Frequency (Doppler effect for a moving black body)

Description

Black-body radiation is the thermal electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body). It has a specific spectrum and intensity that depends only on the body’s temperature, which is assumed for the sake of calculations and theory to be uniform and constant.

The total power (energy/second) the Sun is emitting is given by the Stefan–Boltzmann law.

The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is given by this equation.

Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction α of this energy where α is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction 1 − α of the Sun’s light, and reflects the rest.
a
The relativistic Doppler effect causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f’ ,where v is the velocity of the source in the observer’s rest frame, θ is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and c is the speed of light. This can be simplified for the special cases of objects moving directly towards (θ = π) or away (θ = 0) from the observer, and for speeds much less than c.

Related formulas

Variables

f 'observed wave frequency (Hz)
flfrequency of light (Hz)
vis the velocity of the source in the observer's rest frame (m/s)
cSpeed of light
θangle between the velocity vector and the observer-source direction (rad)