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<Input><Text-field style="Text" layout="Normal"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">JSFH</Equation></Text-field>
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<Input><Text-field style="Text" layout="Normal" alignment="centred"><Font size="26" bold="true" encoding="UTF-8">Cap\303\255tulo 20-Movimento planet\303\241rio</Font></Text-field>
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O movimento planet\303\241rio \303\251 um bom exemplo para se estudar, pois \303\251 poss\303\255vel argumentar que Newton inventou o c\303\241lculo para resolver problemas de movimento planet\303\241rio.

Esta \303\251 uma planilha de demonstra\303\247\303\243o sem exerc\303\255cios.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font size="16" bold="true" encoding="UTF-8">1-Parametriza\303\247\303\243o do movimento el\303\255ptico</Font><Font encoding="UTF-8">

A primeira abordagem ao movimento planet\303\241rio da classe \303\251 tentar tentar parametrizar os caminhos el\303\255pticos dos planetas. Queremos come\303\247ar com a parametriza\303\247\303\243o do movimento circular e sua modifica\303\247\303\243o.

Se queremos mover um c\303\255rculo de randius re per\303\255odo omega, a parametriza\303\247\303\243o mais f\303\241cil \303\251

x (t) = a * cos (2 * Pi * t / \303\264mega), y (t) = a * sin (2 * Pi * t / \303\264mega).

Para a Terra, r = 93 em milh\303\265es de milhas e \303\264mega = 1 em anos.</Font></Text-field>
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Em seguida, queremos levar em conta que a \303\263rbita da Terra \303\251 achatada em uma elipse. A excentricidade da \303\263rbita da Terra \303\251 0,02.

Isso significa que a soma \303\251 deslocada c = e * a = 0,02 * 93 milh\303\265es de milhas fora do centro.

Isso tamb\303\251m significa que o eixo curto \303\251 b = sqrt (a ^ 2-c ^ 2) = a * sqrt (1-e ^ 2) milh\303\265es de milhas.</Font></Text-field>
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Isso leva a uma nova parametriza\303\247\303\243o do movimento da Terra.</Font></Text-field>
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Ainda parece um c\303\255rculo a olho nu.

Se olharmos para a \303\263rbita de Merc\303\272rio, a \303\251 36 milh\303\265es de milhas e e \303\251 0,21.
Tamb\303\251m precisamos calcular o per\303\255odo, que ser\303\241 (36/93) ^ (3/2). (O per\303\255odo \303\251 o tamanho da \303\263rbita, medido em AUs (o tamanho da \303\263rbita da Terra) aumentado para a pot\303\252ncia 3/2.)</Font></Text-field>
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Ainda est\303\241 muito pr\303\263ximo da circular em \303\263rbita, mesmo que notavelmente fora do centro.

Para o cometa de Halley, os n\303\272meros s\303\243o que a = 1680 e e = 0,97.</Font></Text-field>
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Essa \303\263rbita \303\251 claramente uma elipse. Vale notar que nosso modelo teria o per\303\255odo de Merc\303\272rio em 88 dias e o cometa de Haley em 77 anos.

</Font><Font size="16" bold="true">2-Movimentos relativos</Font></Text-field>
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A pr\303\263xima quest\303\243o de interesse \303\251 tra\303\247ar o movimento relativo de dois planetas. Na forma param\303\251trica, simplesmente tomamos a diferen\303\247a das duas parametriza\303\247\303\265es. N\303\243o que queremos que nosso per\303\255odo seja pelo menos o mais longo dos dois per\303\255odos.

Considere Merc\303\272rio e a Terra juntos.</Font></Text-field>
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</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEldG1heEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZBc3NpZ247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSwxLjk5OTk5OTk5OUYnRjlGOQ==">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV0bWF4R0YoJCIrKioqKioqKio+ISIqNyNGLg==</Equation></Text-field>
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Olhando de um planeta para o outro, parece que o planeta est\303\241 parando e come\303\247ando com alguns backups tamb\303\251m.

Considere o que acontece se tentarmos o cometa Terra e o Halley.</Font></Text-field>
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</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEldG1heEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZBc3NpZ247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSwxNTMuNTU2OTU2NEYnRjlGOQ==">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV0bWF4R0YoJCIrayZwYmAiISIoNyNGLg==</Equation></Text-field>
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O movimento est\303\241 obviamente errado. Negligenciamos o fato de que os planetas se movem em velocidades diferentes em suas \303\263rbitas. Eles v\303\243o mais r\303\241pido quando est\303\243o perto do sol.

&gt;

</Font><Font size="16" bold="true">3-Encontrar um caminho determinado pela gravidade</Font></Text-field>
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Se quisermos observar o movimento planet\303\241rio a partir do c\303\241lculo, o que sabemos \303\251 que a \303\272nica for\303\247a que atua nos planetas \303\251 a gravidade e est\303\241 na dire\303\247\303\243o do sol com uma for\303\247a inversamente proporcional \303\240 dist\303\242ncia.

Em termos de equa\303\247\303\265es diferenciais, isso se torna:

x = 'x = r ^ 3, y' '= -y / r ^ 3.

(Observe que x ^ 2 + y ^ 2 = r ^ 2 e que sqrt ((x '') ^ 2+ (y '') ^ 2) = 1 / r ^ 2.)

Trabalhamos apenas com campos vetoriais que s\303\243o equa\303\247\303\265es diferenciais de primeira ordem e esse sistema \303\251 de segunda ordem. Lidamos com esse problema definindo u = x 'e v = y' para obter um sistema de 4 equa\303\247\303\265es de primeira ordem.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Em seguida, podemos ver onde o planeta est\303\241 \303\240s vezes Pi / 2, Pi, 3Pi / 2 e 2Pi. </Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8"> Isso permite que, com 6 pontos decimais de precis\303\243o, o caminho seja um c\303\255rculo com o per\303\255odo 2Pi. </Font>Podemos representar graficamente os resultados para ver isso mais claramente. </Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZBc3NpZ247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIxRidGOUY5">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoIiIiNyNGLg==</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJQ3hHRigvLUkieEdGKDYjIiIhJCIiIyEiIjcjRi4=</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJQ3ZHRigvLUkidkdGKDYjIiIhJCIrKysrK0khIio3I0Yu</Equation></Text-field>
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Vemos que obtemos uma elipse para a \303\263rbita.

Agora estamos prontos para retornar \303\240 quest\303\243o das \303\263rbitas relativas.

Considere primeiro o problema da terra e Merc\303\272rio. Vamos medir em unidades astron\303\264micas.</Font></Text-field>
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</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjYWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSkmQXNzaWduO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMUYnRjlGOQ==">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNhZUdGKCIiIjcjRi4=</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0bWF4ZUdGKCwkSSNQaUdGJiIiIzcjRi4=</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVElSUN4ZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZBc3NpZ247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiM2Ji1GLDYlUSJ4RidGL0YyLUY2Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZYLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRidGOS8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGPUY5RjlGOS1GNjYtUSkmZXF1YWxzO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRmpuNiRRJTAuOThGJ0Y5Rjk=">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJQ3hlR0YoLy1JInhHRig2IyIiISQiIykqISIjNyNGLg==</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJQ3ZlR0YoLy1JInZHRig2IyIiISQiK2hTPz81ISIqNyNGLg==</Equation></Text-field>
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Observe a ligeira mudan\303\247a no caminho relativo dos dois planetas. Alguns dos loops internos s\303\243o claramente menores que outros.

Agora tentamos o mesmo truque com o cometa Halley.</Font></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjYWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSkmQXNzaWduO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMUYnRjlGOQ==">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNhZUdGKCIiIjcjRi4=</Equation></Text-field>
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Algumas coisas a serem observadas nos c\303\241lculos.

No ponto mais pr\303\263ximo, o cometa Halley est\303\241 dentro da \303\263rbita da Terra e se move quase duas vezes mais r\303\241pido.

Se um ano \303\251 2Pi, o cometa de Halley orbita uma vez a cada (482 / (2Pi) = 76,7 anos.

Vale a pena examinar a parte da \303\263rbita mais pr\303\263xima.</Font></Text-field>
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A raz\303\243o para a forma estranha do caminho \303\251 que temos o cometa e a Terra mais pr\303\263xima do sol ao mesmo tempo e do mesmo lado. Isso faz com que o cometa passe pela terra por dentro.

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