E3 Thermal Radiation and Spectrum

Topic

The central element of this model is the temperature of an astrophysical surface and the blackbody spectrum that this single temperature—on its own—completely determines.

Thermal radiation is the most fundamental radiative process in astrophysics because any surface at a given temperature emits a blackbody spectrum whose shape depends solely on that temperature. Wien's law dictates that the peak of this spectrum is fixed by that same temperature: the Sun, at just under six thousand degrees, emits its peak in visible green light, whereas a cooled neutron star, at a temperature of one million degrees, emits its peak in soft X-rays.

The Stefan-Boltzmann law, in turn, relates the total flux emitted by the surface to the fourth power of its temperature; thus, doubling a surface's temperature causes the emitted flux to increase sixteenfold. A star's total luminosity is the integral of this flux over its entire surface, meaning that luminosity increases with both the square of the star's radius and the fourth power of its temperature.

Pogson's magnitude scale—which is logarithmic and inverted—converts this total flux into the magnitude value typically used to catalog stars; it was calibrated so that a difference of five magnitudes always corresponds to a factor of one hundred in flux, regardless of where on the scale the difference is measured. Thus, from the astrophysical surface temperature with which this document begins to the magnitude scale with which it concludes—encompassing Wien's law, the Stefan-Boltzmann law, and total stellar luminosity—these three pillars (Planck, Wien, and Stefan-Boltzmann) enable stellar photometry and spectrophotometry to systematically translate any surface temperature into a measurable spectrum, flux, and magnitude.

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