GEOL212 Due 9/24/18 Homework 4

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1 GEOL212 Due 9/24/18 Homework 4 General instructions: Although you are allowed to discuss homework questions with your classmates, your work must be uniquely your own. Thus, please answer all questions in your own intelligible words. (When in doubt, use complete sentences.) If calculations are involved, show all work (so that I will have some basis for giving partial credit.) Be sure to use appropriate units and significant digits, where appropriate. Part I: Kepler s third law: Kepler predicts that the square of orbital period p is proportional to the cube of semimajor axis a. Let s see if this holds true for a range of orbiting objects. 1.) Calculate p 2, a 3, and their ratio for the planets, below. (The solution for Earth is provided. In this case, we assume that measurements are in Earth years and AU, so there s no need to write the units in every cell. Watch significant figures, however.) Planet Period p Semimajor p 2 a 3 p 2 /a 3 (Earth yr.) axis a (AU) Earth Mercury Venus Mars Jupiter Saturn ) Is Kepler s Third Law a good approximation of the relationship between p 2 and a 3 for all the planets you examined? 3.) Kepler only considered planets in orbit around the sun. Does the third law work for objects orbiting other primaries? To see, calculate p 2, a 3, and their ratio for the Galilean moons of Jupiter. (To simplify matters, orbital periods and semimajor axes are given in what we ll call Jupiter system astronomical units (JSAU) based on the orbital period and semimajor axis of Io. This GREATLY simplifies calculations.) Moon Period p (Io orbital periods IOP = days) Semimajor axis ( JSAU = 421,700 km) P 2 a 3 P 2 /a 3 Io Europa Ganymede Callisto

4.) Does Kepler s Third Law hold up in the Jupiter system, as well? Part II: The Hohmann Transfer Orbit: Kepler s Third Law allows us easily to solve useful problems.

2 4.) Does Kepler s Third Law hold up in the Jupiter system, as well? Part II: The Hohmann Transfer Orbit: Kepler s Third Law allows us easily to solve useful problems. For example, suppose we want to get a spacecraft from Earth to Mars as efficiently as possible. We would want to place it into an orbit whose Rp (periapsis) is 1 AU and whose Ra (apoapsis) is AU. (The semimajor axes for Earth and Mars respectively.) This would allow us to burn just enough fuel to get the spacecraft to Mars. This is called the Hohmann transfer orbit, after Wolfgang Hohmann who first described it in Like any other orbit, the Hohmann transfer orbit has an orbital period and semimajor axis. 5.) In the case of the transfer orbit between Earth and Mars, we know that its periapsis is AU and apopasis is What is the semimajor axis (a) of the transfer orbit? (Watch significant figures and include units in your answer.) (For all calculations, attach extra pages if necessary!) 6.) But what is p, the orbital period? By Kepler s Third Law, the square of the orbital period is proportional to the cube of its semimajor axis. Thus: a 3 = (your answer to q. 5) cubed = p 2 thus p= (a 3 ) 0.5 Calculate p for the Earth/Mars Hohmann transfer orbit. (Again, watch significant figures and include units in your answer.)

3 But wait! To get from Earth to Mars you would only complete half of the full orbit! Divide by two to get the duration of the trip in Earth years. Does your answer sound reasonable? (Consider that the Curiosity rover was launched on 11/25/11 and arrived on 8/6/12.) 7) Imagine that you are a flight engineer on a future human colony on Callisto. A robot spacecraft is to be launched from your base and enter orbit around Io. Using the Hohmann transfer orbit, how long (in Earth days) should this trip take. To simplify calculations, use the IOP and JSAU units from question 3. p (Earth Days) a (km) Io ,700 Callisto p (Io orbital periods - IOP) a (Io semimajor axes - JSAU) Io Callisto a) Calculate the semimajor axis of the transfer orbit in JSAU. 7b) Calculate the full orbital period of the transfer orbit in IOP. (Do this the same way as in question 6.) 7c) What will be the spacecraft s actual travel time between Callisto and Io (not a full orbit) in IOP?

4 7d) Finally, translate the final answer above into Earth days. (Multiply by (1.769 days/1 IOP) canceling out units where appropriate.) 7e) Compared with the orbital periods of Callisto ( Earth days) and Io (1.769 Earth days), does this seem reasonable? 8) Frozen in Callisto s ice, your team finds a derelict alien spacecraft inhabited by a monster so destructive and terrifying that you agree that the only safe thing is to destroy it by dropping it into Jupiter s atmosphere. Using the exact methods above, calculate the semimajor axis and orbital period of the necessary transfer orbit. Note: In this case the only difference is that apoapsis (Ra) will be JSAU and periapsis (Ra\p) will be JSAU. Use the same methods that you used in Q7. 8a) Calculate the semimajor axis of the transfer orbit. 8b) Calculate the total orbital period of the transfer orbit 8c) For how many Earth days must the monster be kept anesthetized between launch and destruction? (Remember, you DON T want to bring the monster back to Callisto!) 8d) Compared with the orbital period of Callisto ( Earth days), do these answers seem reasonable?

6 Part III: Newtonian Gravitation: Newton s Laws of Gravity enable us easily to determine either the gravitational force exerted by an object of known mass or the mass of an object exerting a known gravitational force, as revealed by the period and semimajor axis of an object orbiting it. To calculate gravitational force (g) we use: g = G MP / RP 2 Where: MP and RP are the mass and radius of the planet and G is the universal gravitational constant: * m 3 kg -1 s -2 To calculate g for Earth: g = (( * m 3 kg -1 s -2 ) * * kg )/ (6.371 *10 6 m) 2 = (( * m 3 /kg s 2 ) * * kg) / *10 13 m 2 = ( * m 3 * * kg) / (10 11 kg s 2 * *10 13 m 2 ) = ( * m 3 * * kg) / ( *10 24 m 2 kg s 2 ) canceling where appropriate we get: = ( * m * ) / s 2 = m) / s 2 = m s -2 (rounded to four significant figures) This is a satisfactory approximation of the textbook value of 9.8 m s -2. Yay! Using this method to calculate the force of gravity on the surfaces of : 9a.) Mercury (mass = 0.33 * kg, mean radius = 2.44 * 10 6 m) Attach extra sheets if necessary.

7 9b.) Mars (mass = * kg, mean radius = 3.39 * 10 6 m) 10.) If your calculations are correct, you should find that Mars and Mercury s surface gravities are similar. In one complete sentence, what do these answers suggest to you about Mars and Mercury s relative densities?

AST101: Our Corner of the Universe Lab 8: Measuring the Mass of Jupiter

AST101: Our Corner of the Universe Lab 8: Measuring the Mass of Jupiter Name: Student number (SUID): Lab section number: 1 Introduction Objectives In a previous lab, we measured the mass of the Earth with