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.
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.
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.
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.
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