Optically thin clouds and the Eddington Limit

As part of my independent study I’m going through “Radiative Processes in Astrophysics” by George B. Rybicki and Alan P. Lightman as a way to inform my work. This is in tandem with the other things I’m doing like the literature review and working with HYPERION so this is a pretty light introduction. I plan on working on one problem for each chapter that seems particularly informative or relevant. This first problem is from chapter 1 Fundamentals of Radiative Transfer and it’s problem 4.

1.4a] Show that the condition that an optically thin cloud of material can be ejected by radiation pressure from a nearby luminous object is that the mass to luminosity ration (M/L) for the object be less than, \kappa/(4\pi Gc)
where G = gravitational constant, c = speed of light, k = mass absorption coefficient of the cloud material (assumed to be independent of frequency)

To begin let’s assume that the luminous object is spherically symmetric. In that case we can use the standard definition of flux,

F=\dfrac{L}{4\pi r^2}

From equation (1.34) the force per unit mass is given by,

f=\dfrac{1}{c}*\int \dfrac{\kappa L}{4pi r^2}dv

Since cloud is independent of frequency of the frequency dependencies falloff and we’re left with,

f=\dfrac{\kappa L}{c4pi r^2}dv

In order for material to get objected the radiative pressure needs to be larger than the force of gravity being exerted from the object to the cloud,

\dfrac{GM}{r^2} < \dfrac{\kappa L}{c4pi r^2}

Then simple algebra shows that,

\dfrac{M}{L} < \dfrac{k}{4 \pi c G}

1.4b] Calculate the terminal velocity v attained by such a cloud under radiation and gravitational forces alone, if starts from rest and a distance R from the object.

Terminal velocity is when the gravitational force is equal to the counter force (for skydivers it’s drag, for this example it’s the radiation pressure). So let’s make a force called Geff that’s the radiation pressure subtracted from the gravitational force. This is the force that the cloud actually feels. We can set the Geff force equal to the general velocity equation and solve for v.

1.4c] A minimum value for kappa may be estimated for pure hydrogen as that due to Thomson scatter off free electrons, when the hydrogen is completely ionized. Show that the maximum luminosity that a central mass M can have and still not spontaneously eject hydrogen by radiation pressure is,

L_{EDD}=\dfrac{4\pi Gcm_H}{\sigma_T}

This limit is called the Eddington limit! And it’s really cool! It is usually spoken about the context of the accretion disks of black holes. But do you know what else has accretion disks? Pre-main sequence stars such as RR Tau. Basically, when you have a black hole (or star or what have you) with an accretion disk there are photons coming off the disk. This is why Quasars are so bright. Let’s think about why accretion disk have energy to create photons in the first place. An accretion disk forms when an object with a deep gravity well starts sucking objects toward it. The object it has gravitational potential energy which then gets converted to kinetic energy as it gets closer to the object. This is why accretion disks spin and why they’re luminous. The Eddington Limit basically sets a limit for the rate of accretion. The reason is clear. In our everyday life we’re not knocked over by the light coming off of our lamps even though the the photons coming from our lamps have radiation pressure it just simply isn’t enough to affect us in any significant way. But the the combined pressure of photons from an accretion disk can add up to be enough to start counteracting the force of gravity. So when the radiation pressure is larger than the gravitational force the Eddington Limit has been exceeded. When radiation pressure is equal to gravity that is the Eddington Limit. The answer to this problem is simple. You just make the solution to part a an equality as sub in the mass scattering coefficient for kappa.

Literature Review #4: Flaring and Shadowing

On the interplay between flaring and shadowing in diss around Herbig Ae/Be stars
B. Acke, M. Min, M.E van den Ancker, J. Bouwman, B. Ochsendorf, A. Juhasz, and L. B. F. M Waters

Meeus et al. (2001) used SEDS of a sample of Herbig stars to determine that members of the class can be split up into two groups; group I have much larger far-infrared excess than those of group II. This paper focuses on the proposed physical reason for this discrepancy which is that stars in group II have an outer disk that is protected from stellar radiation from the “puffed up” inner disk. However there has been evidence that if the disks of group II have a large flaring angle (meaning not a steep wall) then the outer disk is not as protected and will have IR excess comparable to the stars of group I.

This paper is actually a letter to the editor with a follow up paper planned. But it presents the preliminary results of 33 Herbig Ae/Be sources with the goal of better understanding the disk geometry and the nature of that geometry.

This was a fun paper to read after the last because it compliments the previous paper in focus but is much more digestible. The last paper was about the changes in dust size and structure as the disks around these stars age and this paper is about how the dust size affects the structure of the disk. The authors did this by comparing their spectra to self-consistent passive-disk models (I don’t know what that means). They found evidence that in the inner disks (~1 AU from the star) and the outer disks (~10s of AU) of the sources in their sample are related to each other. In an attempt to understand the physical cause they made disk models using the radiative transfer code MCMax (Min et al. 2009). Here is a list of their input parameters:
1.) Central star is a main sequence of spectra type A0 (T_eff = 10,000 K, R=2R_sun, M_*=2.5 M_sun)
2.) Assumed hydrostatic equilibrium and thermal coupling between dust and gas
3.) gas-to-dust ratio = 100
4.) Dust is made of astronomical silicates with a grain size of 0.12 microns

They used these parameters to compute models at different dust masses (m). They also varied the surface density power law and the inclination of the disk system (i).

They found that different dust masses gives the most change and spread in the models. The surface density and inclination give either inconclusive pro negligible results.

The authors conclude that the geometry of the outer disk is dependent on the inner disk and that the degree of flaring is determined by the mass of small dust grains and the degree of shadowing is dependent on inner disk scale height.

Literature Review #3: Characterizing dust

c2d SPITZER IRS SPECTRA OF DISKS AROUND T TAURI STARS. I.
SILICAT EMISSION AND GRAIN GROWTH

As I’ve written about previously, this term I will be working on a model of the circumstellar environment around RR Tau. Both Hugh and Bernadette suggested looking at Spitzer data as a way to inform my work. Spitzer is a space telescope that is run by NASA that observes in the infrared. Apparently infrared is where you want to look when you want to characterize dust.

This is the first paper that I read specifically for my tutorial. A lot of it was a bit over my head and I think that I’ll have to keep coming back to it as I learn more. Overall, though, it had some really great information that I think will help my project greatly.

It’s the first paper in a series of data presenting the data from 40 solar mass T Tauri stars and 7 intermiediate mass Herbig Ae stars in the ~5-35 micron region. And, yes, RR Tau just happens to be one of the stars included in their set. The team was able to resolve both the 10 an 20 micron features. The main goal of the paper is to characterize the dust grains in circumstellar disks and determine properties of the disks from the dust grains. This is important because changes in grain size and composition have been linked to disk properties and planet formation but the rates of grain growth and processing in disks is not understood completely. Not to mention the underlying mechanism instigating all of that.

The authors of the paper linked grain features to disk features by comparing the spectroscopic results with synthetic results given by models and doing a statistical analysis over the whole set. Most of their observational focus seems to be on silicates. It’s not clear to me why silicates are so important other than maybe that just happens to be what circumstellar disks are made out of (?). Since it’s possible to determine grain size, shape, and composition from the strength, peak, and shape of an emission line the authors modeled the opacity of the sample grains various, shapes, sizes, and compositions. They then compared the grain opacities to the observed emission without the influence of continuum.

The authors go on to interpret the changes in the line shape and strengths as source-to-source variations in grain size. This is apparently a unique way of doing this but it seems sensible to me. What (I think) this means is that they cannot find a general rule for the type of grain in certain stars. Anything like, “In this type of star you’ll find this type of dust grain”. They attribute this to fast grain growth and turnover (processing). Basically, whatever the process that makes silicates grow and go from amorphous to crystalline has to be relatively quick in order for there to be such disparate source-to-source variations. Some conclusive things they did find is that the equivalent width of H-alpha is not correlated with disk age. On this they say, “This suggests the importance of turbulence and regeneration of small (micron-sized) grans on the disk surface.” and that M stars show much flatter silicate features, which means larger grain sizes, than A/B stars. They tentatively attribute this to the differences in disk temperature which affects where in the radius the emission line comes from. This suggests that the grain size differs as you move through the radius.

September Project Update

I got the proposal submitted on August 18th! I will put the full proposal up when I get the chance to add a wiki element to this site

O I narrowband filters are commercially made so I had to give up on that idea.

The primary mirror at the Bennington observatory is having Issues. The coating doesn’t seem to be taking at all. Worst case scenario it has to be repolished. I bet we could get some volunteers at Stellafane to help us out.

Speaking of Stellafane I should get in touch with Wayne again.

Maria Mitchell as been MIA as far a communication. I’m hoping to hear back from Gemini soon so I have something new to tell the MMO Folks.

Official start of the term is tomorrow!

Fall Tutorial – Finalized

After putting a lot of thought into my fall tutorial the plan is all set (mostly) and the paper work is filled out and going down the line of approval. I decided to use HYPERION which is a radiative transfer program written by T. Robitaille. It’s really new. The paper introducing the program came out in 2011 and the actual code library was released just within the past few months. I think it will be good for a few reasons 1.) It appears to cull the best parts of former radiative transfer libraries and combine into one package 2.) It has a python wrapper which I, the user, use to make my models 3.) The documentation seems really thorough especially given how new it is.

Instead of just “playing in parameter space” I’m going to actually try to make a model of the circumstellar environment of RR Tau. HYPERION produces synthetic SEDs and images as its results so it might be something that could be useful in conjunction with the observational data. Still, the primary goal is to learn about radiative transfer and understand these environments is greater detail which I think HYPERION will be perfect for.

I spoke with Bernadette today about this and she was very helpful in getting me started with how to approach this project. One of things we talked about is how current SED models of Herbig Ae/Be stars have dips in the SEDs (I’ll find a plot of this to put here to make this clearer) and that has been used as an indication for gaps and holes in the disk. The problem with suppositions like this from SED models is that they are not unique solutions so there is an uncomfortable level of uncertainty. The reason for this is because SEDs are the result of blackbody modeling so the inputs are distance from the material and the density of the material. From the way Bernadette described it the models pick the temperature distribution, I’m unclear of the specifics but basically the lack of constraints prevents the models from being rigorous.

One of the interesting things about HYPERION in light of this is that many of the inputs for the models get into the composition of the material in the system. It will be interesting to see how the results I get compare to previous SEDs produced from models.

Another thing that Bernadette brought up is that for SEDs of disks my optical monitoring won’t be useful in constraining the model because disks fall in the IR region. So I’ll have to see what data is already available from places like ISO and Spitzer.

CLOUDY came up in our conversation as a possible code library to look into. I actually have worked with CLOUDY before when I was doing work on quasar absorption lines at UMass. I was using it to get metallicity values from column densities but Bernadette said that it’s good for understanding winds. That’s definitely something I’m interested in but I think for just the sake of time (and sanity) I’ll focus on HYPERION for now.

Also, I got many many many papers and authors to read. I’ll be starting up Literature Reviews again with a vengeance.

Tutorial thoughts and O I doublets

Joe and I moved back from Hawaii to Bennington and I finally met with Hugh. We’ve been meeting semi-regularly the past month or so. He’s been giving me feedback on my proposal for Gemini time as well as helping me sort out the specifics of my tutorial in the Fall. The earliest I’ll be able to start the actually work on my project is probably Field Work Term because I probably won’t have data, or at least a good amount of data, until then. I still want to be doing something with the project in the Fall though. As I’ve said before I want to understand more about the physics of these environments. The current idea for how to go about that is I work with already built radiative transfer code and essentially I’ll play within the parameter space of the program.

I’ll have to some sort of contained project relating to this, it’s not enough to just “learn”. So I think I’ll write a report detailing how changes in the parameters influenced the synthetic observations and why. It will be like a project I did for my Modern Astrophysics class at UMass where we had to change the physics of a stellar equilibrium program and write a report about how that changed the nature of the star being modelled.

My proposal is just about finished. I’m still waiting for approval from Bernadette. I also sent it to Gary and Vladimir. I haven’t heard from Vladimir yet but Gary emailed me back asking what kind of O I filter I want. What kind of bandpass and central wavelength. Until he asked me these questions I didn’t even consider that O I is a doublet. I have no idea if it makes a difference what line we choose to center the filter on. I emailed Bernadette about it but she’s busy with the Gemini Science meeting in SF right now.

Disks and things

Paddling has been wearing me out. We have practice every day for about an hour and half. On Thursday and Friday last week we paddled 4.5 and 3.5 miles respectively and now we’ve gotten into to doing half mile sprints. I have gotten to the point where whileI’m not horribly sore the next day which is really satisfying it still takes a lot out of me. Our regatta is this Sunday and then that’s the end of it. It’s been a ton of fun be out on the bay every day but I’m looking forward to be back to normal energy levels.

I spoke with Bernadette again on Monday. I updated her on the papers I have been reading and the conversation I had with Vladimir. She was particularly interested in the Grinin paper that proposed the binarity of RR Tau. I need to email her the reference so she can take a look at it. I have a few questions on that paper and it would be good if she can answer them.

For most of our conversation we talked about what kind of data I want to get focusing on what wavelength region I should be interested in. In her 2002 paper Bernadette detected about 15 features over a large wavelength range. We decided we wanted to focus on a smaller wavelength range, and thus less features, so we can get higher resolution data. My job is to play with the Integration Time Calculator and decide the observing parameters: the best grating, wavelength range, etc…Then I’m going to write the proposal for time despite the fact that it doesn’t need to be submitted until August when RR Tau finally comes back up. I’ve never written a proposal before so it seems best to start as soon as possible so I can get feedback and make sure everything is correct so when it’s time of submission I can do it immediately.

Bernadette sent me a research statement she wrote a few years back about some high resolution spectra of RR Tau she obtained but never published.

Two Fe II lines overlaid in velocity space.

In the statement Bernadette reiterates the key point made in her 2002 paper, that the spectral signature of RR Tau changes with continuum variations. When the star is at its brightest there are strong metallic absorption lines and at its dimmest there are faint metal emission lines that are associated with the stellar photosphere and an optically-thin disk atmosphere respectively. Understanding this is completely tied to understanding the nature of the obscuring material. What is it made of? How does it move? Evolve? What is it’s structure?

The general outline of discerning the fundamentals of the obscuring material is to use the characteristics of the light curve with spectral information to constrain models which has been fairly rudimentary. I think it will be a real step up to use narrow band filters along with spectroscopy. Also, depending on the resolution we’re able to obtain we might be able to resolve the disk in greater detail than what has been previously managed.

Literature Review #2: Is RR Tau a binary?

On the Nature of Cyclic Light Variations in UX Ori Stars
Grinin et al., Astronomy Letters

Vladimir Grinin is a name that often pops up when looking into UXors. He has been studying them for decades which is quite an advantage in variable star research. He had his his co-authors conducted decades long photometric monitoring on many UXor type stars that produced a series of papers. This paper is the first in the series and includes the results for RR Tau.

They observed RR Tau for 673 nights spanning 30 years. They found evidence of possible periodicity on timescales of ~8.6 and ~3.4 years. As far as I know this is the first discussion of long photometric variability in UXors and something I haven’t really seen discussed until the Maria Mitchell paper. It is interesting to me that our timescales are so different. The long term variability that we discussed in our paper was on the order of 100 days but Grinin et al. are talking about many years. I doubt that this has anything to do with the mechanism of variability that I want to look into for my thesis but it is fascinating to add another layer of complexity to this stellar system.

The change in continuum light over time.

The above figure is from a small review paper they wrote summarizing all their results.

The authors attribute the variation to binarity mostly because it is periodic and because the percentage of binary stars among Herbig Ae/Be is high. From what I can tell from their paper that they don’t directly attribute the periodicity to the companion star passing infront but rather the companion star is disrupting the the column density of the circumstellar disk. They go onto assert that the a binary system accounts for the short term activity that characterizes UXors also because of disruptions. This possibility could be relevant to my investigation on the medium term timescale.

I don’t understand the differing timescales they report, I don’t see that in the light curve. I’m going to read the other papers in the sereis and see if they shed so light on it.

Boats

There haven’t been any updates lately since I’ve been busy and tired with paddling. I’ve been reading a lot of fun papers and had another conversation with Bernadette but I haven’t felt up to writing about them. But here is a picture of the boat we paddle in, it’s very cool.

Sprinted a mile today in this. Picture by Joe.

First Steps in Monitoring RR Tau

When I’m busy my life becomes a positive feedback loop as I do more things not only my ability but my desire to do those things increases. I’ve been busy these last couple of weeks with my normal work coming to an end. There is some sense of urgency to get as much done before I’m done. But also, getting my RR Tau project off the ground while maintaining this site as documentation.

AND I just joined a novice outrigger canoe racing team with Petra and Joe. We have a race in two weeks with practice every day. Holy cow is it fun though.

Conversely, the opposite also tends to be true. When I have nothing to do it’s very easy for me slip into a hole of naps and cake making/eating. I had last Friday off because it was Good and I managed to surprise myself in how much I was able to get done. Most of it consisted of a nice back-and-forth between Vladimir and me.

I mentioned in my previous post Bernadette and I were curious if MMO had been monitoring RR Tau since the paper was published last fall so I emailed Vladimir to ask. He responded saying that they stopped monitoring RR Tau, they’ve moved on to white dwarfs and binary systems, but that he would be willing to continue monitoring the star if I had a solid, new idea that would warrant the effort. He also expressed doubts on my ability to get time on Gemini.

It was good to have someone question the merits of the project because I really needed to think of the best way to justify it. Vladimir seemed to be coming around to the idea but he was right in pointing out that this cannot just be a monitoring project. There has to be an underlying astrophysical idea that guides the project. I’ve got the beginnings of that with my focus on the [O I] line and trying to associate that with the long term variability but I want to go much deeper. I really need to read more papers that go into the physics of dust and gas evolution in stellar environments so I can get a better idea of what direction I should look in.

I’ve also started to look into the process of applying for time on Gemini. I downloaded the application package and started going through it but I can’t actually apply for anything until RR Tau is above the horizon which isn’t until late August.