The difference between the length of a mean solar day and a sidereal day is approximately .......... minutes.

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?

What is parallax? Explain the parallax method to measure the distance between the Earth and a planet.

Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes $0.01 \, cm^3$ of oleic acid per $cm^3$ of the solution. Then you make a thin film of this solution (monomolecular thickness) of area $4 \, cm^2$ by considering $100$ spherical drops of radius $\left(\frac{3}{40 \pi}\right)^{1/3} \times 10^{-3} \, cm$. Then the thickness of the oleic acid layer will be $x \times 10^{-14} \, m$. Where $x$ is ...... .

$A$ man wishes to estimate the distance of a nearby tower from him. He stands at a point $A$ in front of the tower $C$ and spots a very distant object $O$ in line with $AC$. He then walks perpendicular to $AC$ up to $B$,a distance of $100 \; m$,and looks at $O$ and $C$ again. Since $O$ is very distant,the direction $BO$ is practically the same as $AO$; but he finds the line of sight of $C$ shifted from the original line of sight by an angle $\theta = 40^{\circ}$ ($\theta$ is known as 'parallax'). Estimate the distance (in $m$) of the tower $C$ from his original position $A$.

$(a)$ 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.