The nearest star to our solar system is $4.29$ light years away. How much is this distance in terms of parsecs? How much parallax would this star (named Alpha Centauri) show when viewed from two locations of the Earth six months apart in its orbit around the Sun?

Calculate the solid angle subtended by the periphery of an area of $1 \, cm^2$ at a point situated symmetrically at a distance of $5 \, cm$ from the area. (in $, sr$)

Explain the concepts of accuracy and precision in measurement with an example.

$A$ $LASER$ is a source of very intense,monochromatic,and unidirectional beam of light. These properties of a laser light can be exploited to measure long distances. The distance of the Moon from the Earth has been already determined very precisely using a laser as a source of light. $A$ laser light beamed at the Moon takes $2.56 \; s$ to return after reflection at the Moon's surface. How much is the radius of the lunar orbit around the Earth?

The principle of 'parallax' is used in the determination of distances of very distant stars. The baseline $AB$ is the line joining the Earth's two locations six months apart in its orbit around the Sun. That is,the baseline is about the diameter of the Earth's orbit $\approx 3 \times 10^{11} \; m$. However,even the nearest stars are so distant that with such a long baseline,they show parallax only of the order of $1''$ (second of arc) or so. $A$ parsec is a convenient unit of length on the astronomical scale. It is the distance of an object that will show a parallax of $1''$ (second of arc) from opposite ends of a baseline equal to the distance from the Earth to the Sun. How much is a parsec in terms of metres?

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