The red supergiant Betelgeuse has a surface temperature of 3600 K.

a) Using the fact that the Sun's surface temperature is about 5800 K and that its continuous spectrum peaks at a wavelength of 500 nm, find the peak wavelength of Betelgeuse's continuous spectrum. (Please enter your answer in units of nanometers)

To find the peak wavelength of Betelgeuse's continuous spectrum, we can use Wien's displacement law. This law states that the peak wavelength of the continuous spectrum of an object is inversely proportional to its temperature.

The formula for Wien's displacement law is:

λ_max = b / T

Where λ_max is the peak wavelength, b is a constant equal to 2.898 × 10^-3 meters kelvin (m*K), and T is the temperature in kelvin.

Given the surface temperature of Betelgeuse (T = 3600 K) and the peak wavelength of the Sun's continuous spectrum (λ_max = 500 nm = 500 × 10^-9 meters), we can use the equation to find the peak wavelength of Betelgeuse's continuous spectrum.

First, let's convert the peak wavelength of the Sun's continuous spectrum from nanometers to meters:

λ_max = 500 × 10^-9 meters

Now, we can plug the values into the equation:

λ_max = (2.898 × 10^-3 m*K) / (3600 K)

Calculating this, we get:

λ_max ≈ 8.05 × 10^-7 meters

To convert the peak wavelength back to nanometers, we can multiply by 10^9:

λ_max ≈ 805 nm

Therefore, the peak wavelength of Betelgeuse's continuous spectrum is approximately 805 nanometers.