Astronomy 300 HOMEWORK #3 Spring 2020 Due date: Thu, Feb. 20 1. Order-of-magnitude estimate (no calculators!!) Show all your steps in detail. As a result of the interaction between the Moon and the tidal bulges on Earth, the Earth’s spin rate is slowing by about 2 milliseconds per century and the Moon is receding from Earth by about 3.8 cm/year. (i) How long will it be before we can get an extra hour of sleep per night (25 hr days)? (ii) When the Moon is about 8% farther away than it currently is, total Solar eclipses will no longer be possible. How long do you have to catch a total eclipse before it’s too late? Calculators may now be used. Watch those significant figures J 2. Orbiting the Moon In 2013-14 NASA’s LADEE mission sent a spacecraft to the moon to study the structure and composition of the thin lunar atmosphere (dust in particular). For details, see: http://www.nasa.gov/mission_pages/ladee/main/ LADEE orbited the Moon at an altitude of only about 40 km. How fast must the LADEE spacecraft be going to maintain a circular orbit at this altitude? 3. Moons of Jupiter Take a look the Solar System data in Appendix C of Carroll and Ostlie. Notice the information it provides about Jupiter and four largest its moons (the ones discovered by Galileo back in 1610). Assume the moons all have circular orbits. (i) Considering Jupiter’s mass and Io’s distance from Jupiter, make a prediction of what Io’s circular velocity should be. (ii) Considering Io’s orbital period, and the distance it has to travel in its orbit around Jupiter, compute Io’s velocity. Compare the result you get to the prediction you made in (i). How well do the two values agree? Do they match within the uncertainties? (iii) Using ratios, predict circular velocities for the remaining 3 “Galilean” satellites of Jupiter. (Do NOT recompute these values from scratch; start from your result in part (i) and use ratios to compute the vcirc values for the other three large moons. (iv) As you did in part (ii), compute the circular velocities for the remaining Galilean satellites and compare their values to your predictions from (iii). 4. Mass of the Milky Way (i) Compute the velocity of the Sun’s orbit around the Milky Way. Assume a circular orbit. Use the values given in Ch. 1 of Croswell for the Sun’s distance from the Galactic center and its orbital period around the Galaxy. (ii) Using the velocity in part (i), compute the mass of the Milky Way interior to the Sun’s orbit. Express your result first in kg and in then in solar masses. (iii) The mass of the supermassive black hole at the center of the Milky Way is measured to be 4.1 +/− 0.6 million solar masses. What fraction of all the mass interior to the Sun’s orbit is this? (iv) Considering your result in part (iii), would it affect the Sun’s orbit in the Galaxy much if no central black hole were not present? Explain. 5. Oumuamua In 2017 an object was discovered moving quickly through the solar system. At perihelion, this object was 0.255 AU from the Sun and moving at a speed of 87.7 km/s. (i) Compute the velocity of an object that would execute a circular orbit 0.255 AU from the Sun. (ii) Compute the escape speed for an object (0.255 AU from the Sun) (speed required to leave the solar system and never come back). (iii) Compare your result in (ii) to the speed that Oumuamua had at perihelion and comment. (iv) Would you expect Oumuamua to be on an elliptical, parabolic, or hyperbolic orbit? Explain. How Oumuamua Got its Name: The object was officially named 1I/2017 U1 by the International Astronomical Union (IAU), which is responsible for granting official names to bodies in the solar system and beyond. In addition to the technical name, the Pan-STARRS team in Hawaii who discovered it dubbed it ‘Oumuamua (pronounced oh MOO-uh MOO-uh), which is Hawaiian for “a messenger from afar arriving first.” 6. Croswell essay (1 page, please type, hand in on separate sheet with name at top): On p.19, in discussing the work of Jacobus Kapteyn, Ken Croswell says: “Yet to no other astronomer was the Galaxy more cruel.” What does Croswell mean by this? Describe the methods that Kapteyn used to construct a model of the Milky Way. What assumptions did he make, and on what basis did he make these assumptions? What results did he obtain, and how did they compare to the results of William Herschel? How do they differ from what we now know about the shape of the Milky Way, and why? 7. Extra-credit problem Please separate this from other problems and turn this in with your essay. Example 2.2.1 in Carroll and Ostlie shows the steps involved in demonstrating that the gravitational effect of a spherically symmetric mass distribution is identical to that of a point mass at the center of the sphere. (i) Read this example carefully and then do the problem for yourself, start to finish, filling in all the steps. Be sure to show all the details of the integration over the angle theta, which C&O leave out! (But do use their suggested change of variables.) Be sure to include a picture that clearly identifies relevant distances and angles. (ii) Redo this calculation for the case where r < R, i.e. where small m is inside the sphere. Consider a single shell of material at radius R, and show that the net force at r due to that shell is zero. Again, be sure to include a drawing. Hint: exactly the same approach may be taken for this problem as for (i)… the only thing that will change is a limit on the integrand. .