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Consider a sun like a star at a distance of $$2 parsec$$ . When it is seen through a telescope with magnification, what should be angular size of the star? Sun appears to be $$ \left(\dfrac{1}{2}\right)^{o} $$ from the earth due to atmospheric fluctuations, eye cannot resolve objects smaller than $$1\ arc\ minute $$.

Solution
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$$1\ A.U.$$ subtends angle of sun's diameter $$ =\left(\dfrac{1}{2}\right)^{o} $$ As the distance from sun increases angle subtended in the same ratio.
So $$ 2 \times 10^{5} $$ A.U. (one parsec) distant star will form an angle of $$ \theta=\left(\dfrac{1 / 2}{2 \times 10^{5}}\right)^{0}=\left(\dfrac{1}{4 \times 10^{5}}\right)^{0} $$ so angle subtended by $$1\ parsec$$ on earth of size (diameter of sun) is :
$$ \dfrac{1}{4 \times 10^{5}} $$ as the sunlike star's diameter is same as sun.
If sun like star is at 2 parsec then angle at earth by star becomes :
$$ =\dfrac{1}{4 \times 10^{5}} \times \dfrac{1}{2}=\dfrac{1}{8} \times 10^{-5}=\left(1.25 \times 10^{-6}\right)^{0} $$
Angle $$ =\left(1.25 \times 10^{-6}\right) $$ minute$$ =\left(75 \times 10^{-6}\right) \min $$
When these sun like star is seen by telescope of magnification $$100$$ , then angle formed by sun like star becomes
$$ 75 \times 10^{-6} \times 100=75 \times 10^{-4}=7.5 \times 10^{-3} $$ minute which is very less than 1minute as eyes cannot resolve object or image smaller than one minute.
So cannot be observed by given telescope.

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