Question

Use the Distance Modulus Formula, M=5logr−5
(where M
is the distance modulus and r
is the distance from Earth to a star in parsecs), and the fact that 1
parsec is approximately 3.3
light years, to find the distance in light years from Earth to a star with distance modulus 0.62.

Answer 1

To find the distance in light years from Earth to a star with a distance modulus \( M = 0.62 \), we can use the Distance Modulus Formula:

\[
M = 5 \log r - 5
\]

First, we will solve for \( r \) (the distance in parsecs). Rearranging the formula, we get:

\[
M + 5 = 5 \log r
\]

Dividing both sides by 5 gives:

\[
\frac{M + 5}{5} = \log r
\]

Now, we can exponentiate both sides to solve for \( r \):

\[
r = 10^{\left(\frac{M + 5}{5}\right)}
\]

Substituting \( M = 0.62 \):

\[
r = 10^{\left(\frac{0.62 + 5}{5}\right)} = 10^{\left(\frac{5.62}{5}\right)} = 10^{1.124}
\]

Now, calculating \( 10^{1.124} \):

\[
r \approx 10^{1.124} \approx 13.34 \text{ parsecs}
\]

Next, we need to convert the distance from parsecs to light years. Given that \( 1 \) parsec is approximately \( 3.3 \) light years, we can convert the distance:

\[
\text{Distance in light years} = r \times 3.3 = 13.34 \times 3.3 \approx 44.0 \text{ light years}
\]

Thus, the distance from Earth to the star is approximately **44.0 light years**.