Bert's blog

## Testing the stability of supersonic ionised Bondi accretion flows with radiation hydrodynamics

Last week, a scientific paper that has been in my scientific pipeline for over a year got accepted for publication in Monthly Notices of the Royal Astronomical Society (Vandenbroucke et al., 2019), and this seems a good opportunity to explain what this paper is about. It also allows me to tell you some more about the machinery behind the paper, which made extensive use of workflow management systems.

# Introduction

The paper itself is theoretical in nature, but its content relates to the very active field of star formation. It seems a bit odd that a scientific field that is literally named after stars (astronomy derives from astron, the ancient Greek word for star) still hasn’t figured out completely how stars form, but this is the case to a certain extent. We know that stars form in dense clouds of cold interstellar gas when small cores within these clouds fragment and collapse under the clouds gravity, but there are many details about this process that still need to be ironed out. The main problems are that these dense clouds are incredibly hard to observe (they are opaque to most wavelengths of radiation) and that there are many physical processes that contribute to the formation and collapse of these clouds, making it extremely challenging to model the star formation process theoretically.

One of the big outstanding questions in star formation deals with the end of the star formation process. We know that the collapse of a protostellar cloud is halted when the density and temperature in the core of the star reach the extremely high values needed for nuclear fusion, the process that makes stars shine. This fusion process releases a huge amount of energy that leaks out of the star as light, and on its way out causes a huge counter-pressure that counteracts the gravitational attraction that tries to make the star collapse even more. This explains nicely why a protostar stops collapsing, but does not tell us why it stops growing: material from the edge of the cloud can still accrete onto the central core and make the star more massive. At some point, some process is responsible for stopping this accretion, and this will ultimately stop the formation of the star and determine its final mass. Since the mass of a star is the main parameter that sets the stellar properties (like its brightness and how long it lives), getting a handle on the final mass of a star is quite important.

Together with collaborators in Brazil and additional collaborators around the globe, my group in St Andrews is trying to find out if the radiation emitted by a protostar can shut down the accretion onto that protostar. We are particularly interested in massive stars. These stars are the most short-lived and we know that they are already very bright while they are still accreting material, making it very likely that the pressure caused by high-energy radiation will affect the final stages of the accretion onto the protostar.

In 2002 and 2003, Harvard professor Eric Keto wrote two papers in which he investigated the accretion process onto a massive star in the case where the star emits ionising radiation. This is radiation that has sufficient energy to ionise the interstellar medium, i.e. strip neutral hydrogen and helium atoms in the gas surrounding the protostar from its electrons to turn it into a hot plasma. We know that this photoionisation process transfers a lot of energy to the gas, turning it from a cold gas with temperatures around $100-500$ K into a hot gas with a temperature of $\sim{}8,000$ K. At constant density, the pressure in the gas linearly depends on its temperature ($P\sim{}T$), so that this hot plasma has a much higher pressure than the surrounding cold, neutral gas. In a normal scenario, the ionising radiation will create a hot bubble around the star, and the higher pressure inside this bubble will cause the bubble to expand into the interstellar medium. We call this an HII region (HII is the spectral name of ionised hydrogen, the main gas component inside the hot bubble).

This expanding bubble scenario works well when the radiation of the star is the only thing that exerts a force on the gas. And that is not quite the case for a massive forming star that is pulling in surrounding gas because of its strong gravitational pull. This gravitational force is stronger close to the star. Based on earlier work by Hermann Bondi and Leon Mestel in the 40s and 50s of the 20th century, Eric Keto investigated what happens when an expanding HII region is confined to a small spherical volume close to a massive accreting protostar. He showed that when the HII region is very small, the gravitational force completely dominates, and is much stronger than the pressure force that would normally cause the expansion of the HII region. The result is that the HII region does not expand, but stays confined around the protostar: a trapped HII region. Even though the star heats the surrounding gas, the accretion onto the star can still continue.

As the star keeps growing through the trapped HII region, its mass and hence its brightness (or what astronomers call luminosity) keeps increasing as well, so that the HII region slowly grows. Once its size exceeds a critical limit, the pressure force in the hot bubble starts to win from the star’s gravitational pull, and the bubble will manage to expand. The expanding bubble will then push away the gas surrounding the star and shut down the star formation process. That is at least Eric Keto’s theory.

There are a few issues with this theory. First of all, these models were based on a static mathematical description of the problem, and do not actually contain the dynamic evolution of the hot bubble. This means that Eric Keto only showed that this scenario is possible, not that it will actually also happen. Secondly (and more importantly), these models crucially assume a spherical HII region surrounding the star. Observations of star-forming regions almost unanimously favour a different scenario, whereby a star forms in a flattened, disc-like structure called an accretion disc. In this scenario, the polar regions (perpendicular to the disc) have a much lower gas density, so that the shape and evolution of an HII region in such a scenario will be different from the nicely spherical hot bubble described above.

To address these issues, we decided to model the development and evolution of these HII regions surrounding massive protostars in more detail, using complex numerical simulations. These simulations are idealised models of the formation of a massive star that study what happens to interstellar gas that is subjected to a strong gravitational pull and photoionising radiation. The study of the dynamical evolution of gas is called hydrodynamics, and when the effect of radiation on hydrodynamics is modelled, we call this radiation hydrodynamics. It is important to realise that radiation and hydrodynamics are tightly coupled: the pressure changes caused by radiation heating cause movements in the gas and this in its turn causes changes in the gas density that change the way the radiation interacts with the gas.

My paper is part of a series of (currently) three papers that describe our models. The first paper in this series was written by my colleague Kristin Lund and models the spherically symmetric models that Eric Keto came up with. She confirms his original theory and shows that our state-of-the-art radiation hydrodynamics method can accurately model this scenario. The paper by my Brazilian colleague Nina Sartorio uses the same technique (with my simulation code CMacIonize) to model the more realistic scenario where the star forms inside an accretion disc. Her results seem to suggest that in this case radiation is not able to shut down accretion, provided that the HII region is trapped at the moment when the protostar starts emitting ionising radiation. My paper (historically the second in the series, but due to the randomness of the scientific review process the first to be accepted for publication) deals with a theoretical detail of the spherically symmetric case, that I will describe in more detail below.

The accretion onto a massive protostar is a highly dynamic event: gas at large distances is pulled in by the gravitational attraction of the protostar and accelerates towards the protostar. The gas is hence constantly moving. Nevertheless, we can write down a mathematical description for this event that does not change over time. To do this, we have to make three assumptions: (1) that we have an infinite reservoir of gas with a constant density at large distances from the accreting protostar, (2) that the properties of the protostar (like its mass) do not change when gas is accreted onto it, and (3) that the accretion rate onto the protostar is constant in time and space. To understand the latter, picture a large sphere surrounding the protostar. The radius of the sphere should be large enough so that it comfortably fits the entire protostar and its immediate surroundings. We will assume that the radius of the sphere does not change over time. While gas is being accreted, it will cross the surface of the large sphere. We can count how much gas crosses the surface during some time interval $\Delta{}t$, and call this $\Delta{}M$. The accretion rate is then defined as

Our assumption hence means that $\dot{M}$ is a constant number, both when we track the accretion across the surface of the same large sphere over time, or when we do the same for a large sphere with a different radius. The assumption that the accretion of gas does not alter the mass of the protostar is clearly not very realistic, but it is nevertheless a good approximation if the accretion rate is low enough: if the mass of the protostar is $20~{\rm{}M}_\odot{}$ (20 times the mass of our Sun, a typical mass for a massive protostar) and the accretion rate is $\dot{M} = 10^{-5}~{\rm{}M}_\odot{}~{\rm{}yr}^{-1}$ (meaning the star accretes a mass equivalent to 0.001% of our Sun every year), then it will take 20,000 years before the mass of the protostar changes by 1% of its value. In that same time, the gas can cover more than 10,000 times the distance between the Earth and the Sun!

When we make these three assumptions, we can find a so called steady-state solution for the dynamics of the gas. This solution is an example of a dynamic equilibrium: it is a stable solution that does not change over time, while at the same time describing gas that is moving. The steady-state solution for spherically symmetric accretion onto a massive object is called Bondi accretion, since Hermann Bondi was the first one to derive this steady-state solution. Spherically symmetric in this case means that we also assume that the gas properties only change when the gas is closer or further away from the star (we only allow radial changes), and that they are the same for gas that is at a constant distance from the star.

When we want to describe a trapped HII region, we need to make additional assumptions about how the radiation heats the gas. Leon Mestel proposed to assume a two temperature solution in which ionised, hot gas has a constant temperature $T_i$, while neutral, cold gas has a constant temperature $T_n$ (and $T_i > T_n$). He further assumed that the transition from ionised to neutral gas (what we call an ionisation front) is very sharp and happens at a distance $R_I$ from the star.

Using these assumptions, Leon Mestel and later Eric Keto were able to show that a new steady-state solution exists for small enough values of $R_I$. However, their solution was numerical, which means that they were able to show that the solution exists, and could write down computer code that can generate the solution (e.g. to make images). In my paper, we extended this work by actually deriving a mathematical equation that describes the two temperature steady-state solution. The equation itself is not very elegant (it involves a special function called the Lambert-W function), but nevertheless is useful, as it gives us a good way to compute the gas density and fluid velocity at any arbitrary distance from the star. We will use this for our simulations later on.

Another issue that neither Leon Mestel nor Eric Keto discussed in their original work was the stability of the two temperature steady-state solution. Recall that steady-state means that the solution is not supposed to change over time: the gas density and fluid velocity remain the same at any given radius, although the gas itself is constantly moving. In a perfect world (or rather, a perfect Universe), this would mean that our analytic expression is valid for eternity. Unfortunately however, our Universe is not perfect, and real stars will never have a perfectly constant accretion rate (one of the assumptions we made above). This means that we can realistically expect to have small deviations from the constant accretion rate we assume, and hence small changes in the two temperature steady-state solution (the solution won’t be perfectly steady-state).

Stability analysis deals with what happens when a steady-state solution is perturbed by small deviations in the assumptions on which it is based. There are two possible scenarios, called unconditionally stable and conditionally or marginally stable solutions. The easiest way to understand these is to think about a landscape with hills and valleys in which you place a ball that is free to roll under the force of gravity. When the ball is on a slope, gravity will pull it down towards the lower end of the slope; this is clearly not a stable position for the ball to be in. When the ball is on the top of a hill or in the bottom of a valley however, the force of gravity cannot affect it, and it will happily stay there; these are stable positions.

When the ball is at the bottom of a valley, and for some reason it is pushed out of the valley over a small distance (a small deviation from its stable position), it will be pulled back towards the bottom of the valley, since that is the direction of gravity. The valley is an unconditionally stable position: the ball will always roll back to the bottom of the valley (as long as it does not move too far away from it). For the top of a hill position, things are very different: in this case gravity pulls away from the top of the hill, and hence away from what was originally a stable position. The top of a hill is a marginally stable position: any small deviation from it inevitably leads to the ball rolling away towards a more stable other position.

We can try to perform a similar stability analysis for the two temperature steady-state solution, by introducing a small perturbation in one of the fluid variables. We chose to do this by introducing a small bump in the density at a radius larger than $R_I$. This bump will move with the local flow of the gas, which is in the direction of the protostar, and will eventually cross the ionisation front at radius $R_I$. When this happens, the density inside the HII region will increase compared to what it should be in the two temperature steady-state solution, and as a result, the HII region will shrink in size. We show this mathematically, but it is not that hard to understand why this happens: when the radiation of the star ionises an atom of the gas, a tiny fraction of the radiation is absorbed and can no longer be used to ionise another atom. The size of the HII region is nothing more than the volume around the protostar that contains enough gas to absorb all the ionising radiation the star emits. If we suddenly add some more gas to this volume because of the density perturbation, then the radiation of the star will be spent before it can ionise the entire volume of the original HII region, and the HII region will shrink in size.

The question in stability analysis is not so much what happens when a perturbation is introduced, but rather whether the system (in this case the size of the HII region) eventually is restored to its original state. Or in other words: whether we are on the bottom of a valley or on top of a hill. We argue that it has to be the latter: since the density increases steeply as the gas gets closer to the protostar, a small perturbation in density far away from the star is compressed and grows while it is accreted onto the star. This means that the HII region can only shrink further, and will not grow again to its original size, as long as the original density perturbation is present. Unfortunately however, the mathematics to actually rigorously prove this are very complicated (you can ask my colleague Nina Sartorio who spent many months trying to figure this out). This means that we need numerical simulations to prove this. I will tell more about this below.

Additionally, we were also interested in what happens when the original density perturbation is accreted onto the star. Or rather, my colleague Kristin Lund discovered that something really weird happens (this is why we started looking into this in the first place) and we wanted to understand what this is. You might expect that when the original perturbation disappears, the system would be reset to its original state, as if we were putting the ball in our example back on top of the hill. We found that this is not the case: while the perturbation is being accreted, the HII region shrinks and this causes the density outside the now smaller $R_I$ to adapt to the Bondi solution for a neutral gas (which is lower than for an ionised gas). When the perturbation eventually disappears, the radiation from the protostar is suddenly no longer absorbed in the whole volume between its current ionisation front and the original ionisation front radius $R_I$ and ionises this volume again. But it does not stop at the original ionisation front radius $R_I$ either: the amount of gas in this volume is lower than what it should be for the steady-state solution (which has a high density, ionised gas). This means that the ionisation front will end up being outside the original ionisation front radius: the HII region is larger than at the start.

Unfortunately, this larger HII region is not a steady-state solution, so the HII region will eventually shrink again as the density inside the newly ionised gas adapts again to the higher density, ionised case. We show that while this happens, the dynamics will inevitably create new density perturbations that will cause the instability of the ionisation front. Eventually, we end up in a sort of stable scenario, whereby the ionisation front shrinks while density perturbations are accreted, and then expands (too far) again when these perturbations reach the central star. The HII region is still trapped, but is always evolving.

# Simulations

Since we found it to be impossible to mathematically show that the two temperature steady-state solution of Eric Keto is only marginally stable, we had to resort to numerical simulations. To this end, I used a simple 1D, spherically symmetric hydro toy code that I wrote while preparing lectures about numerical hydrodynamics. The code is pretty much a textbook finite volume solver that uses a 1D radial grid of cells and accounts for spherical symmetry by including some correction terms. To perform radiation hydrodynamics, I assumed a very basic model for the radiation, that ties in well with Leon Mestel’s two temperature approach: I assumed a constant ionising luminosity for the protostar, and then numerically computed how much volume could be ionised at any time by that radiation by accounting for the actual hydrodynamical density in each cell of the numerical grid. Inside that volume, I artificially multiplied the pressure the gas would have if it where neutral with a factor 32 (this is roughly the difference in pressure you expect between neutral and ionised gas).

We performed a number of simulations to show (a) how good this approach is, and (b) to show that the theoretical scenarios we proposed actually occur. We were for example able to show that if we set up the two temperature steady-state solution for which we derived a mathematical expression, our RHD code was able to keep this stable for a while, before numerical perturbations (caused by the finite precision of our computers) triggered the instability of the ionisation front. We also showed that a neutral accreting gas with a constant accretion rate will eventually settle into the two temperature steady-state solution, provided that we artificially keep the ionisation front radius $R_I$ fixed (to prevent the instability from developing).

The most tricky and most important thing however was to show that the instability is really catastrophic and cannot be undone once it occurs, and leads to the weird scenario that Kristin Lund discovered. To do this, we had to show that our numerical simulations were converged: we had to demonstrate that the results we found are a direct result of the physical equations we were solving numerically, and were not caused by the numerical code we used. I mentioned before that the finite precision of our computers can already trigger the instability: when this happens, the simulation results always depend on the numerical code and it is impossible to tell whether they are real or not. We had to come up with a way to prevent numerically seeded instabilities from dominating our result.

Eventually, we had to do two things: first of all, we made sure to seed the instability. Just as we introduced a density perturbation in our stability analysis above, we introduced an actual density perturbation in our simulations to manually trigger the instability. If this is done early enough (before numerical issues trigger instabilities), then the seeded perturbation will dominate the simulation, and allow us to study its evolution. By changing the size of the perturbation, we could show various scenarios whereby the HII region changed shape in different ways (but never restored to its original size).

To study the late-time behaviour of the HII region (after the first perturbation disappeared), we had to also soften the ionisation front transition. In the Mestel and Keto models, the ionisation front represents a sudden jump from ionised to neutral gas that happens at a single radius $R_I$. This kind of sudden jumps cannot be represented exactly on a computer, which again results in simulation results that depend on the details of the simulations. By making sure the ionisation transition happens more gradually (and hence softening it), we made it possible to represent it on a computer. This eventually allowed us to show that the Lund scenario is real and not just a numerical issue.

The computationally interesting aspect of the simulations was setting up appropriate initial conditions for the simulations with a softened ionisation front, since we did not have a mathematical expression for a softened two temperature steady-state solution. We instead had to generate our initial conditions: we ran a separate simulation that started out from a neutral accreting gas with a constant accretion rate at large distances from the star, and evolved this dynamically until it settled into a steady-state solution, by imposing a constant ionisation front radius and a constant soft transition size. We then used the final state of these simulations as initial conditions for simulations where we no longer kept the ionisation front radius fixed.

To keep track of all the different initial conditions and simulations, I used a scientific workflow management system (Makeflow), as discussed in a previous post. In this case, the scientific workflow contained everything needed to run and analyse the simulations; when I changed something to the code (which happened quite a lot of times), I simply reran the entire workflow on my computer and all figures that eventually ended up in the paper were automatically regenerated. When the reviewer asked to include an additional set of simulations, I simply had to copy the instructions for some of the existing simulations in the Python script that generated my Workflow and rerun the workflow. Makeflow even made sure that only those simulations that needed to be rerun were actually run again.

# Conclusion

In my paper, we show that spherically symmetric trapped HII regions are not static, but are constantly evolving in size and shape. This result is somewhat counter-intuitive and not really what we expected when we first started modelling these HII regions. Scientifically, it is less relevant, as we know that real HII regions are not spherically symmetric and show a very different evolution.

Our results are very interesting from a numerical point of view as well. Radiation hydrodynamics is an upcoming field in astrophysical modelling, and having simple toy models like the ones we discuss in my paper is useful to understand future algorithms and techniques that try to model real astrophysical scenarios for which we don’t have a clear mathematical description. The instability Kristin Lund discovered by accident made us doubt the accuracy of our methods, and only by rigorously investigating what was actually going on do we know we can still trust them. The same instability makes it very much impossible to model spherically symmetric ionised accretion in 3D simulations: a different numerical triggering of the instability in different directions triggers the formation of star-shaped HII regions that look very different from what we expect theoretically. Now we understand why.

Personally, I really enjoyed working on this paper because it gave me a perfect excuse to start using Makeflow. All 1D simulations in the paper could be run very easily on my desktop computer at work, so that I did not have to deal with any of the network complications that still make it very hard for me to use workflow management systems for large-scale simulations. On top of that, the large number of simulations and reasonable complexity of my workflow made this a very nice example of how workflows are useful; I already included it in a presentation about workflows I gave last year.

Professional astronomer.