$A$ far-off planet is estimated to be at a distance $D$ from the Earth. If its diametrically opposite extremes subtend an angle $\theta$ at an observatory situated on the Earth,the approximate diameter of the planet is .............

Match List-$I$ (Event) with List-$II$ (Order of the time interval for happening of the event) and select the correct option from the options given below the lists:

List-$I$	List-$II$
$(1)$ Rotation period of earth	$(i)$ $10^5\, s$
$(2)$ Revolution period of earth	$(ii)$ $10^7\, s$
$(3)$ Period of light wave	$(iii)$ $10^{-15}\, s$
$(4)$ Period of sound wave	$(iv)$ $10^{-3}\, s$

The Sun's angular diameter is measured to be $1920^{\prime \prime}$. The distance $D$ of the Sun from the Earth is $1.496 \times 10^{11} \ m$. What is the diameter of the Sun?

$(a)$ How many astronomical units $(AU)$ make $1$ parsec?
$(b)$ Consider a sun-like star at a distance of $2$ parsecs. When it is seen through a telescope with $100$ magnification,what should be the angular size of the star? The Sun appears to be $(\frac{1}{2})^{\circ}$ from the Earth. Due to atmospheric fluctuations,the eye cannot resolve objects smaller than $1$ arc minute.
$(c)$ Mars has approximately half of the Earth's diameter. When it is closest to the Earth,it is at about $\frac{1}{2} AU$ from the Earth. Calculate what size it will appear when seen through the same telescope.

$1'' = \dots \text{rad}$

What is meant by the angular diameter of the moon?