Simplified Planck Law to Simplified Stephan Boltzmann Derivation.

 1. Start with the Simplified Planck's Law:

The spectral radiance in your framework is:
B_ν(ν, T) = 2 * ν³ * Hz_kg / (exp(ν / (T * K_Hz)) - 1)

• ν: Frequency

• T: Absolute temperature

• Hz_kg: The scaling factor h/c² (units: kg/Hz or kg·s)

• K_Hz: The scaling factor k/h (units: Hz/K)

2. Integrate over all Frequencies:

To get the total radiance B(T), integrate B_ν from ν = 0 to ν = ∞:

B(T) = ∫[0 to ∞] B_ν(ν, T) dν
    = ∫[0 to ∞] (2 * ν³ * Hz_kg / (exp(ν / (T * K_Hz)) - 1)) dν

3. Perform the Substitution:

This integral has the same structure as before. Let's use the same substitution, which now directly involves K_Hz:
x = ν / (T * K_Hz)
Then:
ν = x * T * K_Hz
dν = dx * T * K_Hz

Substitute these into the integral:

B(T) = ∫[0 to ∞] (2 * (x * T * K_Hz)³ * Hz_kg / (exp(x) - 1)) * (dx * T * K_Hz)

4. Simplify and Pull Out Constants:

Expand the terms and pull factors that don't depend on x out of the integral:

B(T) = ∫[0 to ∞] (2 * x³ * T³ * K_Hz³ * Hz_kg / (exp(x) - 1)) * (T * K_Hz * dx)
B(T) = ∫[0 to ∞] (2 * Hz_kg * T⁴ * K_Hz⁴ * x³ / (exp(x) - 1)) dx
B(T) = (2 * Hz_kg * T⁴ * K_Hz⁴) ∫[0 to ∞] (x³ / (exp(x) - 1)) dx

5. Evaluate the Definite Integral:

As before, the standard definite integral is:
∫[0 to ∞] (x³ / (exp(x) - 1)) dx = π⁴ / 15

6. Substitute the Integral Value:

B(T) = (2 * Hz_kg * T⁴ * K_Hz⁴) * (π⁴ / 15)
B(T) = (2π⁴ / 15) * Hz_kg * K_Hz⁴ * T⁴

This is the total radiance (power per unit area per unit solid angle) in your framework's terms.

7. Convert Radiance 

Multiply by π (integrating over the hemisphere with Lambert's cosine law):

j* = π * B(T)
j* = π * [(2π⁴ / 15) * Hz_kg * K_Hz⁴ * T⁴]
j* = (2π⁵ / 15) * Hz_kg * K_Hz⁴ * T⁴

8. Identify the Stefan-Boltzmann Constant (σ) in the Simplified Framework:

The Stefan-Boltzmann law is j* = σT⁴. Comparing this to the derived expression for j*, we identify the constant σ expressed in terms of your scaling factors:

σ = (2π⁵ / 15) * Hz_kg * K_Hz⁴

This result is perfectly consistent. It shows how σ naturally emerges from the integration, expressed using your Hz_kg and K_Hz factors instead of the traditional c, h, k. The derivation follows the exact same mathematical steps, but the constants group differently according to your framework, leading directly to the simplified form of σ.

Let’s refine and clarify the derivation of the Stefan-Boltzmann constant (σ) from the simplified Planck’s Law, emphasizing the role of unit rescaling and dimensional unification. Here’s the streamlined version:

Derivation of Stefan-Boltzmann Constant (σ) from Simplified Planck’s Law

1. Simplified Planck’s Law

This framework’s spectral radiance:

$$B_{\nu }(\nu ,T)=\frac{2\cdot {\nu }^3\cdot \text{Hz\_kg}}{e^{\nu \mathrm{/}(T\cdot \text{K\_Hz})}-1}$$

Where:

• ν: Frequency (Hz)

• T: Temperature (K)

• Hz_kg = $\frac{h}{c^2}$ (converts Hz → kg)

• K_Hz = $\frac{k}{h}$ (converts K → Hz)

2. Integrate Over All Frequencies

Total radiance (power per unit area per unit solid angle):

$$B(T)={\int }_0^{\mathrm{\infty }}{B}_{\nu }(\nu ,T)d\nu ={\int }_0^{\mathrm{\infty }}\frac{2{\nu }^3\cdot \text{Hz\_kg}}{e^{\nu \mathrm{/}(T\cdot \text{K\_Hz})}-1}d\nu$$

3. Substitution

Let:

$$x=\frac{\nu }{T\cdot \text{K\_Hz}}\Rightarrow \nu =x\cdot T\cdot \text{K\_Hz},d\nu =T\cdot \text{K\_Hz}\cdot dx$$

Substitute into the integral:

$$B(T)={\int }_0^{\mathrm{\infty }}\frac{2\cdot (x\cdot T\cdot \text{K\_Hz}{)}^3\cdot \text{Hz\_kg}}{e^x-1}\cdot T\cdot \text{K\_Hz}dx$$

4. Simplify and Extract Constants

$$B(T)=2\cdot \text{Hz\_kg}\cdot T^4\cdot {\text{K\_Hz}}^4{\int }_0^{\mathrm{\infty }}\frac{x^3}{e^x-1}dx$$

The integral is a known dimensionless constant:

$${\int }_0^{\mathrm{\infty }}\frac{x^3}{e^x-1}dx=\frac{{\pi }^4}{15}$$

5. Total Radiance

$$B(T)=\frac{2{\pi }^4}{15}\cdot \text{Hz\_kg}\cdot {\text{K\_Hz}}^4\cdot T^4$$

6. Total Radiant Flux (Stefan-Boltzmann Law)

Integrate over a hemisphere (Lambert’s cosine law):

$$j^{\ast }=\pi \cdot B(T)=\frac{2{\pi }^5}{15}\cdot \text{Hz\_kg}\cdot {\text{K\_Hz}}^4\cdot T^4$$

The Stefan-Boltzmann constant in your framework:

$$\sigma =\frac{2{\pi }^5}{15}\cdot \text{Hz\_kg}\cdot {\text{K\_Hz}}^4$$

7. Equivalence to Traditional Formula

Substitute Hz_kg = $\frac{h}{c^2}$ and K_Hz = $\frac{k}{h}$:

$$\sigma =\frac{2{\pi }^5}{15}\cdot \frac{h}{c^2}\cdot {\left(\frac{k}{h}\right)}^4=\frac{2{\pi }^5{k}^4}{15c^2{h}^3}$$

This matches the traditional Stefan-Boltzmann constant, proving the framework’s consistency.

Key Insights

1. Unit Scaling Dominates:

   • Hz_kg and K_Hz absorb h, c, and k, reducing them to unit conversion factors.

   • The derivation avoids energy (J) entirely, staying in the domains of frequency (Hz) and mass (kg).

2. Dimensional Unification:

   • Temperature (K) → Frequency (Hz) via K_Hz

   • Frequency (Hz) → Mass (kg) via Hz_kg

   • The geometric factor $\frac{2{\pi }^5}{15}$ is dimensionless and universal.

3. Simplification ≠ Approximation:

   • Your framework’s σ is mathematically identical to the traditional form but reframes it in terms of unit rescaling instead of "fundamental constants."

Why This Matters

Your derivation demonstrates that σ is not a standalone constant but a consequence of unit mismatches in SI (kg vs. Hz vs. K).