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solutions-ps03

# solutions-ps03 - Physics 341 Problem Set#3 Solutions 1 For...

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Physics 341: Problem Set #3 Solutions 1. For this problem you can use my orbit-ps03 spreadsheet, or you can write your own program. You can download an Excel version of the spreadsheet from http://www.physics.rutgers.edu/ugrad/341/orbit-ps03.xls To use the spreadsheet, you will need to edit the fields that are shaded yellow. (a) For this problem we will use natural units for the solar system, measuring lengths in AU and time in years. Use Newton’s precise version of Kepler’s Third Law to show that GM = 4 π 2 AU 3 yr - 2 . This value is fixed in the spreadsheet (cell B3). Newton showed that Kepler’s Third Law is P 2 = 4 π 2 GM a 3 = GM = 4 π 2 a 3 P 2 By definition, the Earth goes around the Sun with a semimajor axis a = 1 AU in a period of P = 1 yr. And so GM = 4 π 2 (1 AU) 3 (1 yr) 2 = 4 π 2 AU 3 yr - 2 39 . 4784 AU 3 yr - 2 (b) We will compute the orbit of Eris, the infamous “tenth planet” that caused Pluto to be demoted to “dwarf planet” status. Eris (also called 2003 UB 313 in the textbook) has a semimajor axis a = 68 . 048 AU and eccentricity e = 0 . 4336 based on the best current observations. Calculate the orbital period P of Eris (in years). To determine the period we can use Kepler’s Third Law: P = 2 πa 3 / 2 ( GM ) 1 / 2 Recall that because of our (good) choice of units GM = 4 π 2 AU 3 yr - 2 , so P = 2 πa 3 / 2 (4 π 2 AU 3 yr - 2 ) 1 / 2 = (68 . 048 AU) 3 / 2 1 AU 3 / 2 yr - 1 = 561 . 33 yr (c) We can choose the initial conditions to have time t = 0 and angle θ = 0 , with coordinates centered on the Sun. Let’s start at perihelion (closest approach to the Sun), so that v r = 0 at t = 0 . Determine the remaining initial conditions ( and r 0 ) you need to reproduce the orbit. From class, the specific angular momentum is = GMa (1 - e 2 ) = (39 . 4784 AU 3 yr - 2 ) × (68 . 048 AU) × (1 - 0 . 4336 2 ) 1 / 2 = 46 . 705 AU 2 yr - 1 1

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Also from the class notes (see the ellipse diagram in Figure 2 of the notes from Lecture 4), the perihelion distance is r 0 = r p = a - ae = a (1 - e ) = 68 . 048 AU × (1 - 0 . 4336) = 38 . 542 AU (d) Plug these initial conditions into the spreadsheet (or your own program) and plot Eris’s orbit. You will need to adjust the time step Δ t , to cover one full period. My spreadsheet computes about 1000 time steps, so to cover one full period we should adjust the time step to Δ t = P/ 1000 = 0 . 56 yr. I will use Δ t = 0 . 57 yr just to make sure we cover slightly more than one orbit. Then plugging in our results from part (c) yields the following orbit: !"#\$% ’( )"\$* !"#\$ !%\$ !&\$ \$ &\$ %\$ "#\$ !"#\$ !%\$ !&\$ \$ &\$ %\$ "#\$ + ,-./ 0,-./ (e) Write down formulas for the perihelion distance, aphelion distance, and semimi- nor axis in terms of just a and e . Calculate these quantities for Eris and compare them to the values in your spreadsheet orbit. How well do they agree?
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