(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 8001123, 146800] NotebookOptionsPosition[ 7912488, 145445] NotebookOutlinePosition[ 7912901, 145462] CellTagsIndexPosition[ 7912858, 145459] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Integrating boson stars using the shooting method", "Title", CellChangeTimes->{{3.8297286765208035`*^9, 3.829728686429947*^9}},ExpressionUUID->"587ba005-a7a7-493b-8901-\ 80051c1f8e29"], Cell["\<\ In this notebook we integrate the differential equations for a spherically \ symmetric boson stars. We divide the notebook into three different cases: \ Boson stars, vector boson stars, and charged boson stars\ \>", "Text", CellChangeTimes->{{3.829728730539654*^9, 3.8297287339740286`*^9}, { 3.8297288275147734`*^9, 3.8297288394296274`*^9}, 3.829731233530972*^9, 3.8297327342662554`*^9, {3.8297327706333513`*^9, 3.8297327936710625`*^9}, { 3.829732864813074*^9, 3.829732906899147*^9}},ExpressionUUID->"7dda5b79-b1e9-4359-85f4-\ 5a16b36e36ad"], Cell[BoxData[ RowBox[{"Quit", "[", "]"}]], "Input", CellChangeTimes->{{3.8297480401193333`*^9, 3.8297480412205095`*^9}}, CellLabel-> "In[704]:=",ExpressionUUID->"700bdd20-8212-47cd-bcb1-ee5def0d78fd"], Cell[CellGroupData[{ Cell["Scalar boson stars", "Subsection", CellChangeTimes->{{3.8297329142680187`*^9, 3.8297329403901396`*^9}, { 3.8297383432577667`*^9, 3.8297383465307035`*^9}},ExpressionUUID->"a54d74e0-6558-4944-86bf-\ 342161d7369c"], Cell["\<\ Here we integrate the equations for spherically symmetric mini boson stars \ (potential with only the mass term). 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We require \ regularity. We also set A(0)=1. While this choice makes the spacetime \ different from the standard Schwarzschild one at infinity, we can later one \ rescale the time coordinate to obtain the solution in the Schwarzchild \ coordinates. The simplest case is to use (B(0),A(0),\[Phi]0(0),\[Phi]0\ \[CloseCurlyQuote](0))=(1,1,\[Phi]c,0). We can also consider an expansion to \ improve convergence.\ \>", "Text", CellChangeTimes->{{3.829738352999463*^9, 3.8297384374075537`*^9}, { 3.8297386764015493`*^9, 3.829738751473749*^9}, {3.829740528357664*^9, 3.829740548269641*^9}, 3.8297407674307766`*^9},ExpressionUUID->"bf997bda-f0ba-447e-b73b-\ c121fc3aeca1"], Cell[BoxData[ RowBox[{ RowBox[{"BOUNDARY", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"B", "[", "ror", "]"}], "\[Equal]", "1"}], ",", RowBox[{ RowBox[{"A", "[", "ror", "]"}], "\[Equal]", "1"}], ",", RowBox[{ RowBox[{"\[Phi]0", "[", "ror", "]"}], "\[Equal]", "\[Phi]c"}], ",", RowBox[{ RowBox[{ RowBox[{"\[Phi]0", "'"}], "[", "ror", "]"}], "\[Equal]", "0"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.8297387569234605`*^9, 3.829738800242407*^9}}, CellLabel->"In[8]:=",ExpressionUUID->"289de4a3-391a-4034-a484-6114e949ef29"], Cell["\<\ Let us give the expression for the scalar field at infinity. \ \>", "Text", CellChangeTimes->{{3.8297407716046777`*^9, 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The result of this integration will be the \ scalar field in a region far from the center. How far depends on how the \ scalar field grows or decays. 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We can therefore use this to find other solutions using as guess the \ initial one. To do so, we can also define the radius of the star to serve as \ a guide for the numerical infinity. 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