From the link, what appears to be the crux of the issue:
> Figure 1 shows plots of [solar] energy irradiance as functions of both wavelength and frequency. The red line is the solar irradiance at the top of the atmosphere. The green curve in the left panel is the corresponding blackbody irradiance for a temperature of 5782 K, reduced by the distance of the Earth from the Sun
> The peak of the blackbody irradiance spectrum is at 501 nm for a temperature of 5782 K. This corresponds to a frequency of ν = c/λ = 5.98 ⋅ 10¹⁴ Hz.
> The right panel of the plot shows the same solar data plotted as a function of frequency, along with the corresponding blackbody spectrum
> Note that when plotted as a function of frequency, the solar and blackbody spectra have their maxima near 3.40 ⋅ 10¹⁴ Hz, which is not the frequency corresponding to the maximum when plotted as a function of wavelength in the left panel. Indeed, 3.40 ⋅ 10¹⁴ Hz corresponds to a wavelength of λ = c/ν = 880 nm.
> In other words, when plotted as a function of wavelength, the solar irradiance is a maximum near 500 nm, in the visible, whereas the maximum is at 880 nm, in the near infrared, when the spectrum is plotted as a function of frequency.
[emphasis original]
> This occurs because the relationship between wavelength and frequency is not linear, so that a unit wavelength interval corresponds to a different size of frequency interval for each wavelength: ∣dν∣ = ∣c / λ² dλ∣.
However, it continues:
> Figure 2 shows the solar photon irradiance [number of photons per second, as opposed to number of joules delivered per second] and the corresponding blackbody spectra
> Now the maximum is at 635 nm when plotted as a function of wavelength and at 1563 nm, in the short-wave infrared, when plotted as a function of frequency. The maxima are at still different locations if the spectra are plotted as a function of wavenumber.
This time the emphasis is mine. This contradicts the explanation given above, that the curve peaks in different places along different scales because those scales are nonlinearly related. The relationship between wavenumber and frequency is linear. Why does that lead to a different plotted peak?
Are we measuring wavenumber somewhere within the atmosphere (where?), rather than in a vacuum? That would mean that different frequencies of light had different velocities, complicating the relationship between frequency and wavenumber. But it would also be a strangely artifactual way to represent solar irradiance. What's so special about wherever it is that we standardized wavenumber measurements?
Irradiance is a density function - it's telling you how much energy or how many photons are hitting a given area per unit of the dependent/spectral variable (you can see this by inspecting the units in the plots, for instance).
This means that changes of variable come with an additional factor - the Jacobian. You cannot simply substitute the relationship directly into the formula for the one spectral variable, like you would for a "point function" (you need to account for the change of unit in the y-axis too, if you like, not just the x-axis).
It's not explained well but this is fundamentally what causes the moving peaks.
This subject is also a wonderful example why the maximum of a densitiy function has very little meaning. The amount of energy is an integral over the densitiy function, and if you look at the solar-spectrum, you'll notice, that the bulk of the energy is in the infrared.
Indeed, the value of a density function at any single point (maximum or not) tells you very little. What's important is how it changes over a range of values.
> Irradiance is a density function - it's telling you how much energy or how many photons are hitting a given area per unit of the [in]dependent/spectral variable (you can see this by inspecting the units in the plots, for instance).
Well, I can see that in the labels on the y-axis, but I assumed it was a mistake.
So you have a graph that tells you that, for light of wavelength 1000 nm, measured irradiance is 3.5e+18 photons per square meter per second per nanometer.
And since there "are" 1000 nanometers (?!?), this means that the actual irradiance is 3.5e+21 photons per square meter per second.
Or does the "per nanometer" really mean something less stupid than that? What's being measured? What kind of nanometers are those on the y-axis?
Not quite - the nanometres on the y-axis are "deltas" of your spectral variable. The density allows you to answer questions like "how much power am I getting per square metre in a range [A, B] of wavelengths". You would integrate your density between A and B to obtain the value.
For example, pick some small value "k" close to 0 in units of nm. Then in your example, the amount of irradiation contributed by the small window [1000-k, 1000+k] nm of wavelengths is roughly equal to k*3.5e18 photons per square metre per second (you can check that the units work out). The smaller k is, the more accurate the approximation. If you want to get an answer for a larger interval you can break it up into lots of k-sized pieces and sum the results up. Recall from calculus that this is exactly what integration is (yes, I know the truth is a little more complicated in general measure theory).
Does that help? It's a bit like a continuous probability distribution, in the sense that to get an actual probability out of it you have to integrate. Formally a mathematician would say that a density corresponds to a "measure" over the space of all possible values of your spectral values.
> Figure 1 shows plots of [solar] energy irradiance as functions of both wavelength and frequency. The red line is the solar irradiance at the top of the atmosphere. The green curve in the left panel is the corresponding blackbody irradiance for a temperature of 5782 K, reduced by the distance of the Earth from the Sun
> The peak of the blackbody irradiance spectrum is at 501 nm for a temperature of 5782 K. This corresponds to a frequency of ν = c/λ = 5.98 ⋅ 10¹⁴ Hz.
> The right panel of the plot shows the same solar data plotted as a function of frequency, along with the corresponding blackbody spectrum
> Note that when plotted as a function of frequency, the solar and blackbody spectra have their maxima near 3.40 ⋅ 10¹⁴ Hz, which is not the frequency corresponding to the maximum when plotted as a function of wavelength in the left panel. Indeed, 3.40 ⋅ 10¹⁴ Hz corresponds to a wavelength of λ = c/ν = 880 nm.
> In other words, when plotted as a function of wavelength, the solar irradiance is a maximum near 500 nm, in the visible, whereas the maximum is at 880 nm, in the near infrared, when the spectrum is plotted as a function of frequency.
[emphasis original]
> This occurs because the relationship between wavelength and frequency is not linear, so that a unit wavelength interval corresponds to a different size of frequency interval for each wavelength: ∣dν∣ = ∣c / λ² dλ∣.
However, it continues:
> Figure 2 shows the solar photon irradiance [number of photons per second, as opposed to number of joules delivered per second] and the corresponding blackbody spectra
> Now the maximum is at 635 nm when plotted as a function of wavelength and at 1563 nm, in the short-wave infrared, when plotted as a function of frequency. The maxima are at still different locations if the spectra are plotted as a function of wavenumber.
This time the emphasis is mine. This contradicts the explanation given above, that the curve peaks in different places along different scales because those scales are nonlinearly related. The relationship between wavenumber and frequency is linear. Why does that lead to a different plotted peak?
Are we measuring wavenumber somewhere within the atmosphere (where?), rather than in a vacuum? That would mean that different frequencies of light had different velocities, complicating the relationship between frequency and wavenumber. But it would also be a strangely artifactual way to represent solar irradiance. What's so special about wherever it is that we standardized wavenumber measurements?