Maurice Wilson's

Astronomy Research and Code

I am developing this site with the purpose of accumulating all of my research experience into one place. I hope that this site serves you well in getting to know me as well as understanding exactly what makes my research significant and exciting.

Below is my research blog. I write about my research and all astronomy related things on my mind. I teach a large variety of scientific concepts that have been relevant to my education as a student with a concentration in astronomy. I write so that those outside of the scientific community may enjoy and understand these scientific concepts just as I did many years before becoming an astronomy PhD candidate. Also, I've found that writing about or teaching a subject helps solidify my knowledge of that subject. Ironically, teaching science helps me learn science.
Note that I began this blog during my undergraduate years on a different site so feel free to visit there as well.

Dusty Disk Temperature Profile

I have already discussed how important radiative transfer and dust density are to a protoplanetary disk's temperature. Now, it's time to apply the radiative transfer method to the protoplanetary disk structure I derived in my Dusty Disk Density Derivation post and illustrated in my Dusty Disk Density Distribution post. This will give us an estimate for the temperature profile of the dust in the protoplanetary disk.

I used the RADMC-3D software to create the dust density structure of the disk and subsequently initiate the radiative transfer simulation. This simulation starts by making my (invisible) central protostar emit half-a-million photons in every direction. As mentioned in my Protoplanetary Disks post, these photons can at random either be absorbed by the dust, absorbed and immediately emitted by the dust, scattered by the dust, or simply never meet any dust grains along its trajectory. After all of the photons finish their unpredictable journeys, the software records which regions of the disk witnessed dust continuum emission due to a photon bombarding a dust grain. Consequently, those are the regions where a temperature can be estimated. Unfortunately, this means that there are regions of dust that will not have a temperature estimate either because photons never interacted with that dust or because photons were merely absorbed by the dust. Regions such as these will not have a color assigned to them in my visualizations. They'll just be empty spaces, as you can see in this next video.

Figure 1: Dust temperature profile derived from simulated dust continuum emission. Note the empty spaces seen here in comparison to the true structure of the disk shown in my previous post. If the limited number of photons could have traveled everywhere in the disk, this visualization would display the same structure as shown in my previous video, except with different colors.

Figure 2: Dust temperature profile of the disk's interior. I simply cut the disk in half so that you may see the interior temperature estimates.

The vast amount of yellowish-green regions suggest that the temperature for the outer regions of the disk is about 40 K. Another noteworthy feature is that there seems to be a hot "ring" of dust around the center. A gap separates that ring from the rest of the disk. This is certainly not a feature of the structure I forged in my previous post, so it must be that the dust in that gap did not release dust continuum emission. A 2D plot should be able to clarify this matter a little further.

Figure 3: Dust temperature profile of the disk's midplane. Remember that the empty/white regions merely mean that no dust continuum emission derived from those areas during this simulation.

The dust immediately around the center is hot simply because it is very close to the central star. However, there is more to this that meets the eye. Recall that the disk I made has a high density all along the midplane. The high density immediately around the center means that less stellar radiation is able to travel through it. With such a high density of dust, the chances of the photons eventually getting absorbed or redirected are very likely. This is why there is a small "ring" that goes from hot (red) to cold (blue) to immeasurable (empty/white). Starting from the central star, many photons approach the closest dust grains, heat them up, and initiate dust continuum emission. This causes the hot red inner circle. There are some photons however that do not get absorbed or redirected here. These leftover photons still traveling straight along the midplane eventually run out of luck and approach some dust grains. Since these are far fewer than the initial amount, these do not heat up the dust grains as much as first batch. This causes the cool green circle. Again, there are still a few photons left traveling in the midplane. They reach the outskirts of the "ring" and finally collide with some dust grains. Since there are so few photons remaining, this only heats the dust a little and these regions are left as a cold blue. From here, there are no more photons left traveling specifically in the midplane to initiate dust continuum emission from the surrounding dust. This results in an empty gap in the temperature profile.

Figure 2 clearly shows that there was still dust continuum emission along the midplane beyond the empty gap. So how can this be, when all of the photons traveling straight along the midplane had already been absorbed or redirected within the central high density "ring" of dust?
The answer to this mystery is that the stellar photons must have been redirected back into the midplane at distances beyond the empty gap. Photons that traveled above or below the midplane eventually were scattered or re-emitted in the direction toward the midplane.

Overall, there are two lessons you should learn from the information in Figure 2. For one, you now see how the incredibly high density in the midplane near the protostar can block a plethora of stellar photons from heating up the rest of the midplane—hence the vast amount of empty/white space. And two, according to the blue and turquoise spaces shown, the temperature of the midplane is between 5 and 10 K.

Just like the dust density distribution plots, we should look at the interior structure from a side view now. This will help you understand the horizontal and vertical behavior of the temperature profile better.

Figure 4: 2D plot of Figure 2.

As expected, there is a lot of empty/white space about the midplane in Figure 4. Thanks to the previous plots, you already know why that is though. What is interesting here is that there seems to be similar characteristics between this plot and the plot of Figure 3 in my previous post.

In regards to the top and bottom boundaries of the disk, they look identical. The top and bottom boundaries are hot near the center but those boundaries seem cooler at further distances away from the center. You can even see the temperature decrease with distance away from the star if you look at every given height (or "layer"). (See also the radial temperature \(T(r)\) equation in my derivation post.) Within the boundaries there is a fair amount of green, or ~20 K, regions. This, along with the fact that there is a lack of dust continuum emission throughout the middle, implies that the temperature does seem to transition from hot to cold as you go vertically from one boundary to the midplane.

The reason why the vertical component of the temperature profile seems to transition like that is because of the limited amount of direct star light that the dust grains receive. The following diagram is a good example of how this works.

Figure 5: Illustration of how the boundary and outermost regions of the flared disk receive more radiation from the star than the regions near the (black) midplane. (NAOJ)

This goes back to the explanation of how the stellar photons take a journey through the disk and randomly get absorbed or redirected. Many of the photons that initially reach the disk's boundary get either absorbed or redirected back outside of the disk. For this reason, there are fewer photons to heat up the inside of the disk and even fewer to heat up the midplane of the disk.

I think that is enough to absorb for today. I recommend that you go back to my Protoplanetary Disks post and read about the characteristics that most protoplanetary disks share, so that you can compare that information with the specific results of this simulated disk of dust that I've created in these videos and plots. My hope was that my example disk along with its videos and plots would help that information be easier for you to understand. Hopefully, you learned a lot from this and had fun as well.

Posted: September 1, 2017

Dusty Disk Density Distribution

The equations in my previous post show that my dust density distribution function \( \rho(r,z) \) cannot be used unless we have information about the protoplanetary disk (e.g., \( M_{disk}, R_c\) and \(\gamma \)) and its central star (e.g., \(M_{\star}, R_{\star}\) and \(L_{\star} \)). To make this hypothetical disk realistic, I used values attributing to the star DoAr 25 and its circumstellar disk for the parameters in my equations. If you're interested, these parameter values can be found in "The Structure of the DoAr 25 Circumstellar Disk" paper.

This is the structure of my protoplanetary disk after cranking out those calculations.
Figure 1: Dust density distribution.

I used the VisIt software to visualize the result of my calculations. Before I go into how the laws of physics derived this structure, I will first mention a few important features in this visualization. The colors represent the density of the dust grains. The regions with the greatest density are colored red and the density decreases as the colors change from red to green to blue. Also note that the star at the center of the protoplanetary disk is not actually shown in my depiction of a protoplanetary disk. Instead, the center of my structure has a blank region that is transparent. These next images will help wrap your head around this structure even more.

Figure 2: I cut off half of the disk to reveal the interior 3-dimensional density distribution.

Figure 3: 2-dimensional plot of Figure 2.

Figures 2 and 3 really show how much this disk is flared. Flaring occurs for a variety of reasons but there are two concepts in particular that I used which are responsible for this flaring.
1) The scale height affects the height (above or below the midplane) at which the density becomes zero and no more dust grains exist beyond that height. This scale height gradually increases with radial distance away from the star. (Refer to the scale height \(H\) equation in my previous post.)
2) At any given distance away from the star, the star's gravitational pull is strongest at the midplane than at any other height. (Refer to the cylindrical \(F_{grav}\) equation in my previous post.) This means that dust, at a given distance, will gravitate toward the midplane and accumulate along that plane. This is why in all of my graphics thus far, you've seen that the midplane is super dense, in comparison to the density at other heights.

Now that I've explained a few characteristics of the structure vertically, I should mention how the density is distributed horizontally. If you look closely at Figure 3, you'll notice that at each layer the color changes from blue to green to yellow to red going from the inside out. Apparently, at a given height, the density of the dust grains increases as you get further away from the star. This is, again, due to the scale height. I already mentioned that the scale height increases with radial distance away from the star. Additionally, an increase in scale height increases the density of the dust grains along any given height, and this is exactly what Figure 3 reveals. (Refer to exponential factor \(exp\) in the \( \rho(r,z) \) equation of my previous post.)

Thanks to the equations and imagery I have introduced you to, you have now witnessed one way to deduce the structure formed by the dust grains of a protoplanetary disk. We can now move on to determining the temperature of dust throughout the disk. In my next post, I'll show visualizations for the 3D temperature profile that I'm sure you've been so desperately waiting to see.

to be continued ...

Posted: June 18, 2017

Dusty Disk Density Derivation

Prepare yourself for a lot of equations! I will just zoom through them since there are so many. I won't go into much detail explaining each equation like I usually do.

If you don't care about this analytical solution of the dust density distribution, then you should just wait for my next post to see the visualizations I created that portray a protoplanetary disk. Those beautiful visuals will be worth the wait, I assure you!

In my previous post, I stated that the gas and dust have very different characteristics. Because of this, deriving their distinct temperature profiles leads down two very different paths. For the sake of simplicity, I will guide you through only one of these paths: the simpler, albeit dusty, path. From hereon, everything I discuss will only be about the dust grains.

Also in my previous post, I briefly discussed the concept of radiative transfer. Its equation can be described as \[ \frac{dI_{\nu}}{d\tau_{\nu}}(\tau_{\nu}) = S_{\nu}(\tau_{\nu}) - I_{\nu}(\tau_{\nu}) , \] where \(I_{\nu}\) is the star light's intensity that fluctuates as it goes through the medium and \(S_{\nu}\) is the ratio of the emission and absorption coefficients of the medium. Both vary with radiation frequency \(\nu\) and optical depth \(\tau_{\nu}\). The optical depth due to dust grains, \[ \tau_{\nu} = N_{gr} \sigma_{gr} Q_{ext}, \] depends on their column density \(N_{gr}\), the cross sectional area \(\sigma_{gr}\), and extinction efficiency \(Q_{ext}\). Now, as stated previously, you can see how the density of the dust greatly influences the radiative transfer.

With a few assumptions, we can derive an approximate dust density distribution function. One assumption being that the dust is a blackbody emitter. Considering this, the dust flux is \[F_{cooling} = \sigma_{sb} T^4 , \] where \(\sigma_{sb}\) is the Stefan-Boltzmann constant and \(T\) is the dust temperature. Because the grains are emitting radiation, this action makes them cool down. However, we know that the dust temperature barely fluctuates at all over time. Consequently, it is safe to assume that the stellar flux, which heats the dust, is equivalent to the dust's flux. In the star's case, \[F_{\star} = \phi \frac{L_{\star}}{4 \pi r^2}, \] where the angle between the surface of the star and the disk is \[ \phi \approx \frac{0.4 R_{\star}}{r}. \] \(L_{\star}\) is the stellar luminosity, \(r\) is the distance from the star, and \(R_{\star}\) is the stellar radius. After equating dust flux to stellar flux, the radial temperature profile for a flat disk can be described as \[ T(r) = \bigg(\frac{0.4 R_{\star}L_{\star}}{4 \pi \sigma_{sb} r^3} \bigg)^{1/4}. \] Although this does not account for the disk having a third dimension, it is still a useful approximation for the midplane of a 3-dimensional disk. Using \(T(r)\) assumes that the disk is vertically isothermal. (Note that we've estimated a 2D dust temperature profile in our derivation of the dust density distribution: a density distribution that will let us derive a 3D dust temperature profile of the disk.)

The equation of state \(P = nkT\) can be equated to the pressure of the medium \(P = c_s^2 \rho\) to find the medium's speed of sound: \[ c_s(r) = \sqrt{ \frac{kT(r)}{m} } , \] where \(k\) is the Boltzmann constant and \(m\) is the mass of the grain.

If the disk is vertically in hydrostatic equilibrium, the vertical component of both the grains' pressure gradient and the stellar gravitational force must be equal. The gravitational force in spherical coordinates is \[ F_{grav} = -( sin(\phi)\hat{z} + cos(\phi)\hat{r} ) \frac{GmM_{\star}}{r^2} . \] Therefore, in cylindrical coordinates it is \[ F_{grav} \approx - \bigg(\frac{z}{r}\hat{z} +\hat{r} \bigg) \frac{GmM_{\star}}{r^2} , \] showing us that \[ F_{grav,z} \approx - \frac{z}{r}\frac{GmM_{\star}}{r^2} . \] At hydrostatic equilibrium, \[ \frac{dP}{dz} = \frac{z}{r}\frac{GmM_{\star}}{r^2} \] where \(P\), as previously stated, is proportional to \(\rho\). Therefore, \[ \frac{d\rho}{dz} = \frac{z}{r}\frac{GmM_{\star}}{c_s^2 r^2} . \] The solution to this differential equation is \[ \rho(r,z) = \rho_0(r) exp\big[-\frac{1}{2} (z/H)^2\big] \] where \[ H = \bigg( \frac{r^3 c_s^2}{GM_{\star}} \bigg)^{1/2} . \] The scale height \(H\) is usually equivalent to ~10% of the radial distance from the star.

With \(\rho(r,z)\) we have derived our 3D dust density distribution function! However, we do not know \(\rho_0(r)\), the density in the midplane, yet. This can be found by using the surface density profile, \[ \Sigma(r) = (2-\gamma)\frac{M_{disk}}{2\pi R_c^2} \bigg(\frac{r}{R_c} \bigg)^{-\gamma} exp\bigg[-\bigg(\frac{r}{R_c} \bigg)^{2-\gamma} \bigg] ,\] where \(R_c\) is the outer boundary of the disk indicating the characteristic radius at which the surface density's power law behavior (\(\Sigma = \Sigma_0(r/R_c)^{-\gamma}\)) breaks down. Because \(\Sigma(r) = \int \rho(r,z)dz \), the \(\rho_0(r)\) can be expressed as \[ \rho_0(r) = \frac{\Sigma}{H\sqrt{2\pi}}. \]

After all of this hard work, the disk's structure should look like a flared disk, as mentioned in my previous post. However, keep in mind that this analytical derivation required a few assumptions that, although gives us a decent estimate, were not realistic.

Next, I will show the structure that these equations conjure up. The temperature profile will be calculated and visualizations for both the density and temperature will serve as a nice treat for all of this hard work.

to be continued ...

Posted: May 31, 2017

Protoplanetary Disks

Today I will discuss protoplanetary disks. This is nothing like the previous topics I have written about. But, there's a first for everything, so why not?

Artist's impression of protoplanetary disk orbiting brown dwarf star. (ESO)

Our solar system consists of 8 planets. Over the past ~30 years, astronomers have found a ton of evidence proving that most of the stars we see in the night sky have their own "solar system." These stellar systems vary in the amount and type of planets they contain. However, one similarity that these planets of distinct stellar systems have is that they were born out of protoplanetary disks. A protoplanetary disk is full of gas and dust that serve as the primary building blocks for planets.

The very well-known and beautiful Orion Nebula. (HST)

The gas and dust of a protoplanetary disk comes from a giant molecular cloud. Giant molecular clouds, like the Orion Nebula, are stellar nurseries. A portion of the giant molecular cloud can come together to form its own molecular cloud clump. The clump has its own gravity that makes it contract and its own internal pressure that helps it expand. Once the gravity overcomes the internal pressure and all other outward-pushing forces, the molecular cloud eventually collapses into a very hot and dense core of gas. This core indicates the birth of a new star, i.e. the formation of protostar. The outer region of the molecular cloud that is no longer touching the core is still affected by the gravitational pull of this young star. The remaining nearby gas and dust orbits the young star, forming a circumstellar disk that is more commonly known as a protoplanetary disk. This disk is also referred to as an accretion disk because the star is continuously pulling in and feeding off of the material in the disk.

By observing these disks in micrometer, millimeter and infrared wavelengths, a wealth of information about their structure, age and orbital mechanics have been inferred. One important detail that astronomers have learned is that these disks actually aren't disks. The structure is not flat. The gas and dust vertically disperses more as their distance away from the central star increases. Although the structure is close to being flat, it is actually a flared disk. Another fun fact is that these disks can live up to 10 million years. They are often a few hundred astronomical units (AU) in diameter. The inner portion of these disks can have temperatures above 2000 Kelvin (K) because of the central star. (As a relevant comparison, recall that the Sun's temperature is about 6000 K.) In the midplane of these flared disks, the temperatures are usually between 10 and 30 K in the outer regions of the disks. (In comparison, the temperature in the cold vacuum of empty space is 2.7 K. Apparently, these disks are quite chilly!) These temperatures vary as the disk ages. Over time, the hot radiation from the central star can evaporate the inner region of the protoplanetary disk out to 1 AU. A massive, second star that is nearby might destroy outer regions of this disk via its radiation too.

The temperature of the gas and dust can also depend on how they collide and how often they coagulate. Gas particles are pretty much always colliding with dust grains. If you think of the dust grain as a baseball and the gas as the wind, it is easy to see that gas can affect the speed of the dust grains. The grains succumb to the drag and friction caused by the gas and this affects the temperature of the local area, especially if the dust grains are large. Not all collisions result in the particles sticking together though. Therefore, it can be tough for chemists to determine when a dust grain, moving at a certain speed, will let other gas or dust stick to it. Here, I referred to chemists instead of astronomers because the study of protoplanetary disks heavily depends on knowledge from multiple disciplines. Dust grains are accumulations of molecules (like MgSiO\(_3\) or FeSiO\(_3\)) that may or may not chemically react to the gas and dust that collides with it. This is why astronomers heavily depend on chemists (and astrochemists) who conduct experiments on Earth in a lab. Their experiments with dust grain collisions help us figure out how often the molecules coagulate and how grains—only the size of 0.1 microns (\(\mu\)m)—eventually become the size of baseballs, asteroids and finally planets. (For reference, note that 0.1 microns is about 50 times smaller than the average size of bacteria.)

Simulation of dust collision in protoplanetary disk.

Apparently, the dust collisions affect the protoplanetary disk temperature and the probability of creating a planet. It seems like collisions are a pretty big deal then. Unfortunately, these myriad amount of collisions cannot be directly detected. Despite this, the temperature of the gas and dust can still be determined. This is done by measuring the radiation of dust continuum emission and gas line emisssion. Studying these two sources of radiation requires distinct sets of equations. This is because the velocity of the gas is extremely important for knowing how often the gas collides with other gas. Gas collisions in (cold) protoplanetary disks cause a gas atom to de-excite, i.e. cause an electron of the atom to drop in energy level, which makes the gas emit a photon. The dust, on the other hand, moves so slowly in comparison to the gas that its velocity doesn't significantly alter its temperature. We can ignore the super slow velocity of the dust and still find a good estimate for its temperature. However, the velocity of the gas can affect the temperature of the dust. So, in theory, the dust temperature calculations would be even better if that information were included too.

Although gas and dust have very different characteristics, their important similarity is that they both are sources of radiation, and the light they emit behaves similarly as it travels from the protoplanetary disk to telescopes on Earth. The intensity of the light we measure is affected by the trajectory it took through the protoplanetary disk to reach us. This intensity is described by the radiative transfer equation. Simply put, this equation explains how the intensity changes as the radiation is transferred through a medium, such as dust grains or gas. Once the radiative transfer is understood, the temperature of various regions in the medium can be determined. Calculating this temperature profile is very important because it determines where gas and dust will be too hot and evaporate or where gas will be too cold and freeze/stick onto grains.

During radiative transfer, the intensity can change once the photons are absorbed, scattered or re-emitted by gas and dust. If a photon is absorbed by gas or dust then it won't reach the telescopes on Earth. As a result, this would decrease the intensity of the light we detect. Dust often scatters (blue) light that comes from the central star. This redirects the light's original path. Dust can also absorb then re-emit the star light. This would redirect the photons and alter their intensity. The radiative transfer calculations should account for these three events regardless of whether the photon originated from the central star, the gas, or the dust. In order to calculate this accurately, we need to know how much gas and dust the protoplanetary disk has and how this gas and dust is distributed throughout the disk. For this reason, I will talk about the density distribution of a protoplanetary disk in my next post.

to be continued ...

Posted: May 26, 2017 is developed and managed by Maurice Wilson.
Maurice Wilson