Know that one light-year is equal to 9.5 × 10^15 m

Topic 6.2.2 – Stars

Learning Objective

Students should be able to state that one light‑year is equal to \$9.5 \times 10^{15}\ \text{m}\$ and use this value to compare astronomical distances.

What is a Light‑Year?

A light‑year is the distance that light travels in a vacuum in one Julian year (365.25 days). It is a convenient unit for expressing the vast distances between stars and galaxies.

Deriving the \cdot alue

The speed of light in vacuum is \$c = 3.00 \times 10^{8}\ \text{m s}^{-1}\$. One Julian year contains \$365.25 \times 24 \times 60 \times 60 = 31\,557\,600\ \text{s}\$.

Therefore:

\$\$\text{1 light‑year} = c \times \text{seconds in a year}

= (3.00 \times 10^{8}\ \text{m s}^{-1})(31\,557\,600\ \text{s})

\approx 9.5 \times 10^{15}\ \text{m}\$\$

Using Light‑Years in Calculations

When converting between metres and light‑years, the following relationships are useful:

• \$1\ \text{ly} = 9.5 \times 10^{15}\ \text{m}\$
• \$1\ \text{m} = 1.05 \times 10^{-16}\ \text{ly}\$ (obtained by taking the reciprocal)

Example Problem

Question: The nearest star to the Sun, Proxima Centauri, is about \$4.0 \times 10^{16}\ \text{m}\$ away. Express this distance in light‑years.

Solution:

1. Write the known distance in metres: \$d = 4.0 \times 10^{16}\ \text{m}\$.
2. Divide by the length of one light‑year:

   \$\text{distance in ly} = \frac{4.0 \times 10^{16}\ \text{m}}{9.5 \times 10^{15}\ \text{m/ly}} \approx 4.2\ \text{ly}\$

3. Therefore, Proxima Centauri is approximately \$4.2\$ light‑years from the Sun.

Typical Astronomical Distances

Key Points to Remember

• A light‑year is a measure of distance, not time.
• Use \$c = 3.00 \times 10^{8}\ \text{m s}^{-1}\$ and \$1\ \text{yr} = 3.15576 \times 10^{7}\ \text{s}\$ to derive \$1\ \text{ly} = 9.5 \times 10^{15}\ \text{m}\$.
• When converting, keep track of powers of ten to avoid arithmetic errors.
• Most stellar distances in the Milky Way are expressed in light‑years; inter‑galactic distances are often given in millions or billions of light‑years.