QuestionDecember 17, 2025

If Star A has three times the surface temperature of the Sun but has the same luminosity as the Sun, the radius of star A must be __ the radius of the Sun. 1/3 3 times 1/9 81 times

If Star A has three times the surface temperature of the Sun but has the same luminosity as the Sun, the radius of star A must be __ the radius of the Sun. 1/3 3 times 1/9 81 times
If Star A has three times the surface temperature of the Sun but has the same luminosity as the Sun, the radius of star A must be __ the radius of the Sun. 1/3 3 times 1/9 81 times

Solution
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Answer

1/9 Explanation 1. Write the luminosity formula L = 4\pi R^2 \sigma T^4 2. Set up the ratio for equal luminosity \frac{L_A}{L_{Sun}} = \frac{R_A^2 T_A^4}{R_{Sun}^2 T_{Sun}^4} = 1 3. Substitute given values (T_A = 3T_{Sun}) \frac{R_A^2 (3T_{Sun})^4}{R_{Sun}^2 T_{Sun}^4} = 1 4. Simplify the equation \frac{R_A^2 \cdot 81 T_{Sun}^4}{R_{Sun}^2 T_{Sun}^4} = 1 \implies \frac{R_A^2 \cdot 81}{R_{Sun}^2} = 1 5. Solve for R_A/R_{Sun} R_A^2 = \frac{R_{Sun}^2}{81} \implies R_A = \frac{R_{Sun}}{9}

Explanation

1. Write the luminosity formula<br /> $L = 4\pi R^2 \sigma T^4$<br />2. Set up the ratio for equal luminosity<br /> $\frac{L_A}{L_{Sun}} = \frac{R_A^2 T_A^4}{R_{Sun}^2 T_{Sun}^4} = 1$<br />3. Substitute given values ($T_A = 3T_{Sun}$)<br /> $\frac{R_A^2 (3T_{Sun})^4}{R_{Sun}^2 T_{Sun}^4} = 1$<br />4. Simplify the equation<br /> $\frac{R_A^2 \cdot 81 T_{Sun}^4}{R_{Sun}^2 T_{Sun}^4} = 1 \implies \frac{R_A^2 \cdot 81}{R_{Sun}^2} = 1$<br />5. Solve for $R_A/R_{Sun}$<br /> $R_A^2 = \frac{R_{Sun}^2}{81} \implies R_A = \frac{R_{Sun}}{9}$
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