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The Schwarzschild radial coordinate is measured in units of ", StyleBox["M", FontSlant->"Italic"], ", so that", StyleBox[" M=1", FontSlant->"Italic"], " in the following formulae.Throughout the variable ", StyleBox["u=1/r ", FontSlant->"Italic"], "is used. Here is a list of what you must set to run the program:\n \ (1) The angular momentum: \[ScriptL].\n (2) The energy parameter: \ \[ScriptCapitalE] = (", Cell[BoxData[ \(TraditionalForm\`e\^2\)]], "-1)/2.\n (3)The starting radius for an orbit which is not bound: ", StyleBox["rst.", FontSlant->"Italic"], "\n (4)The number of orbits to be computed if the orbit is bound: ", StyleBox["norbit", FontSlant->"Italic"], ".\nThese are set by editing the definition statements at various places in \ the program. ", StyleBox["You have to make sure these parameters are set so the orbit is \ classically allowed and doesn't start at a position where the value of the \ effective potential that is greater than \[ScriptCapitalE]. ", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell["The potential:", "Subsection"], Cell[TextData[{ "The effective potential for radial motion ", Cell[BoxData[ \(TraditionalForm\`V\_\(\(eff\)\(\ \)\)\)]], "given in (9.28) is here denoted simply by ", StyleBox["V", FontSlant->"Italic"], ". To be slightly more general a parameter ", StyleBox["signewt", FontWeight->"Bold"], " is introduced which multiplies the non-Newtonian ", StyleBox["1/", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`r\^3\)]], "term in the potential. Set it equal to 1 for general relativity, 0 for a \ Newtonian ", StyleBox["1/r", FontSlant->"Italic"], " potential, and an appropriate value for a Newtonian ", StyleBox["1/r", FontSlant->"Italic"], " potential withan additional quadrupole moment term. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(signewt = 1. \)], "Input"], Cell[BoxData[ \(1.`\)], "Output"] }, Open ]], Cell[BoxData[ \(V[u_, \[ScriptL]_] := \(-u\) + \[ScriptL]^2\ \ u^2/2\ - signewt\ \ \[ScriptL]^2\ \ u^3\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Specifying the orbit:", "Subsection"], Cell[TextData[{ "It takes four numbers to specify an orbit: (1) the angular momentum \ \[ScriptL], (2) the energy parameter \[ScriptCapitalE], (3) the starting \ radius ", StyleBox["rst", FontSlant->"Italic"], ", and (4) the number of orbits to be calculated, ", StyleBox["norbit.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ StyleBox["(1) ", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Specify the ", FontWeight->"Plain"], "angular momentum \[ScriptL] :" }], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptL] = 5\)], "Input"], Cell[BoxData[ \(5\)], "Output"] }, Open ]], Cell[TextData[{ "This is a graph of the 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JOoo003ooomYool00?ooofWoo`00ooooJOoo003ooomYool00?ooofWoo`00ooooJOoo003ooomYool0 0?ooofWoo`00ooooJOoo0000\ \>"], ImageRangeCache->{{{91.5625, 320.938}, {238.5, 97.1875}} -> {-53.4546, \ 0.0981873, 0.362777, 0.00195938}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ From this information you can see what kind of orbit it will be \ \[LongDash] bound orbit, plunge orbit, or scattering orbit. \ \>", "Text"], Cell["\<\ Now the program evaluates the three turning points associated with \ this value of the energy parameter:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(soln = \ NSolve[V[tp\[ThinSpace], \[ScriptL]] \[Equal] \[ScriptCapitalE], tp]\)], "Input"], Cell[BoxData[ \({{tp \[Rule] 0.011659741119405955`}, {tp \[Rule] 0.08506961684645648`}, {tp \[Rule] 0.4032706420341376`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tp1 = tp /. soln[\([1]\)]\)], "Input"], Cell[BoxData[ \(0.011659741119405955`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tp2 = tp /. soln[\([2]\)]\)], "Input"], Cell[BoxData[ \(0.08506961684645648`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tp3 = tp /. soln[\([3]\)]\)], "Input"], Cell[BoxData[ \(0.4032706420341376`\)], "Output"] }, Open ]], Cell[TextData[{ "(3) Specify the ", StyleBox["starting radius rst.", FontWeight->"Bold"], " This together with the energy parameter determines what kind of obrit is \ computed \[LongDash] bound, asymptotic from and to infinity, plunge, \ emerging from ", StyleBox["r=2M", FontSlant->"Italic"], ", etc. depending on where it is set relative to the above turning points. \ However, only ifthe orbit goes off to infinity is the ", StyleBox["value ", FontSlant->"Italic"], "of ", StyleBox["rst", FontSlant->"Italic"], " used. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(rst = 20. \)], "Input"], Cell[BoxData[ \(20.`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ust = 1/rst\)], "Input"], Cell[BoxData[ \(0.05`\)], "Output"] }, Open ]], Cell[TextData[{ "(4) Finally specify the ", StyleBox["number of orbits, norbit , ", FontWeight->"Bold"], "if the orbit is bound. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(norbit = 3\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Computing the Orbit", "Subsection"], Cell[TextData[{ "We are now goint to integrate {(\[ScriptL]/", Cell[BoxData[ \(TraditionalForm\`2\^\(1/2\)\)]], ")[\[ScriptCapitalE]-", StyleBox["V[u,\[ScriptL]]", FontSlant->"Italic"], "]^(-1/2) to find the orbit in the form \[Phi](r). The integrand diverges \ near the turning points, so we specify a parameter", StyleBox[" eps", FontWeight->"Bold"], " which determines how close the numerical integration comes to the turning \ points. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eps = .00000001\)], "Input"], Cell[BoxData[ \(1.`*^-8\)], "Output"] }, Open ]], Cell[TextData[{ "Next the program picks the appropriate radii to start and end the \ integration of the orbit given the energy parameter and starting radius. \ There are four kinds of orbits.\n(a) Bound obits which start at the outer \ turning point. \n(b) Orbits which come in from infinity and go out again. \ These start at ", StyleBox["rst.", FontSlant->"Italic"], "\n(c) Orbits which start close to ", StyleBox["r=2M ", FontSlant->"Italic"], "and fall back to it. \n(d) Plunge orbits which start at rst and end at ", StyleBox["r=2M", FontSlant->"Italic"], ". For bound the starting radii are the \n\nTests for bound orbits:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(testa = \ \((\[ScriptCapitalE] < 0)\) && \((ust < tp2)\)\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(If[testa, \ u1 = tp1 \((1 + eps)\); u2 = tp2 \((1 - eps)\)]\)], "Input"], Cell[BoxData[ \(0.08506961599576031`\)], "Output"] }, Open ]], Cell[TextData[StyleBox[ "Tests for orbits coming from and going to infinity:"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(testb = \((\[ScriptCapitalE] > 0)\) && \((\[ScriptCapitalE] < vmax)\) && \((ust < tp2)\)\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[BoxData[ \(If[testb, \ u1 = ust; \ \ u2 = tp2 \((1 - eps)\)]\)], "Input"], Cell[TextData[{ "Tests for orbits which start from ", StyleBox["r=2M", FontSlant->"Italic"], " and go back to it:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(testc = \((\[ScriptCapitalE] < vmax)\) && \((\ ust > tp3)\)\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[BoxData[ \(If[testc, \ u1 = .5; \ u2 = tp3 \((1 + eps)\)]\)], "Input"], Cell[TextData[{ "Tests for plunge orbits coming from infinity going to ", StyleBox["r=2M", FontSlant->"Italic"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(testd\ = \ \((\[ScriptCapitalE] > vmax)\)\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[BoxData[ \(If[testd, \ u1 = ust; \ u2 = .5 \((1 - eps)\)]\)], "Input"], Cell[TextData[{ "If the parameters are set correctly only one of the four options should be \ ``true'' andthe other three should be ``false''. The values of ", StyleBox["u1", FontSlant->"Italic"], " and", StyleBox[" u2 ", FontSlant->"Italic"], "are displayed below:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(u1\)], "Input"], Cell[BoxData[ \(0.011659741236003366`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(u2\)], "Input"], Cell[BoxData[ \(0.08506961599576031`\)], "Output"] }, Open ]], Cell[TextData[{ "A number of functions are defined, the chief of which is the angle ", StyleBox["theta", FontWeight->"Bold"], " which is the angle \[Phi] swept out from\nthe innermost turning point. \ This is given by the following numerical integral:" }], "Text"], Cell[BoxData[ \(theta[u_, \[ScriptCapitalE]_, \[ScriptL]_, u1_] := NIntegrate[\((\[ScriptL]/ 2^\((1/2)\))\)\ \((\[ScriptCapitalE] - V[w, \[ScriptL]])\)^\((\(-1\)/2)\), \ {w, u1, u}]\)], "Input"], Cell[TextData[{ StyleBox["delphi", FontWeight->"Bold"], " is the total angle \[Phi] swept out between two endpoints of the orbit:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(delphi = theta[u2, \[ScriptCapitalE], \[ScriptL], u1]\)], "Input"], Cell[BoxData[ \(3.7360408472380526`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["z ", FontSlant->"Italic"], "is a parameter which varies from 0 to ", StyleBox["norbit.", FontSlant->"Italic"], " " }], "Text"], Cell[BoxData[ \(n[z_] := IntegerPart[z]\)], "Input"], Cell[BoxData[ \(zf[z_] := FractionalPart[z]\)], "Input"], Cell[BoxData[ \(ua[z_] := u1 \((1 - 2\ \ zf[z])\) + u2\ \ \ 2\ \ zf[z]\)], "Input"], Cell[BoxData[ \(ub[z_] := \ u1 \((2\ zf[z] - 1)\)\ + 2\ \ u2\ \((1 - zf[z])\)\)], "Input"], Cell[BoxData[ \(u[z_] := If[zf[z] < .5, ua[z], ub[z]]\)], "Input"], Cell[BoxData[ \(phia[z_] := 2\ \((n\ [z])\) delphi\ + \ theta[u[z], \[ScriptCapitalE], \[ScriptL], u1]\)], "Input"], Cell[BoxData[ \(phib[z_] := 2\ \((n\ [z] + 1)\) delphi\ - \ theta[u[z], \[ScriptCapitalE], \[ScriptL], u1]\)], "Input"], Cell[BoxData[ \(accphi[z_] := If[zf[z] < .5, phia[z], phib[z]]\)], "Input"], Cell[BoxData[ \(x[z_] := 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