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UNIT 3 UNIT 3 THE PLANETARY ORBITS Goals After mastery of this unit, you should: a. be able to use the small angle equation. b. have an understanding of the gravitational force, including the inverse square law. c. have an understanding of the general structure of the solar system. d. know the general history of ideas of the structure of the solar system. e. know Kepler’s laws, Newton’s laws, and Newton’s version of Kepler’s laws. ➤ Text References 1.2; 2.2; 3.1 through 3.4; 5.1 through 5.5; A.1 through A.2; A.4; A.6 See the online Text Guide to DOSC to find the links to the listing by objectives. ➤ Mastery will be evaluated in two Parts: Part I will be a 15-question multiple-choice evaluation with 13 correct responses required. Part II will be performing the laboratory “Orbits and Positions of the Planets.” Part I will be based upon the following: ➤ Objectives You should be able to recognize: +1. the difference between linear measure and angular measure; 2. that a circle can be divided into 360 degrees, each degree into 60 minutes of arc, and each minute into 60 seconds of arc, and that the symbols (°, ', '') are used for degrees, minutes, and seconds, respectively; 3. the result of multiplying and dividing numbers written in powers-of-ten notation; 4. and use the small angle equation (Optional Basic Equation I); +5. the following about significant figures: (a) the meaning of the term significant figures (b) the correct significant figures in a simple calculation +6. from a list of the planets (or any subset of planets) the correct order of increasing or decreasing distance from the Sun; 3-1 +7. the symbols commonly used to denote the Sun and the planets; +8. from a list a planet’s approximate: (a) distance from the Sun in AU (b) radius relative to the Earth’s radius 9. a description of: (a) Ptolemaic model and problems with that model (b) Occam’s razor 10. the names of the five key figures in the Copernican Revolution and the major contributions of each; +11. the definition of an ellipse and a description of an ellipse’s eccentricity. Given a drawing of an ellipse, you should be able to label the semi-major and semi-minor axes; +12. by choosing an ellipse with the largest or smallest eccentricity, when given two or more ellipses with equal semi-major axes and unequal semi-minor axes; 13. from either words or a sketch, among an ellipse, a parabola, and a hyperbola and under what conditions of launch from the Earth, which of these three would be the resulting orbit; +14. that an ellipse, a parabola, and a hyperbola are conic sections, and why the term conic section is used; +15. the meaning of the following dealing with planetary position: (a) opposition and conjunction (b) synodic and sidereal period (c) perihelion and aphelion (d) inferior and superior planets 16. a description of Kepler’s three laws and to associate each with its number. Also recognize the name often given to the third law and its equation form; 17. a description and the approximate value of the Astronomical Unit and the proper abbreviation for the Astronomical Unit; 18. the following for Galileo Galilei: (a) made basic investigations into physical motions (b) was among the first to use a telescope for astronomical observations 19. a statement, and associate a number with, Newton’s three laws of motion, examples of simple applications, and a statement of Newton’s law of gravitation; 20. a comparison of Kepler’s empirical approach to Newton’s deductive approach; +21. the following about Newton’s modification of Kepler’s Laws: (a) a description of the concept of center of mass for a two-body system (b) that, using his three laws of motion and the gravitational force law, Newton could deduce modified versions of Kepler’s three laws (c) that Newton modified Kepler’s first law, putting the center of mass of the Sunplanet system at the focus of the ellipse and allowed parabolic and hyperbolic orbits (d) that the second law of Kepler was unchanged by Newton 3-2 THE PLANETARY ORBITS UNIT 3 (e) that the third law of Kepler was expanded by Newton to include the sum of the masses of the two gravitating bodies and that a proper choice of units eliminates the constants and the appropriate choice of units for the solar system and for the Earth-Moon system (f) the equation form of the third law as modified by Newton, and be able to use it with some simple numbers 22. The evidence that the heliocentric theory is correct, i.e., the Earth orbits the Sun. ➤ Streaming Videos — Unit 3, Prelab Unit 3 — Meeks videos “The Motions of Attracting Bodies” and “Planetary Motion and Kepler’s Laws” NOTES–If there are no notes, all the information is in the relevant sections of the text. Obj. 1. Linear measure is the straight-line distance between two points; it is commonly measured in meters or kilometers. Another example of a linear measure is the length of an object. Angular measure is a measure of the difference in direction between two points as seen from a specified third point. Angular measure is commonly specified in degrees, minutes, and seconds of arc. Obj. 2. 60 seconds of arc (60'') = 1 minute of arc (1'); 60 minutes of arc (60') = one degree (1°) 60' = 21,600', 360°. ______ 3600" = 1,296,000'' 1 circle = 360° = 360° × ___ 1° 1° Obj. 3. Multiplying and dividing powers of ten is relatively simple if we follow five rules. Rule 1. Process Pose problem Example (6 × 105) × (9 × 10− 2) __________________ Collect all numbers together and all powers of ten terms together. 6×9× _____ 4 4 × 102 (105 × 10− 2) __________ 102 −2 (10 × 10 ) __________ 5 (13.5) × 2. Multiply and divide the numbers. 3. If powers of ten are multiplied, add exponents. 10 10 (13.5) × _____ 13.5 × ___ 102 102 4. If powers of ten are divided, subtract exponents. (numerator exponent minus denominator exponent). 10 = 13.5 × 103 – 2 = 13.5 × 10 13.5 × ___ 2 10 5. Use the powers of ten rules to put the first number into a form between 1 and 10. 5−2 2 10 3 3 (13.5) × 10 = (1.35 × 10) × 10 = 1.35 × 102 Obj. 4. The small angle equation is discussed in Section 3.3 of DOSC (the online text). For the form we will use, the angular size must be expressed in seconds of arc, and the diameter and distance both must be in the same units. Whenever d is much less than D (we write this as d << D) the results are the arc S which is nearly d in Fig. SG 3-1. For larger angles a measures S which is larger than d. This difference can be corrected. Figure SG 3-1 S is the arc length while d is the straight line. 3-3 For this objective, you will also need the skills of rearranging equations discussed in Unit 1. In other words, with powers of ten notation, the equations can be written in three forms: d α " = 2.06 × 105 × __ Equation (1) D α" × D d = __________ Equation (2) 2.06 × 105 d D = 2.06 × 105 × ___ Equation (3) α" are all the same, each being solved for a different quantity. To use the small angle equation, it is necessary to know, or be given, two of the three quantities (α ", d, and D) that appear in the equation. If we know d and D and want α ", Equation (1) is the useful form. If we know α " and D and want d, Equation (2) is the most useful form. If we know α " and d and want D, Equation (3) is the most useful form. The sample questions for this objective give further examples. In each case, draw a figure similar to SGI Fig. 3-1 to help visualize what is desired. Obj. 5. Physical measurements can never be exact. Usually the last significant figure is uncertain by 1 or more; this is often expressed by giving the uncertainty. Thus if a wooden block has a length of 3.52 ± 0.02 cm (read as 3.52 plus or minus 0.02 cm), it probably is at least 3.52 − 0.02 = 3.50 cm long and probably no longer than 3.52 + 0.02 = 3.54 cm. The ± 0.02 is termed the uncertainty. If no uncertainty is given, all physical quantities in this material are assumed to be good to ±0.5 in the last significant figure. That is, if I say the angular size of the Moon is 0°.50, I mean 0°.50° ± 0°.05. The use of significant figures will be of primary importance in multiplication and division. The number 3.1 has two significant figures. The number 0.0034 has two significant figures. The number 345 has three significant figures. The number 2.5 × 107 cm, has two significant figures; while 2.50 has three significant figures. Now, some examples using area = length × width: (a) (3.06 × 103 cm) × (1.2 × 102 cm) = 3.7 × 105 cm2 (2 figures) (b) (3.05 × 103 cm) × (4.2 × 102 cm) = 1.3 × 106 cm2 (2 figures) (c) (0.0012 m) × (0.0122 m) = 1.5 × 10–5 m2 –2 (d) (0.0421 m) × (1.034 m) = 4.35 × 10 m 2 (2 figures) 2 (3 figures) 2 The cm is read as square centimeters and the m as square meters. Both denote an area measurement. In cases (a)–(c), since only two figures are known for one of the lengths, keep only two significant figures in the final answer. In case (d), three and four figures are known for the lengths, so the answer is known to only three figures. Taking example (b), we can justify this rule as follows: 3.05 × 103 is probably between (3.045 3 2 2 and 3.055) × 10 cm, and 4.2 × 10 cm is probably between (4.15 and 4.25) × 10 cm. The extremes occur when we multiply 3.045 by 4.15 or 3.055 by 4.25). The former gives 1.26 × 106 cm; the latter gives 1.29 × 106 cm. Thus the 1.3 × 106 cm2 [which means (1.30 ± 0.05) × 106 cm2] includes the probable answer. 3-4 THE PLANETARY ORBITS UNIT 3 Division is treated in the same fashion. For example: 2 7.9 cm = 2.1, ________ 225 cm = 1.9 × 102 cm, ________ 3.71 cm 1.2 cm and 8.52 cm = 2.07 ________ 4.12 cm However, if you are asked to find 1/2 of 3.71, this is 1.86 not 2, as the 1/2 is assumed to be known perfectly. For all evaluation questions dealing with significant figures, the number of significant figures left is the same as the least number given for any physical quantities (that is, excluding numerical factors such as 2, which are assumed to be known perfectly). Note from the above examples that the ratios of numbers measured in the same units have no units. The correct significant figures on a calculator are obtained by rounding to the number of figures the calculator produces using the given rules. For example: If you have three significant figures and the number calculated is 1.273 give the answer 1.27 but if you have three significant figures and the number calculated is 1.278, give the answer 1.28. Round off — do not just drop extra numbers. Obj. 6-8. Questions will be based on the following table: Planet Symbol Distance from Sun (AU) Radius of Planet in Earth Radii Mercury C 2/5 2/5 Venus D 3/4 1 Earth O 1 1 Mars E 3/2 1/2 Jupiter F 5 11 Saturn G 10 9 Uranus or N 20 4 Neptune I 30 4 In 2006 the IAU (International Astronomical Union) defined the word “planet.” 1. The orbit must be circular enough not to cross the orbit of another planet. 2. A planet must also have enough mass to become nearly spherical. 3. And a planet must be able to sweep other small astronomical objects out of its orbital path. Using this definition, Pluto is not a planet. In 2008 objects satisfying 2 & 3 and usually outside Neptune’s orbit were named Plutoids. Don’t forget that planet radius _______________ planet diameter _____________ = Earth radius Earth diameter Obj. 10. Copernicus’ proposal of a heliocentric theory provided the first explanation of retrograde motion by relative motion of the Earth and planet. Kepler used Brahe’s observational data to get his empirical laws. Obj. 11 and 12. There are a number of ways of defining ellipses. The simplest is to take two points (each point is called a focus) and define an ellipse as the set of all points such that the sum of the distance from each focus is a constant. This is shown in Fig. SG 3-2. Two numbers are sufficient to describe an ellipse. Often we use the semi-major axis and the semi-minor axis. These are shown in Fig. SG 3-3. They are also the quantities labeled “a” 3-5 Figure SG 3-3 Showing semi-minor and semimajor axis of an ellipse. Figure SG 3-2 Definition of an ellipse. F + G = F’ + G’ = F” + G” = constant. The foci are indicated by the *’s. and “b” in Fig. SG 3-2. That is, “2a” is the major axis and “2b” is the minor axis. Another way is to use the semi-major axis and the eccentricity, ε. For planetary motion, the semimajor axis is the time average distance of a planet from the Sun. The eccentricity is a number between 0 and 1 that describes the flatness of the ellipse. A circle has an eccentricity of 0. As the eccentricity increases, the ellipse becomes flatter. As the eccentricity approaches one, the ellipse becomes more and more like a parabola. Obj. 14. Each of the four possible orbits (circle, ellipse, parabola, and hyperbola) may be generated by passing a plane through a right circular cone (imagine an inverted ice cream cone). If the plane is exactly perpendicular to the axis, we produce a circle. If the plane is nearly perpendicular, we produce an ellipse. If the plane is parallel to one edge of the cone, we produce a parabola. A more steeply inclined plane produces a hyperbola (see Fig. SG 3-5, on the next page, for two of the four cases). Obj. 15a. Once we admit that the Earth circles the Sun like the other planets some new terminology and different interpretation of old terminology become necessary. For example, in Fig. SG 3-6, if a planet is in position A relative to the Sun, it is said to be in opposition. If it was in position B the planet is said to be in conjunction. Obj. 15b. Since the Earth is moving about the Sun we must be careful when we refer to a period of a planet. An obvious period would be from opposition to opposition. (The synodic period of the Moon, for example, is the interval between full Moons.) This period, with respect to the Sun, is called a synodic period. The other period is the sidereal period that is the interval with respect to the fixed stars. These two periods differ because the Earth is moving in orbit about the Sun. Obj. 15c. If the planetary orbits are ellipses with the Sun at one focus, then the distance between the Sun and a planet varies as the planet moves about the Sun. When the planet is closest to the Sun, it is said to be at perihelion. When the planet is most distant from the Sun it is said to be in aphelion. Obj. 15d. Planets closer to the Sun than the Earth are termed inferior planets, while planets further from the Sun than the Earth are superior planets. Obj. 16 and 17. Summary of Kepler’s three laws: 1. Orbits are ellipses with Sun at one focus. 2. Line joining planet and Sun sweeps out equal areas in equal times. 3. Squares of sidereal periods are proportional to cubes of semi-major axis: P2 = K a3 3-6 K = universal constant, P the sidereal period, and a the semi-major axis. THE PLANETARY ORBITS UNIT 3 Exercise: Take a length of thread and pierce it with two pins, one near each end. (If you don’t need to know the length of the thread between the pins, you may knot the thread about the pins. Poke two pins vertically into a soft board (or cardboard) with a pin separation less than the length of the thread between the pins. Draw an ellipse with a pencil by using the pencil to make the thread taut against the pins. Move the pencil on the board, keeping the thread taut. Remove pencil and start at a new point to keep the thread from wrapping around the pins. See Fig. SG 3-4. Each pin is a focus of the ellipse and the length of thread between the pins is the quantity “A + B” of Fig. SG 3-1. Try again with the pins closer, then with the pins farther apart. Figure SG 3-4 Drawing an ellipse. Figure SG 3-5 Conic sections. Figure SG 3-6 Showing opposition (at A) and conjunction (at B). Define the Astronomical Unit as follows: 1 Astronomical Unit (AU) is the average distance of the Earth from the Sun: 1 AU = 1.5 × 1011 meters Now make the following choice of units: time — Earth years; distance — Earth-Sun distances (AU) then Kepler’s third law for the Earth becomes: 2 3 (1 yr.) = K × (1 AU) so (1 year)2 K = ________ (1 AU)3 In these units, K has a numerical value of unity. Thus we may write P =a 2 3 if P is sidereal period in years and a is distance in AUs. 3-7 Consider two examples of using Kepler’s third law in these units: Suppose an object is orbiting the Sun with a 4.0 AU semi-major axis, then P 2 = a3 = (4)3 = 64 = (8)2, so P = 8 years. Suppose an object is orbiting the Sun with a period of 5.0 years, then P 2 = a3 = (5)2 = 25 = (2.9)3, so a = 2.9 AU. If the square root or cube root is not an integer, the answers will be spaced so you should be able to estimate and then choose the correct answer. For the second example, choices might be 1.0, 2.9, 5.8, 29, and 36. The three laws describe the following: 1. The shape of a planet’s orbit. 2. How a planet moves in its orbit. 3. The relation of the average speed to the size of the orbit. If possible, go to a planetarium or use an orrery to help you visualize these laws. Obj. 20. Kepler’s approach to the problem of planetary motion was empirical. That is; he had the data on the observed motions of the planets. By a process of systematic trial and error on one set of data (Mars), he arrived at laws that fit that data. He then checked them against the data for the other planets. Newton’s approach, on the other hand, was deductive. He first postulated certain basic laws of the motion of matter, and from these deduced the laws of planetary behavior. Newton began with basic ideas of space (length) and time. These were absolute concepts that were to exist even if matter did not exist. For our purposes, you can take the intuitive idea of space and time that you probably now have, also adding the idea of force. (Weight is a good example of force.) This way Newton arrived at his three laws of motion and the gravitational law. From these he deduced the modified forms of Kepler’s law. Obj. 21. For the application of Newton’s laws to planetary motion, we need the idea of center of mass. For two bodies this is a point located between the two such that the mass of one body times its distance from the center of mass is equal to the mass of the other body times its distance from the center of mass. In terms of the diagram in Fig. SG 3-7, the center of mass is such that if the separation is r = r1 + r2, then r1 m1 = r2 m2 and v1 r2 = v2 r1 where v1 and v2 denote the motions of m1 and m2 relative to some observer. Figure SG 3-7 Showing how center of mass is related to the mass of two bodies and their separation. 3-8 THE PLANETARY ORBITS UNIT 3 From his laws Newton deduced that the center of mass of an isolated system is not accelerated (Newton’s first law), so he could best describe the motion of the planets in terms of motion about center of mass. This leads to Kepler’s laws as modified by Newton. Newton’s essential changes from Kepler’s laws are as follows: First Law: Planets and the Sun move in ellipses about a common center of mass. This center of mass is at one focus. Orbits can be ellipses, parabolas, or hyperbolas. (See Chapter 4, Objective 14 for a discussion of these conic sections.) In the solar system, the Sun is so massive that the center of mass always lies inside the surface of the Sun. Second Law: Unchanged Third Law: If two bodies revolve mutually about each other, the sum of their masses times the square of their sidereal period of mutual revolution is in proportion to the cube of the semi-major axis of the relative orbit of one about the other. As an equation: (M1 + M2)P 2 = A r 3 where A is a universal constant. If we measure mass in units of Solar masses, P is in years and r is in AUs, we can eliminate A as was done with Kepler’s version of this law and get: (M1 + M2)P 2 = r3 Suppose there was a planet of 0.1 Min orbit 1.1 AU from the Sun, then (1.1)3 r3 P 2 = ________ = ________ so P 2 = (1.1)2, and P = 1.y1 M1 + M2 (1 + 0.1) Within our solar system only Jupiter with an MJupiter = 0.001 M0 produces an easily measured motion of the Sun. But when we look outside the solar system, there are many star systems with more than one star. In those cases the modification of Newton is very important. We will return to this in Unit 27. EV tip: Think of Kepler’s three laws as a concept cluster. Do the same for Newton’s three laws and Newton’s version of Kepler’s three laws. You may ask for scratch paper to write these as you take your EV. ➤ Concepts from Unit 3 Objectives, Notes, and Text linear measure semi-major axis of an ellipse inferior planets angular measure eccentricity of an ellipse Sun as a star small angle equation parabola 16a. solar system parallax hyperbola third law of motion significant figures astronomical unit center of mass Kepler planet in opposition transits of planets Newton mass inverse square law Galileo weight law of gravitation Copernicus retrograde motion 3-9 Astronomers, like all scientists, are storytellers in the classical sense. We weave our observations into an explanatory picture of the universe. 3-10 THE PLANETARY ORBITS UNIT 3 UNIT 3 THE PLANETARY ORBITS ➤Word Practice 1. __________ used an empirical method to discover three laws of planetary motion. 2. Newton’s __________ __________ __________ __________ states that for every action there is an equal and opposite reaction. 3. The measurement in the difference in direction between two points as seen from a third point is the __________ measure. 4. __________ used a deductive method to discover three laws of planetary motion. 5. __________ was a Polish astronomer who proposed the heliocentric theory. 6. __________ is a measure of the amount of matter in a body. 7. The Sun is a __________. 8. A satellite launched from Earth with a velocity greater than escape has an orbit that is a __________. 9. The shift in apparent position of a star, due to the motion of the Earth around the Sun, is called the __________ of the star. 10. __________ is a force. 11. __________ figures are the number of digits known for certain in quantity. 12. The __________ __________ equation relates the apparent angular size of an object to the object’s true size and its distance from the observer (if the distance is large). 13. The __________ __________ __________ of the Earth-Sun lies along a line joining them. It lies very close to the center of the massive Sun. 14. The __________ motion of the planets can most easily be understood by having the Earth revolve about the Sun. 15. The __________ __________ is comprised mostly of the Sun and its planets. 16. __________ was the first known astronomer to use the telescope. 17. The __________ __________ of an ellipse is half its long dimension. 18. __________ of inferior planets occur when they pass between the Earth and the Sun. 19. The __________ measure is the distance between 2 points. 20. The __________ __________ is the average distance of the Earth from the Sun. 21. Newton proposed the __________ __________ Law for the strength of the gravitational interaction. 22. Newton’s __________ __________ __________ together with his three laws of motion, enabled him to predict planetary motion. 23. A planet in __________ rises at sunset. 24. The __________ is a measure of the “flatness” of an ellipse. 25. A spacecraft launched at exactly escape velocity moves along a __________. 26. Planets closer to the Sun than the Earth are called __________ planets. 3-11