Question

1 parsec is equal to
A. 2.3 light years
B. 3.3 light years
C. 4.3 light years
D. 4.3 light years

Answer 1

Hint: Parsecond and light years are units of length. The distances between planets and stars in outer space are measured in these units. One parsec is the distance, which subtends a parallax angle of one arcsecond. One light year the distance that light travels in one year.

Formula used:
$\tan \alpha ={\displaystyle \frac{1AU}{d}}$
$1\text{arcsec}={\left({\displaystyle \frac{1}{3600}}\right)}^{\circ }.{\displaystyle \frac{\pi }{{180}^{\circ }}}$
$\text{distance = speed }\times \text{ time}$

Complete step by step answer:

Parsec is a unit of length. 1 parsec is a huge distance, due to which the unit is used in the measure of large distances in the study of astronomy. Parsec is derived using trigonometry and parallax. One parsec is the distance which subtends a parallax angle of one arcsecond.
The radius of earth’s orbit (mean distance between earth and sun) is equal to one astronomical unit (AU). One AU is equal to 150 million kilometres, which is equal to $1.5\times {10}^{11}$ metres.
The parallax angle is denoted by $\alpha$.
The parallax angle in terms of one AU and the distance (d) that subtends the parallax angle $\alpha$ is given as,
$\tan \alpha ={\displaystyle \frac{1AU}{d}}$. …….. (i).
Since the parallax angle is very much small, $\tan \alpha$ is approximately equal to $\alpha$.
i.e. $\tan \alpha \approx \alpha$. ……… (ii).
From equations (i) and (ii) we get that,
 $\tan \alpha \approx \alpha ={\displaystyle \frac{1AU}{d}}$.
We can write the above equation as $d={\displaystyle \frac{1AU}{\alpha }}$. …….. (iii).
Now, by the definition of one parsec, one parsec is the value of d when $\alpha$ is equal to one arcsecond.
One arcsecond is equal to ${\displaystyle \frac{1}{3600}}$degrees. We know that $1^{\circ }$ is equal to ${\displaystyle \frac{\pi }{{180}^{\circ }}}$.
Therefore, $1\text{arcsec}={\left({\displaystyle \frac{1}{3600}}\right)}^{\circ }.{\displaystyle \frac{\pi }{{180}^{\circ }}}=\alpha$ .
Substitute the value of 1 AU and $\alpha$ in equation (iii).
Therefore, $d={\displaystyle \frac{1.5\times {10}^{11}m}{{\left({\displaystyle \frac{1}{3600}}\right)}^{\circ }.{\displaystyle \frac{\pi }{{180}^{\circ }}}}}=3.1\times {10}^{16}m$. ………(1)
Therefore, the value of one parsec in metres is equal to $3.1\times {10}^{16}$ metres.
To convert 1 parsec to light years, first, let us convert 1 light years into metres.
One light year is the distance that light travels in one year. We know that light travels about $3\times {10}^8$ metres per second.
$1\text{year}=365\times 24\times 60\times 60\text{seconds}$.
So, by using the formula $\text{distance = speed }\times \text{ time}$, the total distance (D) that light travels in one year is
$D=(3\times {10}^{8}m{s}^{-1})\times (365\times 24\times 60\times 60\text{seconds})\approx 9.46\times {10}^{15}m$ ------ (2).
Therefore, one light year is approximately equal to $9.46\times {10}^{15}m$.
To convert one parsec to light years, simply divide (1) by (2).
i.e. $1\text{parsec}={\displaystyle \frac{3.1\times {10}^{16}}{9.46\times {10}^{15}}}=0.33\times 10=3.3\text{light years}$
Hence, the correct option is B.

Note: Units like parsec and light years are very large units of length. These units cannot be used in our day-to-day life because on earth, the distances we deal with are in metres, kilometres, and parsec and light years are very much larger than kilometres. Astronomers and scientists doing research on outer space use parsec and light years.