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

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.

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 \(1449\mathrm{~nm}\)

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.

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

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.