Two laws that are one curve read twice

Wien’s displacement law and the Stefan–Boltzmann law are usually introduced as two facts about thermal radiation: the peak moves as $1/T$, and the total flux goes as $T^4$. Both predate Planck’s formula, both were found empirically, and both are presented as independent.

They are not independent. They are two readings of one curve, and the second follows from the first plus one more fact about the same curve.

Why the shape is universal #

Write Planck’s law in terms of $x = hc/\lambda kT$:

$$B_\lambda = \frac{2hc^2}{\lambda^5}\frac{1}{e^x - 1} = \frac{2k^5T^5}{h^4c^3}\cdot\frac{x^5}{e^x - 1}.$$

Every dependence on temperature has been collected into the prefactor, and the remaining function $x^5/(e^x-1)$ depends on $x$ alone.

That is the whole structure. Planck’s law is one universal shape in $x$, scaled by $T^5$, and everything else follows by asking what a universal shape does under a scaling.

• Its peak is at a fixed $x$, so $\lambda_{\max}T$ is a constant. That is Wien.
• Its peak height is $T^5$ times the shape’s own maximum. That is a fifth power.
• Its width in $x$ is fixed, so its width in $\lambda$ goes as $1/T$.
• Its area is the height times the width, which is $T^5 \times T^{-1} = T^4$. That is Stefan–Boltzmann.

The last step is where the two laws become one. A peak five powers high on a curve one power narrow encloses four powers of area, and no additional physics enters between them.

Solving for the peak #

The peak’s position needs one equation solved. Setting $dB_\lambda/d\lambda = 0$ gives

$$5(1 - e^{-x}) = x,$$

a transcendental equation whose non-trivial root is $x = 4.9651142$. The Wien constant is then $hc/kx = 2.897772$ mm K, and the figure obtains it by bisection rather than by quotation.

The equation has no closed form and the root is not a memorable number, which is the one aesthetic blemish in an otherwise clean derivation. It also has a companion: doing the same in frequency rather than wavelength gives $3(1 - e^{-x}) = x$ and a root of 2.8214, so the peak in frequency is not at the same place as the peak in wavelength.

That is not a paradox and it is worth being clear about. A spectral density is a density with respect to something, and changing the variable changes the density by a Jacobian. The Sun’s spectrum peaks at 502 nm per unit wavelength and at 882 nm per unit frequency, and neither number is “where the Sun emits most” — that phrase has no meaning until the variable is named.

What the two laws were before Planck #

Both were empirical, both were derived from thermodynamics without any theory of the spectrum, and the manner of each derivation is worth having because it shows how much can be got from a scaling alone.

Stefan’s law came from measurements and Boltzmann’s from thermodynamics. Josef Stefan noticed in 1879 that Tyndall’s measurements of a heated platinum wire fitted a fourth power. Five years later Boltzmann derived it, by treating radiation as a gas with a pressure equal to a third of its energy density and applying the second law to a Carnot cycle on a cylinder of it. The derivation needs the pressure relation — which is electromagnetism — and nothing about the spectrum.

Wien’s law came from an adiabatic argument. Consider radiation in a slowly expanding cavity with perfectly reflecting walls. Each mode’s wavelength stretches with the cavity, and the entropy is conserved, which forces the spectral distribution to keep its shape while sliding. That gives $\lambda_{\max}T =$ constant and more: it gives the full statement that $B_\lambda$ must have the form $\lambda^{-5}f(\lambda T)$ for some function $f$ nobody could determine.

That last result is the universal shape, sixteen years before Planck found $f$. Wien’s displacement law is a corollary of it, and so is Stefan–Boltzmann: integrating $\lambda^{-5}f(\lambda T)$ over all wavelengths and substituting $u = \lambda T$ gives $T^4$ times a constant, whatever $f$ is.

So the two laws were known to be related long before either could be derived from a spectrum, and the relation was known to follow from thermodynamics alone. What Planck supplied was the constant, and the constant is where $h$ lives.

The radius a colour cannot give #

The practical reason the two laws are worth having together is that one of them is a shape and the other is an absolute quantity.

A colour index gives a temperature, and a temperature is a property of the spectrum’s shape — it is scale-free, and a star twice as big at the same temperature has exactly the same colour. So a colour can never give a size.

Stefan–Boltzmann is different. $F = \sigma T^4$ is a flux per unit area, so combining it with a measured total flux gives an area:

$$L = 4\pi R^2\sigma T^4 .$$

With $T$ from the colour and $L$ from the apparent brightness and the distance, $R$ follows. For the Sun at 5,772 K receiving 1361 W/m² at one astronomical unit, the arithmetic returns $6.957\times10^{8}$ m against a measured $6.957\times10^{8}$.

That agreement is a construction rather than a test, because the Sun’s effective temperature is defined as the temperature of the blackbody with the same total flux and the same radius. What is being checked is the arithmetic and the constants, not the physics.

For any other star it is a genuine measurement, and it is the standard way a stellar radius is obtained. The alternative — measuring the angular diameter directly by interferometry — is available for a few hundred nearby giants and for essentially nothing else.

That insensitivity has a practical consequence for every curve drawn here. A fitted exponent that came back as 3.97 rather than 4.000 would mean the quadrature was wrong or the prefactor was not $T^5$, and there is no third possibility — no amount of numerical noise or sampling can move it, because it is a ratio of two exact evaluations of the same analytic function.

How the errors propagate #

That steepness is the practical difficulty of the method, and it is worth stating quantitatively.

From $L = 4\pi R^2\sigma T^4$, at fixed luminosity,

$$\frac{\delta R}{R} = -2\frac{\delta T}{T}.$$

A temperature good to one per cent gives a radius good to two. A temperature good to three per cent — which is where colour photometry sits for a cool star, once the systematics of reddening and metallicity are included — gives a radius good to six.

And the luminosity carries its own error, dominated by the distance: $\delta L/L = 2\delta d/d$, so $\delta R/R = \delta d/d$. A parallax good to two per cent gives a radius good to two per cent from that term alone.

So the radius is usually limited by the temperature rather than by the distance, which was not true before Gaia and is a recent inversion. For most of the twentieth century distances were the limiting term in almost every stellar measurement; now they frequently are not.

The bolometric problem #

There is a term quietly assumed above that is the hardest part in practice. $L$ is the total luminosity across all wavelengths, and no instrument measures that.

What is measured is the flux in one passband. Converting it to a total requires knowing what fraction of the star’s output that passband admits, which is a bolometric correction — and the correction depends on the temperature, which is what is being measured.

The whole visible window, 400 to 700 nm, admits 36.6 per cent of the Sun’s output. It admits 15.6 per cent of an M giant’s, because most of that star’s light is infrared, and it admits a rapidly shrinking fraction of a hot star’s, because most of that output is in the ultraviolet where the Earth’s atmosphere is opaque. The bolometric correction therefore has a maximum near A and F types and grows in both directions.

The bolometric correction is therefore largest exactly where the temperature is least well known, and it is a model-dependent quantity in both cases. A bolometric correction is computed from a model atmosphere, so a radius obtained this way inherits the atmosphere model’s assumptions.

Reading a radius off a spectrum alone #

There is a variant of the radius measurement that avoids the distance entirely, and it is worth separating because it measures something different.

The observed flux at the Earth from a star of radius $R$ at distance $d$ is $F = (R/d)^2\sigma T^4$. The factor $(R/d)^2$ is the solid angle the star subtends, so fitting an observed spectral energy distribution against a blackbody gives the angular diameter and the temperature together, with no distance needed.

That is the standard way an angular diameter is estimated photometrically, and it agrees with interferometric measurements to a few per cent for stars where both are available. Multiplying the angular diameter by a parallax distance then gives the physical radius, which is the same quantity by another route — and the two routes fail differently, since one is limited by the bolometric correction and the other by the parallax.

The angular diameter is the more robust of the two quantities, and it is worth knowing which measurements need only it. It is also the quantity that improves fastest with instrumentation, because it is set by the photometry and the model atmosphere rather than by a parallax, and a better model atmosphere is cheaper to obtain than a better astrometric mission. A transit depth gives a planet’s radius as a fraction of its star’s, so a planet radius is a stellar radius and inherits whatever the stellar one is worth. A surface brightness — flux per unit solid angle — needs the angular diameter and nothing else, which is why it does not fall with distance.

Exponents fitted off the curves rather than quoted #

The exponents are fitted off the drawn curves rather than taken from a textbook, which is the check that the universal-shape argument holds: if the peak height did not go as $T^5$ to three decimals, the prefactor would not be $T^5$ and the whole derivation would fail.

The Wien constant is solved by bisection on $5(1-e^{-x}) = x$, and it agrees with the measured 2.897771955 mm K to nine figures — which is unsurprising, since both are now derived from defined constants.

The Stefan–Boltzmann constant is likewise exact by definition: $\sigma = 2\pi^5k^4/(15c^2h^3)$, and since 2019 all four of those are defined rather than measured, so $\sigma$ is an exact rational multiple of powers of defined quantities.

And the quadrature is checked against the closed form. Integrating the Planck curve numerically over a logarithmic wavelength grid and comparing against $\sigma T^4/\pi$ agrees to one part in $10^9$ at every temperature drawn, which is the check that the units and the quadrature are both right — a version with the wavelength in the wrong unit would be out by a factor of $10^9$ and would look perfectly reasonable on a normalised plot.

Nothing here is a star #

Nothing here is a star. The curves are ideal blackbodies, and a stellar spectrum departs from one by absorption lines, by a Balmer jump, and in cool stars by molecular bands that remove whole regions. The effective temperature is defined so that Stefan–Boltzmann holds exactly for the star, which pushes the departure into the shape rather than the total.

The peak positions drawn are wavelength peaks. The frequency peaks sit at longer wavelengths by a factor of 1.76 and are not marked, which would make a cluttered figure and would require an axis the plot does not have.

And the area under each curve is not visible. The plot is logarithmic in both axes, so an area on it is not an area under the function, and the $T^4$ has to be taken from the fitted exponent rather than read off the picture. That is a real limitation of a log–log plot and the reason the exponent is printed rather than shown.

Where the effective temperature comes from #

The quantity in $L = 4\pi R^2\sigma T^4$ deserves one section of its own, because it is defined rather than observed and the definition is doing work.

A star is not a blackbody and has no single temperature. Its photosphere spans a range of depths at a range of temperatures, and the spectrum emerging is a superposition. The surface a star stops at is itself a convention — the depth at which the optical depth reaches two thirds — chosen so that the emergent flux equals the local source function.

The effective temperature is defined as the temperature of the blackbody that would emit the star’s total flux per unit area at the star’s radius. So Stefan–Boltzmann is true of a star by construction, and any departure of the star from a blackbody is pushed entirely into the spectrum’s shape.

That is why an effective temperature and a colour temperature are different numbers for the same star. The first is defined through the total flux; the second is the temperature of the blackbody matching the observed colour. For the Sun the two differ by about 10 K; for an M dwarf with strong molecular bands they can differ by several hundred, because the bands remove light from one passband and not the other.

The convention is a good one and it is worth saying why. It makes the quantity in the radius relation exactly the quantity the relation needs, and it moves every complication into a correction — the bolometric correction and the colour–temperature relation — that can be computed once per spectral type and tabulated.

A pattern worth recognising #

The structure — a universal shape scaled by a power, from which every moment’s exponent follows — recurs whenever a problem has one dimensionless variable.

It is why a Maxwell–Boltzmann distribution’s mean speed, most probable speed and root-mean-square speed differ only by pure numbers, all going as $\sqrt{T/m}$ — which is why the gas a planet keeps is decided by a ratio of two speeds and not by either separately. It is why an isothermal sphere’s every property is a power of one radius. It is why the whole family of Planck curves can be drawn as one curve with two axis labels.

The general statement is that a one-parameter family has one shape, and the parameter can only stretch it. Every “law” relating two of its moments is then arithmetic on exponents rather than new physics, and discovering such a law empirically — as Wien and Stefan both did, decades before Planck — is discovering the scaling rather than the function.

That is why the two laws were found first. The scaling is visible in data long before the shape is, since it survives any systematic that multiplies rather than distorts, and both men were working with photometry that could not have determined the curve.

Still open: nothing in the laws, and a great deal in applying them #

Both laws are exact for a blackbody and the derivation has no loose ends. Every quantity in them is now a defined constant or a solved root, which puts them in a very small class: most numbers in this subject are measurements, and these are not. What is not settled is the step from a star to a blackbody, and that step has two parts.

The first is the effective temperature’s definition, which is exact but not observable — it is defined through the total flux and the radius, and measuring either requires the other unless the star is resolved.

The second is the temperature scale itself. Converting an observed colour into an effective temperature requires model atmospheres, and different model grids disagree by 50 to 150 K for cool stars. Those differences propagate into every stellar radius, every luminosity and every age derived from them, and they are the dominant systematic in stellar astrophysics.

From here: the assumption made about every colour so far #

Everything above has assumed that the observed colour is the star’s own. It is not, and three separate effects shift a colour index by more than any modern photometer’s error bar.

Two of them shift it in opposite directions, which means they can cancel — and a reddened metal-poor star can have exactly the colour of an unreddened solar-metallicity one, with nothing in two magnitudes to tell them apart.