The true temperatures of the thermal radiation of stoichiometric hafnium, titanium, and zirconium carbides are defined from the generalized Wien displacement law. It is shown that Wien's displacement law for the investigated stoichiometric carbides decreases linearly with increasing temperature. The uncertainties in the determination of the true temperature are no greater than 1 %. For determining the true temperature of stoichiometric carbides, the experimental values of the position of the maximum of the spectral density power are needed. By extrapolating the generalized Wien displacement laws in the ultra-high-temperature region, the positions of the maximum of the normal energy density of hafnium, titanium, and zirconium carbides at melting temperatures are obtained. Thermodynamics of thermal radiation of stoichiometric carbides is constructed by using the temperature dependences of the generalized Stefan-Boltzmann law. The calculated values of the normal total emissivity for the investigated carbides at different temperatures are in good agreement with experimental data. For determining the true temperatures of the thermal radiation of stoichiometric carbides, experimental values of either the normal total emissivity or the normal total energy density are needed. The temperature dependences of the Helmholtz free energy, entropy, heat capacity at constant volume, pressure, enthalpy, and internal energy of the thermal radiation of stoichiometric carbides at high temperature are obtained. It is shown that thermodynamic function values increase with increasing temperature as a power law.Keywords Enthalpy · Entropy · Generalized Wien's displacement and Stefan-Boltzmann's laws · Heat capacity · Helmholtz free energy · A. I. Fisenko (B) · V. Lemberg

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Use formulas to describe the monopole and dipole spectra of the Cosmic Microwave Background (CMB) radiation, the exact expressions for the temperature dependences of the radiative and thermodynamic functions, such as the total radiation power per unit area, total energy density, number density of photons, Helmholtz free energy density, entropy density, heat capacity at constant volume, pressure, enthalpy density, and internal energy density in the finite range of frequencies v 1 ≤ v ≤ v 2 are obtained. Since the dependence of temperature upon the redshift z is known, the obtained expressions can be simply presented in z representation.Utilizing experimental data for the monopole and dipole spectra measured by the COBE FIRAS instrument in the 60 -600 GHz frequency interval at the temperature T = 2.728 K, the values of the radiative and thermodynamic functions, as well as the radiation density constant a and the Stefan-Boltzmann constant σ are calculated. In the case of the dipole spectrum, the constants a and σ, and the radiative and thermodynamic properties of the CMB radiation are obtained using the mean amplitude T amp =3.369m K. It is shown that the Doppler shift leads to a renormalization of the radiation density constant a, the Stefan-Boltzmann constant σ, and the corresponding constants for the thermodynamic functions. The radiative and thermodynamic properties of the Cosmic Microwave Background radiation for the monopole and dipole spectra at the redshift z ≈ 1089 are calculated.

The general analytical expressions for the thermal radiative and thermodynamic properties of a realbody are obtained in a finite range of frequencies at different temperatures. The frequency dependence of the spectral emissivity is represented as a power series. Stefan-Boltzmann's law, total energy density, number density of photons, Helmholtz free energy density, internal energy density, enthalpy density, entropy density, heat capacity at constant volume, pressure, and total emissivity are expressed in terms of the polylogarithm functions. The general expressions for the thermal radiative and thermodynamic functions are applied for the study of thermal radiation of liquid and solid zirconium carbide. These functions are calculated using experimental data for the frequency dependence of the normal spectral emissivity in the visible-near infrared range at the melting (freezing) point. The gaps between the thermal radiative and thermodynamic functions of liquid and solid zirconium carbide are observed. The general analytical expressions obtained can easily be presented in wavenumber domain.

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Using polylogarithm functions the exact analytical expressions for the radiative and thermodynamic properties of blackbody radiation, such as the Wien displacement law, Stefan-Boltzmann law, total energy density, number density of photons, Helmholtz free energy density, internal energy density, enthalpy density, entropy density, heat capacity at constant volume, and pressure in the finite range of frequencies are constructed. The obtained expressions allow us to tabulate these functions in various finite frequency bands at different temperatures for practical applications. As an example, the radiative and thermodynamic functions using experimental data for the monopole spectrum of the Cosmic Microwave Background (CMB) radiation measured by the COBE FIRAS instrument in the 60 -600 GHz frequency interval at the temperature T = 2.725 K are calculated. The expressions obtained for the radiative and thermodynamic functions can be easily presented in wavelength and wavenumber domains.

Using the three-component spectral model describing the FIRAS average continuum spectra, the analytical expressions for the temperature dependences of the thermodynamic and radiative functions of the galactic far-infrared radiation are obtained. The COBE FIRAS instrument data in the 0.15 -2.88 THz frequency interval at the mean temperatures T = 17.72 K, T = 14 K, and T = 6.73 K are used for calculating the radiative and thermodynamic functions, such as the total radiation power per unit area, total energy density, total emissivity, number density of photons, Helmholtz free energy density, entropy density, heat capacity at constant volume and pressure for the warm, intermediate-temperature and very cold components of the Galactic continuum spectra. The generalized Stefan-Boltzmann laws for the warm, intermediate-36 G total   T I . Other radiative and thermodynamic properties of the galactic far-infrared radiation (photon gas) for the Galaxy are calculated. The expressions for astrophysical parameters, such as the entropy density/Boltzmann constant, and number density of the galactic far-infrared photons areobtained. We assume that the obtained analytical expressions for thermodynamic and radiative functions may be useful for describing the continuum spectra of the far-infrared radiation for outer galaxies.

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