(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 835480, 15556] NotebookOptionsPosition[ 824567, 15385] NotebookOutlinePosition[ 824920, 15401] CellTagsIndexPosition[ 824877, 15398] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Constructing the secular Hamiltonian", "Title", CellChangeTimes->{{3.760982927725913*^9, 3.760982934723608*^9}, { 3.761043463444079*^9, 3.761043469800853*^9}, {3.761046509900992*^9, 3.761046513273294*^9}},ExpressionUUID->"a4beaef9-cdb8-4bce-b6f3-\ 6c1b47e87331"], Cell[CellGroupData[{ Cell["Evaluating integrals of multi-pole terms", "Chapter", CellChangeTimes->{{3.761046814771792*^9, 3.7610468372828817`*^9}},ExpressionUUID->"5ac74caf-9d66-4103-8504-\ 80bf43d80637"], Cell[TextData[{ "Here I consider the secular evolution of a test particle interior to a \ (possibly eccentric) perturber, ", Cell[BoxData[ FormBox["p", TraditionalForm]],ExpressionUUID-> "c5edfa4a-50ec-49bc-8407-21df118f7182"], ". The Hamiltonian governing secular interactions is given by the \ \[OpenCurlyDoubleQuote]double-averaged\[CloseCurlyDoubleQuote] potential:\n\t", Cell[BoxData[{ FormBox[ RowBox[{ SubscriptBox["\[ScriptCapitalH]", "sec"], "=", " ", RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"4", SuperscriptBox["\[Pi]", "2"]}]]}], RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", "0"], RowBox[{"2", "\[Pi]"}]], RowBox[{ RowBox[{"\[DifferentialD]", "l"}], RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", "0"], RowBox[{"2", "\[Pi]"}]], RowBox[{ RowBox[{"\[DifferentialD]", " ", SubscriptBox["l", "p"]}], FractionBox[ RowBox[{"G", " ", SubscriptBox["m", "p"]}], RowBox[{"|", RowBox[{"r", "-", SubscriptBox["r", "p"]}], "|"}]]}]}]}]}]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"\t", RowBox[{"=", " ", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"G", " ", SubscriptBox["m", "p"]}], RowBox[{"4", SuperscriptBox["\[Pi]", "2"], SubscriptBox["a", "p"]}]]}], RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", "0"], RowBox[{"2", "\[Pi]"}]], RowBox[{ RowBox[{"\[DifferentialD]", "l"}], RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", "0"], RowBox[{"2", "\[Pi]"}]], RowBox[{ RowBox[{"\[DifferentialD]", " ", SubscriptBox["l", "p"]}], " ", RowBox[{ SuperscriptBox[ SubscriptBox["\[Sum]", RowBox[{"j", "=", "2"}]], "\[Infinity]"], " ", RowBox[{ SuperscriptBox["\[Alpha]", "j"], " ", SuperscriptBox[ RowBox[{"(", FractionBox["r", "a"], ")"}], "j"], SuperscriptBox[ RowBox[{"(", FractionBox[ SubscriptBox["a", "p"], SubscriptBox["r", "p"]], ")"}], RowBox[{"j", "+", "1"}]], RowBox[{ SubscriptBox["P", "j"], "[", RowBox[{"cos", "(", "\[CapitalPhi]", ")"}], "]"}]}]}]}]}]}]}]}]}]}], TraditionalForm]}],ExpressionUUID-> "f5f87e3d-203a-4b37-9aad-ae44095b89e0"], "\nwhere the second line gives the multi-pole expansion of the potential in \ Legendre polynomials. \n\nI will use a coordinate system in which the ", Cell[BoxData[ FormBox[ RowBox[{"x", "-", "y"}], TraditionalForm]],ExpressionUUID-> "54233830-7dba-4931-86b6-03671532998f"], " plane is the orbital plane of the perturber and the ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "0b3ba169-5a53-483a-b384-d97f752ab739"], " direction is towards the pericenter of the perturber. In this coordinate \ system\n\t", Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{"cos", "[", "\[CapitalPhi]", "]"}], " ", "=", " ", RowBox[{ FractionBox["1", RowBox[{"1", "-", RowBox[{ SubscriptBox["e", "p"], RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}]}]}]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}], "-", SubscriptBox["e", "p"]}], ",", RowBox[{ SqrtBox[ RowBox[{"1", "-", SuperscriptBox[ SubscriptBox["e", "p"], "2"]}]], RowBox[{"sin", "[", SubscriptBox["u", "p"], "]"}]}], ",", "0"}], ")"}], "\[CenterDot]", RowBox[{"(", RowBox[{"X", ",", "Y", ",", "Z"}], ")"}]}]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"\t ", RowBox[{"=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", RowBox[{ SubscriptBox["e", "p"], RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}]}]}], ")"}], RowBox[{"-", "1"}]], "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}], "-", SubscriptBox["e", "p"]}], ")"}], "X"}], " ", "+", " ", RowBox[{ SqrtBox[ RowBox[{"1", "-", SuperscriptBox[ SubscriptBox["e", "p"], "2"]}]], RowBox[{"sin", "[", SubscriptBox["u", "p"], "]"}], "Y"}]}], "]"}]}]}], TraditionalForm]}],ExpressionUUID-> "e8ef5589-9af1-45ec-9143-00517b1e5f35"], "\nwhere ", Cell[BoxData[ FormBox[ SubscriptBox["u", "p"], TraditionalForm]],ExpressionUUID-> "47b61b1b-efc4-4df5-be2b-bb2f41c2f5e5"], " is the perturber\[CloseCurlyQuote]s eccentric anomaly and (", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"X", ",", "Y", ",", "Z"}], ")"}], TraditionalForm]], ExpressionUUID->"16627c12-bb1b-414e-9823-f793e6a7373c"], " are the coordinates of the test particle\[CloseCurlyQuote]s position unit \ vector (explicit expressions to be given below.)\n\n" }], "Text", CellChangeTimes->{{3.7610434872954197`*^9, 3.7610451459485292`*^9}, { 3.761046535170583*^9, 3.761046804455917*^9}, 3.76104687169481*^9, 3.7613870830715027`*^9},ExpressionUUID->"fb739ac8-f006-49a0-9a4b-\ 7aa7593245e9"], Cell[CellGroupData[{ Cell[TextData[{ "Integral of ", Cell[BoxData[ FormBox[ SubscriptBox["l", "p"], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "51de1173-cde5-451c-8ade-86baf5221511"] }], "Subchapter", CellChangeTimes->{{3.761046879077654*^9, 3.761046884528615*^9}},ExpressionUUID->"cf72ef44-3bd6-4a21-bcae-\ 2141e50676ab"], Cell[TextData[{ "Integration over ", Cell[BoxData[ FormBox[ SubscriptBox["l", "p"], TraditionalForm]],ExpressionUUID-> "585f66ef-c6d1-451a-bee9-009ca06ebb01"], " can be done as follow:\n\t- First we will use ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DifferentialD]", SubscriptBox["l", "p"]}], "=", RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{ SubscriptBox["e", "p"], RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}]}]}], ")"}], RowBox[{"\[DifferentialD]", SubscriptBox["u", "p"]}]}]}], TraditionalForm]],ExpressionUUID-> "3bc918bd-0557-421e-97c9-ba20faa4cc89"], "to do the integral over eccentric anomaly\n\n\t- Note that ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SubscriptBox["r", "p"], SubscriptBox["a", "p"]], "=", RowBox[{"1", "-", RowBox[{ SubscriptBox["e", "p"], RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}]}]}]}], TraditionalForm]],ExpressionUUID-> "6992cdf8-5b09-4d1a-8f61-f4e0b264296e"], "\n\n\t- From the expression for cos[\[CapitalPhi]] it is clear that the \ integrand is a series of terms of the form:\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", RowBox[{"e", " ", RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}]}]}], ")"}], RowBox[{"-", "n"}]], RowBox[{ SuperscriptBox["cos", SubscriptBox["k", "c"]], "[", SubscriptBox["u", "p"], "]"}], RowBox[{ SuperscriptBox["sin", SubscriptBox["k", "s"]], "[", SubscriptBox["u", "p"], "]"}]}], TraditionalForm]],ExpressionUUID-> "7302bf0a-fc68-4437-88b5-348c9dcda8a4"], "\n\n\t- If ", Cell[BoxData[ FormBox[ SubscriptBox["k", "s"], TraditionalForm]],ExpressionUUID-> "bc69773c-514b-4d51-80c8-59a37a1213d9"], " is odd then the integrand term is an odd function and its integral=0. \n\n\ \t- If ", Cell[BoxData[ FormBox[ SubscriptBox["k", "s"], TraditionalForm]],ExpressionUUID-> "00721f2d-8e08-4369-b8a2-fd1d75149ec9"], " is even, make the replacement ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"sin", "[", SubscriptBox["u", "p"], "]"}], "2"], "=", RowBox[{"1", "-", SuperscriptBox[ RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}], "2"]}]}], TraditionalForm]], ExpressionUUID->"7cf7e3fd-fbc5-4554-9908-03b454008ee7"], ". Thus, all non-zero integrand terms can be expressed in the form:\n\t\t ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", RowBox[{"e", " ", RowBox[{"cos", "[", SubscriptBox["u", "p"], "]"}]}]}], ")"}], RowBox[{"-", "n"}]], RowBox[{ SuperscriptBox["cos", "k"], "[", SubscriptBox["u", "p"], "]"}]}], TraditionalForm]],ExpressionUUID-> "6dee81ed-5e62-44d2-bbe2-e325882d997c"], "\n\nDefining \n\t ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["I", RowBox[{"n", ",", "k"}]], "(", RowBox[{"a", ",", "e"}], ")"}], "=", RowBox[{ FractionBox["1", RowBox[{"2", "\[Pi]"}]], RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", "0"], RowBox[{"2", "\[Pi]"}]], RowBox[{"\[DifferentialD]", "u", " "}]}]}]}], TraditionalForm]], ExpressionUUID->"62a96f6b-f682-447a-8932-031266e2507d"], " ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "-", RowBox[{"e", " ", RowBox[{"cos", "[", "u", "]"}]}]}], ")"}], RowBox[{"-", "n"}]], SuperscriptBox[ RowBox[{"cos", "[", "u", "]"}], "k"]}], TraditionalForm]], ExpressionUUID->"0c4a0fd8-9a93-4a71-a74f-5c8de2c5ac03"], "\nthe recursion relations:\n\t ", Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ SubscriptBox["I", RowBox[{"1", ",", "0"}]], "(", RowBox[{"a", ",", "e"}], ")"}], "=", FractionBox["1", SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "-", SuperscriptBox["e", "2"]}]]]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["I", RowBox[{"n", ",", "0"}]], "(", RowBox[{"a", ",", "e"}], ")"}], "=", RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"n", "-", "1"}]]}], RowBox[{ FractionBox["\[DifferentialD]", RowBox[{"\[DifferentialD]", "a"}]], RowBox[{ SubscriptBox["I", RowBox[{ RowBox[{"n", "-", "1"}], ",", "0"}]], "(", RowBox[{"a", ",", "e"}], ")"}]}]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["I", RowBox[{"n", ",", "k"}]], "(", RowBox[{"a", ",", "e"}], ")"}], "=", RowBox[{ FractionBox["1", RowBox[{"n", "-", "1"}]], 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