91Ó°ÊÓ

Use the relation between velocity and time period.

Formula:

i)$v=r\mathrm{Ó\neg }=2\mathrm{Ï€}r/T$

ii)$N=\frac{\text{totaltime}}{\text{timeforonerevolution}}$

Where,

v is velocity, r is radius, T is time, N is number of revolutions and $\mathrm{Ó\neg }$ is angular velocity

First, convert$2.3×{10}^4\text{ l²â}$into km as follows:

$r=2.3×{10}^4\text{ly}×\frac{9.46×{10}^{12}\text{ k³¾}}{1\text{ }\text{ly}}=2.18×{10}^{17}\text{km}$

Now,

$v=r\mathrm{Ó\neg }=2\mathrm{Ï€}r/T$

$T=\frac{2\mathrm{Ï€}r}{v}$

$t=\frac{2\mathrm{π}(2.18×{10}^{17})}{250}$

$t=5.5×{10}^{15}\text{sec}$

Hence, it takes $5.5×{10}^{15}\text{ s±ð³¦}$ for the sun to make one revolution about the galactic center.

Now, number of revolution as follows:

First, convert time$4.5×{10}^9\text{year}$into sec,

$4.5×{10}^9×\frac{\text{365 d²¹²â²õ}}{\text{1year}}×\frac{24\text{hour}}{1\text{ d²¹²â}}×\frac{3600\text{​ s±ð³¦}}{1\text{ h´Ç³Ü°ù}}=1.422×{10}^{17}\text{sec}$

Now, number of revolutions are,

$N=\frac{\text{totaltime}}{\text{timeforonerevolution}}$

$N=\frac{1.422×{10}^{17}}{5.5×{10}^{15}}=25.8$

In two significant figures,

$N=26\text{ }\text{revolutions}$

Hence,$26$revolutions have been completed since the sun was found about$4.5×{10}^9$years ago.

Therefore, using the formula for velocity, the time can be found. Using this time, the number of revolutions of the sun can be found.