(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 27674818, 938829]*) (*NotebookOutlinePosition[ 27675485, 938852]*) (* CellTagsIndexPosition[ 27675441, 938848]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["More on the orbits--- extra problems from 5 March 08 session", \ "Subtitle"], Cell[CellGroupData[{ Cell["Halley's Comet", "Subsubsection"], Cell["\<\ As suggested, let's animate Halley's comet. But first, let's carefully draw \ the Earth's orbit. Choose parameters by convention:\ \>", "Text"], Cell[BoxData[ \(c\ = \ 1; \ m\ = 1;\)], "Input"], Cell["\<\ Take as inputs the period T (measured in years) and the eccentricity \ \[Epsilon], e.g. for Earth,\ \>", "Text"], Cell[BoxData[ \(T\ = \ 1; \ \[Epsilon] = .017;\)], "Input"], Cell["To convert to the parameters L and E we use Kepler's law:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s\ \ = \ Solve[T^2\ \[Equal] \ \[Pi]^2\ m\ c^2/\((\(-2\)\ e^3)\), e]\)], "Input"], Cell[BoxData[ \({{e \[Rule] \(-\(\((\(-\[Pi]\))\)\^\(2/3\)\/2\^\(1/3\)\)\)}, {e \[Rule] \ \((\(-\(1\/2\)\))\)\^\(1/3\)\ \[Pi]\^\(2/3\)}, {e \[Rule] \(-\(\[Pi]\^\(2/3\)\ \/2\^\(1/3\)\)\)}}\)], "Output"] }, Open ]], Cell["We want the real solution:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(e0\ = \ e /. \ s[\([3]\)]\)], "Input"], Cell[BoxData[ \(\(-\(\[Pi]\^\(2/3\)\/2\^\(1/3\)\)\)\)], "Output"] }, Open ]], Cell["To get L:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s = Solve[\[Epsilon]\ \[Equal] \ Sqrt[1\ + \ 2\ e0\ L^2/\((m\ c^2)\)], L]\)], "Input"], Cell[BoxData[ \({{L \[Rule] \(-0.5418477561635603`\)}, {L \[Rule] 0.5418477561635603`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(L0\ = \ L /. s[\([2]\)]\)], "Input"], Cell[BoxData[ \(0.5418477561635603`\)], "Output"] }, Open ]], Cell["With L in hand we can define the effective potential:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Ueff[r_]\ = \ \(-\ c\)/r\ + L0^2/\ \((\ 2\ m\ \ r^2)\)\)], "Input"], Cell[BoxData[ \(0.14679949542974255`\/r\^2 - 1\/r\)], "Output"] }, Open ]], Cell["We can draw a line on the graph with:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{Ueff[r], e0}, {r, .1, 2}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 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