08:04:14 
So we'll get started less than a minute but I know Jess would be scolding us for not following her own rules.

08:05:02 O
kay well according to my clock hits five after the hour, and it's time to get into our tutorial.

08:05:09 
And I will say we're scrambling a little bit right now, because Jess Werk was supposed to be hosting this session. And there's been a big power outage in Seattle.

08:05:20 
So she can't do it, Ben Oppenheimer is going to do the hosting but I'm going to introduce the tutorial concept, and then go ahead with mine so there'll be three tutorials today.

08:05:30 
Some weeks there might be more, but having only three for this two hour period allows us to space them out a little bit.

08:05:37 
So I will be talking about what is the real temperature Halo, and then Cameron Hummels will go, and then Prochaska will go.

08:05:48 
And we have a dual purpose here we're giving tutorials, but we're also trying to demonstrate the wide range of things that could be tutorials. Mine will be almost like a blackboard talk about, you know, a relatively simple concept.

08:06:03 
Cameron will be instructed to use a simulator CGM simulated data analysis package.

08:06:10 
And then the last one will be about, you know, a topic in research, you know, research results.

08:06:19 
So, when mine is over. I'm going to invite you to join me in a breakout room, because we're putting these things kind of back to back a bit. And we won't allow for opening discussion so I'll go.

08:06:37 
And then I'll invite you to a breakout room where I'm willing to talk about anything you want to talk about. And shortly after that, Cameron will go and then he'll go to breakout room.

08:06:45 
So this is how we're trying to maximize our ability to give tutorials and have discussion that doesn't interfere with, you know, someone taking the next step.

08:06:55 
So that's how it's going to run. And we are asking people to pitch their tutorial ideas to us for all weeks of the workshop so we can learn as much as possible.

08:07:06 
Okay, so I thought I'd talk about what is the virial temperature of a halo, and, you know, in a way.

08:07:14 
Perhaps you know all this already. I'm not trying to insult your intelligence, but I.

08:07:21 
There's a lot of misconceptions that I see in literature about the idea of virial temperature, and I want to go through what I think about when I think about virial temperature, and hopefully it will help some people fall into misconceptions less.

08:07:38 
So, a basic, you know I'm sure you all know this, but one can write down and approximate real theorem, with similarity signs, the kinetic energy of a real life system is supposed to be half the absolute value of the gravitational energy.

08:07:53 
And if we equate the kinetic energy with approximately, the number of particles times kt.

08:08:12 
Then you can say well, a kt is going to be roughly the circular velocity squared over two times the mass particle, which is like the GM over to our. And so, if you're just doing some estimates.

08:08:16 
That's often good enough.

08:08:18 
But when you want to do better.

08:08:21 
It's not good enough.

08:08:34 
Another thing is, in order to determine the virial temperature.

08:08:27 
You need to

08:08:31 
someone's, yes, say yes I can see in the chat that a virus has crept into my title. I'll fix that later. But yes, auto correct doesn't like the word virial.

08:08:45 
And now I see right.

08:08:47 
So you have to, you know, you need to know mass and radius in order to do theoretically calculated virial temperature, and that gets us into the topic of mass and radius.

08:08:59 
And so the reason it's thorny is this this is we're talking about don't actually have a good boundary, the mass trails off gradually with radius. And so in order to find virial mass or radius.

08:09:11 
We have to set some sort of arbitrary boundary.

08:09:15 
And what's most often done is it's set with respect to some over density. Sometimes it's an over density with respect to the critical density, some is over density with respect to the background density.

08:09:28 
And so there's a Delta factor that says we're going to define the virial radius, so that the mean mass density inside it is some multiple of say the critical density, and the delta is chosen to roughly divide the part that has kind of done gravitation

08:09:45 
relaxation from the part that's still infalling

08:09:51 
up the way, you know, if you do a little algebra.

08:09:54 
You can rearrange that definition to what I think is a convenient way of relating the circular velocity at a radius to the radius. And so the relationship, it's almost like a Hubble's law sort of for virial radius, where the the ratio of circular velocity

08:10:16 
it to radius is really a multiple of, Hubble constant at that time. And whatever that multiple is depends on what you chose for delta. So in a way, it's like saying the orbital time velocity of radius is like some of the age of the universe at a given radius.

08:10:37 
Okay. But the problem.

08:10:41 
I think isn't literature.

08:10:43 
There's a lot of different was that are used.

08:10:46 
There's too many definitions, there's a lot of justification for using each one.

08:10:53 
But it is complicated our task of trying to relate, our works to one another because of this multiplicity of definitions.

08:11:03 
What halos is to lowest order. They had a constant circular velocity. So that says that the circular velocity of a halo is more definite than whatever we call faster recovery or virials, which means that the temperature is going to be more definite.

08:11:23 
Also, in a way, I like to think of halos more as having, you know, virial temperatures, or circle velocities, then you know if you're a masses.

08:11:34 
Another thing is there's something, sometimes called pseudo evolution. The first time I learned this term was in a paper by David Morin Kratsov, where if you just have a halo that's not evolving at all it's made it's just sitting there.

08:11:48 
The fact that you're measuring massive radius with respect to some sort of multiple the critical density means that Halo mass, or via messenger your radius are changing with time, even though the Halo is not changing at all.

08:12:02 
So that can be another point of confusion that comes in, when you're trying to understand what the significance of virial mass and radius are so you're cautionary tales there.

08:12:13 
But the real topic is virial temperature.

08:12:17 
So, in the virial theorem for an isolated system, people are used to thinking about something like this. The kinetic energy is one half the magnitude of the gravitational energy, and then you click it, just like three halves kt particle, and that would

08:12:43 
you a gravitational temperature. That's like the gravitational energy divided by three times in particles. Okay. and that's okay for an isolated system

08:12:47 
with pure gravitational confinement. Right.

08:12:51 
But there aren't isolated systems out there that we're talking about. If you actually have some boundary pressure and this is if you go into the halo you draw a radius and saying inside of this is my real life system, outside of it isn't it, there's actually

08:13:08 
particles out there, moving around which represented pressure.

08:13:13 
And so you have to modify the virial theorem if you have don't have a zero pressure boundary.

08:13:21 
And that leads to another term in this relation, a pressure term, and to, you know, to note me Why can I just write that now while you can look at that and say, what what happens if I just send gravitational energy to zero.

08:13:36 
If I do that, what I obtain is nkt equals P times v. OK, so if you have a boundary pressure. This way of writing down the virial relation can allow you to segue smoothly from a system that's mostly gravitation they can find once mostly pressure can find.

08:13:59 
Okay. And so an example of this is if we just asked what is a singular isothermal sphere like a gravitation potential that has constant circular velocity.

08:14:13 
Um, if you, you know, take a single iso thermal sphere and say what's the gravitational energy, once we draw radius.

08:14:20 
That's mass times VC squared.

08:14:23 
At an investable what's the pressure out there, just what I associate with particle motions. It's one half times the math times b c squared.

08:14:32 
And that means that the temperature of the system is the magnitude of the gravitational energy divided by two n.

08:14:43 
Whereas, you know, if now we've, we've accounted for bowtie pressure, what we think of as the virial temperature is actually 50% greater than the 90 virial theorem.

08:14:55 
So, don't forget that.

08:15:01 
Next thing.

08:15:05 
When we're talking about the gas part, the gases it self is gravitating. So, you know, we can use the veil theorem to say what's, what are the particle motions like of the collision list stuff.

08:15:18 
But actually, usually when we're talking about our attempt you're talking about the gas, especially for talking about the gaseous Halo. And in that case, it's best to start with the idea of hydrostatic equilibrium.

08:15:29 
And again, I like to rewrite the hydrostatic equilibrium equation.

08:15:34 
If I want to know the temperature I can rewrite it in terms of the circular velocity and.

08:15:40 
And this logarithmic pressure gradient.

08:15:45 
And that means we can define something I think its gravitational temperature. In other words, if I have this relation is my expression of hydrostatic equilibrium.

08:15:57 
Then, if I define T sub phi gravitational temperature, like this.

08:16:03 
Then, if my system is actually a single isothermal sphere, then the hydrostatic virial temperature is equal to T fi. Alright, so the single highest levels fear is the closest parallel model, we have to what cosmological halos are like.

08:16:24 
And so this shows this expression particular shows that what's really determining the temperature of gas and a gaseous Halo is this gravitational temperature, and whatever the pressure gradient is so in order to change the temperature of the gas fundamentally

08:16:40 
what I need to do is change the pressure gradient, it's independent of the absolute magnitude of pressure.

08:16:48 
What I want to actually do is changing the slope somehow, if I'm going to change the temperature.

08:16:53 
Okay, So, what implications as, have I pulled this figure out of a paper from Goulding et al, 2016, where they looked at Chandra archival data for galaxies and the massive sample galaxies the Atlas 3d sample.

08:17:13 
And so they had good velocity dispersion measurements of the central galaxy, and they're measuring the gas temperature in the central galaxy within the effect of radius.

08:17:23 
Okay. And you can graph, velocity dispersion, and temperature, and then say, is the relationship well obviously gas temperature is well correlated with velocity dispersion.

08:17:39 
But if you, a kind of us.

08:17:43 
It kind of naive virial argument.

08:17:46
 One way to capture that virial of your argument is something extra astronomers called beta spec. beta spec is defined to be the mean mass per particle, times the velocity dispersion one dimensional last version square divided by kt.

08:18:00 
And that can be rewritten as T five over ti put an approximate equal to here. It's an exactly equal to if this velocities version, as I said tropic, but maybe the last inspirations and I said tropic a bit, which case there be some sort of correction.

08:18:17 
But when you expect things to be like a singular isothermal sphere, then you expect beta spec, to be one. Okay. And these magenta lines are the beta spec equals one lines.

08:18:31 
And you can see that, you know, the relationship between the last dispersion temperature has approximate the rate slope, but it appears to be elevated in temperature.

08:18:42 
Okay. And so this orange dotted line I mean, orange for the purpose of this figure is up at beta spec of point six, which says the gas appears to be hotter than you expect from a simple virial argument.

08:18:57 
Okay.

08:18:58 
And there's been a lot of. So this particular paper is looking at, you know, how it depends on a halo mass how it depends on radio power, because you know if the gas is hotter maybe it was heated.

08:19:09 
Okay.

08:19:11 
It also looks at you know how much it depends on rotation.

08:19:15 
But really the main thing that's going on here, is it has a shallow pressure gradient, something has happened these two systems to make the pressure gradient shallower.

08:19:24 
And at the perfect grades shallower, you're going to have a higher temperature for the Griffin gravitational temperature, it's not a mystery. Some people don't look at this and say, Oh, this must be out of hydrostatic equilibrium because the gases in

08:19:36 
the virial temperature, but all that's happened is there's been some heating process, or cooling process could be either that has changed the pressure gradient.

08:19:45 
Okay, it's changed the mapping of gravitational temperature on to gas temperature. And so, it's not a mystery.

08:19:53 
It's just how things behave when they're close to hydrostatic equilibrium. Okay.

08:20:00 
So, what are the theoretical implications.

08:20:02 
First one virial temperature is, it's better way of thinking halos then mass of your radius, because the problems I outlined in the beginning of the talk, sometimes we do want to know mass, but there's a lot of ambiguities in contradictions

08:20:17 
in definitions of mass.

08:20:32 
virial temperature isn't even really what you want to be talking about, we talked about the gas because virial temperature refers to a system that's self gravitating, the gas is not gravitationally determine, it's responding to a gravitational potential.

08:20:35 
Okay.

08:20:35 
And when you move the gas around the potential doesn't change. Okay, so I prefer to talk about is the, you know, is the gas at the gravitational temperature higher or lower.

08:20:46 
Then the question is all right. I can calculate the gravitational potential the Halo.

08:20:50 
But if I don't know what the gas temperature is the pressure gradient comes into play and not the magnitude the pressure gradient it's just the slope.

08:20:58 
What does that mean that means if I have a cooling flow.

08:21:03 
That's operating and the pressure gradient stays the same.

08:21:11 
The temperature doesn't even have to change. Okay. In a cooling flow and isothermal potential. The gas is being compressed, as it's losing entropy, and the temperature stays the same, so you can have a cooling flow, and yet the temperature doesn't change.

08:21:23 
Okay, it's mapping to the gravitational temperature. It also means that if you heat the gas gradually, and it expands but you haven't changed the pressure gradient, then it just expands without changing the temperature.

08:21:36 
Okay, so I've written my hobbyhorse around a little bit.

08:21:43 
to let you know how what summons gets me going. When I see some misconceptions and literature. And I just wanted to do this tutorial I think for the most part I probably reminded you of things you already knew.

08:22:00 
But maybe don't always remember when you're trying to understand why gas has the temperature that it does in a gaseous Halo.

08:22:07 
And so with that, I will stop sharing and head over to my breakout room, if you want to talk about this further I see there been a lot of comments in the chat, but I'm going to hit the breakout room.

08:22:22 
Thank you for your time and Cameron will be next.

08:22:30 
Okay. Thank you, Mark.

08:22:32 
And we have a break now so I if you want to go and discuss virial temperature further.