$2($a$)$ The Event Horizon Telescope $($EHT$)$ project utilizes several radio telescopes so as to have a large effective area. Assume that the effective diameter of EHT is about ${10}^{4}km$ and that its operating wavelength is $1\mathrm{.}3$ millimetre.

$($i$)$ Estimate the angular resolution of EHT in units of arc$-$second.

$($ii$)$ The supermassive black hole $(SMBH$ lying at the centre of the Andromeda galaxy, which is at a distance $2\mathrm{.}5{10}^6$ light$-$years from Earth, weighs about $2{10}^8{M}_{o.}.$ Verify through calculations whether the EHT would be able to resolve the region at a distance $2-5R_S$ from the centre of the SMBH$.$ Here $R_S$ is the Schwarzschild radius of a black hole.

$($iii$)$ A star is going around this SMBH in a circular orbit of radius ${10}^{17}cm.$ Estimate the speed of the star.

$($iv$)$ In a supernova explosion that occurs at a distance $7{10}^{16}cm$ from the SMBH$,$ the resulting neutron star that is formed is ejected with a speed $75k\frac{m}{s}$ relative to the $SMBH.$ Will the neutron star be able to escape from the gravity of the $SMBH?$

$($b$)$ A hymn in a Vedic text prescribes that a vedic ritual of carrying out Yajna by lighting up the sacred fire be conducted when Kritika Nakshatra $($i$.$e$.$ Pleiades constellation$)$ rises on the horizon during the Vernal Equinox $($i$.$e$.$ Spring Equinox$).$ Explain qualitatively how one may infer that this Vedic hymn was composed around $2334$ years BCE.

$3($a$)$ Using Virial Theorem, obtain a relation between average temperature, mass and radius of a self$-$gravitating gas cloud.

Show that when such a gas cloud loses energy to the environment, its average temperature increases. What happens if energy from outside is added to such a self$-$gravitating gas cloud and why?

$($b$)$ Giving reasons, explain why during the Spring Equinox as well as