Question

use a calculator to solve each problem. round your answers to the nearest hundredth. what is the absolute magnitude of a star that has a period of 62 days?

Explanation:

To determine the absolute magnitude \( M \) of a star with a period \( P \) (in days), we use the Cepheid variable star period - luminosity relation (also known as the Leavitt law), which is often expressed as:

\[ M = -2.76(\log_{10} P - 1) - 4.16 \]

Step 1: Calculate \( \log_{10} P \)

Given \( P = 62 \) days, compute \( \log_{10}(62) \). Using a calculator:
\[ \log_{10}(62) \approx 1.7924 \]

Step 2: Substitute into the formula

Substitute \( \log_{10} P \approx 1.7924 \) into the Leavitt law:

\[

$$\begin{align*} M &= -2.76(1.7924 - 1) - 4.16 \\ &= -2.76(0.7924) - 4.16 \\ &\approx -2.187 - 4.16 \\ &\approx -6.347 \end{align*}$$

\]

Step 3: Round to the nearest hundredth

Rounding \( -6.347 \) to the nearest hundredth (two decimal places) gives \( -6.35 \).

Answer:

\( \boldsymbol{-6.35} \)