(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 133242, 4540]*) (*NotebookOutlinePosition[ 134164, 4569]*) (* CellTagsIndexPosition[ 134120, 4565]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Date[]", "Input"], Cell[BoxData[ \({2003, 9, 2, 10, 10, 8}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Solving the coupled non-linear TOV differential \nequations for \ pure neutron stars with a non-\ninteracting Fermi gas Equation of State.", "Title", FontFamily->"Bold", FontSize->12, FontWeight->"Bold"], StyleBox[" ", "Title", FontSize->12], "\n\n\tWe will work in units of Msun = 1 and km, so that curlyMbar(r) \nis \ dimensionless and curlyMbar(R) gives the star mass in units\nof Msun. (R is \ the distance where the pressure p(r=R) = 0.) \nEnergy density and pressure, \ epsilon(r) and p(r), have units \nMeV/fm^3 or ergs/cm^3 or Msun*c^2/km^3. \ However, it is more\nconvenient to switch to dimensionless forms for these \ quantities. \n\n\tThe equation for p(r) is\n \n dp(r)/dr = \ -Rschw*epsilon(r)*curlyMbar(r)/r^2\n \ntimes three dimensionless \ correction factors from GR.\nHere Rschw = G * Msun/c^2 = 1.48 km is ", StyleBox["HALF", FontWeight->"Bold"], " the Schwartzschild radius \nof the Sun. In CGS units, to check that:" }], "Text"], Cell[CellGroupData[{ Cell["\<\ bigG = 6.67 10^(-8) (* dyne-cm^2/g^2 *); Msun = 1.989 10^33 (* gram *); c = 2.998 10^10 (* cm/sec *); Rschw = bigG Msun/c^2 (* cm *); Rschw = Rschw/10^5 (* km *)\ \>", "Input"], Cell[BoxData[ \(1.476037393841836`\)], "Output"] }, Open ]], Cell["\<\ \tWhat is the central pressure for a star of constant energy \ density, mass = Msun, and radius 10 km? From Aidan Parker's eq. (6),\ \>", "Text"], Cell["Off[General::spell1]", "Input"], Cell[CellGroupData[{ Cell["\<\ bigR = 10.0 10^3 10^2 (* 10 km radius, in cm *); p0 = 3 bigG Msun^2/(8 Pi bigR^4) (* dynes/cm^2 *) p0 = p0 / 10.0 (* in pascals = kg m/s^2 /m^2 *); p0 = p0 / 10^9 (* in gigapascals *) \ \>", "Input"], Cell[BoxData[ \(3.1497551536298097`*^34\)], "Output"], Cell[BoxData[ \(3.14975515362981`*^24\)], "Output"] }, Open ]], Cell[TextData[{ "\tThe second equation needed to complete the solution is\n \n d \ curlyMbar(r)/dr = (4", StyleBox["\[Pi]", FontFamily->"Symbol"], "/Msun)*r^2*", StyleBox["\[Epsilon]", FontFamily->"Symbol"], "(r)\n \nwithout any further dimensionless factors. Epsilon must, at this \ \npoint, be in units of Msuns/km^3 for this equation to have overall \nunits \ of 1/km. We will make it dimensionless in a bit.\n\n\tFor an Equation of \ State we take the fitted Fermi gas values,\n \n epsilon(pbar) = \ eps0*epsbar(pbar), epsbar and pbar dimensionless\n p = eps0*pbar\n \ eps0 = mneutron^4 c^8/(3 Pi^2 hbar^3 c^3)\n \n epsbar(pbar) = \ a1*pbar^(3/5) + a2*pbar\n a1 = 2.4216, a2 = 2.8663\n ", StyleBox[" ", "Text"] }], "Text"], Cell[CellGroupData[{ Cell["\<\ {hbar = 1.055 10^(-27), (* erg-sec *) c = 2.99 10^10, (* cm/sec *) mneutron = 1.67 10^(-24)} (* grams *) eps0 = mneutron^4 c^8/(3 Pi^2 hbar^3 c^3) {a1 = 2.4216, a2 = 2.8663}\ \>", "Input"], Cell[BoxData[ \({1.0549999999999998`*^-27, 2.9900000000000004`*^10, 1.67`*^-24}\)], "Output"], Cell[BoxData[ \(5.346178403625608`*^36\)], "Output"], Cell[BoxData[ \({2.4216`, 2.8663`}\)], "Output"] }, Open ]], Cell["\<\ The first differential equation after dividing out eps0 on both sides then becomes d pbar(r)/dr = -alpha*epsbar(pbar(r))*curlyMbar(r)/r^2 alpha = Rschw where alpha has, at this point, units of km. The second DEqn is d curlyMbar(r)/dr = beta*r^2*epsbar(pbar(r)) where beta = 4*Pi*eps0/(Msun*c^2) must have units of 1/km^3 so the equation itself has units 1/km. \ \>", \ "Text"], Cell[CellGroupData[{ Cell["alpha = Rschw", "Input"], Cell[BoxData[ \(1.476037393841836`\)], "Output"] }, Open ]], Cell["\<\ Convert the CGS value of eps0 (g/cm^3) into units of \ Msun/km^3:\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ {Msuncsqd = Msun*c^2, km3 = (10.0^5)^3} (* ergs and cm^3 *) eps0sunkm = eps0*km3/Msuncsqd (* units Msun/km^3 *)\ \>", "Input"], Cell[BoxData[ \({1.7781858900000004`*^54, 1.`*^15}\)], "Output"], Cell[BoxData[ \(0.0030065351624320937`\)], "Output"] }, Open ]], Cell["\<\ \tSo, what is beta for this value of eps0? beta = 4*Pi*eps0/(Msun*Kbar^(1/gamma)) = 4*Pi*eps0sunkm/Kbar^(1/gamma) in units of 1/km^3. \ \>", "Text"], Cell[CellGroupData[{ Cell["beta = 4*Pi*eps0sunkm", "Input"], Cell[BoxData[ \(0.03778123511622424`\)], "Output"] }, Open ]], Cell["\<\ which is probably OK, beta being not too small. \ \>", "Text"], Cell["\<\ \tWe will integrate these two equations using a length scale unit of 1 km. \t d pbar(r)/dr = -alpha*(pbar(r))^(1/gamma)*curlyMbar(r)/r^2 d curlyMbar(r)/dr = beta*r^2*(pbar(r))^(1/gamma) \ \>", "Text"], Cell[CellGroupData[{ Cell["{alpha, beta, rstart = 0.001}", "Input"], Cell[BoxData[ \({1.476037393841836`, 0.03778123511622424`, 0.001`}\)], "Output"] }, Open ]], Cell["\<\ \tNow see if we can solve the coupled DE's given a value pzero for the energy density at the center of the star. We need a starting value of pbar(r=0) = pzero. If a neutron star has an energy density of \t = 3 Msun*c^2/(4 Pi Rnstar^3), Rnstar = 10 km, \t then\t \t = /eps0, \t \t = Kbar*^gamma, \t might be a good starting value.\t \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Rnstar = 10.0 (* km *) epsilonave = 3/(4 Pi Rnstar^3) (* Msun/km^3 *) epsbarave = epsilonave/eps0sunkm (* dimless *) pbarave = epsbarave/5 \t\t\t(* rough guess *)\ \>", "Input"], Cell[BoxData[ \(10.`\)], "Output"], Cell[BoxData[ \(0.000238732414637843`\)], "Output"], Cell[BoxData[ \(0.07940449778233222`\)], "Output"], Cell[BoxData[ \(0.015880899556466443`\)], "Output"] }, Open ]], Cell["\<\ That is, we should try values of pzero ~ 0.02 to get values of neutron star radii that are physically reasonable. \ \>", "Text"], Cell["\<\ First, we'll integrate the Newtonian case, i.e., without the three additional factors in the TOV equation which come from GR.\ \>", "Text"], Cell["epsbar[r_] := a1*pbar[r]^(3/5) + a2*pbar[r]", "Input"], Cell["\<\ Arhs[r_] := -alpha*(a1*pbar[r]^(3/5) + a2*pbar[r])*curlyMbar[r]/r^2 Brhs[r_] := beta*r^2*(a1*pbar[r]^(3/5) + a2*pbar[r])\ \>", "Input"], Cell[CellGroupData[{ Cell["{alpha, beta}", "Input"], Cell[BoxData[ \({1.476037393841836`, 0.03778123511622424`}\)], "Output"] }, Open ]], Cell["\<\ RMlist = {}; Do[{pzero = 10.0^(-i), Do[{ \ts1 = NDSolve[{pbar'[r] == Arhs[r], pbar[rstart] == pzero, \t\t\tcurlyMbar'[r] == Brhs[r], \t\t\tcurlyMbar[rstart] == 0.0}, \t\t\t{pbar, curlyMbar},{r,rstart,x}, \t\t\tMaxSteps->20000]; \ty = Re[pbar[x]/.s1]; \tz = Re[curlyMbar[x]/.s1]; \tIf[y[[1]] < 0, Break[], {capR = x, capM = z[[1]]}] \t}, {x,1.0,50.0,0.1}]; RMlist = Append[RMlist,{pzero, capR, capM}]; }, {i,2,2}] RMlist;\ \>", "Input"], Cell[CellGroupData[{ Cell["\<\ Print[\" pzero R(km) M(Msun)\"] Print[\" \"] Do[Print[ScientificForm[RMlist[[i,1]],{12,2}],\" \", \tPaddedForm[RMlist[[i,2]],{8,1}], \tPaddedForm[RMlist[[i,3]],{10,4}]], \t{i,1}]\ \>", "Input"], Cell[BoxData[ \(" pzero R(km) M(Msun)"\)], "Print"], 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There are three factors,\nall functions of r:\n\n\t factorpbyeps = 1 + \ p/epsilon = 1 + pbar/epsbar\n\t \n\t factorpbyM = 1 + (4 Pi/c^2) r^3 \ p/curlyM\n\t = 1 + (4 Pi eps0/(Msun c^2)) r^3 pbar/curlyMbar\n\t\ \n\t factorgravity = 1 - 2 bigG curlyM/r\t= 1 - 2 Rschw curlyMbar/r\n\t \n\ This middle factor looks singular at r = 0, since curlyMbar(0) = 0.\nThe r^3 \ in the numerator, however, cancels the singularity since \ncurlyMbar(r) near \ r = 0 is approximately 4 Pi r^3 epsbar(0)/3.\nThus, we can handle this \ problem by setting\n\n\t factorpbyM = 1 + (3 eps0/(Msun c^2)) \ pbar(r)/epsbar(0)\n\t \nwhen, say, r < 0.1 km, and otherwise using \n\n\t \ factorpbyM = 1 + (4 Pi eps0/(Msun c^2)) r^3 pbar(r)/curlyMbar(r)\n\t \nfor \ all other values of r. We will do this using the If[...] function:\n\n\t \ If[r > 0.1, (1 + (4 Pi eps0/(Msun c^2)) r^3 pbar(r)/curlyMbar(r)),\n\t \n\t \ \t\t\t, (1 + (3 eps0/(Msun c^2)) pbar(r)/epsbar(0))]\n\t \nIt appears \ necessary to include all these factors ", StyleBox["within", FontWeight->"Bold"], " the NDSolve block.\t\n \nFirst, just include the factorpbyeps and \ factorgravity factors:\n" }], "Text"], Cell["\<\ RMlistGR = {}; Do[{pzero = 10.0^(-i), epsbarzero = a1*pzero^(3/5) + a2*pzero, Do[{ \tsGR = NDSolve[{pbar'[r] == Arhs[r]* \t\t\t\t(1 + pbar[r]/epsbar[r]) / \t\t\t\t(1 - 2 Rschw curlyMbar[r]/r), \t\t\tcurlyMbar'[r] == Brhs[r], \t\t\tpbar[rstart] == pzero, curlyMbar[rstart] == 0.0}, \t\t\t{pbar, curlyMbar},{r,rstart,x}, \t\t\tMaxSteps->20000]; \ty = Re[pbar[x]/.sGR]; \tz = Re[curlyMbar[x]/.sGR]; \tIf[y[[1]] < 0, Break[], {capR = x, capM = z[[1]]}] \t}, {x,1.0,50.0,0.25}]; RMlistGR = Append[RMlistGR,{pzero, capR, capM}]; }, {i,1,4}] RMlistGR;\ \>", "Input"], Cell[CellGroupData[{ Cell["\<\ Print[\" pzero R(km) M(Msun)\"] Print[\" \"] Do[Print[ScientificForm[RMlistGR[[i,1]],{12,2}],\" \", \tPaddedForm[RMlistGR[[i,2]],{8,1}], \tPaddedForm[RMlistGR[[i,3]],{10,4}]],{i,1,4}]\ \>", "Input"], Cell[BoxData[ \(" pzero R(km) M(Msun)"\)], "Print"], Cell[BoxData[ \(" "\)], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{ TagBox[ InterpretationBox[\("1."\[Times]10\^"-1"\), 0.10000000000000001, AutoDelete->True], (ScientificForm[ #, {12, 2}]&)], "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", TagBox[ InterpretationBox["\<\" 8.5\"\>", 8.5, AutoDelete->True], (PaddedForm[ #, {8, 1}]&)], "\[InvisibleSpace]", TagBox[ InterpretationBox["\<\" 0.8391\"\>", 0.83913827707100241, AutoDelete->True], (PaddedForm[ #, {10, 4}]&)]}], SequenceForm[ ScientificForm[ 0.10000000000000001, {12, 2}], " ", PaddedForm[ 8.5, {8, 1}], PaddedForm[ 0.83913827707100241, {10, 4}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{ TagBox[ InterpretationBox[\("1."\[Times]10\^"-2"\), 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