Turbulent Beginnings: A Predictive Theory of Star Formation in the Interstellar Medium

[Music] [Applause] [Music] incredibly pleased and honored today to be able to introduce Blakeslee Burkhardt who is a colleague and a friend who I have known for a while now and I first met Blakeslee when Alex Lazar Ian who some of you know who if you know him has very very high standards said have this incredibly brilliant student who you have to meet and you have to say that in a very excited voice with a very Russian accent that I'm not going to try to do it but just imagine it ok and that student turned out to be Blakesley who i pleasantly met at a meeting in Okanagan hosted by Eric grosse illowsky and so I always have pleasant memories not just a flake sleep enough our first meeting and we spent a really intense amount of time talking about star formation and what she was doing for her thesis and I was pretty surprised that somebody so young could be so incredibly well-versed in turbulence and magnetic fields which are two subjects that make many people uncomfortable I'll tell you a little bit more about those subjects in a minute but I should include that that Blakesley is from Kentucky not everybody knows that and she actually went to the University of Louisville undergraduate and then she went to graduate school in Wisconsin with Alex Lazar Ian and many of you who know her here at the CFA know that she won a number of postdocs at the same time so first she was an Einstein fellow and now she's a combination ITC SMA fellow and we will probably lose her but anyway maybe should come back someday um and so Blakeley has written so many papers that I cannot possibly begin to describe her career even though she's so young except to say that most of them have something to do with magnetic fields or turbulence or both on many many scales and then in her spare time she does things like help out with the breakthrough starshot initiative just for fun just to remind you how hard this subject is there's a quote that I bet many of you have heard but I wrote it down because I didn't want to get it wrong because it's from Heisenberg and actually it's probably apocryphal it probably never even said this but he might have so it said but he said allegedly when I meet God I'm going to ask two questions why relativity and why turbulence I really believe he will have an answer for the first one so Blakesley is now going to tell us why turbulence the second one might hold some really key answers in the puzzle of star formation another big unsolved question in astrophysics oh please thank you Wow thank you for that amazing introduction Alyssa so I am actually extremely excited to be giving the CFA colloquium today and really honored actually to tell you about progress that we've made on the subject of star formation and how turbulence actually plays a key role in in that problem and also in its solution I'd like to acknowledge many collaborators on this work ELISA included as well as students former students and current students here at Harvard to hope chin Phillip motes Nia Maura who's also a postdoc here as well as others and so this collaboration list really does represent a range of expertise both from theory observations as well as numerical simulations which as I will motivate here is really key for solving the star formation problem and really any problem that touches on the issue of turbulence and so as ELISA very nicely introduced turbulence ends up affecting many problems in astrophysics in particular magnetized turbulence really you can't separate out the effect of the magnetic field on the fluid dynamics whose ultimately we have partially ionized plasmas or fully ionized plasmas in Astrophysical environments and so the two are ultimately coupled I've listed some of my favorite problems here this is not an exhaustive cartoon list today I'll be talking about star formation but I really do want to emphasize this turbulence problem extends to many different areas in astrophysics many interesting problems that we all many of us work on here so reconnection in sand the solar corona and in planetary magnetic fields cosmic ray acceleration the interplanetary medium the intergalactic medium touches on galaxy evolution accretion discs etc so there's many interesting problems that we could talk about in the context of turbulence and I think this is reflective in the literature recently I have been looking into the past citation counts in terms of papers that cite the words MHD turbulence either in the title or the abstract and there is all there's over three hundred thousand papers going back since the 70s and you could say well you know there are just more papers nowadays people are writing more papers but actually it's about 30% currently our papers that mention something about turbulence or magnetic fields at either the title or the abstract which I took to be well okay that's a pretty conclusive part of that paper so this sort of proves my point that the fluid dynamics are now becoming an essential part of astrophysics and I swear I did not show Alyssa this slide you know the turbulence problem is a is an actual it's a problem it's a problem why is it so hard why is it a problem it's because it's unsolved it's in fact as richard fineman very nicely said it's the most important unsolved problem in classical physics the nature of fluids that transition from a smooth flow to a turbulent flow this is a very much an unsolved problem and in the problem that's a millennium prize problem so of course Heisenberg this this quote is attributed to him when I meet God I'm going to ask him two questions why relativity and why turbulence I really believe he'll have an answer for the first so I like to think of this quote because in this era of Lego where relativity is is now becoming more of a solved question we still do not have a full solution to the Navy or Stokes equations in the limit of high Reynolds numbers so this is a problem I'm gonna say is that should worry all of us at night as astrophysicists because if we don't understand fluid turbulence and fluid dynamics how can we hope to have any real understanding of our various Astrophysical problems that are affected by turbulence so I like to think about solutions to this problem and while I don't have a full solution just yet I would like to present to you all a new simulation database which I'm hoping will bring together different Astrophysical communities thinking about the problem of turbulence I've called this database the catalog for Astrophysical turbulence simulations cats now you can probably guess if I'm a dog or a cat person just from this I also thought about database of Godunov simulation so if I was a dog person I would have gone with that so cats is a database that includes 3d MHD simulations from different groups so currently we have contributions from a rep Oh Enzo and a godunov code as well as links to visualization tools statistical analysis tools etc so I'm hoping this will at least push the community towards thinking more about how we can solve this turbulence problem okay so for my talk I want to present sort of in two parts first part what is Astrophysical turbulence I keep talking about this turbulence thing but I really want to hone in on a definition of that what do we mean when we talk about turbulence and then it's an unsolved problem so you could sort of say well why don't we just throw up our hands and and go home right well I want to address some particular ways of how should we study it how should we think about studying turbulence in Astrophysical situations even though it is still somewhat of an unsolved problem and then part two I want to hone in specifically on the star formation problem I could have chose to talk about maybe some different aspects of turbulence but in particular I'm really excited to show you a new theory of star formation in which turbulence sets the seed density distribution for collapse of stars now once that collapse begins I'm gonna motivate to you that it's ultimately gravity that then sets this Universal IMF and the star formation rates and efficiencies now in order to solve an unsolvable problem we really need to bring out all our heavy artillery right we need to connect theory observations and numerix all together and so throughout my talk I'm going to be showing you connections between analytic Theory observations and numerical simulations and in particular this star formation Theory I'll show you I think this theory can really explain a number of observations which were problematic for previous theories so one is that the star formation rate is observed to accelerate so this new theory can explain that star formation rates in the old old theories and I'll motivate this had to invoke very high levels of turbulence to explain the observed star formation rates and so there's no need now to invoke those very high levels of turbulence and furthermore it's been observed that lower star formation efficiencies tend to have higher velocity dispersions and that's not predicted in previous models but it is predicted here and I think the most important thing of this this theory I'll show you this turbulent star formation theory is that it can remove a lot of knobs which have plagued previous star formation theories and that's going to be key for testing with observations in the future and so this will be primarily based on two different papers one which is on the archive now and then one that's still in prep okay so part one what is this turbulence business well I like to think of it in terms of an energy cascade if you ask a engineer they might tell you about a Reynolds number if you ask a mathematician they might start telling you about the Navy or Stokes equations but I really think of it as an energy cascade so a physical process which moves energy from one scale to another so in this cartoon here this is a cartoon of the 4ei power spectrum so you have kinetic energy amplitude on your y-axis and you have spatial scale and units of wave number here on the x-axis so the large scales correspond to small case so usually the way a turbulent cascade works is energy is injected typically on a large scale and then this this cascades in a self-similar way down an inertial range and because it's self similar to get this very nice power law kind of scaling and then finally at some smaller scale the energy is then dissipated through heat or some sort of viscous type of dissipation you can have an inverse cascade actually where you inject energy on a smaller scale and it cascades back up to larger scale but this is actually sort of a rare and special case typically when energy is injected on one scale it cascades down to two smaller scales and we interact with turbulence all the time right in our daily lives so when we put milk into a coffee cup you know we don't usually wait for the molecular diffusion to do the job for us which would take several months right so we stir our coffee induce turbulence and then we have milk mixed in our coffee in a matter of seconds right and similarly turbulence it has this mixing effect in the interstellar medium in the intergalactic medium and this ends up being very useful for for mixing metals for for mixing elements in the galaxy another point is that turbulence is not just chaos right so it's not just sure random motions there's statistical order in these motions and one can see that when one calculates things like a Fourier spacial power spectrum or other statistical Diagnostics which I'll motivate later so there is some order in this chaos and then finally you can also think of turbulence in terms of a pictorial representation there's a very nice poem by Lewis fry Richardson big whirls have little whirls that feed on their velocity and little whirls have lesser whirls and so on to viscosity so you can think of this eddies breaking into smaller Eddie's and so on down to the if skinned now you might say well what's the motivation for turbulence existing in galaxies and in particular in the is M well again you can think of this I mentioned this Reynolds number what is the Reynolds number it's the ratio of the inertial terms to the viscous terms in your force equation so if you take that ratio as a dimensionless number so you have the velocity of the system its characteristic length scale and a characteristic viscosity and so that dimensionless number can tell you if roughly if you should expect a flow to be laminar or turbulent if the Reynolds number is very high so very high means greater than a thousand or so you should expect to have a turbulent flow and if you kind of compute typical velocities of the interstellar medium typical object sizes you can easily have Reynolds numbers in excess of 10 to the 10 so we definitely expect even just from a back-of-the-envelope calculation that the is M should be turbulent so if we want to study turbulence typically we do so with the help of numerical simulations I would say numerical simulations have really revolutionized the study of turbulence in Astrophysical settings simply because it's very difficult to analytically work with turbulent fluids and of course with observations we have a number of difficulties like the line of side effects and so on so simulations have been instrumental for this I'm showing you one of my turbulent box simulations this is a movie here of a fluid simulation which should have similar characteristics what we might expect in the interstellar medium namely it's magnetized and it's supersonic so right there there's two interesting parameters to keep in mind that are important for turbulence simulations for the interstellar medium one is the sonic Mach number so this parameter here which is the ratio of the flow velocity to the Sound speed so in this particular simulation and I'm showing you here density rendering so the purple blobs here that are moving around or the high density features and the yellow is sort of a low diffuse density because the sonic Mach number in this simulation is around seven there's shocks there strongly shocked motions so you get these nice density enhancements in the simulation now there's also a magnetic field and the second parameter that's important there is the al feign Mach number so the Alpha Mach number is the ratio of the flow velocity to the L feign speed you can also think of that as the ratio of the turbulent kinetic energy to the magnetic energy so it's sort of characterizing the relative importance of those those two effects and here the magnetic field is strong so the Alpha Mach number is low and you can see the effect of the magnetic field if you squint threading the low-density material so in this particular density rendering you're looking parallel or sorry you're looking perpendicular to the mean field and therefore you can see the the parallel field lines running through the box and then finally another important parameter I've already been describing it really is is that energy injection the the spectrum right so actually pumping turbulent energy into the simulation and doing that continuously because turbulence likes to of course dissipate in damp and so we have to continuously drive these turbulent simulations in order to keep energy in the box I'm another way to think about the Mach number are the ratio of the Alpha Mach number the Solomonic squared is the plasma beta so it's just another way to think of those parameters so the ratio the gas pressure to the magnetic pressure well these parameters I would argue are very important for many areas of astrophysics and I've put up a few areas where people constantly think about measuring the sonic Mach numbers the elfin Mach numbers etc and of course this simulation here is without gravity so if we wanted to add in gravity and of course for star formation ultimately we need to add in gravity another parameter that's important is the virial parameter which is telling you the relative kinetic to gravitational terms how important the kinetic energy to the gravitational energy is ultimately hi virial parameter systems you don't expect because you expect thermal and kinetic support and if the virial parameter is low so low being like less than two or one or so then you do expect the system to be collapsing so I should say that many many groups perform these sorts of MHC simulations with different initial conditions with different turbulent parameters with gravity without gravity etc and I think this is a really healthy situation for this field to be in because many different groups can test different parts the parameter space can can test their different codes against each other etc okay so how should we go forward studying turbulence right especially for the star formation problem for any other problem and again I'd like to argue that we really do need crosstalk between all these different communities people that do observations simulations and analytic theory but how how can we get that crosstalk ultimately we need statistics right so we need statistics to test and communicate observations to theory to test the scaling laws of theory against observations we need statistics to test the quality of our numerical simulations right if we make a prediction for the scaling of the turbulent power spectrum from analytics does that appear in our numerical simulations and if if not what does that mean does that mean our simulations are too low resolution or or what and then we also need statistics to connect numerix and observations and I like to call this synthetic observations ELISA was calling this taste testing for a long time it's the same thing it's like you know taking the numerical simulations and running the post-processing adding noise adding theme smoothing radiative transfer and then seeing how they compare with the observations okay so what are these statistics well there's a few of them right and and I've told you a little bit about the power spectrum and how important that is but there's many others and here in this cartoon I'm highlighting different statistics in different bubbles and then underlined here is the type of data that you could potentially apply this statistic to and all these different turbulence statistics have different dependencies so here I'm showing you statistic that depend on the sonic Mach number so the power spectrum depends on the sonic Mach number as well as several other Diagnostics there's also of course the elephant Mach number which again is measuring the relative importance of the kinetic and magnetic energy there's also statistics that depend on the Alpha Mach number and you can apply those to column density or position position velocity radio cubes or synchrotron data etc so I won't be telling you all about these different statistics but I do want to talk about one of them today and in some sense it's the simplest one the probability distribution function of turbulence and you can say Wow okay simple sounds you know like a little boring but actually I'll argue that this simple probability distribution function can tell us so much about turbulence and even help us solve this issue of star formation so from there I'll transition to talking specifically about how turbulence impacts the star formation problem and I really you know can't overemphasize how important star formation is in astrophysics and I think most of us realize that star formation and stars in particular touch on nearly every Astrophysical problem from planet formation to stellar evolution and element formation to energy injection into the intergalactic medium all of these different processes are very much linked to the evolution of stars but at the same time we still don't have a complete theory for how stars are formed and so this is a problem where we need to think about the dynamics of the interstellar medium and so the interstellar medium being turbulent and in particular being supersonically turbulent this really has two effects on star formation this is this is essentially our modern understanding of the problem there's these two effects so on one hand supersonic turbulence can enhance star formation because as you show I was of showing you in this movie early saw these density features moving around right so the density is not uniform right we have a fluctuating density field so turbulence can enhance that density field and those density density enhancements can collapse on their local free fall time at the same time turbulence provides additional pressure support right if you think of the virial parameter if you add in extra kinetic energy either through a thermal term or through a kinetic turbulent term you can increase the virial parameter and you have additional global support so there's this dual role that turbulence plays in the star formation problem so now the density distribution ends up being important for enhancing star formation or particularly for supporting and making star formation less efficient so what is that density distribution there's been a lot of attention paid to this in the last decade or so basically the consensus now is that the probability distribution function of the density field of a turbulent flow a supersonic turbulent flow even is log normal so I'm showing you here a movie by price and Federer in 2010 where they initialize turbulence in the turbulent box simulation so they're driving the turbulence the formation of shocks density features can be seen here and then corresponding the evolution of the density power or the density PDF is on the right so you can see that this interaction of multiple shocks that forms and these simulations produces this nice log normal density distribution okay so this has been noticed in basically everyone's simulations that when you have turbulence you get the formation of this log normal form and then cirrus said okay great this is the form of the PDF and from turbulence we know the is M is turbulent therefore we can write down various analytic series of star formation based on this log normal form of the PDF and that's been done in a number of studies this is not even remotely an exhaustive list but basically the initial mass function the star formation efficiency the star formation rate the Kennecott Schmidt relation all of these various studies have been done that are all based on a log normal PDF and how does that work well it's actually pretty simple simply you you know your form of the density PDF and you want to know some critical density over which collapse will begin and you have some critical density and maybe you write down some description for that and it varies for four different studies but okay you once you have your critical density you can then just integrate over this log normal form and you then you have your total density that collapses into stars modulating that by a freefall time and you can then normalize everything here to have a unitless star formation rate per freefall time and then of course there's some fudge factors as always so there's this epsilon here that accounts for feedback so not all of this gas ends up becoming star forming so some of it gets ejected out and then maybe you have another fudge factor here for for modulating this this freefall time which can also depend on on different densities as they collapse and so you can write down the star formation rate analytically then because you know the form of the PDF is log normal and so then you can easily write down star formation rate which depends on the properties of the PDF as well as that critical density that you define so this has been tested and quite a number of papers both numerical as well as some observational works and essentially I've put up here three of these more popular log normal based star formation theories and these were tested in a paper by Federer Fink lesson in 2012 where they took all these different series and also even modified those theories and then test them against simulations changing things like the sonic Mach number the virial parameter and the star formation rate and so basically all these theories they all use the log normal PDF the only real difference here is they have different prescriptions for this critical density over which collapse can happen okay so there's some modulations with the Mach number the virial parameter and then some of these other parameters which I won't go into but to say they all have different critical Penh cities there are some Universal predictions though because they all take the same form of the PDF so some of these predictions include higher star formation rates per free fall time with increased turbulence constant star formation rates if all these different turbulent parameters stayed the same right so the star formation rate should be roughly constant given all these parameters lower star formation rates for higher virial parameters and magnetic field okay so that that should make sense and then this critical density for collapse which depends on a number of various parameters of order unity or so so you can see those predictions most of them represented here so for example higher Mach numbers so Mach numbers on this x-axis here higher Mach number corresponds to increasing star formation lower lower viola parameters also corresponding to higher star formation okay so those are the general predictions of those theories and why for example do you have higher star formation rates with higher Mach numbers in these theories well it's actually easy to see pictorially what the Mach number does to the PDF and I'll show you that here so here's a simulated box is actually one of my simulations with a Mach number of 0.5 so here the density fluctuations are very weak so the corresponding PDF of density is very narrow it's a very narrow log normal because you don't have strong shocks so if you had some critical density for star formation maybe this simulation the density distribution is so narrow that it might not even cross that but if I increase the Mach number and just now you you have say a Mach number of 2 you have stronger density fluctuations right so you have higher density peaks from the shocks and you also have more low-density material out at in these vacated regions and the corresponding PDF gets wider right and so now there's maybe some amount of material that crossed that critical density can that canal for stars so you can keep playing this game of increasing the sonic Mach number and you see the corresponding width of the PDF just gets wider and wider and wider you have stronger shocks as Mach 4.5 which produce stronger density fluctuations and pushes that higher density region further past the critical density okay so this is a Mach 8 so it has the widest PDF of all so people have noticed this and say oh okay this this is nice to connect to these analytic models because the width of the PDF we can now write down in terms of physical parameters so things like the sonic Mach number and also there's a parameter B which is related to how we how we drive the turbulence how we force the turbulence so if it's supernovae that's driving turbulence this could be very strongly compressive motions so B is higher and if it's more gentle sort of whirling eddies this can be like solenoidal motions and so B is then lower but basically the Mach number and this B parameter control the width of of the distribution now this has been claimed to have been observed in a number of papers so these are now observations showing you PDF width as measured from dust extinction data corresponding Mach numbers on the x-axis measured from C oh this is study by Konya Leinen and collaborators in 2013 and there's a general trend that you can see here with higher PDF width corresponding to higher Mach numbers I will throw a point out that it's very difficult to measure the low pollen density part of the PDF the log normal PDF so this could be subject to some line of side effects but people certainly have have claimed to see this effect with wider PDFs corresponding to larger sonic Mach numbers the diamonds I think those are different studies so these are different clouds so they call in this study by Kanye Lane and they they have cloud ad HG so these are individual clouds in their study okay so this is all well and good but we should really be asking ourselves is this probability distribution function of density truly log normal because if it's not then we really do need to revise all these various analytic theories so to that end we undertook a study to look into Perseus with different tracers so looking at the atomic hydrogen with the golf ax survey so this is an Arecibo h1 Survey of Perseus that was first study first conducted by min Yong Li in 2015 so so min looked at the total h1 in and around Perseus as well as the total gas and dust using to mass and IRS data and we revisited that data in 2015 and we found something very interesting then in fact the PDF in the log normal is primarily built up of the atomic medium and the diffuse molecular medium and that slightly past the h1 to h2 transition point is where that log normal starts to form into what people now call in the literature a power-law tail so it's no longer really a log normal so much out of the highest densities but it looks to be more like a power law and so it seems more like the atomic medium and the diffuse molecular medium is tracing that log normal but really the dense molecular gas somehow takes a different form of the PDF looks more like a power law and we've seen that not only in Perseus but Nia Amara and I took a look at several other local clouds so a few 'kiss Oh Ryan Manar to California and in all of those clouds we looked at the h1 and the total gas and dust so the h1 shown here in red shows this nice log normal form in all cases and then the molecular gas shows what looks to be a power law and I should point out that the h1 is tricky right because there's so much h1 the line-of-sight and so over plotted here is the whole line of sight h1 but when we bend the h1 in terms of velocity bends that correspond to the co then this this log normal form appears and this is actually not only seen in observations but also in numerical simulations here I'm showing simulations from Collins at all and burkhardt at Collins Liz Arian 2012-2015 and in this simulation it's very similar to the turbulent box simulation I showed you earlier except now we turn on gravity so this starts off at T of zero as just supersonic turbulence with a magnetic field but then past T of zero we switch on gravity and so there you can start to see shock to density features begin to collapse on themselves this was the movie I was playing at the very start of my talk and I hope you got a good look at it because of course the obvious thing here is that the densest features collapse fastest right on their own freefall time but what maybe is less obvious is that a lot of these features don't collapse some of them are just moving around under the influence of the ambient turbulent driving and so you still have this log normal form of the PDF and even as the materials collapsing you still have that log normal but what you see at high densities is you begin to build up a power-law tail and the power law tells slope becomes shallower and shallower as the collapse proceeds okay this has also been observed in column density so I was showing you a density PDF in that movie before but again this is seen in column densities so here's Herschel observations from Schneider at all 2014-2015 see this nice log normal form but as you go out to the highest column density values you begin to see the formation of this power law tail and if we look at our numerical simulations in terms of their column density we see the same thing we see a nice log normal form that's due to turbulence and then as gravity is switched on we see the formation of this power law tail which becomes shallower and shallower as gravity proceeds to collapse things and this of course has been not only observed and in my simulations but as as well as in other people's simulations and then of course there's other observations that I could point you to that all observe this power law tale form okay so what produces this power law while gravity is an obvious culprit but it turns out the magnetic field has a role as well the magnetic field ends up slowing things down and here you can see that very clearly this is a plot of the power law slope which I'll start calling alpha as a function of time in one of my collapsing turbulent box stimulation so there are units of the freefall time and once gravity is turned on the power law tail slope forms very rapidly definitely less than a freefall time but the slope saturation very much depends on the magnetic field so the highest magnetic field case is these black points here you can see that it definitely lags behind the lowest magnetic field case which is the blue points now there's analytic work that suggests that this collapse of this this power law tail should saturate to about 1 to 1.5 and that seems to happen in about less than what less than a free fall time for sure now I've over plotted here the power law values for different clouds these are clouds from the Sneijder 2014-2015 papers and just for fun I looked up for these clouds the rough number of young stellar objects or ys o--'s and it turns out there's a clear evolutionary trend the highest number of YS Oso NGC 3603 it has the shallowest power law that's observed while Orion B has a little bit less and Aquila has even less number of Weiss's so there seems to be a Volusia nary trend here with star formation and this the formation of this power law tail and a recent paper by hope chen has really put this whole picture together right and this is especially this perspective of the observations of column density in terms of the probability distribution function of the the column density the log normal seems to be mostly the diffuse atomic gas or diffuse molecular gas which is very difficult to observe there's a lot of line of side effects that sort of hinder the observations of that log normal but what's robust is the transition point as well as the formation of this power-law tail which is built up by individual dense gravitating structures okay so this is our state of the art of the PDF our full understanding that's that's at least our current understanding of the density distribution that tribulus plus gravity seems to form now if we revisit all of these analytic models we have to start questioning the form of the PDF here now as I said all of these studies are based on a log normal but I hopefully convinced you that that log normal is now out so we really do need to revisit the calculation for the for the star formation rate and I should also mention that there's some authors even one of them in the room that just said molecular clouds have power law probability distribution functions there is no log normal at all so this really riled me up wow this is so interesting we have to revisit everything here and so that's indeed the the topic that I'm a recent paper that I've been focusing on which is updating all the analytic star formation calculations to now include this power law form of the PDF so if we use this corrected density form what do we what do we find and in particular I'd like to very much focus on comparison with observations and I'd like to show you three observations and one philosophical point where I think the previous analytic models had a lot of tension with with observations in particular so I'll go through each of these different points so one is that observations actually for quite a while since 2000 since this study by by Paulo installer in 2000 people have noticed that star formation is not constant star formation is a time varying phenomena in particular star formation can accelerate so this is from this classic paper or in the Orion Nebula showing as a function of time now the increase in in stars in Orion is predicted to be quite quite dramatic they studied several different clouds in this in this paper and came to the same conclusion in all of them that the star formation rate should be accelerating so the lognormal theories would very much predict constant star formation for a given turbulence parameters other observations and I would say this is a big point of tension show higher star formation efficiencies for decreasing velocity dispersions this is directly counter from all the lognormal theories which say that the higher the Mach number the wider that PDF should be which means more material past that critical density which means more star formation that's not what's observed this is a plot from a recent paper by Adam Leroy and collaborators showing the star formation efficiency on the y-axis and the velocity dispersion on the x-axis and you can clearly see there is a decrease in the star formation efficiency as you get to higher velocity dispersion right so lhasa dispersion is proportional to the mach number so it's a big point of tension i would say another point is that the lognormal turbulent theories predict very large turbulent kinetic energies to explain the scatter and a lot of extra galactic star formation laws so things like the Kennecott Smit relation or the mass star formation relation so to get really the full scatter in those in those relations a lot of times those theories have to invoke very large mach numbers and so when one includes the power law i would say you don't need to do that and then finally this is more of a philosophical point these different log-normal only series have a lot of free parameters right so I would say maybe as many free parameters as greek letters right so this is a bit of a problem for observers to connect to those theories and so I'll show how when you want to includes the log normal then you can alleviate all four of these points okay so how do we construct this new theory well first consider a piecewise density PDF all right so we know we have this log normal at low densities we've a power law at high densities and there's some transitional density between the two we'll call s of T so s s being the log of the density field and if you assume this PDF is continuous and differentiable then you can you have two equations and you have two unknowns here you have the amplitude at which the power law joins the log normal and then you also have the transition density so you can solve for both of those and it turns out the amplitude where the power law joins the log normal depends on the width and the slope of the power law and the transitional density also depends on the slope of the power law and the width of the log normal so if you say okay well I remember the width of this PDF is proportional to the Mach number of the turbulence then you can also write down this transitional density in terms of the Mach number so then we can revisit our star-formation calculation so in some sense nothing really changed we still have this PDF starts off as a nice log normal from turbulence but now we will include the power law and so to do that we can just take a piecewise this piecewise PDF and split the integral into two parts so we have the integral from the transition density or sorry from the critical density to the transition density over the log normal portion and then from the transition density to infinity over the power law so we multiplied by the normalized density and the normalized freefall time to get a star formation rate per free fall so now this expression will depend on the properties of both the log normal and the the power law but we can play around a bit here we change the Mach number which is controlling the width we can also change the power loss slope which we know should get shallower as the cloud starts to collapse right and as it becomes shout as is this power law becomes shallower we get more area under the curve and so we should expect the star formation rate to accelerate to increase so we can go out to even very shallow power laws or very yeah so alpha is about 1.5 we see we get a lot of additional area under the curve which will increase our star formation over just a log normal so we can compare those two models so here I'm showing you a plot of the star formation rate of the log normal and power law over the star formation rate that's just the log normal would predict okay so as you shallow the power loss slope so these different colors are for shallower and shallower power laws you naturally get more and more star formation and you can easily get up to factors of 10 over just a pure log normal okay so how can this new theory alleviate all these these four different tensions that I mentioned well first I'd like to address this issue of all the parameters right all these different parameters that these star formation rate calculations depend on you can actually reduce quite a number of them and that's because these transitional density introduces a mathematically defined critical density so there's no need to really invoke an additional critical density how does that work okay well again you have this piecewise log normal and power law form and now let's say all they dense gravitating gas is in the power law and that makes sense because what gravity does is it produces power loss it produces this self-similar collapse all our diffuse supported turbulent gases in the log normal and now we have a mathematically defined transition point based on continuity and differentiability of this PDF which separates those two so we can then write down a dense gas fraction right so this is this is pretty simple right it's just a all the gas in the power-law / the total gas so the gas in the lognormal + the gas in the power loss that's our dense gas fraction so one can then write down an expression for this and then you can make it even simpler if you say well that power loss saturation to about 1 or 1.5 happens in less than a freefall time cloud lifetimes are much longer cloud lifetimes they're 2 to 10 freefall times so this dense gas fraction ends up being a function of just the slope of the power law and the width of the PDF but if you take alpha to be 1.5 it only depends on the width of the PDF okay so the width of the PDF is something that you can maybe estimate from the velocity dispersions or maybe you can even directly measure it in some situations but for the most part this is a very I would say simple definition of the the dense gas fraction just simply based on on this width and I think the reason why this works out the magic here is that this transitional density acts as a critical density in the sense that mathematically the transitional density in the limit of alpha is 1.5 works out to be roughly the post-shock density which many authors in the previous studies were arguing is indeed the critical density for collapse this post-shock density is basically the density where the ambient turbulent and thermal pressure can no longer hold the cloud up against the gravity and so this this actually works out mathematically very nicely okay and then you can write down a star formation law now you have the dense gas fraction in hand so you can write down some star formation efficiency per free fall is the dense gas fraction oh and sorry there's still one parameter left we still don't know what feedback does to this of course not all the gas maybe ends up making it into stars therefore you still have some epsilon not which is characterizing the ejection from from feedback but basically once you have this dense gas fraction you can then write down a nice star formation law from there so what kind of gas fractions do we get when we just plug in different Mach numbers right this only depends on the Mach number and the power loss slope while you get gas fractions that are quite reasonable so for alphas of 1.5 which is generally what we should saturate to we get gas fractions that are roughly 50 to 30% dense gas fractions so again this depends on just a few parameters the width of the PDF okay so you can observational e constrain that with velocity dispersions so for example Co observations the slope of the power law tail okay so you can say roughly for most of the cloud lifetime that's constrained or you can even maybe directly measure it in in the case of local clouds and the effective feedback through an epsilon naught so this is this is really a truly free parameter but hopefully I will be able to constrain that okay so we can reduce the parameterization which I think is a really powerful thing here for connecting to observations now the three other points of tension I'd like to address so for example accelerated star formation this is actually baked into this from the start right because we have now this shallowing power law as the cloud collapses so we no longer expected constant star formation rate based on just the turbulent parameters we now also have to take into account the shallowing power law tale which which should give us an accelerated star formation but we can do better we can compare with observations here and I'd like to show you a comparison of the theory to two different data set compilations by law de Lombardi and Alvis 2010 and lombardi Alvez a lot of 2015 so in these two papers they took the same set of molecular clouds with the number of ys o is measured so one can measure a star formation rate based on the number of whiles the cloud masses are also constrained and then in a later in the later study the authors also measured the slopes of the power law tails and they made a very nice argument in this paper that the width of the log normal is very difficult to observational e constrain but this power loss slope is actually a robust observational parameter that one can measure so these two papers give us what we need we have the slope of the power law tale and we have the star formation efficiency per free fall because we have the star formation rate the cloud mass and density so here's what this looks like so now I'm showing you power law hill slope on the x-axis and the star formation efficiency per free fall on the y-axis and the different points are labeled here for the different clouds you can see there is definitely an increase in the efficiency of star formation as the power law hail slope becomes shallower and within the bounds of the model that seems to match pretty well the exact Mach numbers and so forth are not really constrained at this point but I think this is certainly in agreement with the trend that the model would predict okay so the the new theory can also alleviate another point of tension and that's meaning to invoke very large kinetic energies to explain the scatter and the kind of catchment relation so here I'm showing you a plot from Fedorov 2013 which is using these log normal turbulence models to investigate the chemical might relationship here's the surface density of star formation the star formation rate and the gas surface density divided by the free fall time and the different points are all different observations and overlayed are different log normal models and so invoking different widths of the PDF from mach number 2 to mach number of a hundred can explain the observations but mach number of a hundred is maybe a little too extreme we typically measure mach numbers of around 30 or so in this nco in the Milky Way so 30 or a hundred might be a bit too high so here I'm showing you data from Charlie lotta and collaborators in 2010 and 2012 and then Chris facies very nice compilation of NGC 300 from 2014 these authors measured cloud mass and star formation rate so this is directly measuring the molecular mass and comparing that to the star formation rate in local Milky Way clouds NGC 300 so this was part of Chris Chris's thesis and then going all the way up to the spirals and even very star-forming galaxies like ulirgs at the highest mass range so you see this beautiful linear relationship between the galactic clouds going all the way up to to the galaxies and of course theorists would like to connect to these observations and so one way to connect to that is is calculating the star formation rate from the law of normal models and as I said to do that you do need to invoke very different Mach numbers to get different PDF widths so Mach numbers of five are invoked to explain the the lower star forming systems while higher Mach numbers are needed to explain these higher star forming systems correspondingly the depletion time would be predicted to be much shorter for the higher Mach number systems well if you include the power law tail so now I'm showing you over plotted in the colored lines the updated model that includes the power law tail so four different power laws but also including just a Mach number of five which is actually maybe even a little low nevertheless you can explain all this scatter and correspondingly different depletion times when you include the power law tail and so there's no need now to invoke Mach numbers of a hundred or very very high turbulent velocity dispersions all of this can be easily explained in this star formation rate increase with the power law again another point of tension the observations show higher star formation efficiency with decreasing velocities version this one's really interesting because the theory the other theories predict directly the opposite now if you think again about this gas fraction and the definition of the transitional density which is which is defined by the continuity and differentiability of the PDF this makes sense because this is proportional to the width of the PDF squared so as the Mach number increases the PDF gets wider this transitional density also increases so here I'm showing you this pictorially I've increased the Mach number from two to seven the PDF gets wider but the transitional density also moves down to higher densities and so you end up getting less area under this curve right and that's because this PDF needs to stay continuous and so this is why this theory would predict naturally this this anti-correlation with Mach number so we can test that against the pause data this is a very nice survey from Leroy at all in particular the data I'll use here is from Leroy at all 2017 this is a plateau to Bureau arcsec at whirlpool Survey and here I'm showing you a movie of the co distribution and then 51 so the red points here are individual GMCs and the other colored are just the the rest of the molecular gas in em 51 so these observations have the surface density the gas surface density star formation rates measured star formation efficiencies so it provides a very nice observational data set to connect the bottle and one thing that they see in the data very clearly is when one looks at the star formations surface density as a function of the gas surface density per free fall to this very nice velocity gradient so this is all the different colored points here a velocity dispersion of the individual regions and m51 these are about 40 parsec beams and you see the low dispersion systems on this side and the high dispersion systems over here towards higher values of gasps surface density for freefall and this is in the direction of lower star formation efficiencies so this data is telling us something it's telling us that actually for higher Mach numbers because Mach number is proportional to velocity dispersion that we have lower star-forming efficiencies if you over plot predictions from the model where we include the power law so that's what I've done here in these lines these are Mach numbers of ten roughly this blue line here and the red line here Mach numbers of 30 so we get the right direction right we would also predict the direction of this this plot would go also towards lower star formation efficiencies with higher Mach numbers another way to look at that is in terms now of the velocity dispersion on the x-axis and the star formation efficiency now directly plotted on the y-axis the black triangles here are the co data from pause and the different colored points are the model so this is now a Monte Carlo of the analytic model for different values of the velocity dispersion one gets roughly the right range of star formation efficiencies which is about 1% in the in the pause data and then of course there's still the free parameter here of what feedback is doing the star formation feedback so this is the epsilon naught which we can constrain to be we would predict roughly two percent to eight percent so the prediction is that maybe we would go back into the pause data and see some differences here in terms of the IMF in terms of high mass star forming regions or low mass star forming regions so this will be certain something certainly interesting to do to connect to the model okay so I've I'll go ahead and summarize here and I'll show you a couple takeaway points and also I think some exciting future work that can be done in terms of turbulent studies in particular star formation studies so the main point I want you to walk away with is we're really in this new era of studying is turbulence not only it's connecting that to the star formation problem but also to other interesting problems in astrophysics and I think we can really do that now because we have a big tool set in which to do it and so there's a lot of work ahead I would say applying these different diagnostic tools so I really focused on the probability distribution function and all the things that can do for star formation but there's certainly a big list of other diagnostics that'll be interesting to apply both for star information and for other problems more connections between the simulations and and data I think are gonna be huge especially now with with Alma doing polarization Sofia in the future date JWST there's still quite a bit more to do in terms of analytic work and numerical work on the feedback especially in the star formation problem what is the role of feedback and how can that change change this picture and something I think that's very exciting is that star formation is is probabilistic it's ultimately dependent on the structure of the density distribution which maybe sounds complicated but in fact is much simpler than what previously was invoked in terms of ambipolar diffusion and and all this different plasma physics it really just comes down to how turbulence sets the density distribution and how gravity interacts with that so this is probabilistic and so then we can include things like this new star formation model that I showed you in cosmological simulations and I think that's nice because it ultimately has very few parameters which is also again important for observers to connect you and so I'll show now just my my summary here of what I showed for the star formation theory it's the same as my introduction slide my summary slide and I'll just leave this up here and then take your questions thank you [Applause] I'm wondering if there's another diagnostic method of testing this and making you us through but if the energy in turbulence is being dissipated in chalks that's going to have a number of consequences one is chemical it's going to change the chemistry of the gas and it will change the excitation in a lot of species you might not equally think that the ratio for example in Co of the month is zero to the 1 3 to 2 will be different so I'm just curious how much you've explored the chemical Diagnostics but the theory would predict great so that's that's a great point indeed that it has been observed and especially in higher J transitions of co the effective shock dissipation in my only numerical simulations I've run a lot of radiative transfer for four different J transitions as you know and we do see the same we do see the shocks very prominently in those synthetic observations so that's I think a very important direction of the research to connect to connect the simulations where we can resolve the shocks and then do post processing for different lines and then connect with the observations the chemistry will change with the B field because the magnetic field as it increases serves to the cushion the post shocked region and so you get more molecular gas the higher the B field more atomic gas the lower the B field so yeah and we see that effect very clearly cushioning effect there's a magnetic field on the shocks yeah that's something that definitely requires a more study but it's a very good point take my driving test both of you but what about the increased actual circulation and you need from that is that even more dangerous for the chemistry well I would imagine that it depends on the environment you're looking at yes I mean UV field will destroy all the hills that these shocks foster but presumably you know you look over a large enough region and you do not have to account for that I mean look turbulence is everywhere so if you look away from where young stars are being formed in a region you know like Taurus or Perseus presumably you can test this idea ideally you test them in a region of low mass or or or loathe star formation currents confirmation I think what's the Uni really picks on the feedback for ionization and all of these things that can shut the party down real quick right so star formation can be rapidly shut off the molecular gas will transition back to atomic and the cloud will disperse so I'm what time skills that happens relative to this yes it's the next step can you go from the PDF or the n PDF of nearby star forming regions to predict the mean critical the critical density for star formation I mean we know from other studies you don't get much deficiency of star formation about to see that be predicted by it absolutely it absolutely can so you can predict that roughly from the the sonic scale and the post-rock density so that 10 to the 3 10 to the 4 actually falls out from that and as I showed this model actually gives you the post-rock density roughly in this limit that alpha is 1.5 so you don't even need to necessarily go out and measure the Mach number and measure all these things if you have the Alpha measured from the molecular clouds the local clouds but they have different outfits but roughly you can measure the post-rock density need to expect there and you can test that with with going out and measuring the Mach number and so forth but that does seem to be the critical density for collapses it's roughly the close shot density and we also see that in numerical simulations now the catch here is to be thinking of what about the magnetic field right so when the only case where that's not really true where it's not the postdoc density is in the case of the strong field and then if you have if you have a master flux of lower than one so the magnetic field should support the cloud we still see collapse but it's no longer at the protracted it has more to do with with the properties of asymmetric shocks collapsing long strong field lines but it without within the limit of trans or super authentic turbulence pojat density seems to be the critical density and you can nail that down - no two counters that match up yes so you must be aware that there's another community studying this problem namely the fluid mechanics community and they have an entirely different approach which involves no turbulence Jack studied turbulent in general and you presume no that they would see this derivation event and say it's all parameter food you I hope that everybody in the room understands that there is another way of solving this problem with no zero free parameters zero epicycles it starts with the table of atomic scattering cross-sections and the entire star formation picture could be done that way in a different community however they don't introduce all of these parameters for star formation efficiency for this this epsilon parameter that you define all of those are epicycles to a theory which can be enormously simplified because the fluid academy has worked this out and none of the papers that you reference reference the classical hydrodynamics orbital mechanics community okay we should definitely talk that sounds very interesting if anyone has actually applied that now not in the star formation community because everything in our community is simulations and arbitrary fitting the parameters epicyte so I actually showed very few simulations so I define tomorrow which is based on flute what we know from fluid dynamics how supersonic turbulence phase density fluctuations and that's I would say is very well-established and then I showed how that mellow kinetic observations actually I didn't show very many numerical simulations in terms of the connection of the model simulation result says plots then all our suppliers we're models and observations I should emphasize that really that those are not numerical simulation results alright so my question is how do you compute the viscosity of hydrogen gas I mean this is not relevant for the model you discuss use everything the turbulence no I mean you finally would place they continue this later because like that's not anybody that's just Josh you had a question right yeah yeah that's that's a great question resection of your power right right so I mentioned the magnetic field has the effect that it slows down the formation of the power law together now the simulations I've shown for that that was actually that result was a simulation as well it provides additional pressure support right so it slows it makes sense slows down to collapse but those simulations had still fairly weak magnetic fields recently Phillip most who was a grad student recently here and myself and several others we looked at cases where we had different magnetic fields drinks including a very strong field case and in that case we still got collapsed and we still see the showering of the power loss slope it just takes a lot longer but this interesting thing there is the structures of star formation are very different which would show up maybe in polarization observations like home observations or SMA observations so we still get collapsed we still see the power loss health shallow but the overall structures can be can be different so that's another and testing that let's be a little bit modest those simulations did a really nice job of matching the alcohol the medical field population so there's something like sparkly dude that guy's charged in that nice diagram you happy to show the evolution of the power of wasp opposite because the lower our competencies part of that was a simulation all part of it I think is analytic so my question is as you make the problem of slope shell which other you're taking wall dancing material presumably cover leaked a high vessel right yet it seemed as if the the PDF was not changing on the low density Iceland cetera right so that consistency there was that just yeah so so that's it that's a great point so in the current model what I'm assuming is that there is continuous mass accretion from the diffuse I assume into the log normal into the power law so this would be like an D Burkett's bathtub where you have this continuous accretion if you didn't want to include that and you wanted to say for example let's compare to a simulation where we have mass conservation then the mean density would also shift over and so that's also can be included in the model as well right but that's normalized on your x axis is right so for what I show that's that's normalized out but one can also include that there's an expression we can write down the expression rules everything the shift to total mass of increase right exactly right so you need some inflow if you want to assume mass conservation then you can also write down the normalization condition for the x-axis it's getting a little bit late so let's just go Tony has one urgent question we wrote the last 20 30 question and then we'll thank you like saying that that's the more question back so this the source of the energy for the turbulence there's certainly two possible sources one is when you forming the molecular cloud out of the material you galaxy about Spang together and so on largest scale you have an injection of energy that originated in presumably the rotational energy of galaxies as a whole and the other is once we start forming stars start injecting energy from the solar winds and things can you tell the difference between those two sources of energy injections and how important each one is very points yes that's the subject that I love - yes you can there's different ways ultimately what you want to do is measure the kinetic energy power spectrum right so you can there there are turbulence techniques that I like to get listed in that cartoon that can give you that from radio position position velocity tubes so you can measure an injection scale and then potentially infer what kind of energy might be injector might be dominant there's other scaling relations for star formation rates as a function of velocity dispersions and gas fractions galaxy-wide properties that can very much depend on what's driving the velocity dispersions I didn't think that those relationships can change as another way to test what's driving turbulence but ultimately the the sort of the consensus is there's many different energy drivers so supernovae is certainly one possibility gravitational instability cloud cloud collisions so those could be large scale drivers of turbulence and sort of the jury's still out on which they're all [Music] [Music]

Channel: CfA Colloquium

Views: 4,714

Rating: 4.9069767 out of 5

Keywords: Stellar Evolution, ISM, MHD, Young Stars, Astrophysics

Id: B0f11I8L_Yo

Length: 76min 25sec (4585 seconds)

Published: Thu Feb 08 2018