Question

The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is 24,000 K and that it radiates energy at a total rate of 1.0 $\times$ 10${}^{25}$ W. Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius B? (b) What is the peak-intensity wavelength? Is this wavelength visible to humans? (c) What is the radius of Sirius B? Express your answer in kilometers and as a fraction of our sun's radius. (d) Which star radiates more $total$ energy per second, the hot Sirius B or the (relatively) cool sun with a surface temperature of 5800 K? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

Short answer

(a) Total intensity: $1.19\times {10}^{10}{\text{W/m}}^2$. (b) Peak wavelength: 121 nm, not visible. (c) Radius: 5840 km or 0.84% of the sun. (d) Sun radiates 38.46 times more energy.

Step-by-step solution

Total Radiated Intensity Calculation

The total radiated intensity, also known as the power per unit area, can be calculated using the Stefan-Boltzmann Law for blackbody radiation: $I=\frac{P}{A}$, where $I$ is the intensity, $P$ is the power, and $A$ is the surface area. First, we find the area using the formula for the surface area of a sphere, $A=4\pi R^2$. Since we need $I$, let's express the formula using the Stefan-Boltzmann Law directly: $I=\sigma T^4$, where $\sigma =5.67\times {10}^{-8}{\text{W/m}}^2{\text{K}}^4$. For Sirius B, $T=24000\text{K}$. Thus, $I=5.67\times {10}^{-8}\times (24000{)}^4{\text{W/m}}^2$. Calculating this yields $I=1.19\times {10}^{10}{\text{W/m}}^2$.

Calculate Peak-Intensity Wavelength

To find the peak-intensity wavelength, use Wien's Displacement Law: ${\lambda }_{\text{max}}=\frac{b}{T}$, where $b=2.897\times {10}^{-3}\text{m}\cdot \text{K}$. Thus, ${\lambda }_{\text{max}}=\frac{2.897\times {10}^{-3}}{24000}=1.21\times {10}^{-7}$ m, or 121 nm. Since the visible spectrum ranges from about 400 nm to 700 nm, this wavelength is not visible to humans; it is in the ultraviolet range.

Determine the Radius of Sirius B

Using the power equation $P=\sigma A{T}^4$, solve for the radius $R$. We have $A=4\pi R^2$, thus $P=\sigma \cdot 4\pi R^2{T}^4$. Rearranging gives $R^2=\frac{P}{4\pi \sigma T^4}$. Plugging in the values yields: $R=\sqrt{\frac{1.0\times {10}^{25}}{4\pi \cdot 5.67\times {10}^{-8}\cdot (24000{)}^4}}\approx 5.84\times {10}^6$ m or 5840 km. Compared to the sun's radius $R_{\text{sun}}=6.96\times {10}^5$ km, the radius of Sirius B is $\frac{5840}{696000}\approx 0.0084$ or 0.84% of the sun's radius.

Compare Total Energy Radiation Per Second

To compare the total energy radiated by the sun and Sirius B, calculate the ratio $\frac{P_{\text{sun}}}{P_{\text{Sirius B}}}$. Sun's power $P_{\text{sun}}$ can be calculated using $P=\sigma A{T}^4=4\pi R^2\sigma T^4$ and given $T_{\text{sun}}=5800\text{K}$. Replace $T$ and $R$ with sun's values to compute $P_{\text{sun}}$. The accurately measured powered by the sun is approximately $3.846\times {10}^{26}$ W. Therefore, $\frac{3.846\times {10}^{26}}{1.0\times {10}^{25}}=38.46$. The sun radiates 38.46 times more energy than Sirius B.

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in physics that describes how energy is emitted by a blackbody. A blackbody is an idealized object that absorbs all light that hits it and re-emits energy perfectly. The law is expressed by the formula $E=\sigma T^4$, where $E$ is the total energy emitted per unit area, $\sigma$ is the Stefan-Boltzmann constant, equal to $5.67\times {10}^{-8}\text{}ext{W/m}^2\text{}ext{K}^4$, and $T$ is the absolute temperature in Kelvin.
Understanding this law helps us determine the "intensity" or the power per unit area of objects like stars. For example, Sirius B behaves like an ideal blackbody with a surface temperature of 24,000 K. Using the law, its radiated intensity is found to be approximately $1.19\times {10}^{10}\text{}ext{W/m}^2$.
This law is widely used in astrophysics to estimate the thermal radiation of stars and other celestial bodies. Its simplicity and precision make it a powerful tool for scientists studying the universe.

• It explains how energy emission relates to temperature.
• Used to calculate total radiative power from stars.
• Helps understand foundational star characteristics.

Wien's Displacement Law

Wien's Displacement Law helps in determining the most significant wavelength of radiation emitted by a blackbody, which corresponds to the peak intensity. This can be mathematically illustrated as ${\lambda }_{\text{max}}=\frac{b}{T}$, where ${\lambda }_{\text{max}}$ is the wavelength at which the emission intensity is highest, $b$ is a constant $2.897\times {10}^{-3}\text{}extm\cdot \text{}extK$, and $T$ is the temperature.
For Sirius B, with a temperature of 24,000 K, this law helps us find that the peak-intensity wavelength is around 121 nm. This wavelength is in the ultraviolet range, which is outside the visible spectrum (approximately 400-700 nm) and therefore not observable by the human eye.
This concept is crucial in astrophysics for understanding the color and type of electromagnetic radiation heavily emitted by stars. By knowing the temperature of a star, Wien's law allows scientists to infer the most energetic wavelength it emits, leading to significant insights into its physical properties and structure.

• Links temperature and dominant emission wavelength.
• Helps determine if emission is within visible spectrum.
• Useful in classifying stars by their radiation type.

Binary Star Systems

Binary star systems contain two stars that orbit around a common center of mass. They are fascinating both in terms of physics and observation because they provide insights into stellar mass and evolution. Sirius, also known as the Dog Star, is one of the most famous binary systems, with Sirius A being a bright main-sequence star and Sirius B as the white dwarf companion.
In binary systems like Sirius, the properties of each star can be studied individually and in relation to each other. For example, in our exercise, Sirius B's properties, such as its smaller radius compared to the sun, illustrate the diversity of stars in binary systems.
Studying binary stars allows astronomers to gain deeper understanding of stellar masses and how stars interact gravitationally. The study of their dynamics can be essential for determining stellar characteristics like mass, which is otherwise difficult to measure.

• Consists of two stars orbiting a common center.
• Provides critical insights into stellar evolution.
• Helps measure stellar mass through orbital dynamics.