Question

Calculate the temperature of the solar surface, using the total solar irradiance \(S\), the solar radius \(R_{s}=6.96 \times 10^{8}[\mathrm{~m}]\) and the sun-earth distance \(R_{s e}=1.50 \times 10^{11}[\mathrm{~m}]\). Use the assumption that in this regard the sun behaves as a black body.

Short answer

The solar surface temperature is approximately 5778 K.

Step-by-step solution

Understanding the Problem

The problem asks us to calculate the temperature of the sun's surface using given parameters related to irradiance. The sun is assumed to behave like a black body, which will allow us to use the Stefan-Boltzmann law to find the temperature.

Write Down Known Values

The given values are: solar radius \(R_s = 6.96 \times 10^8 \text{ m}\), sun-earth distance \(R_{se} = 1.50 \times 10^{11} \text{ m}\), and total solar irradiance \(S = 1361 \text{ W/m}^2\). The Stefan-Boltzmann constant is \(\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4\).

Link Irradiance and Emission

The total solar irradiance \(S\) observed at Earth is related to the sun's emission via the inverse square law: \(S = \frac{L}{4\pi R_{se}^2}\), where \(L\) is the luminosity of the sun.

Find Sun's Luminosity

Rearrange the formula for irradiance to solve for \(L\):\[L = S \times 4\pi R_{se}^2 = 1361 \times 4\pi (1.50 \times 10^{11})^2.\] Calculate \(L\).

Apply Black Body Radiation Law

The luminosity \(L\) is also given by the Stefan-Boltzmann law for a black body: \(L = 4 \pi R_s^2 \sigma T_s^4\). Rearrange to solve for the sun's surface temperature \(T_s\):\[T_s = \left(\frac{L}{4 \pi R_s^2 \sigma}\right)^{1/4}.\]

Calculate Sun's Surface Temperature

Substitute \(L\) and known values into the formula:\[T_s = \left(\frac{1361 \times 4\pi (1.50 \times 10^{11})^2}{4 \pi (6.96 \times 10^8)^2 \times 5.67 \times 10^{-8}}\right)^{1/4}.\]Simplify and compute \(T_s\).

Key concepts

Black Body Radiation

When we talk about black body radiation, we're looking at an idealized physical body that absorbs all incident electromagnetic radiation. This means no radiation passes through it or gets reflected by it.
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• A black body in thermal equilibrium emits radiation based on its temperature.
• Stefan-Boltzmann Law describes the power radiated from the black body.
• The law is given by: \(E = \sigma T^4\), where \(\sigma\) is the Stefan-Boltzmann constant and \(T\) is the temperature of the black body in Kelvin.

In real-world applications like our sun, we assume it to behave like a black body. This helps us to use the mathematical formulas to find properties like the temperature of its surface by analyzing its light emission.

Solar Surface Temperature

The surface temperature of the sun is one critical aspect of understanding its behavior and energy output. We calculate it by assuming the sun behaves like a black body. Using the Stefan-Boltzmann law, the temperature can be found if the luminosity is known.
\[\]The solar surface temperature \(T_s\) can be derived using the formula: \[T_s = \left(\frac{L}{4 \pi R_s^2 \sigma}\right)^{1/4},\] where \(L\) is the sun's luminosity, \(R_s\) is the solar radius, and \(\sigma\) is the Stefan-Boltzmann constant.
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• Knowing the irradiance \(S\) at Earth's orbit helps calculate \(L\), using: \(L = S \times 4\pi R_{se}^2\), where \(R_{se}\) is the distance from the Earth to the sun.
• By solving this equation, we can determine the temperature of the sun's surface, which is crucial for understanding solar phenomena.

Solar Irradiance

Solar irradiance refers to the power per unit area received from the sun in the form of electromagnetic radiation.
\[\]It is critical for calculations involving energy received from the sun and is measured in watts per square meter (W/m²). Also known as the solar constant, it varies slightly as Earth orbits the sun, but is around 1361 W/m² at Earth on average.
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• This value is crucial for determining the sun's luminosity using the formula: \(S = \frac{L}{4\pi R_{se}^2}\).
• This equation helps understand the total energy output of the sun and is pivotal in climate studies, solar panel technology, and more.

Overall, solar irradiance acts as a critical link between the energy emitted by the sun and what we receive here on Earth.

Luminosity

Luminosity is the total amount of energy emitted by a star like the sun per unit of time and is an intrinsic property of the star itself.
\[\]Measured in watts (W), it differs from apparent brightness which is how bright a star appears from Earth. Luminosity provides insight into the sun's energy output.
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• It is calculated using the formula for solar irradiance: \(L = S \times 4\pi R_{se}^2\), where \(S\) is the solar irradiance, and \(R_{se}\) is the distance from the sun to Earth.
• Once \(L\) is known, it can be used to find the star's surface temperature with the Stefan-Boltzmann Law: \(L = 4\pi R_s^2 \sigma T_s^4\).

Understanding luminosity is crucial for astrophysics as it helps determine other astronomical characteristics, such as mass and potential lifespan of the star.