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(A) The area $A$ of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is given by $A = \pi ab$.For an elliptical orbit,the semi-minor axis $

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$T^{4/3}$

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(A) The area $A$ of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is given by $A = \pi ab$.For an elliptical orbit,the semi-minor axis $b$ is related to the semi-major axis $a$ and eccentricity $e$ by $b = a\sqrt{1-e^2}$.Thus,$A = \pi a^2 \sqrt{1-e^2}$.Since $e$ is a constant for a given orbit,$A \propto a^2$.According to Kepler's $III$ law of planetary motion,the square of the time period $T$ is proportional to the cube of the semi-major axis $a$:$T^2 \propto a^3$$a \propto T^{2/3}$Substituting this into the area relation:$A \propto (T^{2/3})^2$$A \propto T^{4/3}$Therefore,the area of the elliptical orbit is proportional to $T^{4/3}$.

Consider a planet moving around a star in an elliptical orbit with period $T$. The area of the elliptical orbit is proportional to ...........

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