The apparent brightness of a star is $1.0 \times 10 ^ { - 12 } \mathrm {~W} / \mathrm { m } ^ { 2 }$ and the peak wavelength in its light is 600 $\mathrm { nm }$. Assuming it is the same size as our sun and that it radiates like an ideal blackbody, estimate its distance from us, in parsecs. The constant in Wien's law is $0.00290 \mathrm {~m} \cdot \mathrm { K }$, and 1 parsec is equal to $3.09 \times 10 ^ { 16 } \mathrm {~m} . \left( R _ { \text {sun } } = 6.96 \times 108 \mathrm {~m} , \sigma = 5.67 \times 10 ^ { - 8 } \frac { \mathrm { W } } { \mathrm { m } ^ { 2 } \cdot \mathrm { K } ^ { 4 } } \right)$
A) $1.2 \mathrm { pc }$
B) $120 \mathrm { pc }$
C) $1200 \mathrm { pc }$
D) $12 \mathrm { pc }$

Question

Quizplus · Accepted Answer

First, use Wien's law to find the temperature of the star: $\lambda_{\text{max}} T = b$, where $b = 0.00290 \, \text{m} \cdot \text{K}$ and $\lambda_{\text{max}} = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m}$. Solving for $T$, we get $T = \frac{b}{\lambda_{\text{max}}} = \frac{0.00290}{600 \times 10^{-9}} \approx 4833 \, \text{K}$.Next, use the Stefan-Boltzmann law to find the luminosity $L$ of the star, assuming it has the same radius $R$ as the Sun: $L = 4\pi R^2 \sigma T^4$, where $\sigma = 5.67 \times 10^{-8} \, \text{W} / \text{m}^2 \cdot \text{K}^4$ and $R = 6.96 \times 10^8 \, \text{m}$. Substituting the values, we find $L$.The apparent brightness $B$ is given by $B = \frac{L}{4\pi d^2}$, where $d$ is the distance to the star. Rearranging for $d$ gives $d = \sqrt{\frac{L}{4\pi B}}$. Substituting the given value of $B = 1.0 \times 10^{-12} \, \text{W} / \text{m}^2$ and the calculated value of $L$, we can solve for $d$ in meters.Finally, convert $d$ from meters to parsecs using the conversion $1 \, \text{pc} = 3.09 \times 10^{16} \, \text{m}$. The calculation will show that the distance is closest to $120 \, \text{pc}$, making option B the correct answer.

The Apparent Brightness of a Star Is 1.0×10−12 W/m21.0 \times 10 ^ { - 12 } \mathrm {~W} / \mathrm { m } ^ { 2 }1.0×10−12 W/m2

The Apparent Brightness of a Star Is $1.0 \times 10 ^ { - 12 } \mathrm {~W} / \mathrm { m } ^ { 2 }$