1. Thermal Radiation Laws:
According to the Stefan-Boltzmann law, the total thermal radiation emitted power per unit surface area, P, of a blackbody (which absorbs all radiation that falls on it) is given by:
P = σT^4
where σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W m^-2 K^-4) and T is the absolute temperature in Kelvin.
The Wien's displacement law states that the peak wavelength of thermal radiation emitted by a blackbody is given by:
λ_max = b/T
where b is Wien's displacement constant (2.898 x 10^-3 m K) and T is the absolute temperature in Kelvin.
The surface area of a sphere of radius R is given by:
A = 4πR^2
Therefore, the thermal radiation emitted power, P, of the sphere is:
P = σT^4 x A = σT^4 x 4πR^2
Substituting the values R = 6 370 000 m and T = 288 K, we get:
P = (5.67 x 10^-8 W m^-2 K^-4) x (288 K)^4 x 4π(6 370 000 m)^2 ≈ 3.99 x 10^17 W
The peak intensity wavelength, λ_max, of the thermal radiation emitted by the sphere is:
λ_max = (2.898 x 10^-3 m K) / 288 K ≈ 1.00 x 10^-5 m = 10 μm
Therefore, the sphere primarily emits infrared radiation.
2. Doppler Calculations:
The Doppler shift law relates the observed wavelength of radiation to the wavelength emitted by a source moving at a speed v relative to the observer:
λ_obs = λ_rest x (1 + v/c)
where λ_rest is the rest wavelength of the radiation, λ_obs is the observed wavelength, v is the speed of the source relative to the observer, and c is the speed of light.
In this case, the rest wavelength λ_rest is 121.6 nm, and the observed wavelength λ_obs is 120.7 nm. We can rearrange the Doppler shift equation to solve for v:
v = c x (λ_obs/λ_rest - 1)
Substituting the values, we get:
v = 3.00 x 10^8 m/s x (120.7 nm / 121.6 nm - 1) ≈ -2.99 x 10^5 m/s
The negative sign indicates that the star is moving away from us.
Therefore, the star is moving away from us at a speed of approximately 299,000 m/s (or 0.1% of the speed of light).