10 points) Answer: If the fragment of the asteroid that has reached the surface of earth, then scientist can find its age by examining the rock with the help of radioactive dating methods. In case the rock contains even trace amounts of uranium, radioactive dating can immensely help. Uranium has 2 radioactive isotopes with half-lives that are long enough. Both these isotopes can decay to form different isotope of lead. “The amount of these uranium isotopes and their daughter nuclei are measured. From the ratios of these amounts, the number of half-lives since the rock was formed can be calculated.” (Physics: Principles and Problems 2004) 6. Asteroids can be classified into two broad groups based on their composition and location: carbon-rich asteroids dominate the outer part of the asteroid belt, whereas metal rich asteroids dominate the inner part of the belt. Analysis of the fragment we have discussed in this project reveals that it contains nearly equal amounts of metals and carbon. Can we conclude that the original asteroid had a similar composition? Form a hypothesis about the origin of this asteroid based on the available information. (20 points) Answer: Carbonaceous asteroids are located in the outer belt while metal-rich asteroids are in the inner belt. However, the fragment under examination has almost equal amounts of carbon and metals. This fact does not conclusively imply that the asteroid, as a whole, has a similar rock composition all over it. Maybe, it is just a coincidence that the given specimen has reached the Earth’s surface after its original composition changed significantly due to abrasion with the atmosphere. So it cannot be stated that from which part of the asteroid belt does our asteroid belong. However, if it really happens to have equal amounts of metals and carbon, then it probably formed in the middle of the asteroid belt when it had sufficient inertial speeds and the action of gravitational pull by bodies like Jupiter was not yet so profound. If perturbations were greater and frequent when the molten material of the asteroid cooled down, then the lighter nonmetal (that is carbon) would have been lost. If that is not the case, then the asteroid is not sufficiently in the inner regions of the inner asteroid belt. References Berlotti, B., Farinella, P., and Vokroutilicky, A. (2008). Physics of the Solar System, New York: Springer Bombelli, L. (2013). Newton: Motion and Gravity. Department of Physics and Astronomy, The University of Mississippi. Retrieved on 27th January 2014 from http://www.olemiss.edu/depts/physics_and_astronomy/ Broholm, C. (1997). Potential and Kinetic Energy in Orbit. Retrieved on 27th January 2014 from http://www.pha.jhu.edu/~broholm/124/node1.htm College Physics (2013). Houston: OpenStrax College Physics: Principles and Problems, (2004). New York: Glencoe/McGraw-Hill Smith, T. (2009). Kepler’s Laws Tutorial. Astronomy Department, University of Washington. Retrieved on 27th January 2014 from http://www.astro.washington.edu/users/smith/Astro150/Tutorials/Kepler/ The Periodic Table (2008). New York: Infobase Read More