Consider a restricted form of the three-body problem in which a body of small mass orbits...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Consider a restricted form of the three-body problem in which a body of small mass orbits two other bodies with much larger masses, such as an Apollo spacecraft orbiting the earth-moon system. We will use a two-dimensional coordinate system in the plane determined by the three bodies, with the origin at the center of mass of the two larger bodies, and the coordinate system rotating so that the two larger bodies appear fixed. The coordinate system is shown in the accompanying diagram, earth, d spacecraft D moon where D is the distance from earth to moon, d is the distance from the center of earth to the center of mass, r, is the distance from earth to space- craft, and ry is the distance from moon to space- craft. The mass of the spacecraft is assumed to be negligible compared with the other masses. By using Newton's laws of motion and gravita- tion, and allowing for the centrifugal and Coriolis forces due to the rotating coordinate system, the motion of the spacecraft is described by the sys tem of second-order ODES -GIM(z+µD)/r +m(a-D)/r +9² +20 =-G My/r+my/rål +²y-20tz, a where G is the gravitational constant, M and m are the masses of earth and moon, p and are the mass fractions of earth and moon, and ? is the angular velocity of rotation of the moon about the earth (and hence of the coordinate system). The numerical values of these quantities are given in the following table: G M m #4² # D d 71 6.67259 x 10 m/(kg ²) 5.974 x 1024 kg 7.348 x 10² kg M/(m + M) m/(m + M) 3.844 x 10 m 4.669 x 10 m. [(x+d)² + ²¹/2 (D-d-a)²+ya 52 2.661 x 10 / Use a library routine to solve this system of ODES with the initial conditions (0)-4.613 x 10, z(0)=0. v(0) -0. v (0) = -1074. Plot the resulting solution trajectory (z(t), y(t)) in the plane as a function of time. Indicate the positions of earth and moon on the graph. Com- pute the solution for at least one complete orbit (i.e., until the spacecraft returns to its original lo cation), which is from t= 0 until approximately t = 2.4 x 10 s. Experiment with various error tolerances to see how much difference they make in whether the orbit is actually closed. Try to monitor the step size used by the ODE routine as the integration progresses. When does the step size become smaller or larger? How close does the spacecraft come to the surface of earth? (Earth's radius is 6.378 x 10 m, so the center of mass of the earth-moon system is actually inside the earth.) Consider a restricted form of the three-body problem in which a body of small mass orbits two other bodies with much larger masses, such as an Apollo spacecraft orbiting the earth-moon system. We will use a two-dimensional coordinate system in the plane determined by the three bodies, with the origin at the center of mass of the two larger bodies, and the coordinate system rotating so that the two larger bodies appear fixed. The coordinate system is shown in the accompanying diagram, earth, d spacecraft D moon where D is the distance from earth to moon, d is the distance from the center of earth to the center of mass, r, is the distance from earth to space- craft, and ry is the distance from moon to space- craft. The mass of the spacecraft is assumed to be negligible compared with the other masses. By using Newton's laws of motion and gravita- tion, and allowing for the centrifugal and Coriolis forces due to the rotating coordinate system, the motion of the spacecraft is described by the sys tem of second-order ODES -GIM(z+µD)/r +m(a-D)/r +9² +20 =-G My/r+my/rål +²y-20tz, a where G is the gravitational constant, M and m are the masses of earth and moon, p and are the mass fractions of earth and moon, and ? is the angular velocity of rotation of the moon about the earth (and hence of the coordinate system). The numerical values of these quantities are given in the following table: G M m #4² # D d 71 6.67259 x 10 m/(kg ²) 5.974 x 1024 kg 7.348 x 10² kg M/(m + M) m/(m + M) 3.844 x 10 m 4.669 x 10 m. [(x+d)² + ²¹/2 (D-d-a)²+ya 52 2.661 x 10 / Use a library routine to solve this system of ODES with the initial conditions (0)-4.613 x 10, z(0)=0. v(0) -0. v (0) = -1074. Plot the resulting solution trajectory (z(t), y(t)) in the plane as a function of time. Indicate the positions of earth and moon on the graph. Com- pute the solution for at least one complete orbit (i.e., until the spacecraft returns to its original lo cation), which is from t= 0 until approximately t = 2.4 x 10 s. Experiment with various error tolerances to see how much difference they make in whether the orbit is actually closed. Try to monitor the step size used by the ODE routine as the integration progresses. When does the step size become smaller or larger? How close does the spacecraft come to the surface of earth? (Earth's radius is 6.378 x 10 m, so the center of mass of the earth-moon system is actually inside the earth.)
Expert Answer:
Related Book For
Thermodynamics An Engineering Approach
ISBN: 978-0073398174
8th edition
Authors: Yunus A. Cengel, Michael A. Boles
Posted Date:
Students also viewed these accounting questions
-
The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moons spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter...
-
An orbiting spacecraft is described not as a zero-g, but rather as a microgravity environment for its occupants and for on-board experiments. Astronauts experience slight lurches due to the motions...
-
Consider two bodies of identical mass m and specific heat c used as thermal reservoirs (source and sink) for a heat engine. The first body is initially at an absolute temperature T1 while the second...
-
In the Walton Bookstore example with a discrete demand distribution, explain why an order quantity other than one of the possible demands cannot maximize the expected profit.
-
What is the APT? In what ways is it both better or worse than the IAPM?
-
This unadjusted trial balance is for Challenger Construction at the end of its fiscal year, September 30, 2023. The beginning balance of the owners capital account was $46,000 and the owner invested...
-
Using \(q_{0.05}=4.339\) for the Tukey HSD method, compare the treatments in Exercise 12.45. Data From Exercise 12.45 12.45 Given the following data from a randomized block design, Blocks 1 234...
-
On January 1, 2017, Xiamen Company made amendments to its defined benefit pension plan that resulted in 60,000 yuan of past service cost. The plan has 5,000 active employees with an average expected...
-
A thin airfoil is measured to have the following linear Cp distributions on its upper and lower surfaces at some small angle of attack, \ alpha . C p u pper = - 1 * ( 1 - x / c ) , C p l ower = 0 . 5...
-
For the beam shown in Figure P4-79, use a computer program to determine the deflection at the mid-span using four beam elements, making the shear area zero and then making the shear area equal 5/6...
-
Explain how SWOT analysis can be used to identify the Opportunities that may arise from project internal Strengths and the Threats that may arise from Weaknesses in risk management.
-
You place two magnets side by side but, instead of attracting, the magnets repel each other. Provide a possible explanation of your observation.
-
How would the following transactions affect UK net capital outflow? Also, state whether each involves direct investment or portfolio investment. a. A British mobile telephone company establishes an...
-
What, according to Keynes, was the main reason why recessions and depressions occurred? As a result of identifying this key reason, what did Keynes suggest was an appropriate policy repose?
-
Explain how the marginal propensity to withdraw affects the outcome of a rise in autonomous expenditure.
-
How would the following transactions affect UK exports, imports and net exports? a. A British art lecturer spends the summer touring museums in Italy. b. Students in Paris flock to see the latest...
-
My fund: Assuming my fund decides to maintain its existing percentage ownership by making an addition cash infusion as described in my answer to Question 13, my fund ' s ownership is of 44.6691% as...
-
Anna, a high school counselor, devised a program that integrates classroom learning with vocational training to help adolescents at risk for school dropouts stay in school and transition to work...
-
Joe Smith, an old-fashioned engineering student, believes that the boiling point of water is best suited for use as the reference point on temperature scales. Unhappy that the boiling point...
-
An ordinary egg can be approximated as a 5.5-cmdiameter sphere. The egg is initially at a uniform temperature of 8°C and is dropped into boiling water at 97°C. Taking the properties of egg to...
-
What is the difference between the specific humidity and the relative humidity?
-
The NCAA is described as a cartel. In what way is it a cartel? What is the product being produced? How does the cartel stay together?
-
What is the cost to a firm in an oligopoly that fails to take rivals actions into account?
-
The cement industry is an example of an undifferentiated oligopoly. The automobile industry is a differentiated oligopoly. Which of these two is more likely to advertise? Why?
Study smarter with the SolutionInn App