Abstract
The Sun is a nuclear fusion reactor with a life expectance of about 4. 5 × 109 years. In its core, at a proton density of 1025 cm−3 and a temperature of 1. 5 × 107 K, protons (p+) are converted by fusion processes and via several intermediate steps into nuclear products, such as helium (2 4He), amongst others [1]. The average energy gain per nucleon in such fusion reactions amounts to several MeV. The total rate of change of the mass deficit of the Sun amounts to \(\dot{m} = 6 \times 10^{9}\,\mathrm{kg/s}\), corresponding to a total power of 3. 6 × 1026 W emitted by the outer surface of the Sun, or an energy flux
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Notes
- 1.
The sunlight arrives at the Earth’s position under the solid angle of \(\varOmega _{\mathrm{Sun}} =\pi (R_{\mathrm{Sun}}/d_{\mathrm{SE}})^{2}\) and—according to the second law of thermodynamics—might be concentrated only up to the flux per solid angle of the source, known also as conservation of the etendue.
- 2.
The total energy of thermal radiation using classical electromagnetic theory via the integral \(\int _{0}^{\infty }u_{\omega }(\hslash \omega,T)d(\hslash \omega ) \rightarrow \infty \), whereas with the Bose-distribution function equivalently with Planck’s approach, the total energy of a thermal radiator gives a finite value (see [4]).
- 3.
The above deduction of Planck’s law for the spectral behavior of thermal equilibrium radiation is based on the density of states of photons, in other words, on the density of three-dimensional modes for stationary solutions of a wave equation resulting in 3D standing waves. Mode numbers and individual wave vectors translate linearly into frequencies and energies of particular photons, and the multiplication of the density of states by the particular photon energy yields the corresponding spectral energy flux. So apart from the Bose term \(\big[\exp (\hslash \omega /\mathit{kT}) - 1\big]^{-1}\), the integrand contains the independent variable ω to a power n equal to the number of dimensions, i.e., (ℏ ω)n. Consequently, the analytically representable solution of the definite integral also contains the number of dimensions, i.e., in the dependence of the total energy flux on temperature ∼ T (n+1). Accordingly, σ SB also depends on the number of dimensions [7], and in general reads
$$\displaystyle{\sigma _{\mathrm{SB}} = 2\pi ^{(n-1)/2}\frac{\varGamma _{(n+1)}\varsigma _{(n+1)}} {\varGamma _{(n+1)/2}} \frac{k^{n+1}} {(2\pi \hslash )^{n}c^{n-1}}\;,}$$where Γ and \(\varsigma\) are the Gamma and Riemann zeta functions, respectively.
- 4.
The entire flow collected by an ideally absorbing sphere at distance from the Sun d SE is related to the perpendicular projection of the sphere (cross section) and amounts to \(\varGamma _{\epsilon }(d_{\mathrm{SE}})\pi R_{\mathrm{sphere}}^{2}\) whereas for emission the sphere offers the total surface area of \(4\pi R_{\mathrm{sphere}}^{2}\). Here the maximum solar light concentration would correspond to the illumination of the entire surface of the sphere (\(4\pi R_{\mathrm{sphere}}^{2}\)) and vanishing access of the light from the universe.
- 5.
The flux from the Sun Γ ε (d SE ) is defined as solar light flux at Air Mass Zero (AM0) since it is not affected by interactions with the Earth atmosphere; corresponding modifications of the spectral distribution of Γ ε , as well as its magnitude by the atmosphere are designed by AM1, AM2 etc. Here Γ ε suffers from absorption, scattering, and reflection by particles of different kinds, like molecules, water clusters, dust, and so on, qualitatively represented by the path length of the light through the atmosphere.
- 6.
We have to recognize that the solar light even at the large distance from the sun d SE = 1. 5 × 1011 m might not be treated as a plane wave and accordingly concentration is limited by conservation of the photon flux per solid angle (etendue).
- 7.
Instead of balancing the light flows like that of the solar light with its respective concentration reaching the absorber and the light flow emitted by the absorber commonly solid angles, Ω in for the entrance and Ω out for the exit are used, which implicitly contain the factor of sunlight concentration.
- 8.
This approach is valid provided the spatial extension of source and receiver are small compared to their distance.
- 9.
This configuration is commonly realized with solar thermal absorbers and with photovoltaic cells and modules, which both are exposed only by one side to the solar insolation on comparatively small areas for which the photons are assumed to propagate parallel and of which the area for light reception equals that for light emission.
- 10.
The upper limit is again given by thermodynamics, whereas we should not trust ray optics which erroneously would allow for even higher light fluxes of the image than that of the source.
- 11.
In real systems the angle \(\varTheta _{\mathrm{r,max}} = (\pi /2)\) for the collection of photons arriving from the entire hemisphere to which the receiver is exposed to can hardly be realized.
- 12.
Here we neglect the influence of the spectral shift of solar photons when leaving the gravitational field of the Sun or entering that of the Earth. The curious reader may be inspired to estimate the influence of the gravitational fields of the Sun and the Earth.
- 13.
Although the photon density absorbed and providing for the excitation of the electron system (flow in) with respect to spectral distribution and entire density might differ from the spectral photon density emitted (flow out) the emission is determined by the excitation state regardless by which spectral distribution this has been achieved as far as the balance of the rates is met. From this point of view we are allowed to also assume the photon density for absorption to equal that of emission.
- 14.
These two states in CB and VB are arbitrarily chosen; in thermal equilibrium—and analogously assumed under excitation—the individual rates for transitions of absorption and spontaneous and stimulated emission, regardless their respective photon energy, also compensate each other.
- 15.
Due to strong electron-electron interaction and because of efficient electron-phonon coupling within energy bands the electrons after excitation by light undergo a very fast relaxation ((10−13–10−12) s) and occupy a distribution of maximum entropy which allows for the introduction of the magnitude temperature.
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Bauer, G.H. (2015). Sun as Energy Source. In: Photovoltaic Solar Energy Conversion. Lecture Notes in Physics, vol 901. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46684-1_3
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