5th Degree Polynomials

*To*: mathgroup at smc.vnet.net*Subject*: [mg37129] 5th Degree Polynomials*From*: jdhouse4 at mac.com*Date*: Sat, 12 Oct 2002 05:04:54 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hey folks, I have been working on a problem that seems to not lend itself to a solution. The following Mathematica code begins with the expression that I am trying to solve. For the curious, it's a degree 2 zonal and sectoral harmonics problem where I am trying to calculate and plot the geoid of earth as compared to an ellipse to see how well the geoid is approximated as an ellipse. In any case, we have the following relation ship, U =GM/r( 1 - (ae/r)^2 ( J2 (3/2 Sin[t]^2 - 1/2) - 3 Cos[t]^2 (C22 Cos[2 x] + S22 Sin[2 x])); Ur =1/2 we^2 (r Cos[t])^2; W[x_] =U + Ur; In trying to reorder W to become a function r wrt t, that is r[t_], I tried, among others, Solve[W[t], r] which returned ({{r -> Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 - 3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 - 2 W0 #1 ^3 + we^2 Cos[ [t]] ^2 #1 ^5 &, 1]}, {r -> Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 - 3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 - 2 W0 #1 ^3 + we^2 Cos[ [t]] ^2 #1 ^5 &, 2]}, {r -> Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 - 3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 - 2 W0 #1 ^3 + we^2 Cos[ [t]] ^2 #1 ^5 &, 3]}, {r -> Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 - 3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 - 2 W0 #1 ^3 + we^2 Cos[ [t]] ^2 #1 ^5 &, 4]}, {r -> Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 - 3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 - 2 W0 #1 ^3 + we^2 Cos[ [t]] ^2 #1 ^5 &, 5]}} ) which wasn't too much help, though it is a list of 5 Root functions. But in order to plot, I need a function r(t) so I can plot r wrt t...right? ParametricPlot[r[t], {t, 0, Pi}] So, I guess my questions are as follows: 1. How do I get Solve[ ] to output numbers, as //N and NSolve did nothing to Solve[r[t], ...] to get any numbers instead of just r -> Root[...]? 2. Is there a way to use ParametricPlot[ W[t], {t, 0.0, Pi}] instead of using r[t] and negating the whole issue of solving W[t] for r[t]? I have read that Solve only works for up to 4th order polynomials. I have been unable to find anything that works on my problem, having tried SolveAlways[ ] and other, and combination of others. Any help is welcome. I'll be glad to forward my Notebook if someone asks. Thanks ahead of time. Jim Hillhouse jdhouse4 at mac.com Ph.D. Graduate Student Aerospace Engineering University of Texas at Austin jdhouse4 at mail.utexas.edu 512-784-3205