We saw last time a plot just like this:
Linked are two sources:
1) This source is designed for astrophysics graduate students. It explains when the common is useful and why that is so.
2) This source is more user friendly and explains things a little more generally.
where c is the speed of light. Recall that the energy of a single photon with wavelength is:
where h is Planck’s constant.
Now, net flux is defined as the intensity at a given wavelength observed over all directions. In theory, we assume the intensity is isotropic, or the same in any direction. That means the net flux observed in a given wavelength is assumed to be isotropic in all directions, too. This is not necessarily true across the light spectrum though, because this only defines the net flux measurement in one given wavelength!!
This can be mathematically expressed as the following.
Where the intensity is variable on frequency and thus, so is the flux. Integrating over all angles like this gives you the net flux.
To find the total flux observed in a given frequency range (i.e. from frequency v to some other frequency v’ ) in units of ergs/cm/s is
You might be thinking: Well, oh okay, this is the same units as the plot above so we must be done and that’s how we plot spectral energy distributions. Sorry, but you would be wrong! You certainly can plot F vs. v but you wouldn’t be able to look right at the plot and see what frequency ranges dominate the flux density, i.e. what frequencies of light are more abundant from this source than other frequencies.
This is where our funky notation and definition for the spectral flux density comes in!
Note: this is essentially the same as using wavelength (), converting by just using the relationship above using the speed of light. This is also essentially the same as
by making a few rearrangements using the relationship between energy and frequency. In my field of high energy astrophysics, we don’t really talk about photon energies in terms of wavelength or frequency. I don’t really know why – I suppose because frequencies are really large and wavelengths are very, very small in the high energy regime. Instead, we speak of its energy. This is why, in all of my posts, I refer to the X-ray range in energy units. For example, the soft energy range of X-rays (i.e. low energy X-rays) are defined as 0.5-10keV. keV means kilo-electronvolt. It’s just another unit of energy. Any unit of energy can be converted into another. ergs is also a unit of energy. And Joules. And Calories!
(the prefixes here are just referencing the orders of magnitude. They can be Googled easily!)
For good measure,
Get a fancy one in there (i.e. 3/3 = 1 so )
You might need more math to understand this next jump but you can trust me it’s a solid thing to say.
To generalize, recall the slope of a curve is m and is related to the axes by y=mx. In this case, , , and .
You can plot versus the and get a lot more information (over a wider range of frequencies!). There are a lot of special things about this trick but the main one I want to emphasize is that plotting this way, we can see where the total flux is being dominated. Look at the example of another spectral energy distribution (SED) shown below.
You can tell just by looking at the graph – no calculations necessary! This is a huge perk.
From this we can show
because is the photons per area per second, thus is the change with energy, and E and are related. I’ll leave this up to you to ponder (and the pdf linked at the beginning has some extra insight to this!)
More on the logarithmic scales….
The y-axis is plotted from order to more than ergs/cm/s. That’s THREE orders of magnitude!
The x-axis is plotted from 1 MeV to over MeV. That’s EIGHT orders of magnitude!
To put this into perspective, take the ratios. For the y-axis,
and for the x-axis,
These are huge ranges we are trying to plot over. This is exactly why you see the plot axes looking so funky. It’s plotted in logarithmic scale to be able to fit all of this measured data onto one plot. Plotting in logarithm base ten allows us to plot fluxes versus their corresponding energies over a wide range of energies by creating equally spaced axes based on their order of magnitude.