Lab 10 - Gravitation

Section:__________ Group:__________ Name:____________________ Name:____________________ Name:____________________ Physics 183: Laboratory Experiment Gravitation Equipment: White Styrofoam board, 2 push pins, 2 string loops, ruler I. Theoretical Background a. Nicolaus Copernicus (1473 – 1543) In 1543 Copernicus published his model of the solar system in a book titled “De revolutionibus orbium coelestium”. The Copernican model was a heliocentric model in which all the known planets, including the Earth, orbited the Sun. This went against the prevailing idea that the Earth was the center of the solar system (i.e. Geocentric model). Figure 1 shows a diagram from Copernicus illustrating his model of the solar system. Figure 1: Copernican Model One defining characteristic of the Copernican model was the use of circular orbits. This model was fairly accurate in its predictions and had the advantage of being much simpler than previous Geocentric models. b. Johannes Kepler (1571 – 1630) Kepler inherited a massive amount of data on planetary locations from his mentor Tycho Brahe. Though a believer in the Copernican model, Kepler noticed small deviations between the predicted planetary locations from the model, and the observed locations in Tycho’s data, specifically with regards to the planet Mars. Kepler presented his own ideas concerning planetary motion in “Astronomia Nova” published in 1609. In this work Kepler introduces his first two laws of planetary motion. 1 Law I: The planets move in elliptical orbits with the sun at one focus. Law II: The speed of the planet changes at each moment such that the time between two positions is always proportional to the area swept out on the orbit between these positions. Figure 2 is a drawing by Kepler from “Astronomia Nova” analyzing geometric properties of ellipses. Figure 2: Analysis of Ellipse In 1619 Kepler published “Harmonice Mundi” in which he presented his third law of planetary motion. Law III: The ratio of the square of an object’s orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary. This is stated more concisely in mathematical notation as: 𝑇 ! ∝ 𝑎" Where T is the orbital period and a is the semi-major axis. With these three laws of planetary motion Kepler very precisely explained how the planets orbited the Sun, but not why they did. c. Isaac Newton (1643 – 1727) In 1687 Newton published “Philosophiae Naturalis Principia Mathematica” in which he proposed his law of universal gravitation. Newton said gravity was an always attractive force between massive objects with the force magnitude given by: 𝐹= 𝐺𝑚# 𝑚! 𝑟! Using this law of gravity and the innovative mathematical tools he developed Newton was able to derive Kepler’s three laws of planetary motion. Figure 3 shows an illustration from the Principia demonstrating the nature of orbits. 2 Figure 3: The nature of orbits Newton’s law of gravitation didn’t just describe the motion of the planets, it explained why they moved the way they did. Newton’s law of gravity became one of the most successful theories ever developed, explaining everything from the motion of the planets, the origin of the tides, to the motion of stars and galaxies. II. Laboratory Activities c. Kepler’s 1st Law Kepler’s 1st Law states that planets orbit the Sun in ellipses. An ellipse will have a long axis known as the major axis and a short axis known as the minor axis. There are two important fixed points on the major axis known as the foci. For elliptical planetary orbits the Sun is located at one focus point and there is nothing at the other. For any point on the ellipse, the sum of the distance to the two foci is a constant. This gives an easy way to draw an ellipse, which you will do in step 1. 1. Using the graph paper segment below, insert two push pins a distance of six blocks apart. Now use the small piece of circular string to loop around the pins and a sharp pencil to draw an ellipse. Label the major axis, the minor axis, and the foci points. 3 In Kepler’s law of planetary motion, the semi-major axis is mentioned as an important quantity. The semi-major axis is equal to the length of the major axis divided by two. 2. In the ellipse you just drew, what is the semi-major axis? ____________________ cm Another important quantity when analyzing ellipses is the eccentricity of the ellipse. The eccentricity of an ellipse is a measure of how much it deviates from being a circle. One way to define the eccentricity of an ellipse is the distance from a focus to the center of the ellipse divided by the length of the semi-major axis. The eccentricity is then a pure number that ranges from 0 to 1, with a perfect circle having an eccentricity of zero. 3. Calculate the eccentricity of the ellipse you drew in step 1: e = ______________________ When analyzing the orbits of planets around the Sun there are two special points, the perihelion, which is the point of closest approach to the Sun and the aphelion which is the point the planet is furthest away from the Sun. Table 1 gives important orbital parameters for planets around our Sun Object Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Halley’s Comet Perihelion (x 106 km) 46 107.48 147.09 206.62 740.52 1352.55 2741.30 4444.45 4434.99 87.8 Aphelion (x 106 km) 69.82 108.94 152.10 249.23 816.62 1514.50 3003.62 4545.67 7304.33 5270 Eccentricity Table 1: Orbital Parameters 4. Complete Table 1 by calculating the eccentricity of each object and writing it in the space provided. When Kepler was studying Tycho’s data and analyzing orbital information he specifically looked at the orbit of Mars. Here Kepler was lucky, if he had analyzed Venus for instance, he probably wouldn’t have noticed deviations from a circular orbit! 5. On the following piece of graph paper make three ellipses, one with an eccentricity of e = 0, one with an eccentricity of e = 0.5 and one with an eccentricity of e = 0.9. Label them appropriately. Put one foci point at the center of the page and use it for each ellipse. 4 5 d. Kepler’s 2nd Law Kepler’s 2nd law states that planets (or anything) orbiting around the Sun “sweep out equal areas in equal times”. This means that a line drawn from the center of the Sun to the planet will sweep out an area as it orbits. The area swept out in some unit of time (perhaps a day, a week, or a month) will always be the same. This law is a consequence of conservation of angular momentum. Figure 4 below shows the orbit of a planet moving around a star. Figure 4: A planetary orbit 6. Figure 4 shows the orbit of a planet around a star. Its location is shown at three different months. The planet is orbiting in a counterclockwise direction. Find and mark the location of the planet in July. To do this you can draw lines and calculate areas (by counting boxes?) 6 e. Kepler’s 3rd Law About 10 years after publishing his first two laws of planetary motion, Kepler was able to determine a relationship between the orbital periods of planets to their semi-major axis. By analyzing Table 2 you can quickly see that as the semi-major axis increases so does orbital period. However, the exact mathematical nature of this relationship is not obvious. Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Period (T) 0.241 0.615 1.000 1.881 11.863 29.447 84.017 164.791 Semimajor Axis (a) 0.387 0.723 1.000 1.524 5.203 9.537 19.189 30.069 Table 2: Planetary Orbital Data Kepler tried multiple relationships to see if he could discover a pattern. In the spirit of Kepler, let’s try a few relations and see if anything obvious appears. These are a very small sample of the infinite possible relations. Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune a/T a2/T a/T2 a3/T2 Table 3: Possible Relation between T and a 7. Which of the mathematical relations seems to be special? 7 ________________ f. Kepler-62 Kepler-62 is a star that is both smaller and cooler than our Sun. It is located about 980 lightyears (9.272 x 1018 m) from the Earth in the constellation Lyra. In 2013 it was discovered that this star has five planets orbiting it, two of which are in the habitable zone. The two planets in the habitable zone are Kepler-62e and Kepler-62f. It is believed based on observed data that Kepler-62f is a rocky planet. Table 4 below presents some of the orbital parameters of the Kepler-62 planetary system. Planet Mass Kepler-62b Kepler-62c Kepler-62d Kepler-62e Kepler-62f 2.1ME 0.1ME 5.5 ME 4.8 ME 2.8 ME Semimajor axis (AU) 0.0553 0.093 Orbital Period (days) 5.71493 18.16406 0.427 267.29 Radius 1.34RE 0.55RE 1.99RE 1.64RE 1.44RE Table 4: Kepler-62 Planetary System 8. Some of the data in Table 4 is missing, calculate and fill in the missing data. Using Newton’s Law of Gravity we can calculate what the surface gravity would be on the planets in the Kepler-62 system. This is important to know in order to determine if the planet can maintain an atmosphere. The two planets in the habitable zone are Kepler-62e and Kepler62f, calculate the surface gravity on these planets. 9. What is the acceleration due to gravity on the surface of Kepler-62e and Kepler-62f? Kepler-62e surface gravity: __________________ m/s2 Kepler-62f surface gravity: __________________ m/s2 8 10. The fastest spacecraft made so far was the Helios spacecraft launched in the mid-1970s which traveled at a speed of 250,000 km/h. How long would it take this spacecraft to travel from the Earth to the Kepler-62 system? ______________________ years g. A Satellite Orbit It is very difficult to put objects into Low Earth Orbit (LEO), not only are the technical and engineering challenges considerable, the energy required per kilogram is very large. Let’s calculate the required energy per kilogram to insert an object into LEO. (ME = 5.98 x 1024 kg, RE = 6380 km) Suppose we want to place 1 kg into a circular orbit 300 km above the Earth’s surface. 11. What speed, period and radial acceleration must the 1 kg mass have when it is in orbit? Speed: _________________ m/s Period: _________________ min Acceleration: _________________ m/s2 9 12. Calculate the total energy the 1 kg satellite has when it is in orbit. Energy: ____________________ J 13. Calculate the energy the 1 kg satellite has when it is sitting at rest on the Earth’s surface (assume a non-rotating Earth) Energy: _________________ J 14. Calculate the work that must be done on the 1 kg mass to place it into orbit. Work Required: _________________ J This is a large amount of energy that must be provided to each kilogram in order to reach orbit. For perspective, when gasoline is burned it releases on the order of 43 MJ/kg. This tells us that a rocket used to put satellites in orbit must consists mostly of fuel, not payload. Most rocket launch systems are more than 90% fuel by mass at takeoff. 10 15. How much additional work would have to be done to make the 1 kg satellite escape the Earth? (look at the energy calculated in Question 12) Energy Required: __________________ J h. Black Holes In lecture we derived an expression for the escape velocity from a massive body, we got the following: 𝑣=* 2𝐺𝑀 𝑟 If we look at an object located on the surface of the Earth we get an escape velocity of roughly 11,200 m/s. More massive objects tend to have higher escape velocities. In 1783 Rev. John Mitchell noted that if an object was massive enough (with an appropriately small radius) the escape velocity could exceed the speed of light and “all light emitted from such a body would be made to return toward it”. Mitchell became the first person to suggest the existence of what we now call a black hole. If we set the escape velocity equal to the speed of light, v = c we get…. 𝑐=* 2𝐺𝑀 𝑅$ Where RS is called the Schwarzschild radius, if we solve for RS we get… 𝑅$ = 2𝐺𝑀 𝑐! 11 16. Astrophysical theory suggest that a burned-out star wil collapse under its own gravity to form a black hole when its mass is at least three solar masses. If it does, what is the Schwarzschild radius? (MS = 1.99 x 1030 kg) Schwarzschild radius: _____________________ km 17. Though there is no known mechanism for this to happen, what would the Schwarzschild radius be if the Earth collapsed into a black hole? (ME = 5.98 x 1024 kg) Schwarzschild radius: _________________ cm 18. What would the density of the Earth be in question 17? Density: ________________ kg/m3 12