Planck curves

Figure 2.1 Planck curves for black bodies of different temperatures...

Now that we have a simple model for the continuum spectrum of the stars based around the Planck curve, the temperature and the luminosity, we can make some observations and classifications of the stars. There are some constellations that dominate the night sky in both the northern and southern hemispheres and even a casual look should inspire wonder. Star hopping in the night sky should lead to the simplest observation not all stars have the same colour. A high-quality photograph of the constellation of Orion (see page 2 of the colour plate section) shows stars... 

The Planck curves for different T proceed such that those for lower T always lie below the curves for higher r - in other words, the curves never intersect. The... 

Figure 9.39 Slope of Planck curve for various temperatures after normalized to radiant intensity at 1.55 4m. (After Ref [33].)...

All astronomical objects emit infrared radiation. In fact, the peak of the integrated light from all of the stars in a normal galaxy is near a wavelength of 1 /xm. The amount of infrared radiation emitted by a star, planet, or other astronomical object is determined by its temperature, the blackbody or Planck curve, and its emissivity at each wavelength, as given in Eq. (1.2). 

The factor in the square brackets above is the Bose factor it is very close to unity except for wavelengths greater than the peak of the Planck curve, and the effect is normally not considered in performance predictions. However, for a detector whose spectral range is limited to long wavelengths, the factor can be significant. 

In 1989 the Cosmic Background Explorer, COBE, was launched. It made precise measurements of the CBR at wavelengths from a few micrometers out to 1 cm (Fig. 8.1). COBE unambiguously proved the CBR to follow a Planck curve at 2.73 K so there is no doubt that we are seeing the residual radiation left behind from the primordial Big Bang. Tiny variations in the CBR result from the formation of structures during the early evolution of the universe (see also Fig. 8.2). 

Fig. 3. Comet Swift-Tuttle at 9.8 fim. Minimum contour level and level spacing are 10% of peak flux (17 mJy). The sun direction is toward the southwest. Comet Swift-Tuttle was mapped at 8.8, 9.8, 11.7, and 12.5 m (Deutsdi et al. 1992) to study the temperature, composition, spatial extent, and evolution of the dust cloud surrounding the comet nucleus. Spectral points at 8.8, 9.8, and 12.5 fim are well-flt by a Planck curve (T = 364 K). There is no evidence for a 9.8 fim amorphous silicate peak but a clear excess is seen at 11.7 fim.

Figure 4.26. Special radiance versus wavelength for CaAljSi Og glass shocked to 48 GPa and 84 GPa. Best-fitting Planck blackbody curves are shown in relation to the radiance data. (After Boslough and Ahrens (1984). the American Geophysical Union.)...

We wish to show that no points to the leftbb of 2 on the isotherm 62 are accessible from point 1 via any adiabatic path, reversible or irreversible. Suppose we assume that some adiabatic path does exist between 1 and 2. We represent this path as a dotted curve in Figure 2.11a. We then consider the cycle I —>2 —> 1 — 1. The net heat associated with this cycle would be that arising from the last step 1 — 1, since the other two steps are defined to be adiabatic. We have defined the direction 1 — 1 to correspond to an absorption of heat, which we will call qy. From the first law, the net work vv done in the cycle, is given by w = —q, since AU for the cycle is zero. Thus, for this process, iv is negative (and therefore performed by the system), since qy is positive, having been absorbed from the reservoir. The net effect of this cycle, then, is to completely convert heat absorbed at a high temperature reservoir into work. This is a phenomenon forbidden by the Kelvin-Planck statement of the Second Law. Hence, points to the left of 2 cannot be reached from point 1 by way of any adiabatic path. 

A form of the curve of growth more relevant to stellar (as opposed to interstellar) absorption lines is derived from work by E. A. Milne, A. S. Eddington, M. Min-naert, D. H. Menzel and A. Unsold. In the Milne-Eddington model of a stellar photosphere, the continuum source function (equated to the Planck function in the LTE approximation) increases linearly with continuum optical depth rA and there is a selective absorption i]K, in the line, where rj(Av), the ratio of selective to continuous absorption, is a constant independent of depth given by... 

A voltammetric curve can be viewed in electrochemistry as the emission or absorption spectra in spectroscopy. The current density (i.e., the number of charges per unit of time and area) corresponds to the emitted or absorbed light intensity (the number of quanta per unit of time and area). Finally, when multiplied by the Faraday constant, the potential defines the energy of the system and can thus be treated as an analog of the light frequency, which can also gives energy when multiplied by the Planck constant. 

Differences in the physicochemical properties of isotopes arise as a result of quantum mechanical effects. Figure 1.3 shows schematically the energy of a diatomic molecule, as a function of the distance between the two atoms. According to the quantum theory, the energy of a molecule is restricted to certain discrete energy levels. The lowest level is not at the minimum of the energy curve, but above it by an amount 1/2/tv where h is Planck s constant and v is the frequency with... 

Spectral distribution of blackbody radiation. The family of curves is called the Planck distribution after Max Planck, who derived the law governing blackbody radiation. Note that both axes are logarithmic. 

Max Planck, utilizing his quantum theory postulates and modifications of the Boltzmann statistical procedure, established the theoretical formula for the spectral distribution curves of a black body ... 

Fig. 4.2 In the limit the bar graph becomes a curve, the graph of j X) versus X, where f(X) = lim jj = essentially intensity of radiation versus wavelength. Planck s efforts to find...

Planck s explanation of the blackbody radiation curves (1900 [4]) and Einstein s explanation of the facts of the photoelectric effect (1905 [7]) indicated that the flow of energy in physical processes did not take place continuously, as had been believed, but rather jerkily, in discrete jumps, quantum by quantum. The contributions of Planck and Einstein were the signal developments marking the birth of quantum theory and the transition from classical to modem physics. 

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