(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 11.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1064, 20] NotebookDataLength[ 275940, 6123] NotebookOptionsPosition[ 265031, 5778] NotebookOutlinePosition[ 266918, 5842] CellTagsIndexPosition[ 266875, 5839] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Binarias eclipsantes y cruvas de luz", "Title"], Cell[TextData[{ "Guillermo S\[AAcute]nchez. ", ButtonBox["diarium.usal.es/guillermo", BaseStyle->"Hyperlink", ButtonData->{ URL["http://diarium.usal.es/guillermo"], None}, ButtonNote->"/diarium.usal.es/guillermo"] }], "Author"], Cell["\[CapitalUAcute]ltima actualizaci\[OAcute]n: 2016-09-30", "Date"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Paquete becl (Eclipsantes binarias y curvas de luz)" }], "Section"], Cell["\<\ En este cuaderno utilizaremos el paquete becl (binarias eclipsantes y curvas \ de luz), desarrollado por Guillermo S\[AAcute]nchez dentro del proyecto. \ SA130U14 de la Junta de Castilla y Le\[OAcute]n.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Sistemas binarios" }], "Section"], Cell["\<\ De las estrellas que observamos m\[AAcute]s de la mitad son sistemas de dos o \ m\[AAcute]s estrellas. Normalmente no es posible con telescopio distinguir \ las distintas estrellas que forman los sistemas m\[UAcute]ltiples (no \ confundir con binarias visuales que normalmente se trata de estrellas que \ parecen pr\[OAcute]ximas vistas desde la Tierra pero que realmente son \ estrellas de sistemas distintos muy distantes entre s\[IAcute]). La detecci\ \[OAcute]n se suele realizar por el m\[EAcute]todo del tr\[AAcute]nsito que \ consiste en analizar la variaci\[OAcute]n de luminosidad que se produce \ cuando en un sistema binario (o multiple) una estrella se interpone entre la \ otra estrella y el observador en la Tierra (es decir: las \[OAcute]rbitas de \ las dos estrellas y el observador estan en un mismo plano) lo que provoca una \ reducci\[OAcute]n de la luz que llega al observador, las binarias con estas \ carater\[IAcute]sticas se conocen como binarias eclipsantes. \ \>", "Text"], Cell["\<\ Este mismo m\[EAcute]todo tambi\[EAcute]n se aplica a la busqueda de planetas \ extrasolares, en este caso es el planeta el que se interpone entre la \ estrella y el observador en Tierra, naturalmente el cambio de luminosidad es \ mucho menor que los cambios de luz que experimentan las binarias eclipsantes,\ \>", "Text"], Cell[TextData[{ "A continuaci\[OAcute]n vamos a describir como construir un sistema de este \ tipo con ", StyleBox["Mathematica", FontSlant->"Italic"], " y estudiar su luminosidad. Nos basaremos en el art\[IAcute]culo disponible \ en ", ButtonBox["http://www.physics.sfasu.edu/astro/ebstar/ebstar.html", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.physics.sfasu.edu/astro/ebstar/ebstar.html"], None}, ButtonNote->"http://www.physics.sfasu.edu/astro/ebstar/ebstar.html"] }], "Text"], Cell["\<\ La figura de abajo representa un sistema binario (aunque nos referiremos a \ dos estrellas el m\[EAcute]todo es similar si consideramos una estrella y un \ planeta). Las coordenadas del centro son (x1, y1, z1) para la estrella 1 y \ (x1, y1, z1) para la estrella 2 (Al suponer que son esferas \ homog\[EAcute]neas los centros de masa coinciden con los centros \ geom\[EAcute]tricos). Como origen de coordenadas {0, 0, 0} se toma el centro \ de masas del sistema. Las coordenadas se han elegido de forma que el \ observador, que se supone en la Tierra, est\[AAcute] situado en el eje OZ. Es \ decir: el que va desde el centro de masas al observador. Como eje OY se toma \ la perpendicular a OZ en el plano que forma la estrella 1 cuando esta forma \ un \[AAcute]ngulo \[Theta] = 0, lo que ocurre cuando esta pasa por el punto \ m\[AAcute]s pr\[OAcute]ximo al observador. \ \>", "Text"], Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJztnQlcFOX/x0fuQ9MKLC28L/DsJ2r9UfHKH4pKiqmliCBemcppiQeKiqYo hCdqgBwqaHniUVqUWL8ES/DgEEFDpZSfWSk3/v6fnSe2dVnWXdhjdvm+XwOv 2dmZ7z4zO/Pe55mZ5zvtPRdN8DTkOM61Ccf9hD/R+P9ooIEGGmiggQYaaKBB PwaCIAhCHZBpCYIg1A2ZliAIQt2QaQmCINQNmZYgCELdkGkJgiDUDZmWIAhC 3ZBpCYIg1A2ZliAIQt2QaQmCINQNmZYgCELdkGkJgiDUDZmWIAhC3ZBpCYIg 1A2ZliAIQt2QaQmCINQNmZYgCELdkGkJgiDUDZm2EVBUVBQfH79RLnFxcZhN 2yUlCD2FTKsXVFVVlZSUVFZWynz30qVLgwcPNjU1bdKkCcdxBgYGZmZmFjwm JiYcT79+/S5evKjhYhNEY4FMqxekpqbOmTPn5MmTMt8tLi4+ceLErFmzIFtI 1crKysfHJ4Zn8uTJZFqCUDtkWr1g7969lpaWGzZskDMPZIuabSuesLCwNJ6F CxcaGhq2b9/e3d392rVrGiswQTQuyLR6gSKmRc02PT3dzc0NNduuXbv252nT po2Zmdny5cuh2cePH2uswATRuCDT6jiSZwbkm5axePFi7lksLCyio6PVX1KC 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Se expresan en unidades \ arbitrarias, lo normal es tomar como valor unidad la distancia media de \ separaci\[OAcute]n entre las estrellas. En el caso de una \[OAcute]rbita \ circular es una constante, y con el criterio anterior R=1. \ \>", "ItemParagraph"], Cell["\<\ m1, m2 = masa de las estrellas (pueden elegirse unidades arbitrarias pues lo \ importante es que se mantenga la relaci\[OAcute]n q = m2/m1). \ \>", "ItemParagraph"], Cell["\<\ r1, r2 = radios. Por conveniencia tomaremos r1>r2, de hecho en la \ pr\[AAcute]ctica en un sistema binario siempre pueden elegirse la estrella 1 \ y 2 de forma que se verifique esta condici\[OAcute]n. Los radios es \ conveniente expresarlos en fracciones de la longitud del semieje mayor. En el \ caso de una \[OAcute]rbita circular obviamente el semieje mayor es igual al \ menor e id\[EAcute]ntico al radio R.\ \>", "ItemParagraph"], Cell[TextData[{ "\[Theta] = Fase, corresponde al \[AAcute]ngulo que forma la estrella \ respecto a OZ, tomando por \[Theta] = 0 aquel en el que el centro de la \ estrella 2 est\[AAcute] m\[AAcute]s pr\[OAcute]ximo al observador. En un \ sistema binario concreto con orbita circular \[Theta] es el par\[AAcute]metro \ variable (va variando con el tiempo) mientras que los par\[AAcute]metros \ anteriores suelen ser constantes (aunque no siempre es as\[IAcute], por ej.: \ Pueden producirse cambios intr\[IAcute]nsecos de la luminosidad, incluso la \ masa puede experimentar cambios como ocurre cuando se va trasfiriendo parte \ de la masa de una estrella a otra pero en lo que sigue nos referimos al caso \ m\[AAcute]s simple, que adem\[AAcute]s es el m\[AAcute]s frecuente). Se \ dice que \[Theta] es el \[AAcute]ngulo azimutal y cumple ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"\[Theta]", " ", "="}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"2", " ", "\[Pi]", " ", RowBox[{"(", RowBox[{"t", "-", "t0"}], ")"}]}], "P"], TraditionalForm]]], ", donde P es el per\[IAcute]odo orbital." }], "ItemParagraph"] }, Open ]], Cell["\<\ En un sistema binario las coordenadas {x1, y1, z1} y {x2, y2, z2} de los \ puntos en los que est\[AAcute] cada estrella est\[AAcute]n dados por:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"x1", "=", FractionBox[ RowBox[{"-", "x"}], RowBox[{"1", "+", RowBox[{"(", RowBox[{"1", "/", "q"}], ")"}]}]]}], ",", " ", RowBox[{"y1", "=", FractionBox[ RowBox[{"-", "y"}], RowBox[{"1", "+", RowBox[{"(", RowBox[{"1", "/", "q"}], ")"}]}]]}], ",", RowBox[{ RowBox[{"z1", "=", " ", FractionBox[ RowBox[{"-", "z"}], RowBox[{"1", "+", RowBox[{"(", RowBox[{"1", "/", "q"}], ")"}]}]]}], ";"}]}]], "ItemParagraph", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"x2", "=", FractionBox["x", RowBox[{"1", "+", "q"}]]}], ",", RowBox[{"y2", "=", " ", FractionBox["y", RowBox[{"1", "+", "q"}]]}], ",", RowBox[{ RowBox[{"z2", "=", FractionBox["z", RowBox[{"1", "+", "q"}]]}], ";"}]}]], "ItemParagraph", InitializationCell->True], Cell["\<\ Que podemos trasformarlos a coordenadas esf\[EAcute]ricas mediante la \ trasformacion : \ \>", "Item"] }, Open ]], Cell[BoxData[{ RowBox[{ RowBox[{"x", " ", "=", " ", RowBox[{"R", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"y", " ", "=", " ", RowBox[{"R", " ", RowBox[{"Cos", "[", "i", "]"}], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"z", " ", "=", " ", RowBox[{"R", " ", RowBox[{"Sin", "[", "i", "]"}], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ";"}]}], "Input"], Cell["\<\ Entonces la posici\[OAcute]n de cada estrella en funci\[OAcute]n 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Curvas de luz en sistemas binarios" }], "Section"], Cell["\<\ El m\[EAcute]todo del tr\[AAcute]nsito, al que antes nos hemos referido, se \ basa en el estudio de la variaci\[OAcute]n de la lumininosidad que se produce \ cuando una estrella o un planeta se interpone delante de la otra estrella. \ Como se ha indicado, las estrellas binarias raramente pueden ser \ distinguibles visualmente, y menos a\[UAcute]n una estrella y un planeta, lo \ que se observar\[AAcute] ser\[AAcute] una variaci\[OAcute]n de la luminosidad \ aparente. \ \>", "Text"], Cell["\<\ El m\[EAcute]todo que aqu\[IAcute] se describe, con la llegada de las camaras \ astr\[OAcute]nomicas CCD, est\[AAcute] al alcance del buen aficionado a \ astronom\[IAcute]a. Hay alg\[UAcute]n caso donde aficionados han llegado a \ detectar planetas extrasolares empleando esta t\[EAcute]cnica. Es un ejemplo, \ de c\[OAcute]mo puede hacerse ciencia sin necesidad de grandes instalaciones. \ \>", "Text"], Cell[TextData[{ "La luminosidad ", StyleBox["l", FontSlant->"Italic"], " de una estrella est\[AAcute] definida como la cantidad de energ\[IAcute]a \ que escapa de la estrella en la unidad de tiempo. El flujo F es la energia \ emitida por unidad de superficie y unidad de tiempo, esto es F1 = ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["l", "1"], RowBox[{"4", "\[Pi]", " ", SuperscriptBox["R1", "2"]}]], TraditionalForm]]], " y F2 = ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["l", "2"], RowBox[{"4", "\[Pi]", " ", SuperscriptBox["R2", "2"]}]], TraditionalForm]]], " donde ", Cell[BoxData[ FormBox[ SubscriptBox["l", "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["l", "2"], TraditionalForm]]], " = luminosidades (En unidades arbitrarias, lo que importa es que se \ mantenga la relacion ", Cell[BoxData[ FormBox[ SubscriptBox["l", "2"], TraditionalForm]]], "/", Cell[BoxData[ FormBox[ SubscriptBox["l", "1"], TraditionalForm]]], "). " }], "Text"], Cell["\<\ Entonces un observardor ver\[AAcute] al sistema con una luminosidad dada por:\ \>", "Text"], Cell[CellGroupData[{ Cell["l = K (F1 A1 + F2 A2) ", "ItemParagraph"], Cell["\<\ donde A1 y A2 es el \[AAcute]rea del disco de cada estrella tal como es visto \ por el observador y K es una constante que puede ser determinada a partir del \ \[AAcute]rea del detector del observador, que est\[AAcute] en el eje OZ, y la \ distancia entre la Tierra y el sistema binario. \ \>", "ItemParagraph"] }, Open ]], Cell["\<\ Para encontrar estas \[AAcute]reas necesitamos conocer la distancia aparente \ \[Rho] entre las dos estrellas tal como es vista por el observador (situado \ en el eje OZ). La cual est\[AAcute] dada por: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", "=", SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "1"], "-", SubscriptBox["x", "2"]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["y", "1"], "-", SubscriptBox["y", "2"]}], ")"}], "2"]}]]}], TraditionalForm]], "ItemParagraph"], Cell[TextData[{ "La siguiente funci\[OAcute]n calcula \[Rho](R, i, \[Theta], q) siendo: ", Cell[BoxData[ RowBox[{ RowBox[{"x", " ", "=", " ", RowBox[{"R", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], ";", RowBox[{"y", " ", "=", " ", RowBox[{"R", " ", RowBox[{"Cos", "[", "i", "]"}], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ";", RowBox[{ RowBox[{"{", RowBox[{"x1", ",", "y1"}], "}"}], "=", RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"-", "x"}], RowBox[{"1", "+", RowBox[{"(", RowBox[{"1", "/", "q"}], ")"}]}]], ",", " ", FractionBox[ RowBox[{"-", "y"}], RowBox[{"1", "+", RowBox[{"(", RowBox[{"1", "/", "q"}], ")"}]}]]}], "}"}]}], ";", RowBox[{ RowBox[{"{", RowBox[{"x2", ",", "y2"}], "}"}], "=", RowBox[{"{", " ", RowBox[{ FractionBox["x", RowBox[{"1", "+", "q"}]], ",", " ", FractionBox["y", RowBox[{"1", "+", "q"}]]}], "}"}]}], ";"}]], CellChangeTimes->{ 3.50338048299699*^9, {3.503380524497364*^9, 3.503380526332469*^9}, 3.50338116082576*^9, 3.5033815617146893`*^9, {3.526229566069276*^9, 3.526229578705298*^9}}] }], "Item"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"rho", "[", RowBox[{"2", ",", " ", RowBox[{"2", " ", "Pi", " ", RowBox[{"85", "/", "360"}]}], ",", " ", "\[Theta]", " ", ",", " ", RowBox[{"0.1", "/", "0.6"}]}], " ", "]"}]], "Input"], Cell[BoxData[ SqrtBox[ RowBox[{ RowBox[{"0.030384493975583876`", " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"]}], "+", RowBox[{"4.`", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]}]]], "Output"] }, Open ]], Cell[TextData[{ "Con los datos del ejemplo (esto es R =2, i = 2 Pi 85 /360; ", Cell[BoxData[ FormBox[ SubscriptBox["m", "1"], TraditionalForm]]], " = 0.6; ", Cell[BoxData[ FormBox[ SubscriptBox["m", "2"], TraditionalForm]]], " = 0.1) la distancia aparente \[Rho] var\[IAcute]a de acuerdo con la \ siguiente figura. Con el criterio que hemos tomado el menor tama\[NTilde]o \ aparente corresponde al de m\[AAcute]xima interposi\[OAcute]n que se produce \ para \[Theta] = 0, Pi, 2 Pi, ..., ", StyleBox["n", FontSlant->"Italic"], " Pi." }], "Item"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"rho", "[", RowBox[{"2", ",", " ", RowBox[{"2", " ", "Pi", " ", RowBox[{"85", "/", "360"}]}], ",", "\[Theta]", " ", ",", RowBox[{"0.1", "/", "0.6"}]}], " ", "]"}], ",", " ", RowBox[{"{", " ", RowBox[{"\[Theta]", ",", "0", ",", " ", RowBox[{"4", " ", "Pi"}]}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\[Theta]", ",", "\[Rho]"}], "}"}]}]}], "]"}], " "}]], "Input"], Cell[BoxData[ GraphicsBox[{{{}, {}, TagBox[ {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[ 1.], LineBox[CompressedData[" 1:eJw1m3c41e//x63seVD24XCOUYkiFXq9iDZKaRspDaSorESbKJUiSWhZhVBU MpLsMjML2bI3h3Pe3/fnd12/v1zPyzn3+36t+34+zkHF8ay1ExcHB0c8NwfH 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Cell[CellGroupData[{ Cell["\<\ Para calcular A1 y A2 es necesario tener en cuenta la etapa del ciclo en la \ que nos encontramos. \ \>", "ItemParagraph"], Cell[TextData[{ "En las figuras que siguen la estrella de radio mayor (", Cell[BoxData[ FormBox[ SubscriptBox["r", "1"], TraditionalForm]]], ") la representamos en color rojo, y la de radio menor (", Cell[BoxData[ FormBox[ SubscriptBox["r", "2"], TraditionalForm]]], ") en color azul." }], "Item"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", "2"}], "]"}], ",", "Blue", ",", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"4", ",", "0"}], "}"}], ",", "1"}], "]"}]}], "}"}], "]"}]], "Input", CellID->392018628], Cell[BoxData[ GraphicsBox[{ {RGBColor[1, 0, 0], DiskBox[{0, 0}, 2]}, {RGBColor[0, 0, 1], DiskBox[{4, 0}]}}]], "Output", ImageSize->{104, 100}, ImageMargins->{{0, 0}, {0, 0}}, 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Por tanto, es \ cuando la luminosidad ser\[AAcute] m\[AAcute]xima. En esta etapa etapa se \ verifica ", Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", ">", RowBox[{ SubscriptBox["r", "1"], "+", SubscriptBox["r", "2"]}]}], TraditionalForm]]], " con: " }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ SubscriptBox["A", "1"], TraditionalForm]]], " = \[Pi] ", Cell[BoxData[ FormBox[ SubsuperscriptBox["r", "1", "2"], TraditionalForm]]], " y ", Cell[BoxData[ FormBox[ SubscriptBox["A", "2"], TraditionalForm]]], " = \[Pi] ", Cell[BoxData[ FormBox[ SubsuperscriptBox["r", "2", "2"], TraditionalForm]]], " " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["b) Eclipse parcial ", "Subsubsubsection"], Cell[TextData[{ "Ocurre cuando una estrella eclipsa parcialmente a la otra. En esta etapa \ se verifica que ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["r", "1"], "+", SubscriptBox["r", "2"]}], ">", "\[Rho]", ">", RowBox[{ SubscriptBox["r", "1"], "-", SubscriptBox["r", "2"]}]}], TraditionalForm]]], " . " }], "Text"], Cell[TextData[{ "La funci\[OAcute]n de abajo la hemos dibujado utilizando:\n", StyleBox["Graphics[{{Red, Circle[{0, 0}, 1]}, {Blue, Circle[{1.15, 0}, 0.5, \ {-2 Pi/3, 2 Pi/3}]}, {Dashed, Blue, Circle[{1.15, 0}, 0.5]}, Line[{{0, 0}, \ {0.9, 0.425}, {0.9, -0.425}, {0, 0}}], Line[{{0.9, 0.425}, {1.15, 0} , {0.9, \ -0.425} }] }]", "Input"], "\nA la salida generada con la funci\[OAcute]n anterior le hemos \ a\[NTilde]adido leyendas con la herramienta de gr\[AAcute]ficos 2D: [Ctrl]+[D]" }], "Item", CellID->312736669], Cell[BoxData[ GraphicsBox[{ {RGBColor[1, 0, 0], CircleBox[{0, 0}]}, {RGBColor[0, 0, 1], CircleBox[{1.15, 0}, 0.5, NCache[{Rational[-2, 3] Pi, Rational[2, 3] Pi}, {-2.0943951023931953`, 2.0943951023931953`}]]}, {RGBColor[0, 0, 1], Dashing[{Small, Small}], CircleBox[{1.15, 0}, 0.5]}, LineBox[{{0, 0}, {0.9, 0.425}, {0.9, -0.425}, {0, 0}}], LineBox[{{0.9, 0.425}, {1.15, 0}, {0.9, -0.425}}], {GrayLevel[0.], AbsolutePointSize[3.], AbsoluteThickness[0.5], Opacity[1.], Dashing[{}], Arrowheads[0.04], EdgeForm[None], FaceForm[None], StyleBox[InsetBox[Cell["\[Phi]2", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline], {0.19440514842300533, -0.01736564625850368}, { Left, Baseline}, Offset[{24.000000000000014, 14.}, {0., 0.}], {{ 1.5000000000000004`, -9.143013143971876*^-17}, {0., 0.8235294117647055}}], FontColor->GrayLevel[0.]]}, {GrayLevel[0.], AbsolutePointSize[3.], AbsoluteThickness[0.5], Opacity[1.], Dashing[{}], Arrowheads[0.04], EdgeForm[None], FaceForm[None], StyleBox[InsetBox[Cell["\[Phi]1", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline], {1.0355896335807044, -0.02632803803339523}, { Left, Baseline}, Offset[{15., 10.266666666666664}, {0., 0.}], {{ 0.9374999999999996, -2.2988147333415003`*^-16}, {0., 0.6039215686274504}}], FontColor->GrayLevel[0.]]}, {GrayLevel[0.], AbsolutePointSize[3.], AbsoluteThickness[0.5], Opacity[1.], Dashing[{}], Arrowheads[0.04], EdgeForm[None], FaceForm[None], StyleBox[InsetBox[Cell["r2", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline], {0.4283662518037513, 0.2860753324056894}, { Left, Baseline}, Offset[{14.733333333333333, 13.}, {0., 0.}], {{ 1.133333333333333, 0.}, {0., 0.764705882352941}}], FontColor->GrayLevel[0.]]}, {GrayLevel[0.], AbsolutePointSize[3.], AbsoluteThickness[0.5], Opacity[1.], Dashing[{}], Arrowheads[0.04], EdgeForm[None], FaceForm[None], StyleBox[InsetBox[Cell["r1", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline], {1.1157390228818787, 0.1644428726035868}, { Left, Baseline}, Offset[{14.933333333333314, 16.}, {0., 0.}], {{ 1.1487179487179482`, 0.}, {0., 0.941176470588235}}], FontColor->GrayLevel[0.]]}, {GrayLevel[0.], AbsolutePointSize[3.], AbsoluteThickness[0.5], Opacity[1.], Dashing[{}], Arrowheads[0.04], EdgeForm[None], FaceForm[None], StyleBox[InsetBox[Cell["\[CapitalDelta]A1", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline], {0.7347093382807666, -0.03401008812615914}, { Left, Baseline}, Offset[{36., 15.}, {0., 0.}], {{1.2413793103448274`, 0.}, {0., 0.8823529411764703}}], FontColor->GrayLevel[0.]]}, {GrayLevel[0.], AbsolutePointSize[3.], AbsoluteThickness[0.5], Opacity[1.], Dashing[{}], Arrowheads[0.04], EdgeForm[None], FaceForm[None], StyleBox[ ArrowBox[{{1.1264938930117496`, -0.21197758194186767`}, { 0.9421246907854042, -0.012244279529993685`}}], FontColor->GrayLevel[0.]]}, {GrayLevel[0.], AbsolutePointSize[3.], AbsoluteThickness[0.5], Opacity[1.], Dashing[{}], Arrowheads[0.04], EdgeForm[None], FaceForm[None], StyleBox[InsetBox[Cell["\[CapitalDelta]A2", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline], {1.1725861935683357, -0.22990236549165027}, { Left, Baseline}, Offset[{26., 19.999999999999996}, {0., 0.}], {{ 0.8965517241379309, 0.}, {0., 1.176470588235294}}], FontColor->GrayLevel[0.]]}}]], "Text"], Cell["\<\ Para calcular la luz que llega de la estrella eclipsada hay que restar del \ \[AAcute]rea total el \[AAcute]rea eclipsada. Para ello hay quecalcular \ \[CapitalDelta]A1 y \[CapitalDelta]A2 que corresponde a los segmentos \ correspondiente a la estrella 1 y 2, que esta dada por:\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[CapitalDelta]A1", " ", "=", " ", RowBox[{ FractionBox["1", "2"], SuperscriptBox["R1", "2"], RowBox[{"(", " ", RowBox[{"\[Phi]1", "-", " ", RowBox[{"Sin", "[", "\[Phi]1", "]"}]}], ")"}]}]}], ";", " ", RowBox[{"\[CapitalDelta]A2", " ", "=", " ", RowBox[{ FractionBox["1", "2"], SuperscriptBox["R2", "2"], RowBox[{"(", " ", RowBox[{"\[Phi]2", "-", " ", RowBox[{"Sin", "[", "\[Phi]2", "]"}]}], ")"}]}]}], ";"}], TraditionalForm]], "ItemParagraph"], Cell["Las siguiente expresiones nos permiten calcular \[Phi]1 y \[Phi]2", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["R2", "2"], " ", "=", " ", RowBox[{ SuperscriptBox["R1", "2"], "+", SuperscriptBox["\[Rho]", "2"], " ", "-", RowBox[{"2", " ", "R1", " ", "\[Rho]", " ", RowBox[{"Cos", "[", 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{RGBColor[0, 0, 1], DiskBox[{-2, 0}]}}], ",", GraphicsBox[{ {RGBColor[1, 0, 0], DiskBox[{0, 0}, 2]}, {RGBColor[0, 0, 1], DiskBox[{2.5, 0}]}}]}], "}"}]], "Output", ImageSize->{104, 100}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, GraphicsBoxOptions->{ImageSize->{100, Automatic}}] }, Open ]], Cell[TextData[{ "Las dos figuras de la izquierda corresponden al caso en que la estrella 1 \ (roja) es la m\[AAcute]s pr\[OAcute]xima, esto es ", Cell[BoxData[ FormBox[ RowBox[{"z1", ">", "z2"}], TraditionalForm]]], ", entonces la luz que llega al observador ser\[AAcute]:" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["A", "1"], " ", "=", " ", RowBox[{ RowBox[{"\[Pi]", " ", SubsuperscriptBox["r", "1", "2"], " ", "y", " ", SubscriptBox["A", "2"]}], " ", "=", " ", RowBox[{ RowBox[{"\[Pi]", " ", SubsuperscriptBox["r", "2", "2"]}], " ", "-", " ", "\[CapitalDelta]A1", "-", " ", "\[CapitalDelta]A2", " "}]}]}], TraditionalForm]], "ItemParagraph"], Cell["\<\ El dibujo de la 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La funci\[OAcute]n mostrar\[AAcute] el \[AAcute]rea que el observador \ medir\[AAcute]. " }], "Text"], Cell["\<\ Las funciones siguientes estan incluidas en el paquete, aqui se muestran pero \ no se ejecutan.\ \>", "Text"], Cell[TextData[{ StyleBox["Sin eclipse", FontWeight->"Bold"], ".- Ocurre cuando se verifica ", Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", "\[GreaterEqual]", RowBox[{ SubscriptBox["r", "1"], "+", SubscriptBox["r", "2"]}]}], TraditionalForm]]] }], "Text"], Cell["\<\ area[R_, i_, q_, r1_, r2_, l1_, l2_, \[Theta]_] := Module[{A1, A2}, {A1, A2} \ = {Pi r1^2 , Pi r2^2 }; l1 A1/r1^2 + l2 A2/r2^2] /; rho[R, i, \[Theta], q] >= r1 + r2\ \>", "Program", Evaluatable->False, InitializationCell->False], Cell[TextData[{ StyleBox["Eclipse parcial", FontWeight->"Bold"], ". 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"DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 1.5}, {0., 1.}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Determinaci\[OAcute]n del periodo a partir de la curvas de luz" }], "Section"], Cell["\<\ La variaci\[OAcute]n de la luminosidad de una estrella adem\[AAcute]s de a la \ presencia de variables eclipsantes o a la interposici\[OAcute]n de planetas \ puede deberse a distintas razones. Quiz\[AAcute]s la m\[AAcute]s conocida es \ el cambio de luminosidad que se produce en estrellas variables, como las del \ tipo cefeida a las que antes nos hemos referido. Tambien puede estudiarse los \ cambios de luz en otro tipo de astro, por ejemplo: algunos asteroides muy \ irregulares, como Eros, al girar presentan distinta luminosidad asociada a la \ luz que reflejam . Sea la causa que sea la construcci\[OAcute]n experimental \ de una curva de luz consiste en tomar medidas de la luminosidad en distintos \ momentos t y a partir de ellas deducir el periodo con el que se repite el \ ciclo.\ \>", "Text"], Cell["\<\ La siguiente expresi\[OAcute]n simula las medidas de la magnitud en funci\ \[OAcute]n del tiempo (en la pr\[AAcute]ctica no se conoce esta \ funci\[OAcute]n y frecuentemente de lo que se trata es de deducirla a partir \ de datos experimentales. La funci\[OAcute]n elegida es muy simple y la \ utilizamos s\[OAcute]lo con prop\[OAcute]sitos did\[AAcute]cticos)\ \>", "Item"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"mag", "[", "t_", "]"}], "=", RowBox[{"5", "+", RowBox[{"0.14", RowBox[{"Cos", "[", RowBox[{ RowBox[{"0.3", " ", "t"}], " ", "+", " ", "0.1"}], "]"}]}]}]}], ";"}], " "}]], "Input"], Cell[TextData[{ "Suponemos que hemos tomado las medidas en distintos momentos ", Cell[BoxData[ FormBox[ SubscriptBox["t", "i"], TraditionalForm]]] }], "Item"], Cell[BoxData[ RowBox[{ RowBox[{"data", " ", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"t", ",", " ", RowBox[{"Round", "[", RowBox[{ RowBox[{"mag", "[", "t", "]"}], ",", " ", "0.01"}], "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "200", ",", " ", "5"}], "}"}]}], "]"}]}], ";"}]], "Input"], Cell[TextData[{ "Representemos gr\[AAcute]ficamente los datos anteriores. Si solo dispusi\ \[EAcute]semos de los datos ser\[IAcute]a dif\[IAcute]cil darse cuenta cual \ es la funci\[OAcute]n subyacente (pruebe a eliminar ", StyleBox["Joined\[Rule] True", "Input"], "), unimos los puntos pues de esa forma es f\[AAcute]cil observar su car\ \[AAcute]cter c\[IAcute]clico. Lo normal es que falten muchos puntos \ intermedios aqu\[IAcute] representados, lo que dificulta m\[AAcute]s su an\ \[AAcute]lisis." }], "Item"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ListPlot", "[", RowBox[{"data", ",", " ", RowBox[{"Joined", "\[Rule]", " ", "True"}]}], "]"}]], "Input"], Cell[BoxData[ GraphicsBox[{{}, {{}, {}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.012833333333333334`], AbsoluteThickness[1.6], StyleBox[LineBox[CompressedData[" 1:eJxTTMoPSmViYGDQBGIQDQP9h75qxPSLOEB46LSKQ6Et1/XFBcJQvp7Dm8Ad cq2vYXwTB1T9lg6aMSARGN/OoVpknfvDKqj6BkeHs2dAAKbfxaH1NchAqPoG NwePh1VAHTD9nlDzYfq9HZYUgBwE0+/nkJ4GAlD1DgGo/IZAVPUHglDNYwhB tc8hFOpemP4wVPceCEf1D0Mkqn8dolDDoyEaNbwOxKCFZxxqeDvEo+p3SHAQ BTsHyn+QgKq/IdEByAGKQPkKSajheSDJwQ6k3BbKT0iGysPsT3EAG78Oyl+Q AjUfZn+qA9h7MTD7Ux1mzQQBmP1pqOGpkO4QBPJuIMz+dNTwTMhwOAxSfghm fyY8PAFyz6sJ "]], FontFamily->"Arial"]}}, {}, {}, {{}, {}}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 4.846000000000001}, BaseStyle->{FontFamily -> "Arial"}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 200.}, {4.86, 5.14}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output"] }, Open ]], Cell["\<\ El m\[EAcute]todo de an\[AAcute]lisis al que normalmente se recurre es a \ representar el fen\[OAcute]meno anterior en un diagrama de fases\ \>", "Text"], Cell["\<\ Para calcular la fase tenemos que aplicar la ecuaci\[OAcute]n siguiente:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Phi]", " ", "=", RowBox[{"Parte", " ", "decimal", " ", RowBox[{"de", "[", FractionBox[ RowBox[{"t", "-", SubscriptBox["t", "0"]}], "P"], "]"}]}]}]], "ItemParagraph"], Cell["donde: ", "ItemParagraph"], Cell[TextData[{ "\[Phi] representa la fase en ciclos\n", Cell[BoxData[ FormBox[ SubscriptBox["t", "0"], TraditionalForm]]], " (normalmente llamado", StyleBox[" \[EAcute]poca", FontSlant->"Italic"], ") representa el origen de la variable tiempo expresado en fecha juliana \ (DJ)\n", StyleBox["t", FontSlant->"Italic"], " es el momento donde se toma la medida (normalmente se va a medir la \ magnitud aparente, m)\nP es el periodo, es decir el tiempo que tarda en \ repetirse un ciclo." }], "ItemParagraph"], Cell["\<\ Una expresion equivalente a la anterior, que es la que usaremos, es \ \>", "ItemParagraph"], Cell[BoxData[ RowBox[{"\[Phi]", " ", "=", RowBox[{"Mod", "[", FractionBox[ RowBox[{"t", "-", SubscriptBox["t", "0"]}], "P"], "]"}]}]], "ItemParagraph"], Cell[TextData[{ "La siguiente expresion ", StyleBox["Mathematica", FontSlant->"Italic"], " obtiene la fase a partir de los datos iniciales y del per\[IAcute]odo. " }], "ItemParagraph"], Cell["\<\ Se muestra la funcion pero no se evalua, esta funci\[OAcute]n esta \ incorporada en el paquete\ \>", "ItemParagraph", FontSize->9], Cell["\<\ phi[list_, p_] := Module[{data}, data = list; Transpose[{Mod[data[[All, \ 1]]/p, 1], data[[All, 2]]}]];\ \>", "ItemParagraph", Evaluatable->False], Cell["\<\ En este caso hemos supuesto que el per\[IAcute]odo es conocido, pero en la pr\ \[AAcute]ctica es lo que se busca. Podemos comprobar que si utilizamos un \ valor aproximado del periodo la representaci\[OAcute]n es m\[AAcute]s difusa. \ \>", "ItemParagraph"], Cell[TextData[{ "En las observaciones reales el peridod hay que determinarlo y ello no \ siempre es sencillo. La siguiente funci\[OAcute]n nos ayudar\[AAcute] a \ calcularlo. Se tiene en cuenta que en las curvas magnitud-fase se invierte el \ orden en la representaci\[OAcute]n del eje OY. Esto es, las magnitudes se \ ordenan de mayor a menor siguen el orden inverso al habitual, es decir", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", "3"}], ">", RowBox[{"-", "2"}], ">", "0", ">", "1", ">", "2"}]}], TraditionalForm]]], "." }], "ItemParagraph"], Cell["\<\ Se muestra la funcion pero no se evalua, esta funci\[OAcute]n esta \ incorporada en el paquete\ \>", "Item"] }, Open ]], Cell["\<\ curveL[data_, per_, kmax_] := Module[{mag, a, b, k, i}, mag = \ Round[Transpose[data][[2]], 0.1]; Manipulate[ListPlot[Join[phi[data, per + k] \ /. {a_, b_} -> {a, -b}, phi[data, per + k] /. {a_, b_} -> {-(1 - a), -b}], \ AxesLabel -> {\"Per\[IAcute]odo\", \"Magnitud\"}, AxesOrigin -> {-1, \ -Max[mag] - 0.1}, Ticks -> {Automatic, Table[{i, -i}, {i, -Max[mag] - 0.1, \ -Min[mag] + 0.1, (Max[mag] - Min[mag])/10}]} ], {{k, 0, \"Desviaci\[OAcute]n \ respecto el per\[IAcute]odo (d\[IAcute]as)\"}, 0, kmax}]]\ \>", "Program", Evaluatable->False], Cell["\<\ Supongamos un ciclo completo (2 \[Pi]). Utilizamos la funci\[OAcute]n \ anterior para simular las medidas experimentales e incluimos una componente \ aleatoria. 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Recu\[EAcute]rdese que una \ mayor luminosidad num\[EAcute]ricamente implican una menor magnitud. Por ello \ para representar los datos en un diagrama de fase/luminosidad suponemos ", Cell[BoxData[ FormBox[ RowBox[{"k", " ", "=", RowBox[{"-", "1"}]}], TraditionalForm]]], "." }], "Text"], Cell[TextData[{ "Ejecute la siguiente funci\[OAcute]n: ", StyleBox["curveL[Map[Times[{1,-1},#]&,simulation],15, 20]", "Input"], " y desplace el bot\[OAcute]n deslizante hasta que el per\[IAcute]odo sea 5 \ d\[IAcute]as. Comprobar\[AAcute] que la curva es muy similar a la de la \ figura anterior." }], "ItemParagraph"], Cell[CellGroupData[{ Cell["Ejemplo: Curva de luz de la binaria eclipsante NSV 03199", \ "Subsubsubsection"], Cell[TextData[{ "Vamos a aplicar la funci\[OAcute]n desarrollada a los datos de luminosidad \ de la NSV 03199 obtenidos por Garcia-Melendo, E.,Henden, A., A., 1998, IBVS, \ No. 4546. 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Kepler: un sat\[EAcute]lite a la busqueda de planetas extrasolares" }], "Section"], Cell["\<\ Desde 1995 en que se encontr\[OAcute] el primer planeta extrasolar se han \ detectado m\[AAcute]s de 400. \ \>", "Text"], Cell["\<\ Para la localizaci\[OAcute]n de planetas extrasolares se utilizan dos \ procedimientos: \ \>", "Text"], Cell["\<\ a) Detecci\[OAcute]n del movimiento del centro de masas de la estrella.- Un \ planeta que orbita entorno a una estrella provoca un desplazamiento peridico \ del centro de masas del sistema. Visto desde la Tierra se observa una \ desplazamiento minusculo de la estrella provocado por su traslaci\[OAcute]n \ de la Tierra en torno al centro de masas. Estos movimiento s\[OAcute]lo son \ detectables para planetas gigantes pr\[OAcute]ximos a estrellas, \ \>", "Text"], Cell["\<\ b) Tr\[AAcute]nsito del planeta delante de la estrella.- Cuando el planeta \ se interpone entre la estrella y el observador en la Tierra provoca una \ reducci\[OAcute]n de la luz que llega al observador (es el mismo \ m\[EAcute]todo que el utilizado en el estudio de variables binarias \ eclipsantes a las que antes nos hemos referido). Este \[UAcute]ltimo m\ \[EAcute]todo se est\[AAcute] mostrando como el m\[AAcute]s eficaz y es el \ empleado por el sat\[EAcute]lite Kepler.\ \>", "Text"], Cell[TextData[{ "El sat\[EAcute]lite Kepler (", ButtonBox["http://www.nasa.gov/kepler", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.nasa.gov/kepler"], None}, ButtonNote->"http://www.nasa.gov/kepler"], ") lanzado el 6 de marzo de 2009 dise\[NTilde]ado espec\[IAcute]ficamente \ para la localizaci\[OAcute]n de planetas extrasolares. Kepler analiza la \ curvas de luz de mas de 100000 estrellas situdas en la constelaci\[OAcute]n \ de Piscis. En la actualidad (febrero 2012) el Kepler ha detectado mas de 1000 \ estrellas que potencialmente pueden contener sistemas planetarios, \ previsiblemente cuando lea esto la cifra anterior se habr\[AAcute] \ incrementado sustancialmente y se habr\[AAcute] identificado cuales de estas \ estrellas candidatas contienen realmente planetas." }], "Text"], Cell["\<\ Como se ha indicado el trabajo fundamental del Kepler es obtener curvas de \ luz de estrellas. En algunos casos la variaciones de luminosidad \ corresponderan a sistemas planetarios pero en la mayoria de los ocasiones se \ deber\[AAcute] a la observaci\[OAcute]n de sistemas de estrellas binarias \ eclipsantes. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Recursos adicionales" }], "Section"], Cell[TextData[{ "Las curvas de luz, y los datos correspondientes, de las eclipsantes \ binarias obtenidas por Kepler pueden descargarse de ", ButtonBox["http://archive.stsci.edu/kepler/eclipsing_binaries.html", BaseStyle->"Hyperlink", ButtonData->{ URL["http://archive.stsci.edu/kepler/eclipsing_binaries.html"], None}, ButtonNote->"http://archive.stsci.edu/kepler/eclipsing_binaries.html"], "\ny en ", ButtonBox["http://keplerebs.villanova.edu", BaseStyle->"Hyperlink", ButtonData->{ URL["http://keplerebs.villanova.edu"], None}, ButtonNote->"http://keplerebs.villanova.edu"], ". La descripci\[OAcute]n de los par\[AAcute]metros utilizados puede \ encontrarse en R.W. Slawson et al., 2011, AJ, 142, 160 (", ButtonBox["http://arxiv.org/pdf/1103.1659v1.pdf", BaseStyle->"Hyperlink", ButtonData->{ URL["http://arxiv.org/pdf/1103.1659v1.pdf"], None}, ButtonNote->"http://arxiv.org/pdf/1103.1659v1.pdf"], ")." }], "Text"], Cell[TextData[{ "En Demostrations: ", ButtonBox["http://demonstrations.wolfram.com/topic.html?topic=Astronomy", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/topic.html?topic=Astronomy"], None}, ButtonNote->"http://demonstrations.wolfram.com/topic.html?topic=Astronomy"] }], "Text"], Cell[TextData[{ "En Eric Weisstein Word of Astronomy: ", ButtonBox["http://scienceworld.wolfram.com/astronomy", BaseStyle->"Hyperlink", ButtonData->{ URL["http://scienceworld.wolfram.com/astronomy/"], None}, ButtonNote->"http://scienceworld.wolfram.com/astronomy/"] }], "Text"], Cell[TextData[{ "Ejercicios b\[AAcute]sicos de astronomia: Astroex (", ButtonBox["http://www.astroex.org", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.astroex.org"], None}, ButtonNote->"http://www.astroex.org"], ")" }], "Text"], Cell["\<\ Los siguientes libros son unas introducciones excelentes y actualizadas de \ Astronom\[IAcute]a y Astrofisica editadas por Pearson/Addison Wesley: \ \>", "Text"], Cell[TextData[{ "\tAstronomy Today por", StyleBox[" Chaisson y McMillan", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "\tAn Introduction to Modern Astrophysics por ", StyleBox["C. Ostlie", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "Para el seguimiento de objetos de luminosidad variable (estrellas \ variables, seguimientos de supernovas y mucho m\[AAcute]s) el mejor sitio \ es: AAVSO | American Association of Variable Star Observers (", ButtonBox["http://www.aavso.org", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.aavso.org"], None}, ButtonNote->"http://www.aavso.org"], ") " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Carga del paquete: becl (BinariasEclipsantesCurvasLuz). 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