The luminosity of the star Aldebaran is 520 times that of the Sun. The wavelength of light at peakintensity for Aldebaran is 740 nm and the wavelength of light at peak intensity for the Sun is 500 nm.Explain whether Aldebaran is cooler or hotter than the Sun.Calculate the ratio:radius of Aldebaran / radius of the Sun.
Question
The luminosity of the star Aldebaran is 520 times that of the Sun. The wavelength of light at peakintensity for Aldebaran is 740 nm and the wavelength of light at peak intensity for the Sun is 500 nm.Explain whether Aldebaran is cooler or hotter than the Sun.Calculate the ratio:radius of Aldebaran / radius of the Sun.
Solution
The temperature of a star can be determined by its color, which is related to the wavelength of light at peak intensity. This is known as Wien's Law. According to Wien's Law, the peak wavelength of light is inversely proportional to the temperature. This means that stars emitting peak light at longer wavelengths are cooler than those emitting peak light at shorter wavelengths.
Given that the peak wavelength for Aldebaran is 740 nm (longer than the Sun's 500 nm), Aldebaran is cooler than the Sun.
To calculate the ratio of the radius of Aldebaran to the radius of the Sun, we can use the Stefan-Boltzmann Law, which states that the luminosity of a star is proportional to the fourth power of its radius and the fourth power of its temperature.
Let's denote: Ls = Luminosity of the Sun La = Luminosity of Aldebaran Ts = Temperature of the Sun Ta = Temperature of Aldebaran Rs = Radius of the Sun Ra = Radius of Aldebaran
From the Stefan-Boltzmann Law, we have:
La/Ls = (Ra/Rs)^2 * (Ta/Ts)^4
We know that La/Ls = 520 (given in the problem) and Ta/Ts = 500/740 (from Wien's Law).
Substituting these values into the equation, we get:
520 = (Ra/Rs)^2 * (500/740)^4
Solving this equation for Ra/Rs gives us the ratio of the radius of Aldebaran to the radius of the Sun.
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