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trilobite submitted a new PF Insights post
Exploring the Spectral Paradox
Continue reading the Original PF Insights Post.
Exploring the Spectral Paradox
Continue reading the Original PF Insights Post.
Overall, the key point to remember is that one must be careful when interpreting spectral curves, especially their peaks
Yes: be aware of what is being plotted against what! (Do as I say and not as I do. )anorlunda said:Is there a general rule to follow in data analysis that would let us avoid such paradoxes?
The paper by Soffer and Lynch does make a similar point concerning the alternative of using a logarithmic representation. However, the authors state that this method has "no special physical significance" and should not be "singled out as a preferred physical ... representation" for electromagnetic spectra, even though it does cause the wavelength and frequency peaks to coincide. As a wavelength, that peak for solar radiation is approximately 720 nm, according to the paper. Note that that value is very close to Heald's median point of 710 nm, as I suspect the mathematics imply, although Heald does describe the value as "physically meaningful!"NH_EE said:We could avoid this dichotomy by expressing Power Spectral Density as Power Log(spectral) density, i.e., instead of
W/(m^2 * delta(nm)),
use
W/(m^2 * delta(log(nm)))
This should give the same shape curve as
W/(m^2 * delta(log(THz)))
because
log(nm) = log(c)-log(THz),
thus removing the nonlinear mapping between the wavelength and frequency based metrics.
As with the original unit, denominator is area times an interval. Here, the interval in the measurement unit would actually be the difference of the logarithms of the ends of the interval, or the log of the ratio of the ends of the interval.
We don't even have to use logarithms if we force the interval in the measurement to be constant on a log scale, i.e., a constant ratio such as a milli-octave.
I would think it could be a useful exercise to find other situations in Science where a peak in a curve shifts according to the units on the x axis.trilobite said:a detailed exploration into where that sense of paradox is coming from can
Check out the formulae in Hyperphysics and WikipediaWe see that:The important point here is that you get that same value whether you measure the area under the wavelength curve (Fig. 1) or under the frequency curve (Fig. 2), even though the two curves have different shapes!
Reference https://www.physicsforums.com/insights/exploring-spectral-paradox/
hmm, do you have a link to the mathematical proof?sophiecentaur said:The Integral between the limits of λ will be equal to the integral between equivalent limits of ν.
Is a proof needed that the energy between the limits of frequency and the corresponding limits of wavelength is the same? It's the same energy and the same conceptual filter letting it through. I know that Maths is usually required here but this constraint comes before the Maths - not after a proof (I would have thought).tade said:hmm, do you have a link to the mathematical proof?
I managed to figure out how it workssophiecentaur said:Is a proof needed that the energy between the limits of frequency and the corresponding limits of wavelength is the same? It's the same energy and the same conceptual filter letting it through. I know that Maths is usually required here but this constraint comes before the Maths - not after a proof (I would have thought).
tade said:How do we show that the two integrals are equivalent?
##\int \frac{du}{d\lambda}\cdot g \quad d\lambda \quad, \quad \int \frac{du}{d\nu}\cdot g \quad d\nu##
One of the first (or the first?) threads in PF which discusses this paradox was started by me in 2006: "Wien's Displacement law, a paradox".trilobite said:trilobite submitted a new PF Insights post
Exploring the Spectral Paradox
View attachment 197646
Continue reading the Original PF Insights Post.
The spectral paradox is a concept in physics that refers to the discrepancy between the observed spectral lines of stars and the predicted spectral lines based on their temperatures. This paradox has puzzled scientists for centuries and has led to various theories and experiments to try and explain it.
The spectral paradox is explored through various methods, including spectroscopy, which involves analyzing the light emitted by stars to determine their chemical composition and temperature. Scientists also use computer simulations and mathematical models to explore different theories and explanations for the paradox.
There are several proposed explanations for the spectral paradox, including changes in the fundamental constants of nature, the presence of unknown elements in stars, and errors in our understanding of atomic physics. Another theory suggests that the paradox may be an illusion caused by our limited understanding of how light behaves in extreme conditions.
The spectral paradox is important because it challenges our current understanding of the universe and has the potential to lead to groundbreaking discoveries. By exploring this paradox, scientists may uncover new knowledge about the fundamental laws of physics and the nature of the universe.
If the spectral paradox is solved, it could have significant implications for our understanding of the universe and could lead to new technologies and advancements in science. It could also pave the way for future research and experiments that could further our understanding of the laws of physics and the origins of the universe.