Question

Arrange the following lengths in descending order of their magnitudes: Light year, parsec, and Astronomical unit.

Answer 1

Hint:We know the distance traveled by light in one year is called light-year(ly). It is used to measure large distances. A parsec is a distance at which one astronomical unit subtends an angle of one second of arc, and the Astronomical unit measures the distance between the earth and the sun. A light-year is also related to astronomy and parsec. For the arrangement of these lengths in descending order, we calculate the values of each.

Complete step by step answer:
We know that one light-year is equal to the distance that light travels in a year.
$\therefore 1ly=c\times 1yr$
Here c is the speed of light in a vacuum, which is equal to $3\times {10}^8{\displaystyle \frac{m}{s}}$ and 1 ly means one light year. Hence on simplifying the above relation, we get,
$$\Rightarrow 1ly=3\times {10}^{8}m/s\times 365\times 24\times 60\times 60s$$.
$$\Rightarrow 1ly=9.46\times {10}^{15}\mathrm{m}$$
The relation between light-year (ly) and astronomical unit (A.U)
$1(\mathrm{l}\mathrm{y})=9.46\times 10^{15}m$ …… (I) and,
$1\mathrm{A}.\mathrm{U}=1.5\times 10^{11}m$ …… (II)
Here 1 A.U. means 1 astronomical unit.
On dividing equations(I) and (II), we get:
${\displaystyle \frac{1\mathrm{l}\mathrm{y}}{1\mathrm{A}.\mathrm{U}}}={\displaystyle \frac{9.456\times {10}^{15}m}{1.5\times {10}^{11}m}}$
$\Rightarrow {\displaystyle \frac{1\mathrm{l}\mathrm{y}}{1\mathrm{A}.\mathrm{U}}}=6.3\times {10}^4$
$\therefore 1\mathrm{l}\mathrm{y}=6.3\times 10^4\mathrm{A}.\mathrm{U}$
The relation between the light-year (ly) and parsec
Again $1\mathrm{l}\mathrm{y}=9.46\times 10^{15}m$ …… (III)
And $1\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}=3.1\times 10^{16}m$…… (IV)
On dividing equations (III) and (IV), we get
${\displaystyle \frac{1\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}}{1\mathrm{l}\mathrm{y}}}={\displaystyle \frac{3.1\times {10}^{16}m}{9.46\times {10}^{15}m}}$
${\displaystyle \frac{1\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}}{1\mathrm{l}\mathrm{y}}}=3.28$
$\therefore 1\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}=3.28\mathrm{l}\mathrm{y}$
Suppose we saw the actual value of light year, parsec, A.U. (Astronomical unit). In that case, we will observe that parsec has greater value than a light-year, and a light year is greater than the Astronomical unit.
Therefore arrange in descending order of their magnitudes parsec>light year>astronomical unit.
$\Rightarrow {10}^{16}>{10}^{15}>{10}^{11}$

Note:The lengths of a light-year, parsec, and astronomical unit are the same and scientifically proven throughout the universe. When we calculate the large distances and very small distances, we use non-S.I units of lengths for such measurements.