Download Homework #3 MHC Astronomy 100/101/110 Prof. Stage For ALL the

Homework #3 MHC Astronomy 100/101/110 Prof. Stage For ALL the problems in this assignment, be sure to show your work / calculation steps / explain your reasoning. Answers without work will lose points. This assignment is worth 40 points total, with an extra‐credit question worth up to an additional 4 points. You do not need to type this entire assignment. Some of the problems are more easily written out by hand. Just be neat! 1. Units, units everywhere. One of the important parameters about stars is Luminosity, which is the total power of a star or glowing object. Luminosity is measured in Watts (for stars or light bulbs)—but what’s a Watt? In the standard metric MKS system, distance is measured in meters, time in seconds, and mass in kg (hence, M‐K‐S). The unit of force is a Newton, the unit of energy is a Joule, and the unit of power is a Watt. Power is an expression of energy per time. Energy is an expression of force times distance. Force is an expression of mass times acceleration. (Newton’s 2nd law) Acceleration is the change in velocity per time. And velocity is distance per time. a. (4pts) Express each of the five quantities above as a combination of only m, s, and kg. (Hints: start with velocity and work your way up. When you are just working with units, use the formulas, but put in only the units (and ignore numerical constants). For example, velocity is v = distance / time, so to get the units, [v] =[d]/[t] = (m) / (s) = m/s. A typical way to indicate you are just thinking about units is to use [ ]. Keep in mind that you can still multiply, divide, apply powers and square roots, etc. to units. A 5ft by 5ft room needs a 25 ft2 rug). b. (4pts) In class we said we could ignore the units and constants in a3
G
=
( m1 + m2 ) 2
p
4π 2
but only if we put a in AU, P in yrs, and M in solar masses. By converting units on G = 6.67 x 10‐11 m3/(kg s2), Show that €
G
AU 3
=
1
4π 2
M solar × yr 2
(You will need to look up some conversions. Your answer may not be exact depending on how precise your values of the constants are). €
2. (4 pts) Calculate the average velocity of the Earth in its orbit around the Sun, in m/s. 3. (4 pts) A comet is orbiting the Sun and has an elliptical orbit. At perihelion (closest approach to the Sun), it gets as close as 1.5 AU to the Sun. At aphelion (farthest) it is 300 AU Homework #3 MHC Astronomy 100/101/110 Prof. Stage from the Sun. What is the ratio of the comet’s speed at these points? Where is it traveling faster? 4. Calculate the masses of exoplanets. a. (4pts) Gliese 581c is one of four planets orbiting the star Gliese 581, and was the smallest known exoplanet when it was discovered. Gliese 581c has a semi‐major axis of 0.073 AU and an orbital period of 13 days. Calculate the mass of the star Gliese 581 in solar masses. If Gliese 581 is on the main sequence, what kind of star do you expect Gliese 581 to be? Compare your answers with the data for the Gliese system in the NASA‐JPL Planetquest database: http://planetquest.jpl.nasa.gov/index.cfm b. (4pts) What would be the semi‐major axis of the orbit of an exoplanet which takes 3 years to orbit Regulus? Give your answer in AU. (You can find the mass of Regulus in your textbook). 5. Calculate the mass of the black hole at the center of the galaxy. Figure 2 from Schödel et al., Nature, v.417, 2002, p. 694‐696. This figure shows the observed positions and the best‐fit orbit for the star S0‐2 which orbits the galactic center. (Data is a combination of infrared and radio observations using ground‐based telescopes). a. (4pts) Using the best‐fit result given for the semi‐major axis in the image above, and by estimating a period of the orbit from the dates of observations, calculate the mass of the likely black hole in solar masses. You will need to convert the angular measurement of the semi­major axis to AU, as it is given in arc seconds (´´). Measurements on the sky are made by the number of degrees across something (180 Homework #3 MHC Astronomy 100/101/110 Prof. Stage degrees from horizon to horizon). The actual size of the something depends on how far away it is. At the distance of the Galactic Center, 1 arc second = 1´´ =8000 AU. b. Extra‐Credit (up to 4 points): Using the Declination axis for a scale, try to measure the semi‐major axis in arc seconds (you may want to print out the page). What do you get? Can you think of any reason why the distance you measure here might be less than the best‐fit result the scientists obtained? 6. Eclipsing binary stars. (4pts) You are observing an eclipsing binary star system which is composed of a star of B spectral type and a star of K spectral type. The stars are the same radius1. Sketch a light curve for this system. Also draw a set of sketches showing the positions of the two stars in their orbits for key times from your graph. (Hints: a light curve is a graph of luminosity vs. time. Consider Wien’s law. Answers should not be numerical, e.g., “50 solar luminosities at this time”, but should show qualitatively what is going on.) 7. (4pts) How many times greater is the Earth’s gravitational force on the Moon than the Moon’s gravitational force on the Earth? 8. (4pts) Explain how a car is able to drive around a corner, with reference to Newton’s laws. Is acceleration involved? Where are the forces transmitted? What pushes on what? It may be helpful to make a diagram or two. 1 This is possible, but indicates that at least one of the stars is no longer on the main sequence.