(*********************************************************************** 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[ 105624, 3933]*) (*NotebookOutlinePosition[ 106546, 3962]*) (* CellTagsIndexPosition[ 106502, 3958]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Date[]", "Input"], Cell[OutputFormData["\<\ {2003, 1, 30, 9, 45, 7}\ \>", "\<\ {2003, 1, 30, 9, \ 45, 7}\ \>"], "Output"] }, Open ]], Cell["\<\ For a mixture of neutrons, protons, and electrons, satisfying charge neutrality (kFprot = kFelec) and weak interaction equilibrium (muneut = muprot + muelec). The chemical potentials are defined by mu_i = Sqrt[kF_i^2 + m_i^2]. To tabulate p, eps in terms of kFneut, the neutron Fermi momentum -- see Weinberg, Eqns. 11.4.1-2. Let x_i = kF_i/(mneut*c) to get dimensionless integrals. Quantities p and eps in CGS units have dimensions ergs/cm^3. Define dimensionless pbar and epsbar by dividing each by \t eps0 = mneut^4 c^8/(3 Pi^2 hbar^3 c^3), \t which will be the overall factor carrying these units. (Note m c^2 is ergs and hbar*c = erg-cm, giving overall ergs/cm^3.)\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ {hbar = 1.055 10^(-27) (* erg-sec *), c = 2.99 10^10 (* cm/sec *), mprot = 1.67 10^(-24) (* grams *), mneut = (0.93955/0.93826)*mprot (* grams *), melec = 0.911 10^-27 (* grams *)} eps0 = mneut^4 c^8/(3 Pi^2 hbar^3 c^3)\ \>", "Input"], Cell[OutputFormData["\<\ {1.0549999999999998*^-27, 2.9900000000000004*^10, 1.67*^-24, 1.6722960586617784*^-24, 9.110000000000001*^-28}\ \>", "\<\ -27 10 -24 {1.055 10 , 2.99 10 , 1.67 10 , -24 -28 1.6723 10 , 9.11 10 }\ \>"], "Output"], Cell[OutputFormData["\<\ 5.375640625919981*^36\ \>", "\<\ 36 5.37564 10\ \>"], "Output"] }, Open ]], Cell["\<\ First, need to find kFprot for any given kFneut. From the MMa solution of muneut = muprot + muelec, there is an analytic form:\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ {melec=0.000511, mprot=0.93826, mneut=0.93955} (* for now, in \ GeV/c^2 *)\ \>", "Input"], Cell[OutputFormData["\<\ {0.000511, 0.93826, 0.93955}\ \>", "\<\ {0.000511, \ 0.93826, 0.93955}\ \>"], "Output"] }, Open ]], Cell["\<\ kFpbykFn[k_] := Sqrt[(k^2 + mneut^2 - melec^2)^2 \t\t\t\t\t- 2(k^2 + mneut^2 + melec^2)*mprot^2 \t\t\t\t\t \t\t\t\t\t+ mprot^4]/(2*k*Sqrt[k^2 + mneut^2]) (* dimensionless *)\ \>", \ "Input"], Cell[CellGroupData[{ Cell["{kFpbykFn[2.0], 2*kFpbykFn[2.0]}", "Input"], Cell[OutputFormData["\<\ {0.45282506107767556, 0.9056501221553511}\ \>", \ "\<\ {0.452825, 0.90565}\ \>"], "Output"] }, Open ]], Cell["\<\ The second number agrees with that in FindkFpII.nb. The energy density (divided by eps0) is the sum of three integrals. \ \>", \ "Text"], Cell[CellGroupData[{ Cell["\<\ Iepsbyeps0 = 3*Integrate[u^2*Sqrt[u^2+1],{u,0,x}] (* neutron \ term *)\ \>", "Input"], Cell[OutputFormData["\<\ (3*(x*Sqrt[1 + x^2] + 2*x^3*Sqrt[1 + x^2] - ArcSinh[x]))/8\ \>", "\<\ 2 3 \ 2 (3 (x Sqrt[1 + x ] + 2 x Sqrt[1 + x ] - ArcSinh[x])) / 8\ \>"], "Output"] }, Open ]], Cell["\<\ epsbar[x_] := (3/8)*((x + 2*x^3)*(1 + x^2)^(1/2) - ArcSinh[x]) \t\t(* dimensionless *) \ \>", "Input"], Cell[CellGroupData[{ Cell["epsbar[1.5]", "Input"], Cell[OutputFormData["\<\ 5.1293009227506925\ \>", "\<\ 5.1293\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ IepsXYbyeps0 = 3*Integrate[u^2*Sqrt[u^2+y^2],{u,0,x}] \t(* integral for nproton and electron terms, y = m_i/m_n *)\ \>", "Input"], Cell[OutputFormData["\<\ (3*(2*x^3*Sqrt[x^2 + y^2] + x*y^2*Sqrt[x^2 + y^2] + y^4*Log[Sqrt[y^2]] - y^4*Log[x + Sqrt[x^2 + y^2]]))/8\ \>", "\<\ 3 2 2 (3 (2 x Sqrt[x + y ] + 2 2 2 x y Sqrt[x + y ] + 4 2 y Log[Sqrt[y ]] - 4 2 2 y Log[x + Sqrt[x + y ]])) / 8\ \>"], "Output"] }, Open ]], Cell["\<\ epsbarXY[x_,y_] := (3/8)*(2*x^3*Sqrt[x^2+y^2] + x*y^2*Sqrt[x^2+y^2] \ \t\t\t\t\t + y^4*Log[y/(x + Sqrt[x^2+y^2])])\ \>", "Input"], Cell[CellGroupData[{ Cell["epsbarXY[1.5,1]", "Input"], Cell[OutputFormData["\<\ 5.129300922750692\ \>", "\<\ 5.1293\ \>"], "Output"] }, Open ]], Cell["\<\ which agrees with epsbar[1.5], as it should. The total epsbar is \ then the function\ \>", "Text"], Cell["\<\ {melec=0.000511, mprot=0.93826, mneut=0.93955}; (* in GeV *) epsbartot[x_] := epsbar[x] + epsbarXY[kFpbykFn[x]*x, mprot/mneut] \t\t\t\t\t \ \t+ epsbarXY[kFpbykFn[x]*x, melec/mneut];\ \>", "Input"], Cell[CellGroupData[{ Cell["\<\ {x=2.0, kFpbykFn[x]*x, epsbar[x], epsbarXY[kFpbykFn[x]*x, \ mprot/mneut], \t\tepsbarXY[kFpbykFn[x]*x, melec/mneut], epsbartot[x]}\ \>", "Input"], Cell[OutputFormData["\<\ {2., 0.9056501221553511, 14.552095544931529, 0.9040838222808384, 0.5045488494345729, 15.96072821664694}\ \>", "\<\ {2., 0.90565, 14.5521, \ 0.904084, 0.504549, 15.9607}\ \>"], "Output"] }, Open ]], Cell["The following is Weinberg's form for the pressure:", "Text"], Cell[CellGroupData[{ Cell["\<\ Clear[x] Ipbyeps0 = Integrate[u^4/Sqrt[u^2+1],{u,0,x}]\ \>", "Input"], Cell[OutputFormData["\<\ (-3*x*Sqrt[1 + x^2] + 2*x^3*Sqrt[1 + x^2] + 3*ArcSinh[x])/8\ \>", "\<\ 2 3 \ 2 (-3 x Sqrt[1 + x ] + 2 x Sqrt[1 + x ] + 3 ArcSinh[x]) / 8\ \>"], "Output"] }, Open ]], Cell["\<\ pbar[x_] := (1/8)*((-3*x + 2*x^3)*(1 + x^2)^(1/2) + 3*ArcSinh[x]) \t(* also dimensionless *)\ \>", "Input"], Cell[CellGroupData[{ Cell["pbar[1.5]", "Input"], Cell[OutputFormData["\<\ 0.9550668545947896\ \>", "\<\ 0.955067\ \>"], \ "Output"] }, Open ]], Cell["\<\ Sanjay defines pressure slightly differently as the negative of the \ free energy density, eps - mu*N. One does have to be careful with the factor of 3 in the definition of eps0\ \>", "Text"], Cell["mu[x_] := Sqrt[x^2 + 1]", "Input"], Cell["pbarSR[x_] := 3*mu[x]*Integrate[u^2,{u,0,x}] - epsbar[x]", "Input"], Cell[CellGroupData[{ Cell["pbarSR[x]", "Input"], Cell[OutputFormData["\<\ x^3*Sqrt[1 + x^2] - (3*(Sqrt[1 + x^2]*(x + 2*x^3) - ArcSinh[x]))/8\ \>", "\<\ 3 2 x Sqrt[1 + x ] - 2 3 3 (Sqrt[1 + x ] (x + 2 x ) - ArcSinh[x]) ---------------------------------------- 8\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pbarSR[1.5]", "Input"], Cell[OutputFormData["\<\ 0.95506685459479\ \>", "\<\ 0.955067\ \>"], "Output"] }, Open ]], Cell["\<\ So, the two definitons of pressure ARE consistent. We'll continue \ with Weinberg's definition.\ \>", "Text"], Cell[CellGroupData[{ Cell["Ipbyeps0 = Integrate[u^4/Sqrt[u^2+y^2],{u,0,x}]", "Input"], Cell[OutputFormData["\<\ (2*x^3*Sqrt[x^2 + y^2] - 3*x*y^2*Sqrt[x^2 + y^2] - 3*y^4*Log[Sqrt[y^2]] + 3*y^4*Log[x + Sqrt[x^2 + y^2]])/8\ \>", "\<\ 3 2 2 (2 x Sqrt[x + y ] - 2 2 2 3 x y Sqrt[x + y ] - 4 2 3 y Log[Sqrt[y ]] + 4 2 2 3 y Log[x + Sqrt[x + y ]]) / 8\ \>"], "Output"] }, Open ]], Cell["\<\ pbarXY[x_,y_] := (1/8)*((2*x^3 - 3*x*y^2)*Sqrt[x^2+y^2] \t\t\t\t\t\t+ 3*y^4*Log[(x + Sqrt[x^2+y^2])/y ])\ \>", "Input"], Cell[CellGroupData[{ Cell["pbarXY[1.5,1]", "Input"], Cell[OutputFormData["\<\ 0.9550668545947896\ \>", "\<\ 0.955067\ \>"], \ "Output"] }, Open ]], Cell["\<\ also in agreement with pbar[x] at this x value. The total pressure is again the sum of the three contributions.\ \>", "Text"], Cell["\<\ pbartot[x_] := pbar[x] + pbarXY[kFpbykFn[x]*x, mprot/mneut] + \t\t\t\tpbarXY[kFpbykFn[x]*x, melec/mneut];\ \>", "Input"], Cell[CellGroupData[{ Cell["\<\ {pbartot[1.5],pbar[1.5],pbarXY[kFpbykFn[1.5]*1.5, mprot/mneut], \tpbarXY[kFpbykFn[1.5]*1.5, melec/mneut]}\ \>", "Input"], Cell[OutputFormData["\<\ {1.0144703487853803, 0.9550668545947896, 0.01842405235278649, 0.04097944183780429}\ \>", "\<\ {1.01447, 0.955067, \ 0.0184241, 0.0409794}\ \>"], "Output"] }, Open ]], Cell["\<\ Why is the electron pressure bigger than the proton pressure? The contribution of the electrons to the energy density is less than that of the protons. Is it because most of the proton energy comes from its (much greater) mass?\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Do[Print[PaddedForm[pbartot[x],{10,6}],PaddedForm[epsbartot[x],{15,\ 6}], PaddedForm[x,{10,2}]],{x,0.2,5.0,0.2}]\ \>", "Input"], Cell["\<\ 0.000063 0.008106 0.20 0.001951 0.067523 0.40 0.014088 0.242853 0.60 0.055865 0.622608 0.80 0.159434 1.324408 1.00 0.369824 2.496649 1.20 0.744423 4.319044 1.40 1.352410 7.002968 1.60 2.274358 10.791721 1.80 3.601961 15.960728 2.00 5.437893 22.817662 2.20 7.895711 31.702520 2.40 11.099812 42.987673 2.60 15.185404 57.077885 2.80 20.298490 74.410329 3.00 26.595863 95.454594 3.20 34.245102 120.712685 3.40 43.424572 150.719026 3.60 54.323422 186.040457 3.80 67.141586 227.276235 4.00 82.089786 275.058031 4.20 99.389533 330.049928 4.40 119.273127 392.948423 4.60 141.983658 464.482420 4.80 167.775007 545.413235 5.00\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 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