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Orion1
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\[\color{blue}{\text{Symbolic identity key:}}\]
\[\begin{array}{lcl} \text{s} \text{ - spin quantum number} \\ n_{s} \text{ - spin states total integer helicity number} \\ N_{s} \text{ - species total integer number} \\ N_{n} \text{ - total effective degeneracy number} \\ \end{array}\]
\[n_{s} = 2 s + 1 \; \; \; \; \; \; s < 2 \; \; \; \; \; \; m = 0\]
\[n_{s} = 2 \; \; \; \; \; \; s \geq 2 \; \; \; \; \; \; m = 0\]
\[\text{if } n_{s} \geq N_{s} \text{ then } N_{n} = n_{s}\]
\[\text{if } n_{s} \leq N_{s} \text{ then } N_{n} = N_{s}\]
\[\color{blue}{\text{Toy model quantum particle properties chart:}}\]
\[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\]
\[\color{blue}{\text{For massless quantum particles, the transverse modes cannot exist due to Lorentz invariance.}}\]
\[\color{blue}{\text{Only positive and negative helicity states remain. For massless scalar particles, only zero helicity states remain.}}\]
\[\color{blue}{\text{The spin 1 photon is also restricted to its positive and negative helicity states, and has a total effective degeneracy number of 2.}}\]
\[\color{blue}{\text{A massless graviton has only 2 helicity states, and has a total effective degeneracy number of 2.}}\]
\[\color{blue}{\text{Dark energy quinton total effective degeneracy number:}}\]
\[\boxed{N_{\Lambda} = 1}\]
\[\color{blue}{\text{Planck satellite dark energy cosmological composition parameter:} \; (\text{ref. 1, pg. 11})}\]
\[\Omega_{\Lambda} = 0.6825\]
\[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation temperature:} \; (\text{ref. 2})}\]
\[\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3} \right)^{1/4}}\]
\[\boxed{T_{\Lambda} = 35.013 \; \text{K}}\]
\[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\Lambda}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right]}{2 \pi} \left(\frac{45 \Omega_{\Lambda} H_0^2 c^5}{4 \hbar G N_{\Lambda} \pi^3} \right)^{1/4}}\]
\[\boxed{f_{\Lambda} =  2.05837 \cdot 10^{12} \; \text{Hz}}\]
\[\boxed{f_{\Lambda} =  2.058 \; \text{THz}}\]
\[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\]
\[T_{\gamma} = 2.72548 \; \text{K}\]
\[\color{blue}{\text{Cosmic neutrino and sterile neutrino background radiation temperature at present time:} \; (\text{ref. 5})}\]
\[T_{\nu} = \left(\frac{4}{11} \right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}\]
\[\boxed{T_{\nu} = 1.945 \; \text{K}}\]
\[\color{blue}{\text{Dark matter scalaron and sterile neutrino radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\]
\[\boxed{T_{\phi} = T_{s \nu} = T_{\nu} = 1.945 \; \text{K}}\]
\[\color{blue}{\text{Bose-Einstein dark matter cosmic scalaron background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\phi} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\phi}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{\phi} = 1.14366 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\phi} = 114.366 \; \text{GHz}}\]
\[\color{blue}{\text{Fermi-Dirac dark matter sterile neutrino background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{s \nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{s \nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{s \nu} = 126.915 \; \text{GHz}}\]
\[\color{blue}{\text{Solve for cosmic neutrino background radiation spectral radiance peak frequency:}}\]
\[\color{blue}{\text{Neutrino species total effective degeneracy number:}}\]
\[N_{\nu} = 3.046\]
\[\color{blue}{\text{Neutrino radiation energy density Fermi-Dirac distribution:}}\]
\[\epsilon_{\nu} = \frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4}{\left(2 \pi \hbar c \right)^3} \int_{0}^c \frac{E_t \left(v \right)^3}{e^{\frac{E_t \left(v \right)}{E_1 \left(T_{\nu} \right)}} + 1} dv\]
\[\color{blue}{\text{Solve Fermi-Dirac distribution first derivative x-axis zero intercept with respect to frequency:}}\]
\[\frac{d \epsilon_{\nu}}{df_{\nu}} = \frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\nu} \right)}} + 1 \right)} \right) = 0\]
\[\frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 \left(h f_{\nu} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\nu}}} + 1  \right)} \right) = \frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} - \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2} = 0\]
\[\frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} = \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2}\]
\[\frac{h f_{\nu}}{k_B T_{\nu}} = 3 \left(1 + \frac{1}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \right) \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = 3 + \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} + 3\]
\[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = \frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = 3 \; \; \; a = 3 \; \; \; k = 0\]
\[f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h} = 1.26915 \cdot 10^{11} \; \text{Hz} =  126.915 \; \text{GHz}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\color{blue}{\text{Fermi-Dirac cosmic neutrino background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\]
\[\boxed{f_{\nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\nu} = 126.915 \; \text{GHz}}\]
\[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\color{blue}{\text{Photon species total effective degeneracy number:}}\]
\[\boxed{N_{\gamma} = 2}\]
\[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\]
\[T_{\gamma} = 2.72548 \; \text{K}\]
\[\color{blue}{\text{Planck's law:} \; (\text{ref. 6})}\]
\[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)}\]
\[\color{blue}{\text{Solve Bose-Einstein Planck's law first derivative x-axis zero intercept with respect to frequency:}}\]
\[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = 0\]
\[\color{blue}{\text{Planck's law first derivative with respect to frequency:}}\]
\[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = \frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\]
\[\frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\]
\[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\]
\[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\]
\[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\]
\[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\]
\[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\]
\[f_{\gamma} = 160.23 \; \text{GHz}\]
\[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\color{blue}{\text{Photon radiation energy density Bose-Einstein distribution:}}\]
\[\epsilon_{\gamma} = \frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4}{\left( 2 \pi \hbar c  \right)^3} \int_{0}^\infty \frac{E_t \left(\omega \right)^3}{e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1} d \omega\]
\[\color{blue}{\text{Solve Bose-Einstein distribution first derivative x-axis zero intercept with respect to frequency:}}\]
\[\frac{d \epsilon_{\gamma}}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1  \right)} \right) = 0\]
\[\frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 \left(h f_{\gamma} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\gamma}}} - 1  \right)} \right) = \frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\]
\[\frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\]
\[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\]
\[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\]
\[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\]
\[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\]
\[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\]
\[f_{\gamma} = 160.23 \; \text{GHz}\]
\[\color{blue}{\text{Cosmic graviton background radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\]
\[\boxed{T_{G} = T_{\nu} = 1.945 \; \text{K}}\]
\[\color{blue}{\text{Bose-Einstein cosmic graviton background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{G} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{G}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{G} = 1.14366 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{G} = 114.366 \; \text{GHz}}\]
\[\color{blue}{\text{Are Fermilab quantum particle detectors capable of detecting cold dark matter quantum particles?} \; (\text{ref. 7})}\]
\[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\]
Reference:
Planck 2013 results. XVI. Cosmological parameters: (ref. 1)
http://www.ymambrini.com/My_World/Articles_files/Planck_2013_results_16.pdf
Science Forums - Dark energy quinton radiation temperature - Orion1: (ref. 2)
https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=931225
Wikipedia - Lambert W function: (ref. 3)
https://en.wikipedia.org/wiki/Lambert_W_function
Wikipedia - Cosmic microwave background - Importance of precise measurement: (ref. 4)
https://en.wikipedia.org/wiki/Cosmic_microwave_background#Importance_of_precise_measurement
Wikipedia - Cosmic neutrino background - Derivation of the CvB temperature: (ref. 5)
https://en.wikipedia.org/wiki/Cosmic_neutrino_background#Derivation_of_the_CνB_temperature
Wikipedia - Planck's law: (ref. 6)
https://en.wikipedia.org/wiki/Planck's_law
Fermilab - How scientists at Fermilab search for dark matter particles: (ref. 7)
https://bit.ly/3AvGTo9
Science Forums - Toy model calculation versus observation comparison summary - Orion1: (ref. 8 )
https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1112539

 

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Posted (edited)

\[\color{blue}{\text{Toy model quantum particle properties chart:}}\]
\[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\]
\[\color{blue}{\text{Planck's law:}} \; (\color{blue}{\text{ref. 1}})\]
\[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} \pm 1 \right)}\]
\[\color{blue}{\text{A plus sign in the denominator is a Fermi-Dirac distribution, a minus sign in the denominator is a Bose-Einstein distribution.}}\]
\[\color{blue}{\text{+ sign - Fermi-Dirac distribution}}\]
\[\color{blue}{\text{- sign - Bose-Einstein distribution}}\]
\[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 1}})\]
\[114.366 \; \text{GHz} \; \; \; 126.915 \; \text{GHz} \; \; \; 160.229 \; \text{GHz}\]
\[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 2}})\]
\[2.058 \; \text{THz}\]
\[\color{blue}{\text{In this toy model, some quantum particle radiation distributions are embedded within the photon radiation distribution.}}\]
\[\color{blue}{\text{Would a quantum particle radiation distribution embedded within the photon radiation distribution induce cosine anisotropy in the photon radiation?}} \; (\color{blue}{\text{ref. 2}})\]
\[\color{blue}{\text{If cosine anisotropy is a detectable amount, is it possible to map cosine anisotropy amounts verses frequency within the photon radiation distribution?}} \; (\color{blue}{\text{ref. 2}})\]
\[\color{blue}{\text{Based upon the frequency multi-distribution plot, what frequency domain in the photon radiation distribution would you expect to detect cosine anisotropy?}}\]
\[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\]
Reference:
Wikipedia - Planck's law: (ref. 1)
https://en.wikipedia.org/wiki/Planck's_law
Wikipedia - Anisotropy - Physics: (ref. 2)
https://en.wikipedia.org/wiki/Anisotropy#Physics

graph1.jpg

graph2.jpg

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\[\color{blue}{\text{Solar surface photon radiation temperature:} \; (\text{ref. 1})}\]
\[T_{\odot} = 5772 \; \text{K}\]
\[\color{blue}{\text{Bose-Einstein solar surface photon radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\odot}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{\gamma} = 3.39332 \cdot 10^{14} \; \text{Hz}}\]
\[\boxed{f_{\gamma} = 339.332 \; \text{THz}}\]
\[\color{blue}{\text{Calculated solar surface photon radiation spectral radiance peak frequency:} \; (\text{ref. 2})}\]
\[f_{\gamma} = \frac{\left[W_{0}\left(-3e^{-3} \right) + 3 \right] k_B T_{\odot}}{h}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[T_{1} = 6000 \; \text{K} \; \; \; f_{\gamma} = 352.735 \; \text{THz}\]
\[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\]
Reference:
Wikipedia - Sun Sol: (ref. 1)
https://en.wikipedia.org/wiki/Sun
Wikipedia - Wien's displacement law: (ref. 2)
https://en.wikipedia.org/wiki/Wien's_displacement_law#Parameterization_by_frequency
Wikipedia - Lambert W function: (ref. 3)
https://en.wikipedia.org/wiki/Lambert_W_function

graph3.jpg

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