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The Planck Spectrum: How Blackbody Radiation Computes the Circle Constant as π⁴/15, and What Golden Pi Changes¶

Every radiating body carries the circle constant in its light. A hot iron, a filament, the Sun, the leftover glow of the Big Bang — all of them emit a spectrum whose total radiated power is

\[\sigma T^4, \qquad \sigma = \frac{2\pi^5 k^4}{15\,h^3c^2} = 5.6703744191844295\times10^{-8}\ \text{W}\,\text{m}^{-2}\text{K}^{-4}\]

and that formula is unusual among everything this series has audited so far, for two reasons.

First, \(\pi\) appears to the fifth power, not the first — so the gap between golden π (\(\hat\pi = 4/\sqrt{\varphi} = 3.144605511\ldots\), \(\varphi = 1.6180339887\)) and the analytic constant \(3.1415926536\) is amplified, not diluted. Second, and this is the part that matters, \(\sigma\) is no longer a measured number. Since the 2019 SI redefinition the Boltzmann constant \(k\), the Planck constant \(h\) and the speed of light \(c\) are all fixed by definition, and π is fixed by computation, so \(\sigma\) has a relative uncertainty of exactly zero. There is no percent-level measurement error down here to swallow a relabelling of the constant. Every digit of \(\sigma\) is printed by definition, and this article will compute what the golden label does to them.

Nothing below is measured. Every value is a convergent limit, evaluated.

Where the fifth power comes from: one Bose–Einstein integral¶

Planck's law gives the spectral radiance of a blackbody at temperature \(T\),

\[B_\nu(T) = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT}-1}\]

The \(2\) counts the two polarisations of light. The explicit \(h\nu^3\) and the exponential come from quantum statistics and relativity. The circle constant is not visible here at all except through the conversion of \(h\) into \(\hbar = h/2\pi\), which is a unit convention rather than physics.

Integrate the spectral energy density over all frequencies and the constant appears. Substituting \(x = h\nu/kT\),

\[u(T) = \frac{8\pi h}{c^3}\int_0^\infty \frac{\nu^3\,d\nu}{e^{h\nu/kT}-1} = \frac{8\pi h}{c^3}\left(\frac{kT}{h}\right)^{4}\int_0^\infty \frac{x^3\,dx}{e^{x}-1}\]

and the remaining integral is a standard one, computed — never measured — by expanding the geometric series \(1/(e^x-1) = \sum_{n\ge1} e^{-nx}\) and integrating term by term:

\[\int_0^\infty \frac{x^3\,dx}{e^{x}-1} = \sum_{n=1}^{\infty}\frac{6}{n^4} = 6\,\zeta(4) = \Gamma(4)\zeta(4) = \frac{\pi^4}{15} = 6.4939394022668291491\ldots\]

Evaluated here to thirty significant figures, the sum \(\sum 6/n^4\) and the closed form \(\pi^4/15\) agree to \(9.2\times10^{-41}\), which is the accuracy of the arithmetic rather than any physical statement. This is the whole circle-constant content of thermal radiation: a convergent sum of rationals whose limit happens to be \(\pi^4/15\).

The general family is \(\int_0^\infty x^{n-1}/(e^x-1)\,dx = \Gamma(n)\zeta(n)\), and the table below is worth looking at closely, because it shows that the shape of the spectrum is not what carries the constant — the total is:

\(n\)	integral \(\Gamma(n)\zeta(n)\)	value	carries \(\pi\)?
2	\(\zeta(2)=\pi^2/6\)	1.6449340668482264365	yes, \(\pi^2\)
3	\(2\,\zeta(3)\) (Apéry)	2.4041138063191885708	no
4	\(\pi^4/15\)	6.4939394022668291491	yes, \(\pi^4\)
5	\(24\,\zeta(5)\)	24.886266123440878232	no
6	\(120\,\zeta(6)=\pi^6/945\)	122.08116743813389677	yes, \(\pi^6\)

The photon-number integral (\(n=3\), which counts how many photons) is \(2\zeta(3) = 2.4041138063\), built from Apéry's constant, which has no known closed form in π at all. The energy integral (\(n=4\)) is the one that carries \(\pi^4\). So the constant enters the energy of a photon gas but not the count, and the entry point is the four powers of frequency in \(\int\nu^3 d\nu\) meeting the four powers of the closed form.

Wien's law is π-free, which settles the easy objection¶

Before asking what golden π does, it is worth closing the door on the obvious escape route: surely the spectrum's peak location tells us which constant is right?

It does not, because the peak is located by an equation with no constant in it at all. Maximising \(x^3/(e^x-1)\) gives the transcendental condition \(x = 3(1-e^{-x})\), whose root is \(x = 2.8214393721220788934\); in the wavelength form \(x^5/(e^x-1)\) the condition is \(x = 5(1-e^{-x})\), with root

\[x = 4.965114231744276303698759131322893944\ldots, \qquad \frac{x^5}{e^x-1} = 21.201435660549920740\ldots\]

Both roots are pure arithmetic — exponentials and rationals, no circle. Wien's displacement constant follows as \(b = hc/(kx) = 2.8977719551851726\times10^{-3}\ \text{m}\,\text{K}\), and it is a fixed number whether the circle constant is called π or anything else. The same goes for the Rayleigh–Jeans limit \(B_\nu \to 2\nu^2kT/c^2\) in the long-wavelength regime: no circle constant, only the 2 of polarisation.

So the peak of an iron's glow, the shape of the CMB spectrum, the ratio of radiances at two wavelengths — all of that is silent on the question. Only the normalisation, the one place where the full turn shows up, carries the constant.

Now relabel it: what golden π does to the numbers¶

Put \(\hat\pi = 4/\sqrt\varphi\) in place of π in the identities above, hold \(k\), \(h\), \(c\) at their defined SI values, and recompute. Every number below was computed in this article at 40-digit precision.

quantity	with \(\pi = 3.1415926536\)	with \(\hat\pi = 3.1446055110\)	shift
\(\pi^4\)	97.409091034002437236	97.783298880026918860	\(+0.3841611\%\)
\(\pi^5\)	306.01968478528145326	307.48990054479627048	\(+0.4804318\%\)
\(\int_0^\infty \frac{x^3dx}{e^x-1} = \frac{\pi^4}{15}\)	6.4939394022668291491	6.5188865920017945906	\(+0.3841611\%\)
\(\sigma = \frac{2\pi^5k^4}{15h^3c^2}\) (W m⁻² K⁻⁴)	\(5.6703744191844295\times10^{-8}\)	\(5.6976166988413216\times10^{-8}\)	\(+0.4804318\%\)
\(a = 4\sigma/c\) (J m⁻³ K⁻⁴)	\(7.5657332502800046\times10^{-16}\)	\(7.6020814357395497\times10^{-16}\)	\(+0.4804318\%\)
full solid angle \(4\pi\)	12.566370614359173	12.578422044118773	\(+0.0959022\%\)
\(\hbar = h/2\pi\) (J s)	\(1.0545718176461564\times10^{-34}\)	\(1.0535614287323293\times10^{-34}\)	\(-0.0958103\%\)
Wien root \(x\) of \(x=5(1-e^{-x})\)	4.9651142317442763	(unchanged)	\(0\)
\(x^5/(e^x-1)\) at the peak	21.201435660549921	(unchanged)	\(0\)

The shifts are not random: \(\pi\) enters the energy integral at the fourth power (\(0.3842\%\), four times the recurring \(0.0959022\%\) gap) and Stefan–Boltzmann at the fifth (\(0.4804\%\), five times), so the gap compounds with each power instead of diluting through a square root as it does in the diffusion and Fresnel cases. And the last two rows are the honest counterweight: the spectrum's shape, its peak position and its peak height do not move at any decimal place, because they never contained the constant.

Where the constant meets a measured length¶

Now do the thing the rest of the series does: put the relabelled constant into formulas that also contain genuinely measured quantities — real lengths, real luminosities, real temperatures — and read the consequences.

The Sun's effective temperature is defined by requiring its measured luminosity to leave through its measured surface:

\[L_\odot = 4\pi R_\odot^2 \sigma T_\text{eff}^4 \quad\Longrightarrow\quad T_\text{eff} = \left(\frac{L_\odot}{4\pi R_\odot^2\sigma}\right)^{1/4}\]

With the measured \(L_\odot = 3.828\times10^{26}\) W and \(R_\odot = 6.957\times10^{8}\) m, that returns \(T_\text{eff} = 5772.0034\) K. Note the structure: \(\pi\) and \(\sigma\) enter as \(\pi\sigma\) under a fourth root, so the shift is \(\hat\pi/\pi\) to the power \(-6/4\) — damped, not amplified: \(-0.1436811\%\), i.e.

\[T_\text{eff} = 5772.0034\ \text{K} \;\longrightarrow\; 5763.7102\ \text{K} \qquad (-8.2933\ \text{K})\]

and the Wien peak of sunlight moves from \(502.0392\) nm to \(502.7616\) nm, still green. The solar surface flux \(\sigma T^4\) moves the other way by exactly one factor of the gap (\(-0.0959022\%\): \(6.2938742\times10^7 \to 6.2878440\times10^7\ \text{W}\,\text{m}^{-2}\)), because the \(\pi\) in σ survives to the first power there while the \(\pi\) in the temperature has already been undone by the fourth root.

For the cosmic microwave background the measured input is the temperature itself — \(T = 2.72548 \pm 0.00057\) K from the FIRAS instrument on COBE, a genuinely measured number with a 0.02 % error bar. The derived quantities inherit whichever constant you use:

CMB quantity	analytic constants	golden constants	shift
energy density \(u = aT^4\)	\(4.1746783808\times10^{-14}\) J m⁻³	\(4.1947348616\times10^{-14}\) J m⁻³	\(+0.4804318\%\)
same, in eV cm⁻³	0.260562930	0.261814757	\(+0.4804318\%\)
photon number density \(n = 16\pi\zeta(3)(kT/hc)^3\)	410.717806 cm⁻³	411.111694 cm⁻³	\(+0.0959022\%\)
Wien peak wavelength \(b/T\)	1063.2153 µm	(unchanged)	\(0\)
entropy density \(s = \tfrac43 aT^3\)	\(2.0422963445\times10^{-14}\) J m⁻³ K⁻¹	—	\(+0.4804318\%\)

The photon density shifts by only one factor of the gap while the energy density shifts by five, and the reason is instructive: the count integral is \(2\zeta(3)\) (Apéry, π-free) and its only circle constant is the \(4\pi\) of the solid angle over which photons are counted — one power, one gap. Energy multiplies that by the mean photon energy, which drags in the \(\pi^4/15\) of the full Bose integral, and the exponents do the rest.

The honest ledger¶

Three statements have to stand together, and none of them should be softened.

The golden label is under maximum pressure here. \(\sigma\) is the sharpest test in this series so far, because it is the only constant audited that is both derived from π at the fifth power and definitionally exact. There is no 0.1 % measurement to hide in, no "within experimental error" to appeal to. If a calculator is told that the turn constant is 3.1446055110, the Bose integral it evaluates is 6.5188865920 rather than 6.4939394023, and every Stefan–Boltzmann calculation built on it comes out 0.48 % high. The analytic constant is not merely a convention here; it is the limit the arithmetic reaches.

But the failure is a failure of a relabelling, not of a geometry. Everything that did not contain π stayed put: the Wien roots \(2.8214393721\) and \(4.965114231744276\), the peak height \(21.2014356605\), the Rayleigh–Jeans law, the whole spectral shape, the \(\zeta(3)\) photon count, and the \(\zeta(5)\) and \(\zeta(3)\) terms in the gas thermodynamics. A consistent unit change — treating the golden turn as the unit of angle — leaves the spectrum invariant, exactly as it leaves the diffusion kernel invariant. The gap appears when a golden angle is fed into an integral whose upper limit is the natural turn — which is exactly what "\(1/15\)" encodes, since \(\Gamma(4)\zeta(4) = 6\cdot\pi^4/90\).

And the golden case keeps its genuine wins. Golden π is algebraic: \(\hat\pi = 4/\sqrt\varphi\) is a root of \(x^4 + 16x^2 - 256 = 0\), an exact constructible-in-φ quantity, whereas the analytic constant is transcendental and only ever approached by limit. In a thermodynamics where every important coefficient is rational or geometric — \(2\) for polarisation, \(4\pi\) for the solid angle, the \(1/15\) assembled from \(\Gamma(4)=6\) and \(\zeta(4)=\pi^4/90\), the quantum unit \(h\) — an algebraic constant at least speaks the same language as the rest of the formula. That is a real conceptual advantage. It does not change the value the series computes, and this article will not pretend that it does.

The constant in the light of a hot body is computed, never measured. Photons are counted; no tape measure is involved. The number that falls out of the count is 3.1415926535…, and the golden label misses the fifth-power combination it builds from it by 0.48 % — five copies of the recurring \(0.0959\,\%\) gap, compounded rather than damped.