# OUR DEEPEST EXPLORATION OF THE UNIVERSE IN X-RAYS

eROSITA is an X-ray space telescope that was launched on July 13, 2019 by an international collaboration, mainly funded by Germany and Russia. The space telescope took its first ever X-ray image three months after orbiting the Earth the following October and has already released some of the first data collected in the first months of operation as well as a schedule confirming the official first data release by December 2022. Most recently, the Astronomy and Astrophysics peer-reviewed science journal has released a special issue including ~35 publications that analyze new eROSITA data. Given the exciting first light and the already big discoveries the telescope has made including the largest supernova remnant ever discovered in X-rays, I thought it would be appropriate to highlight a little bit more about the telescope on my blog! 😄

The eROSITA telescope flies aboard a large satellite: the Spektrum-Röntgen-Gamma (SRG) space satellite. Along with the primary instrument, eROSITA (extended ROentgen Survey with an Imaging Telescope Array), on the SRG is the Russian ART-XC instrument which can probe higher energy X-rays than eROSITA.

As you have probably guessed, this is an X-ray imaging space telescope. It turns out that the Earth’s atmosphere actually absorbs incoming X-rays (see image below).

This is precisely why all astrophysical X-ray instruments are deployed in space including eROSITA.

eROSITA is made up of seven identical and strategically aligned X-ray Mirror Assemblies (MAs) that are situated on an optical bench. Underneath this is the rest of the supporting structure (see the schematic view below), which includes connecting the MAs to the camera assemblies (CAs), i.e. the mirrors will deflect incoming X-rays from its surface in very tiny incident angles that then focus the incoming X-rays onto the cameras (called the grazing incidence angle and is a common practice for designing sensitive X-ray instruments).

The X-ray “baffles” are used to prevent X-ray photons that are outside of the field of view from contaminating the image being taken at that time. This is particularly important when you need to observe an object that may have bright X-ray sources nearby that can contaminate the X-ray measurements.

The telescope (not SRG, the observatory it is deployed on right now) itself is 1.9 meters wide and 3.2 meters high. For my American readers that is about 6 by 10 feet! 😀 The completed instrument weighs in at a whopping 808 kg or 1781 pounds!

The Field Of View (FOV) of the full instrument (including all seven cameras) is about 1 degree in diameter. To give you an idea of what portion of the sky eROSITA can see at any given time, the full moon is about 1/2 a degree in the night sky, so eROSITA is able to see an area in the sky that is 2 times larger than the full moon.

This FOV is considerably larger than both of the previously most sensitive X-ray space telescopes, Chandra and XMM-Newton. Further, eROSITA will operate optimally for a specific energy range of X-ray photons. You will almost always see X-ray astronomy use kiloelectron volts to describe the X-ray energies,

$1 \text{keV} = 1,000 \text{eV} = 1.6 \times 10^{-12} \text{Joules}$

eROSITA, along with Chandra, XMM-Newton, and several other currently operating (and retired) X-ray instruments, can detect X-ray photons between 0.2 keV and 10 keV (see image below, but don’t freak out 😉)

The above plot is showing the field of view averaged effective area in cm squared as a function of energy. You can think of this as the sensitivity of the instrument as a function of energy. Each line corresponds to a different instrument: eROSITA’s seven modules in solid red, Chandra’s ACIS-I setup in green dot-dashed, another Chandra instrument called HRC-I in purple dashed, XMM-Newton’s 3 cameras with the thin filter on, and the previously retired ROSAT PSCPC instrument.

You can see that eROSITA is just about the most sensitive instrument from energies ~0.5keV to ~2keV which is often referred to as the soft X-ray range which just indicates the lower energy range of the X-ray band. Above 2 keV, the sensitivity of eROSITA drops off at a similar rate as the Chandra instruments, while XMM-Newton wins the sensitivity competition at energies greater than about 2keV. With its large field of view in comparison to Chandra and XMM-Newton, eROSITA will make (and has already demonstrated) significant discoveries to X-ray astronomy.

What separates eROSITA from other current missions like Chandra, in addition to its large field of view and sensitivity, is its angular and energy resolution and most of all — the way it will take data. Chandra and XMM-Newton X-ray telescopes are pointing missions. This means the telescope has to position itself for specific observations in varying parts of the sky. The time gets “shared” among thousands of researchers who request for telescope observations every year. eROSITA, on the other hand, is an all-sky survey.

## It is the first ever X-ray instrument to survey the entire sky from 0.2-10keV in astronomy HISTORY!

ROSAT was also an all-sky survey, but it only imaged soft X-ray photons, so it didn’t detect X-ray photons with energy more than 2.4keV. ROSAT also had a similar field of view of 2 degrees, but by inspecting the above effective area (i.e. sensitivity) plot, we can see that eROSITA will be a much deeper sky survey, by about 4 times!

To visualize this difference, here is a ROSAT view of the Vela supernova remnant (if you are familiar with my work you have seen the ROSAT image before) in the left panel below compared to the Vela SNR image from eROSITA on the right. I’m unable to find more details about the eROSITA image, but I’m guessing that the colors indicate three energy bands: red is likely the softest of X-rays < 0.6 keV, green is probably “medium” X-rays from 0.6 – 1-ish keV, and blue is likely 1-2.3 keV energies. If this assumption is correct, most of the Vela SNR is dominated by soft and medium X-rays (which is indeed the case, see the ROSAT image on the left lol!). We can also see the smaller overlapping supernova remnant Puppis A is bright in this X-ray range in both images, but that “hard” (higher-energy, see eROSITA image) X-rays dominate the observed emission. Additionally, one can easily spot the Vela central pulsar (lots of hard X-rays there in blue, too!) in the near-center of the eROSITA image, and a third supernova remnant in the lower left corner, visible by only a faint circular blue hue. Neither the central pulsar nor the lower-left supernova remnant is resolved in the ROSAT image. Note: do I see the third supernova remnant’s central compact object in the eROSITA image?!

## First light for theeROSITA telescope occurred in mid-October just months after launch

Moreover, eROSITA has already detected 10 times more sources than ROSAT which is about as many as have been discovered by all previous X-ray missions combined. Less than a year after launch, eROSITA has already completed its first all-sky survey, one of eight anticipated full sky surveys.

eROSITA’s first all-sky survey will be released in 2022 (well, the half that the Germans own), reporting already thousands of new sources, most being active galactic nuclei. One of the exciting discoveries includes the largest supernova remnant discovered in X-rays to date which has been nicknamed “Hoinga”. There are a lot of special surprises associated with Hoinga, including its high location with respect to the Galactic plane, an unusual location for supernova remnants to be found.

Hoinga is estimated to have a diameter of about 4.4 degrees. Vela SNR has a diameter of 8 degrees, but it was discovered first in radio, not X-ray.

To conclude, here is a super cool visual graphic about the SRG observatory where eROSITA operates.

I will definitely be on the lookout 👀 for the first data release, although that means I will have to learn (yet another) new software to clean and analyze the data….. 🥵 😅

## A random side note

What I think is extra intriguing about this telescope is the collaboration between Germany and Russia (Just hear me out lol). The terms of the collaboration seem a little unusual. They have defined a German half of the X-ray sky as well as a Russia half of the X-ray sky. Essentially the Western hemisphere of the Galaxy (in Galactic coordinates) is owned by the Germans with unique scientific data exploitation rights and the Eastern hemisphere belongs to the Russians. They have decided to equally share the all-sky surveys, so I suppose the data that has been divided will include individual mission projects i.e. pointed observations for a particular object will have certain proprietary rights depending on its location in the sky. With that being said, only the German half of the sky has been scheduled a public release of data for 2022, and all of the Russian X-ray data and its release schedule is to be determined.

It will be very interesting to see how the data-sharing pans out with this particular method. To be fair, I’m not totally sure if this is a standard practice in international space efforts such as this, but I would be surprised if it is.

# A DISTANT, COMPLEX SUPERNOVA REMNANT G344.7-0.1

Wow, hey guys! It has been quite the hiatus and I do apologize. But, hello!😀 As you can see, I have moved things over to a new web host (WordPress). I lost quite a bit of formatting in many of my old posts so it took me some time to go through and fix it all, though I need to go back (again) to fix how some of the posts appear on mobile devices, so thank you all so much for your patience with me during this time 😇🥰

I thought it was about time to not only come back here to continue my regular blog posts, but to go ahead and conclude the second paper that I started discussing back in September and is available here.

To briefly remind us what we are dealing with:

1. We have a known supernova remnant (SNR) located along the Galactic plane with the Galactic coordinates (G) 344.7 (longitude, in degrees), -0.1 (latitude, in degrees). Hence, the identifier “G344.7-0.1”.
2. Faint, point-like gamma-ray emission, which is detectable at energies > 10 GeV with the Fermi-LAT, overlaps with the Western edge of the remnant, suggesting some kind of connection.
3. Extended very high energy emission above 1 TeV is adjacent to the > 10 GeV emission location, suggesting that whatever is responsible for the > 10 GeV gamma-rays is also responsible for the very high energy gamma-rays above 1 TeV.
4. Analyzing old archival X-ray observations from the XMM-Newton X-ray Space Telescope, we discover the SNR is dominated in this energy band by thermal X-rays with no hint of non-thermal emission in or around the SNR.
5. Finally, there is a notable IR filament that overlaps well into the confidence region for the gamma-ray emission (i.e. the region where the gamma-ray emitting source is most likely to be located).

There are a lot of cool things happening here. Because my first source turned out to be a shock-cloud interaction (probably), I was definitely wondering if I was somehow looking at another one! The evidence seemed to line up in a strikingly similar way to the Vela SNR. But G344.7-0.1 is a lot farther away from us than Vela, so that introduces some extra challenges to trying to figure this out.

In fact, Vela is only about 1,000 light years away from us. That means it takes 1,000 years for LIGHT to travel from the pulsar to us, which is roughly 5,879,000,000,000,000 miles from Earth or 9,461,000,000,000,000 kilometers. But really, as far away as that might sound, that’s still relatively close. Look at some similar systems plotted by distance from us (the Sun/Solar system) with respect to the larger Galaxy we belong to, the Milky Way 🥰.

The above image is borrowed from an analysis on objects like the Vela SNR in the HESS Galactic Plane Survey released in 2018. The free version of the article can be found here. Look how close Vela-X (a component to the Vela SNR) is to us within our Galaxy!

The black outlined star with the initials “GC” represents the Galactic Center (GC) which is about 8 kiloparseconds away from our Solar system. That’s roughly 26,000 light years away from us. That’s right! That means that the light from the center of our own Galaxy has to travel for 26,000 YEARS before it reaches us! And yes, it’s moving as fast as it can!

Now, this is all relevant to G344.7-0.1 for the following reason: Measuring the distance of a far-away SNR like this one can be tricky, so our estimates leave us with some pretty big margins of error, but we estimate this particular system is at least 3 kiloparseconds away (about 10,000 light years) but could be on the opposite side of the Galaxy anywhere between 9 kiloparseconds (kpc for short) and 14 kiloparseconds (or almost 46,000 light years away from us 😧). The uncertainties attached to these huge ranges in distance haunt us throughout our analysis!

It also haunts us in the quality of data that is available for this source. For Vela, we had so much literature and surveys to sift through, which ultimately gave us the “smoking gun” for the shock-cloud interaction, the hydrogen cloud that shared the same shape and location as the higher energy emission we had discovered! But this source, G344.7-0.1, being much farther away, does not have adequate imaging for those same surveys, so we aren’t able to make any firm conclusions about what the SNR could be running into.

This is because the angular resolution of most available surveys for data covering our SNR region was just not gonna cut it to resolve any meaningful connections. This time, we had to really think about the physics involved here to be able to interpret our findings and offer a consistent explanation.

So what did we do? Our most favorite thing! We took all of the data we could measure for this source in light waves — from radio wavelengths to TeV gamma-rays — and we tried to put it on a plot (remember those spectral energy distributions we discussed some time back?). Then, with some software tools like Python, we can apply relevant physics equations to the processes going on and try to predict what we would observe and then compare that to what we actually see. This way, we control the physics and processes that explain the observations so we can make meaningful conclusions on what the possible scenario is going on here.

Recall that we had all of the following wavelength ranges on this source: radio, X-ray, Fermi (MeV-GeV) gamma-rays, and HESS TeV gamma-rays. In the plot above, however, you’ll notice that only the gamma-rays are plotted on our spectral energy distribution (SED) plot. This is because these models assume that the particles radiating the observed emission is all one population with the same characteristics — same radiative mechanisms, same interaction processes, same average energies, etc — and this may not always be a good assumption to make. It is entirely possible that the particles responsible for emitting in radio waves are totally separate from the population responsible for the higher energy emission!

As we plotted all of the radio — gamma-ray data, it became clear that there must be more than one particle population present: one to explain the radio and X-ray data and one to explain the higher energy data. However, our model is limited in this regard. If we need to “add” more particle populations to the model, it gets very complicated, and so we are unable to investigate this further. As a result, we limit ourselves to only trying to characterize the population behind the high-energy emission in the gamma-ray regime. Hence, the SED above only plots the gamma-ray data points.

Now, we are almost there for coming up with a way to understand what we are seeing. We now know that the high energy emission is disconnected in a particular way from the lower-energy (radio and X-ray) emission from our supernova remnant. This is interesting since the radio and X-ray emission is confined to the supernova remnant itself: the radio and X-ray emission fill the SNR, but both the radio and X-ray emission steeply decline just beyond the SNR shell. On the other hand, the gamma-ray emission is located on the Western edge of the SNR, with higher-energy gamma-rays extending to the South-East of the SNR shell. This picture seems consistent with the current model results that these are two separate populations.

Based on the X-ray and radio location and properties, we know the SNR has accelerating (i.e., radiating) particles that are emitting synchrotron radiation largely in radio, but these particles are not the same ones generating the gamma-ray emission. We can also tell by the SED above, that we cannot make any distinction with the data alone to say if it’s more likely to be protons (hadronic scenario) or electrons (leptonic scenario) that is responsible for the observed emission… Did we hit a road block? Is this where our journey ends?

The answer is no. We still have some information to consider. We need to consider now both the morphological properties (aka how the SNR looks in each wavelength) and compare it to our best-fit model and the estimated parameters (properties of the particles) to try and understand the most likely origin for the high-energy emission.

This is a lot to digest though, so you know what time it is! 😉 Next time, we will wrap up with the big picture for this object and I’ll also explain a little more about what measurements we can make from our best-fit model (particularly the right panel shown in the above image).

## Quiz! Take it here.

And as always, check out the free version of the article being summarized here.

# LIGO GRAVITATIONAL WAVE (GW) EVENT GW190521

## May 21, 2019,  the Laser Interferometer Gravitational-Wave Observatory (or LIGO) detected a signal that came from the merging of two black holes.

Wow! Where to begin. One of the world’s first gravitational wave detectors, LIGO, is a pair of laser interferometers located 3,000 km apart or about 1,800 miles. Locations are the Hanford Observatory in Washington and the Livingston Observatory in Louisiana.  This detector is incredibly sensitive, detecting some of the weakest signals known to man. A short video summarizes the first observation made by LIGO told directly by the leading scientists in the endeavor. These scientists, Rainer Weiss, Kip Thorne, and Barry Barish, won the 2017 Nobel Prize in Physics for the discovery. LIGO is supported by the National Science Foundation (NSF), Caltech, and MIT.
We have come a long way with our understanding of the Universe and general relativity. Gravitational waves (GWs) are detected as disturbances in the interferometer. We can measure the duration and the frequency evolution of the signal and from that, we can apply general relativity theory to understand what took place and where for the GW event to occur. In 2015, LIGO reported the first ever detection of such an event, and it was determined to be from the merging of two black holes. These black holes merged to form an ever larger black hole.

Since then, LIGO and the twin observatory, VIRGO in Italy, have made several more observations of GW events! Isn’t that amazing? Most of the observations are consistent with general relativity (GR) modeling for binary black hole mergers with a single binary neutron star merge event, GW170814. All of these events are unique for different reasons but I will focus on the most recent detection yet, GW190521.

GW190521 is another binary black hole merge event, with the observed measurements being completely consistent with GR numerical simulations where they have assumed a quasi-circular compact binary coalescence – this is basically fancy talk describing two black holes slowly spiraling together in their respective circular orbits until collapsing and merging together. So, they were able to determine the most likely scenario to generate a GW signal with the same properties as GW190521 was a binary black hole system that ended with the two black holes merging together to form an altogether larger black hole.

First of all, this is amazing that this kind of research is possible. I just – wow. I mean, right? This is just incredible. The unimaginable has been imagined and then brought to life! Thank you, Albert Einstein. Second of all, this binary black hole (BBH from now on) merge event is special for two main reasons.
1) The two black holes that merged together have been measured to be 85 and 66 times the mass of the Sun, which we denote using M ☉. The primary black hole (BH), the 85 M ☉ one, is especially intriguing. The current understanding of stellar evolutionary theory for very massive stars predict that no black holes should be formed from stellar collapse between about 55 to 120 M ☉. Of course, there are uncertainties surrounding where exactly this mass “gap” really lies but, the physics and modeling all appear to predict the same thing — no black hole remnants should be formed from stars between 55-120 M ☉, give or take. But how much give or take? In the second report characterizing the astrophysical properties of GW190521, they report that the probability one of the black holes that merged together has a mass within the mass gap is 99%.
2) The mass of the final BH from the merge event is estimated to be ~150 M ☉. This is the first strong evidence for the existence of an intermediate black hole. An intermediate black hole is a black hole that ranges in size from ~100s to ~100,000s the times the mass of the Sun. LIGO has detected several BH merge events, but only from stellar-mass black hole systems so far. GW190521 is the most massive BBH merge event observed to date!

The LIGO community announced the detection only just Sept. 2, 2020. Below is the GW detection signal from both LIGO detectors and VIRGO, and the signal represented in the time-frequency domain in the images displayed in the bottom panels. The signal was very short, with a duration of only 0.1 seconds and ranged in frequency from 30-80 Hz.

I believe it was first reported in Physical Review Letters here, of which the above image is adapted from. A really cool short video is shared below that nicely reiterates what I’ve just mentioned (and then some!).
Other cool materials are available here: https://www.ligo.org/detections/GW190521.php including a video of a numerical simulation of a binary coalescence that reproduces GW190521 (which I also display directly below because its COOL).
Above: Numerical simulation of two black holes that inspiral and merge, emitting gravitational waves. The black holes have large and nearly equal masses, with one only 3% more massive than the other. The simulated gravitational wave signal is consistent with the observation made by the LIGO and Virgo gravitational wave detectors on May 21st, 2019 (GW190521).

## A New Frontier: Gravitational Wave Astronomy

Einstein predicted in general relativity theory that changes in the gravitational field will travel through the universe at the speed of light. These are gravitational waves and were only first confirmed in 2015. In 2020, we discover the most massive black hole merge event observed to date, and it happens to be pushing the limits of what we understand about the evolution of massive stars and, the nature and formation of intermediate sized black holes. Today, there are currently no confirmed intermediate black holes (IMBHs). There are several candidates, most of which reside in dwarf galaxies and are associated to low-luminous active Galactic centers. However, having no firmly identified IMBHs leaves many questions unanswered. How do IMBHs form? What conditions are necessary? The final BH responsible for GW190521 may give us clues and perhaps, future detections by LIGO and VIRGO may further reveal the Universe’s secrets.
I’ve been reading into this some and will discuss a short letter that investigates the origin of the primary 85  M ☉ BH of GW190521 in the coming blog post!

# PAPER II: (ANOTHER) NEW GAMMA-RAY SNR

This particular object I began looking more into right around the time I was also preparing for the written qualification exams at Clemson University. PHEW that is a stressful time to look back on. Maybe I’ll write more about the PhD process in the program in another blog post but, for now, let’s focus on the second Fermi object I investigated. I had a task to gather data for 12 newly detected very high energy sources – or VHE, which we will define at an energy E > 50 GeV which is among the highest energy for light, the gamma-ray range. The gamma-ray space telescope, Fermi, had just discovered these sources and were reported in 2016 (free and public pdf version available here). I was looking at multiple sources at once but, one particular source, led me somewhat down a rabbit hole and led to my second paper (in review but hopefully close to acceptance and publishing!) on the multi-wavelength analysis of the supernova remnant, G3447.1-0.1, which is found to be a likely origin for the newly discovered gamma-ray emission at the edge of the remnant. Does this sound familiar? It should! My first report was also on the discovery of gamma-ray emission on the edge of the Vela supernova remnant (SNR). Coincidence? Yes. Lol. Though this time the physics was a little more complicated but, all the more exciting!

## First thing I noticed was this new gamma-ray source was pointing to the SNR G344.7-0.1, pictured below.

Admittedly my only experience so far with gamma-ray objects was the emission we identified as a shock-cloud interaction on the west edge of the Vela SNR and, so far, I was looking at another SNR with gamma-ray emission appearing on the west edge which overlapped with bright radio emission and a strong infrared filament. The bright radio emission can be seen in the left panel above. The green color is showing you where the radio emission brightens which is seen to be on the north and western regions. Similarly, we see brighten infrared emission in the northern-central and western regions as shown in the middle panel. Do you see the red filament in the middle panel? This coincides with the gamma-ray position and bright radio emission. Interesting.

By the way, the above images can be found in the following reports: (left) Eagle et al., 2020 (arXiv pdf version here), (middle) Combi et al., 2010 (arXiv pdf version here), and (right) available here.

So what does this mean, exactly? Well, in general, where one might see increased emission accompanying high energy emission, like this gamma-ray source, points to some kind of interaction that would accelerate these particles to radiate at the varying wavelengths. Each wavelength provides us different information about the particles responsible for the radiation and how they must be interacting with their environment. This also means we can infer what that environment is made up of.

Recall my first report we were able to determine the presence of a small neutral hydrogen cloud that was found to coincide precisely with the western wall of the Vela SNR that paired with the observed gamma-ray emission there. We can plausibly describe the gamma-ray emission as arising between the remnant’s forward shock – that is, it’s outer layer that was blast into the interstellar medium first at the time of explosion – is running into its surroundings and is colliding with and disturbing the material.

Let’s break down my first findings:

• There is bright radio emission to the west of the SNR G344.7-0.1 as can be seen in the left panel above. This may suggest the remnant is interacting with its surroundings, much like the scenario just described.
• There is a bright infrared filament that outlines the west wall of the remnant, overlapping with the bright radio emission seen and also coincides with the X-ray boundary of the SNR (we will see in the next post that the radio emission mostly follows the filled X-ray center appearance of the SNR).
 The infrared emission suggests the remnant has recently swept up dust and shocked it that then radiated in the infrared regime. At this point in time, I was leaning towards thinking it was another shock-cloud interaction. In my defense, the pieces seemed to be falling in a way that a shock-cloud interaction made sense but, also keeping in mind, it was my only exposure so far. I needed to keep an open mind. I still had questions. If it were a shock-cloud interaction, then where is the cloud? Can I prove the presence of one? We knew at the time the SNR was in a dense environment in this part of the Galaxy based on its appearance across the light spectrum and by measurements of the surrounding densities but, the SNR’s distance was too uncertain to reliably pinpoint the existence of gas clouds in the same region of this remnant.
Furthermore, there is even higher energy gamma-ray emission found to the southwest of the remnant, in the TeV range! It was first detected by HESS, a system of Imaging Atmospheric Cherenkov gamma-ray telescopes that operate on the ground. HESS stands for the High Energy Stereoscopic System and more information on it can be found here. There were no other sources that were found to be in the same location as the HESS emission found at that time (which extends over a decent amount of space, it’s not just a point source). They labeled the HESS source as “dark” because they could not identify its origin due to the lack of information at other wavelengths. But now, we have a lead!

It was suspected before that the HESS TeV emission and gamma-ray emission from Fermi were coming from the same source. We plotted their flux data on the same plot and saw the data flowed well together, and could plausibly be from the same emission mechanism (above, left panel). The right panel shows the extent the HESS source has. It was unclear at the time what was unfolding here. If the TeV emission is connected to the supernova remnant, and the remnant is gamma-ray emitting on the western edge, how does it all fit together? What is going on?

# SPECTRAL ENERGY DISTRIBUTIONS

## We saw last time a plot just like this:

We plot the “spectral flux” or “spectral density” or “spectral flux density” versus the energy. Why do we do it this way?
The plot comes straight from this paper published in September 2019 (and I’ve linked the free version!). The y-axis (the vertical) is plotted using $E^{2} \frac{dN}{dE}$ in units of ergs/cm $^{2}$ /s. This is the spectral flux density in units of energy per area per second. But why $E^2 dN/dE$? What does that even mean to us? How does it relate to the total flux from a source at a given frequency? And what are the perks to defining and plotting the spectral flux density?

Linked are two sources:
1) This source is designed for astrophysics graduate students. It explains when the common $\nu F_\nu$ is useful and why that is so.

2) This source is more user friendly and explains things a little more generally.

First, let’s get a handle on one thing: the relationship between frequency and energy.
Recall the relationship between frequency, we will call $\nu$ (or nu, a greek letter), and wavelength, $\lambda$ (or lambda, ​another greek letter):

$c = \nu \lambda$

where c is the speed of light. Recall that the energy of a single photon with wavelength $\lambda$ is:

$E = h \nu = h\frac{c}{\lambda}$

where 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.

$F_\nu = I_\nu Cos[\theta] d\theta d\phi$

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$^2$/s is

$F = \int F_\nu d\nu$

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.

Now, if you plot $F_\nu$ (the net flux over a given frequency) against the frequency and integrate the area under the subsequent data (see the figure below), you simply get back the total flux in that range. That’s really it. There’s no safe way to guess how much of say, the X-ray flux, compares to the gamma-ray flux just by plotting it this way. You’d have to sit down and do the math using the equations above.

Taken from the textbook Radiative Processes in Astrophysics written by George Rybicki and Alan Lightman. I’ve kept the page number and chapter (Bremsstrahlung) for your reference.

## This is where our funky notation and definition for the spectral flux density comes in!

We like to use

$\nu F_\nu\, \text{vs.}\, \nu$

Note: this is essentially the same as using wavelength ($\lambda$), converting by just using the relationship above using the speed of light. This is also essentially the same as

$E^2 \frac{dN}{dE}$

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!

$1\,\text{eV} = 1.6\times10^{-19}\,\text{Joules}$
$1\,\text{keV} = 1,000\,\text{eV}$
(the prefixes here are just referencing the orders of magnitude. They can be Googled easily!)
$1\,\text{eV} = 1.6\times10^{-12}\,\text{ergs}$
For good measure,
$1\,\text{MeV} = 1\,\text{million eV or}\, 1\times10^6\, \text{eV}$

Now this next section will also tie in why we use logarithms (in addition to the huge spans of measurements which I discuss below).

We want

$\nu F_\nu$

$F = \int F_\nu d\nu$

Get a fancy one in there (i.e. 3/3 = 1 so $\frac{\nu}{\nu} = 1$)

$F = \int F_\nu \frac{\nu}{\nu} d\nu$

or

$F = \int F_\nu \nu \frac{d\nu}{\nu}$

You might need more math to understand this next jump but you can trust me it’s a solid thing to say.

$d Log[\nu]= \frac{d\nu}{\nu}$

Such that

$F = \int \nu F_\nu dLog(\nu)$

To generalize, recall the slope of a curve is m and is related to the axes by y=mx. In this case, $F=m$, $y=\nu F_\nu$, and $x=Log[\nu]$.

You can plot $\nu F _\nu$ versus the $Log[\nu]$ 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.

This plot is from the analysis of a pulsar wind nebula in the SNR G327.1-1.1, Temim et al. 2013.
The plot is coming from this paper (again, free version!). The peaks show us where the flux is being dominated. For example, most of the overall flux from this given source above is being dominated in the X-ray regime (where Chandra has measured the spectral flux density in this energy range).

## You can tell just by looking at the graph – no calculations necessary! This is a huge perk. ​

In short, plotting $\nu F_\nu\,\text{vs.}\, \nu$ enables us to immediately understand what part of the electromagnetic spectrum being generated from some source is dominating the observed flux. i.e. how bright is it in one energy range from another?

From this we can show

$\nu F_\nu \propto E^2 \frac{dN}{dE}$

because $N(E)$ is the photons per area per second, thus $\frac{dN}{dE}$ is the change with energy, and E and $\nu$ 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….

Take a look at the plots above (specifically the very first figure). How many magnitudes are we plotting along the y- and x- axes? Let’s see…

The y-axis is plotted from order $10^{-12}$ to more than $10^{-9}$ ergs/cm$^2$/s. That’s THREE orders of magnitude!
The x-axis is plotted from 1 MeV to over $10^{8}$ MeV. That’s EIGHT orders of magnitude!

To put this into perspective, take the ratios. For the y-axis,

$\frac{10^{-9}}{10^{-12}} = 1000$

and for the x-axis,

$\frac{10^{8}}{10^{0}} = 10^8 = 100\, \text{million}$

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​.