lecture-05 - These lecture notes were prepared for Rutgers...

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Unformatted text preview: These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights reserved. c 2011 Lecture 5: The Black Hole at the Center of the Galaxy I. Using Kepler III: Motion Mass Its all well and good that Newton gave a physical explanation of Keplers empirical laws. But what good does it do us? Lets reconsider Kepler III. Rearranging, we can write M = 4 2 a 3 GP 2 This form is useful because the left-hand side is something we may want to know the mass of an astronomical object while the right-hand side involves stuff we can measure the size and period of an orbit. In other words, we know what measurements we need to make, and calculations we need to do, in order to measure mass. A couple of things to keep in mind for later in the course are that M represents the total mass of the system, and that a is the average separation of the two bodies in three dimensions, whereas we typically observe a system projected onto the two-dimensional sky. The key concept is that we can observe motion, interpret it using Newtons laws of motion and gravity, and infer mass. Or, as I will often summarize the principle: motion mass . Example: : Jupiter. We can use the orbits of Jupiters moons (from Appendix C of Carroll & Ostlie) to compute the planets mass. Moon Orbital Period Semimajor Axis M J (d) (10 3 km) (10 30 g) Io 1.769 421.6 1.90 Europa 3.551 670.9 1.90 Ganymede 7.155 1070.4 1.90 Callisto 16.689 1882.7 1.90 1 II. Equations of Motion Redux Lets look again at the complete set of equations of motion for the one-body problem: dr dt = v r d dt = = r 2 dv r dt = d 2 r dt 2 = a r + r 2 =- GM r 2 + r 2 Lets make the finite difference approximation: r dr dt t = v r t d dt t = r 2 t v r dv r dt t =- GM r 2 + r 2 t Now suppose you specify r , , and v r at some time t , along with the angular momentum (which is constant). Then you can compute the changes r , , and v r and take a step: ( t, r, , v r ) ( t + t, r + r, + v r + v r ) Then you can repeat the process to take another step ... and then another ... and so on. This repetitive process sounds tedious, but it is perfect for computers. I have created a spreadsheet to do the calculation, and you will work with it on problem set #3. A word of warning: the finite difference approximation is valid only when the time step t is small. If you want your calculation to be accurate, you cant make t too big. On the other hand, when t is small it can take many, many steps to compute an orbit. Computers do make very small errors in calculations (because they have to round numbers to a large but finite number of digits), and if you take many, many steps those errors can build up....
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This note was uploaded on 10/20/2011 for the course PH 341 taught by Professor Gawiser during the Fall '11 term at Rutgers.

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lecture-05 - These lecture notes were prepared for Rutgers...

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