Question

How many earth years are in 1 light year?

Answer 1

This is a case of a misleading question. Recall what physical quantities are measured with earth years and light years as units. Though both have the word ‘year’ in them, one of them is not a measure of time. Determine which one is the misleading unit and correctly redefine it with an appropriate proof, possibly using the speed of light as a mathematical aid.

Formula Used:
1 earth year = 356 days
1 light year = 94608 $\times 10^7$ km

Complete Solution:
It is a common misconception to consider a light year to be a unit of time because of its terminology. However, light year is a unit that is used to describe the distance that light travels in a span of one earth year.

An earth year is a measure of time, and signifies the duration of one year on earth, which is nothing but the amount of time the earth takes to complete one revolution around the Sun. This is significant while comparing the differential times taken by the other planets in the Solar System.

We know that the speed of light in vacuum or space is a fundamental physical constant that remains unchanged and has a value of:
$c \approx 3 \times 10^{8}\;ms^{-1}$.

Now, a year on earth has about $356$ days and each day has about $24$ hours and each hour is made up of $3600$ seconds. Therefore, the number of seconds in a year is given by:
$t = 365 \times 24 \times 3600 = 31,536,000\;s$

We know that distance can be expressed as a product of velocity and time. Therefore, the distance travelled by light in a span of 1 year, (which is nothing but 1 light year) is given as:
$d = 1\;light\;year = c \times t = 3 \times 10^{8} \times 31,536,000 = 94608 \times 10^{11}\;m$

To summarize, earth year is a measure of time whereas light year is a measure of distance.
1 earth year (yr) = $356$ days
1 light year (ly) = $9.461 \times 10^{12}\;km = 9.461$ trillion km

Note:
Usually, distance between objects within a star system tend to be small fractions of a light year and are usually expressed in astronomical units (au). 1 au is defined as the distance between the Earth and the Sun and is equivalent to $\dfrac{1}{63241}\;ly$.
Units such as light minute, light hour and light day are also used in science publications. Light also travels at approximately one foot in one nanosecond $(10^{-9}\;s)$ and the term light-foot is used to measure time.