Consider a planet moving around a star in an elliptical orbit with period T . area of elliptical orb

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7.Gravitation

hard

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

$T^{4 / 3}$

$T$

$T^{2 / 3}$

$T^{1 / 2}$

Solution

(a)

Area of ellipse

$A=\pi r_1 r_2$

$\because r_1=a-a e=a(1-e)$

$\text { and } r_2=a+a e=a(1+e)$

$A=\pi\{a(1-e)\}\{a(1+e)\}$

$=\pi a^2(1-e)(1+e)$

$=\pi a^2\left(1^2-e^2\right)$

$\because e^2 \ll a^2 \text { then }$

$A=\pi a^2$

$\text { So, } a \propto A^{1 / 2}$

$\text { According to Keplar's $III$ law }$

$T^2 \propto a^3$

$T^2 \propto\left[(A)^{1 / 2}\right]^3$

$T^2 \propto A^{3 / 2}$

$A \propto\left(T^2\right)^{2 / 3}$

$A \propto T^{4 / 3}$

Standard 11

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