Why is spectral flux density calculated per unit wavelength?

[billyharris72]

Hi all:

Question as it sounds really. I'm trying to work out why we do this and what the resulting number actually quantfies. Thinking this through I have:

a) If we want the flux we would simply calculate watts received by a given area ( W m⁻²). This would quantify the absolute amount of energy received in a given time over that area and would be directly proportional to the power of the light source.

b) If for some reason we wanted relative watts "per wave" we could normalise this by frequency (or multiply by wavelength). So wave A of half the length of wave B now has it's value halved, as the energy received over a given period of time is carried by twice as many waves.

So far so starightforward, but why divide by wavelength to get watts per square meter per unit wavelength ( W m⁻² nm⁻¹ )? I assume it can't be because shorter wavelengths correspond to higher energy, since surely that's already captured in our measure of wattage - presumably there's no need to correct for that.

Sorry if this seems a daft question but I'm finding this measure quite counter-intuitive and the book I'm reading (Green and Jones, An introduction to the sun and stars) doesn't explain the rationale.

Thanks for any pointers you can give,

Billy.

[Merlin66]

The black body distribution of energy peaks at a wavelength dependent on temperature... the distribution of the energy therefore varies with wavelength....

[andrew s]

The reason is the flux is not uniform in wavelength so your case b does not work.

Hence, the units used.

Regards Andrew 

[vlaiv]

Simply because energy carried by photon depends on its wavelength.

If you take total energy per unit time across a surface, you will get just that - total energy, but you will not be able to tell its distribution over spectrum (was there more "blue" photons, or "red" photons). On the other hand if you have spectral curve you can integrate it over wavelengths to give you total energy.

 

[billyharris72]

Hi all:

Thanks for the replies but I'm still missing something - I also wasn't clear that I was talking about measuring intensities at points on a spectrum. I'm aware of Wien's displacement law and the shape of a black body curve, and the distribution of energy does vary with wavelength. My question is, if this is the case, why, when we look at a spectrum, why don't we compare the flux at one wavelength directly with that from another if flux is directly proportional to energy?

For example, if a body is of a given temperature is radiating at a given peak wavelength x, and I measure this and another wavelength y that is radiating twice the amount of energy, is that not exactly the same thing as saying that the flux at x is twice that of the flux at y? If that were the case then there would be no point dividing each flux value by the individual wavelength. In fact, if that were the case (I'm open to the possibility it somehow might not be) then dividing by wavelength would get us the wrong answer.

To put it another way, when graphing a spectrum, why not plot wavelength vs flux, rather than wavelength vs flux/wavelength? If I'm understanding the units correctly, the energy radiated by an object at a given wavelength is proportional to flux, and hence not to spectral density at that wavelength.

Apologies ... feel like I'm missing something really obvious here... Possible things I'm running through in my head (but at least 1 feels wrong, probably 2 also).

1) Energy radiated is not proportional to flux (???). This seems way off.

2) Having to normalise by wavelength is something to do with Planck's Law (I don't know the theory here and it could be as simple as the final units that Planck's Law is stated in). But if so then Planck's Law isn't stated in terms of energy radiated but in terms of energy radiated / wavelength?

Or something completely different. Help!

Billy.

[vlaiv]

Let's compare that to probability density, and maybe that will help.

You know Gaussian distribution and curve (bell shape) that represents probability density? Assigning probability to particular number just makes no sense - probability that random variable with that particular distribution will have precisely 0.3422422323
is infinitesimally small. What you can do instead is say, what is probability of random variable being between 0.3 and 0.4 - and take integral of probability distribution between two values and you will have right answer.

Similarly if we observe wavelengths - total flux is bound quantity (not infinite), but there are infinity many wavelengths in any given range, so we can't say what is the energy at precisely 686.2332nm. In order to get meaningful result we need to integrate over certain range of wavelengths. Even when we measure we can't isolate any particular wavelength - there is no such thing as single wavelength filter - it is always filter that passes very narrow band - even solar Ha filters are measured in angstrom (like 0.3A) - such range contains very large number of values (infinite number of real values).

As for your above questions:

1) Flux of what? Flux is defined as some quantity over surface in a given amount of time. So you can say total energy flux - meaning total energy passing in unit time thru unit surface. You can say photon flux - meaning count of photons passing thru unit surface in unit time.

Now these two values are related by Plank's law, but you can't convert one to another without knowing above distribution.

If you observe energy over range of frequencies / wavelengths and assign each "wavelength" certain energy - thus creating spectral curve, you should use above mentioned units. Why? Imagine that you measure energy flux within visible spectrum - 400-700nm. Best you can do to describe energy at 450nm is to assume that we are talking about range of wavelengths between 450-451 and that all wavelengths carry same amount of energy. Thus we say there are 300 such ranges in 400-700nm range and each one is carrying 1/300th of total energy - so we divide with wavelength. Of course this is totally imprecise, but let's continue reasoning like that and instead of measuring total energy flux in range 400-700nm we do it in 400-500,500-600,600-700 with suitable filters and do the same, but this time we are dividing with 100 values in each filter. Ok, so now it's a bit more precise but still far away from curve. Continue this process and in the end use limiting case when measuring interval tends to infinity (and there is infinite number of them next to each other) - you will get the curve. But you see in each step you needed to divide with wavelength range - and as range shrinks in limiting case it tends to single wavelength.

Makes sense?

 

[robin_astro]

You have to express any Y quantity per unit of the X axis. Take for example plotting the distribution of people's  height.  Each person has a unique height so you sum them in one inch intervals for example.  The same applies for photons.  

BTW the shape of the distribution is irrelevant. There are many physical processes other than thermal black body radiation each of which produce a different shape spectral energy distribution.  The distribution is still expressed per unit wavelength (or it could be unit frequency, wave number or even photon energy if you like, whatever X axis you chose)

Robin

[vlaiv]

I think that simplest way to put it why units needs to be like that is to observe what spectral flux density curve is and what you do with it:

In order to get total energy flux between certain wavelengths (in certain band) you need to integrate spectral flux density, and integration is summation of term spectral_flux_density * d_wavelength and result should have units of Wm^-2, but d_wavelength has units of nm so we are "short" of one nm^-1 in spectral_flux_density term and it needs to be Wm^-2nm^-1 in order to produce correct units after integration.

[robin_astro]

BTW,  stars are not black bodies, and nobody measures the temperature of stars by fitting Planck curves to the spectrum despite what you sometimes read.

Robin

[billyharris72]

7 hours ago, vlaiv said:

n order to get total energy flux between certain wavelengths (in certain band) you need to integrate spectral flux density, and integration is summation of term spectral_flux_density * d_wavelength and result should have units of Wm^-2, but d_wavelength has units of nm so we are "short" of one nm^-1 in spectral_flux_density term and it needs to be Wm^-2nm^-1 in order to produce correct units after integration

I can almost hear the click as the penny drops. This is starting to make sense, insofar as my dodgy maths can grasp it. I was thinking of the graph provided in the text as similar to a histogram, where the y values are typically in units independent of x, but for a continuous variation in wavelength plotting density (thinking analogously here to normal probability density, where is noticeable that sd^-1 is part of the units on the y axis) makes more sense.

Just to check I'm roughly on the right path - if I integrate over a range of wavelengths, does that give me the total flux for that range?

Weird how certain things stick in my head and bug me like this - I could have just gone reading and enjoying the book and it wouldn't have made a whole lot of difference:)

Thanks,

Billy.

[vlaiv]

2 hours ago, billyharris72 said:

if I integrate over a range of wavelengths, does that give me the total flux for that range?

Yes, that is correct. It also gives you a tool to convert between photon count and energy (because it can be viewed with appropriate transform as photon density function - transform would be just to apply energy of photon - wavelength relationship).