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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[ 46226, 1713]*) (*NotebookOutlinePosition[ 47150, 1742]*) (* CellTagsIndexPosition[ 47106, 1738]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Date[]", "Input"], Cell[OutputFormData["\<\ {2003, 2, 4, 14, 24, 43}\ \>", "\<\ {2003, 2, 4, 14, \ 24, 43}\ \>"], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Solving the coupled non-linear Newtonian differential \nequations \ for polytropic stars, i.e., white dwarfs.", "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 \nHere Rschw = G * Msun/c^2 = 1.48 \ km is the Schwatzschild radius of\nthe 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[OutputFormData["\<\ 1.476037393841836\ \>", "\<\ 1.47604\ \>"], "Output"] }, Open ]], Cell["\<\ \tWhat is the central pressure for a star of constant energy \ density, mass = Msun, and radius 10,000 km? From Aidan Parker's eq. (6),\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ bigR = 1.0 10^4 10^3 10^2 (* 10,000 km radius, in cm \ *); p0 = 3 bigG Msun^2/(8 Pi bigR^4) (* dynes/cm^2 *) p0 = p0 / 10.0 (* in pascals = N m/s^2 /m^2 *); p0 = p0 / 10^9 (* in gigapascals *) \ \>", "Input"], Cell[OutputFormData["\<\ 3.149755153629809*^22\ \>", "\<\ 22 3.14976 10\ \>"], "Output"], Cell[OutputFormData["\<\ 3.1497551536298096*^12\ \>", "\<\ 12 3.14976 10\ \>"], "Output"] }, Open ]], Cell[TextData[{ "That seems to be a ", StyleBox["very high pressure", FontWeight->"Bold"], ". Is it anywhere near right?\nYes, it is: see Weinberg's (11.3.44)for \ rhocrit ~ 10^6 gm/cm^3, \nconvert to epsiloncrit ~ 10^27 ergs/cm^2 by \ multiplying by c^2, \nthen calculate pcrit using p = bigK epsilon^gamma, \ gamma = 4/3, \nand the bigK calculated below, 5 10^-14 cm/(erg^1/3). One \ gets\nabout 5 10^22 dynes/cm^2 (ergs/cm^3).\n" }], "Text"], 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 \nbut epsilon here must, at this point, be in units of Msuns/km^3 \ for \nthis equation to have overall units, say, of 1/km. We will make it \n\ dimensionless in a bit.\n\n\tAssume a polytropic equation of state,\n \n \ p(r) = bigK*epsilon(r)^gamma . \n \n", StyleBox["For now, we will take ", "Text"], StyleBox["\[Gamma]", "Text", FontFamily->"Symbol"], StyleBox[" = 5/3, which is its value for the largest \nmass white dwarf, \ but leave it as ", "Text"], StyleBox["\[Gamma]", "Text", FontFamily->"Symbol"], StyleBox[" in all formulae to maintain\ngenerality. ", "Text"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["gamma = 5/3", "Text"]], "Input"], Cell[OutputFormData["\<\ 5/3\ \>", "\<\ 5 - 3\ \>"], "Output"] }, Open ]], Cell["\<\ \tWork out bigK for the case m_electron << kF, with gamma = 4/3. This gamma is that for the largest mass white dwarf. From Weinberg's Eq. (11.3.48)\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ {hbar = 197.32 (* MeV-fm *), mnucleon = 939.0 (* MeV *), melectron = 0.511 (* MeV *), mu = 56.0/26} (* assuming star is all Fe-56 *) bigK = hbar^2 (3 Pi^2/(mnucleon mu))^(5/3) /(15 melectron Pi^2) (* fm/MeV^(2/3) *) \ \>", "Input"], Cell[OutputFormData["\<\ {197.32, 939., 0.511, 2.1538461538461537}\ \>", "\<\ {197.32, 939., 0.511, 2.15385}\ \>"], "Output"], Cell[OutputFormData["\<\ 0.45091864194641235\ \>", "\<\ 0.450919\ \>"], \ "Output"] }, Open ]], Cell["\<\ Do it again, but this time in CGS units (using Wbg's Eq. \ (11.3.45)):\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ {hbar = 1.055 10^(-27) (* erg-sec *), mnucleon = 1.67 10^(-24) (* grams *), melectron = 9.11 10^(-28) (* grams *), mu = 56.0/26} (* assuming star is all Fe-56 *) {(hbar c)^2 /(15 Pi^2), (* erg-cm *) mu mnucleon c^2} (* ergs *) bigK = (hbar c)^2 (3 Pi^2/(mnucleon c^2 mu))^(5/3) / \t (15 melectron c^2 Pi^2) \t (* CGS -- cm^2/erg^(2/3)*)\ \>", "Input"], Cell[OutputFormData["\<\ {1.0549999999999998*^-27, 1.67*^-24, 9.11*^-28, 2.1538461538461537}\ \>", "\<\ -27 -24 {1.055 10 , 1.67 10 , -28 9.11 10 , 2.15385}\ \>"], "Output"], Cell[OutputFormData["\<\ {6.757361791857176*^-36, 0.0032329159003076932}\ \>", "\<\ -36 {6.75736 10 , 0.00323292}\ \>"], "Output"], Cell[OutputFormData["\<\ 3.3085664333540937*^-23\ \>", "\<\ -23 3.30857 10\ \>"], "Output"] }, Open ]], Cell["\<\ which has correct dimensions cm/erg^(1/3). [Compare with \ fm/MeV^(2/3), since hbar c = 197.32 MeV-fm) and m_el c^2 and m_N c^2 are in MeV.] \ \>", \ "Text"], Cell["\<\ \tWe will normalize our energy density to be dimensionless, by dividing epsilon by an (arbitrary) factor of eps0, whose value we choose below so that the constants in the DE's are of order 1. pbar(r) = Kbar*epsbar(r)^gamma, (all factors dimensionless) epsbar = epsilon/eps0 pbar = p/eps0 Kbar = bigK*eps0^(gamma-1)\ \>", "Text"], Cell["\<\ We will solve for pbar instead of epsbar, so we need \t epsbar(r) = (pbar(r)/Kbar)^(1/gamma)\ \>", "Text"], Cell["\<\ The first differential equation then becomes d pbar(r)/dr = -alpha*(pbar(r))^(1/gamma)*curlyMbar(r)/r^2 alpha = Rschw/Kbar^(1/gamma) = Rschw/(bigK*eps0^(gamma-1))^(1/gamma) where alpha has, at this point, units of km (Kbar is dimensionless). The second DEqn is d curlyMbar(r)/dr = beta*r^2*(pbar(r))^(1/gamma) where beta = 4*Pi*eps0/(Msun*c^2*Kbar^(1/gamma)) = 4*Pi*eps0/(Msun*c^2*(bigK*eps0^(gamma-1))^(1/gamma)) has units, say, of 1/km^3 so the equation itself has units 1/km. \ \>", \ "Text"], Cell[CellGroupData[{ Cell["{gamma, 1/gamma, gamma-1, (gamma-1)/gamma}", "Input"], Cell[OutputFormData["\<\ {5/3, 3/5, 2/3, 2/5}\ \>", "\<\ 5 3 2 2 {-, -, -, -} 3 5 3 5\ \>"], "Output"] }, Open ]], Cell["\<\ so (for gamma = 5/3) \t alpha = Rschw/(bigK*eps0^(2/3))^(3/5) \t = Rschw/(bigK^(3/5)*eps0^(2/5))\ \>", "Text"], Cell["\<\ Thus we can make alpha whatever we want (in km) by choosing \t eps0 = ((Rschw/alpha)^gamma / bigK)^(1/(gamma-1) \t \t = (Rschw/alpha)^(5/2) / bigK^(3/2) (for gamma = 5/3) \t for Rschw in km. Let me choose alpha = 0.05 km: \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ {alpha = 0.05 (* km *), Rschw, bigK} eps0 = (((Rschw/alpha)^gamma)/bigK)^(1/(gamma-1)) alpha = Rschw/(bigK*eps0^(gamma-1))^(1/gamma)\ \>", "Input"], Cell[OutputFormData["\<\ {0.05, 1.476037393841836, 3.3085664333540937*^-23}\ \>", "\<\ -23 {0.05, 1.47604, 3.30857 10 }\ \>"], "Output"], Cell[OutputFormData["\<\ 2.4880471152489845*^37\ \>", "\<\ 37 2.48805 10\ \>"], "Output"], Cell[OutputFormData["\<\ 0.05000000000000009\ \>", "\<\ 0.05\ \>"], "Output"] }, Open ]], Cell["\<\ Convert this 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[OutputFormData["\<\ {1.7877139956000006*^54, 1.*^15}\ \>", "\<\ \ 54 15 {1.78771 10 , 1. 10 }\ \>"], "Output"], Cell[OutputFormData["\<\ 0.013917478530529348\ \>", "\<\ 0.0139175\ \>"], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Kbar = bigK*eps0^(gamma-1)", "Input"], Cell[OutputFormData["\<\ 281.9760739869947\ \>", "\<\ 281.976\ \>"], "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/Kbar^(1/gamma)", "Input"], Cell[OutputFormData["\<\ 0.005924382199315727\ \>", "\<\ 0.00592438\ \>"], \ "Output"] }, Open ]], Cell["which is not of order 1 but may still be OK numerically.", "Text"], Cell["\<\ \tWe will try integrating 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[OutputFormData["\<\ {0.05000000000000009, 0.005924382199315727, 0.001}\ \>", "\<\ {0.05, 0.00592438, 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 white dwarf has a density of \t = 3 Msun*c^2/(4 Pi Rdwarf^3), Rdwarf = 10,000 km, \t \t epsbar = /eps0, \t \t pbar = Kbar*epsbar^gamma, \t would be a good starting value.\t \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Rdwarf = 10.0^4 (* km *) epsilonave = 3/(4 Pi Rdwarf^3) (* Msun/km^3 *) epsbarave = epsilonave/eps0sunkm (* dimless *) pbarave = Kbar*epsbarave^gamma (* dimless *)\ \>", "Input"], Cell[OutputFormData["\<\ 10000.\ \>", "\<\ 10000.\ \>"], "Output"], Cell[OutputFormData["\<\ 2.3873241463784306*^-13\ \>", "\<\ -13 2.38732 10\ \>"], "Output"], Cell[OutputFormData["\<\ 1.7153424315630175*^-11\ \>", "\<\ -11 1.71534 10\ \>"], "Output"], Cell[OutputFormData["\<\ 3.217093209010801*^-16\ \>", "\<\ -16 3.21709 10\ \>"], "Output"] }, Open ]], Cell["\<\ That is, we should try quite small values of pzero to get values of white dwarf radii that are physically reasonable.\ \>", "Text"], Cell["\<\ Arhs[r_] := -alpha*pbar[r]^(1/gamma)*curlyMbar[r]/r^2 Brhs[r_] := beta*r^2*pbar[r]^(1/gamma)\ \>", "Input"], Cell[CellGroupData[{ Cell["{alpha, beta}", "Input"], Cell[OutputFormData["\<\ {0.05000000000000009, 0.005924382199315727}\ \>", "\<\ {0.05, 0.00592438}\ \>"], "Output"] }, Open ]], Cell["\<\ RMlist = {}; Do[{pzero = 10.0^(-i), Do[{ \ts1 = NDSolve[{pbar'[r] == Arhs[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]/.s1]; \tz = Re[curlyMbar[x]/.s1]; \tIf[y[[1]] < 0, Break[], {capR = x, capM = z[[1]]}] \t}, {x,1000.0,50000.0,10.0}]; RMlist = Append[RMlist,{pzero, capR, capM}]; }, {i,15,16}] 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,2}]\ \>", "Input"], Cell["\<\ pzero R(km) M(Msun) -15 1. \[Times] 10 10620.0 0.3941 -16 1. \[Times] 10 13360.0 0.1974\ \>", "Print"] }, Open ]], Cell["\<\ Are these realistic starting values of pzero? We must have the density much less than rho-critical (where kF = melectron*c). Weinberg's expression for rho-critical is Eq. (11.3.44):\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ {mnucleon, mu, melectron, c, hbar} rhocrit = mnucleon*melectron^3*c^3/(3 Pi^2 hbar^3) (* CGS *) (* lacking a factor of mu *)\ \>", "Input"], Cell[OutputFormData["\<\ {1.67*^-24, 2.1538461538461537, 9.11*^-28, 2.9980000000000004*^10, 1.0549999999999998*^-27}\ \>", "\<\ -24 -28 {1.67 10 , 2.15385, 9.11 10 , 10 -27 2.998 10 , 1.055 10 }\ \>"], "Output"], Cell[OutputFormData["\<\ 978561.4627856967\ \>", "\<\ 978561.\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ epsbarcrit = mu*rhocrit*c^2/eps0 pbarcrit = Kbar*epsbarcrit^gamma\ \>", "Input"], Cell[OutputFormData["\<\ 7.61390483756169*^-11\ \>", "\<\ -11 7.6139 10\ \>"], "Output"], Cell[OutputFormData["\<\ 3.856778228881215*^-15\ \>", "\<\ -15 3.85678 10\ \>"], "Output"] }, Open ]], Cell["\<\ so cases for pzero > 10^-15 do NOT satisfy the condition assumed for this equation of state.\ \>", "Text"], Cell[CellGroupData[{ Cell["capR", "Input"], Cell[OutputFormData["\<\ 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