Question

What changes would you notice if the sun emitted radiation at an effective temperature of \(2000 \mathrm{~K}\) instead of \(5762 \mathrm{~K}\) ?

Short answer

Answer: If the sun emitted radiation at an effective temperature of 2000 K, it would appear much dimmer with the peak wavelength being in the infrared range, and it would emit about 0.072% of the energy it emits at its current temperature, making the Earth significantly colder.

Step-by-step solution

Understand Wien's Law and Stefan-Boltzmann Law

Wien's Law is used to determine the peak wavelength of radiation emitted by a blackbody at a particular temperature. It is given by: \(\lambda_{max} = \frac{b}{T}\) where \(\lambda_{max}\) is the peak wavelength, \(b\) is the Wien's constant (\(2.898\times10^{-3}\mathrm{~m.K}\)), and \(T\) is the temperature in Kelvin. Stefan-Boltzmann Law is used to find the total energy emitted by a blackbody at a particular temperature. It is given by: \(E = \sigma T^4\) where \(E\) is the energy, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{~W.m^{-2}.K^{-4}}\)), and \(T\) is the temperature in Kelvin.

Calculate the peak wavelength at the new temperature

Using Wien's Law, let's calculate the peak wavelength of the radiation emitted by the sun at the new temperature: \(\lambda_{max} = \frac{2.898\times10^{-3}\mathrm{~m.K}}{2000\mathrm{~K}}\) \(\lambda_{max} = 1.449\times10^{-6}\mathrm{~m}\) or \(1449\mathrm{~nm}\)

Determine the effect on visible light spectrum

The visible light spectrum ranges from about \(380\mathrm{~nm}\) (violet) to \(700\mathrm{~nm}\) (red). At the new temperature, the peak wavelength is \(1449\mathrm{~nm}\), which is in the infrared range. Therefore, the sun would emit much less visible light and would appear much dimmer to our eyes.

Calculate the total energy emitted at the new temperature

Using Stefan-Boltzmann Law, let's calculate the total energy emitted by the sun at the new temperature: \(E = 5.67 \times 10^{-8} \mathrm{~W.m^{-2}.K^{-4}} \times (2000\mathrm{~K})^4\) \(E = 4.55 \times 10^{4}\mathrm{~W.m^{-2}}\)

Compare the energy emitted at both temperatures

Now, let's calculate the total energy emitted by the sun at the original temperature: \(E_{original} = 5.67 \times 10^{-8} \mathrm{~W.m^{-2}.K^{-4}} \times (5762\mathrm{~K})^4\) \(E_{original} = 6.30 \times 10^{7}\mathrm{~W.m^{-2}}\) Now, to calculate the ratio of the energy emitted at both temperatures: \(\frac{E}{E_{original}} = \frac{4.55 \times 10^{4}\mathrm{~W.m^{-2}}}{6.30 \times 10^{7}\mathrm{~W.m^{-2}}}\) \(\frac{E}{E_{original}} \approx 7.22 \times 10^{-4}\) At the new temperature, the sun would emit about 0.072% of the energy it emits at its current temperature. This would have a major effect on our planet, making it significantly colder. In conclusion, if the sun emitted radiation at an effective temperature of \(2000\mathrm{~K}\) instead of \(5762\mathrm{~K}\), the sun would appear much dimmer with the peak wavelength being in the infrared range, and it would emit significantly less energy, making the Earth a much colder place to live.

Key concepts

Wien's Law

Wien's Law is a fundamental principle in thermal physics that helps us understand the relationship between the temperature of a blackbody and the peak wavelength of the radiation it emits. The principle is summarized by the formula: \[ \lambda_{max} = \frac{b}{T} \]where \(\lambda_{max}\) stands for the peak wavelength, \(b\) is Wien's constant (approximately \(2.898\times10^{-3}\mathrm{~m.K}\)), and \(T\) is the absolute temperature in Kelvin. This formula implies that as the temperature of a blackbody increases, the peak wavelength of its radiation shifts to shorter wavelengths. For instance, with the sun at its current temperature of \(5762\mathrm{~K}\), the peak wavelength falls within the visible spectrum. However, if the sun cooled to \(2000\mathrm{~K}\), the peak wavelength would shift to the infrared range, altering the sunlight we perceive.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a key concept in understanding how much energy a blackbody emits. According to this law, the energy radiated by a blackbody per unit surface area is directly proportional to the fourth power of its absolute temperature. Mathematically, it is expressed as:\[E = \sigma T^4\]where \(E\) is the energy emitted per unit area, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{~W.m^{-2}.K^{-4}}\)), and \(T\) is the temperature in Kelvin.

• An increase in the temperature results in significantly more energy emitted.
• If the sun's temperature dropped to \(2000\mathrm{~K}\), the total energy emitted would drastically decrease, leading to a much colder Earth.

Understanding this law helps us comprehend the powerful impact of temperature on energy emission and, consequently, on the climate and ecosystems on Earth.

Peak Wavelength

The peak wavelength is the specific wavelength at which the emission of radiation from a blackbody is at its maximum. In simple terms, this is the color of light that you are most likely to see. Factors affecting this are the body's temperature, as articulated by Wien's Law.

• When the sun was at its original temperature (\(5762\mathrm{~K}\)), the peak wavelength was around \(500\mathrm{~nm}\), which is in the visible spectrum.
• If the sun were at \(2000\mathrm{~K}\), the peak wavelength would increase to \(1449\mathrm{~nm}\), which falls in the infrared range.

The shift to infrared means less visible light, appearing dim to human eyes, with major consequences for life and the climate on Earth.

Infrared Range

The infrared range is a part of the electromagnetic spectrum commonly associated with heat radiation. Wavelengths in this range are longer than visible light, generally spanning from \(700\mathrm{~nm}\) to \(1\mathrm{mm}\). Human eyes cannot perceive this radiation; instead, we feel it as heat.

• With the sun at \(2000\mathrm{~K}\), the peak wavelength of \(1449\mathrm{~nm}\) implies that most of the sun's radiation would be infrared.
• This would result in significantly less visible daylight, altering the perception of our environment.

Understanding the infrared range is essential for grasping how variations in temperature can affect the types of radiation emitted by the sun and other celestial bodies.

Thermal Physics

Thermal physics is the study of heat, temperature, and their effects on matter. It combines principles from thermodynamics, statistical mechanics, and kinetic theory to understand energy exchange processes.

• Key principles include Wien's Law, explaining how temperature affects radiation wavelength.
• Stefan-Boltzmann Law describes how temperature impacts energy emission from bodies like stars.

These concepts are crucial in assessing changes in temperatures, such as hypothetical shifts in the sun's surface temperature, and predicting their impacts on Earth's climate and ecosystems. By studying thermal physics, we gain insights into energy transformations in various physical systems.