Become a Creator today!Start creating today - Share your story with the world!
Start for free
00:00:00
00:00:01
46: Earth Irradiated (the Greenhouse Effect)
359 Plays5 years ago
Since time immemorial, blacksmiths have known that the hotter metal gets, the more it glows: it starts out red, then gets yellower, and then eventually white. In 1900, Max Planck discovered the relationship between an ideal object's radiation of light and its temperature. A hundred and twenty years later, we're using the consequences of this discovery for many things, including (indirectly) LED TVs, but perhaps one of the most dangerously neglected (or at least ignored) applications of this theory is in climate science. So what is the greenhouse effect? How does blackbody radiation help us design factories? And what are the problems with this model?
This episode is distributed under a CC BY-SA license. For more information, visit CreativeCommons.org.
[Featuring: Sofía Baca, Gabriel Hesch]
---
This episode is sponsored by
· Anchor: The easiest way to make a podcast. https://anchor.fm/app
Support this podcast: https://anchor.fm/breakingmathpodcast/support
Recommended
Transcript
Introduction to Climate Science Concepts
00:00:00
Speaker
Since time immemorial, blacksmiths have known that the hotter metal gets, the more it glows. It starts out red, then gets yellower, and then eventually white. In 1900, Max Planck discovered the relationship between an ideal object's radiation of light and its temperature.
00:00:15
Speaker
120 years later, we're using the consequences of this discovery for many things, including, indirectly, LED TVs. But perhaps one of the most dangerously neglected, or at least ignored, applications of this theory is in climate science. So what is the greenhouse effect? How does blackbody irradiation help us design factories? And what are the problems with this model? All this and more on this episode of Breaking Math. Episode 46, Earth Irradiated.
00:00:48
Speaker
I'm Sophia. And I'm Gabriel. And you're listening to Breaking Math.
Supporting the Podcast
00:00:52
Speaker
We have a few plugs to go through. We have a Patreon that you can donate as little as a dollar to, and we can mention you on the show if you want in your favorite math concept. You know, just like if your name is Emily and you like spiners, we could say like Emily likes spiners. Nobody's done that yet. You can do the $5 tier where you get access to slightly early episodes when we have them earlier than we are putting them out on the stream.
00:01:18
Speaker
And you also get access to our outlines, which we try to put up every week. And then for the poster tiers, $22.46, I believe, a month. But if you don't want to pay for that every month, you could just buy a poster on Facebook at facebook.com slash Breaking Mouth Podcast.
00:01:38
Speaker
Yeah, just to emphasize that. So the Patreon model is a monthly thing. So that's why on Patreon it says if you give that amount, it's $22.46 per month. But as Sophia just said, if you buy that on our Facebook store, that's only a one-time thing. So you have that option as well.
Investigating Climate Science Beliefs
00:01:57
Speaker
Yeah, and it's $22.46 for a tensor poster on Facebook plus $4.50 shipping and handling. You can also find updates about the show there. We're also on Twitter at Breaking Math Pod and we have a website, breakingmathpodcast.com, which is currently in the process of being renovated. This is a very, very exciting episode. I know it's been a little while since we've put an episode out. It's been around December. We took some time off for the holidays and to do some research. And Sophia, you've done some really serious research since our last episode, haven't you?
00:02:25
Speaker
Oh yeah, like I've been doing research into the math behind climate science because people are always saying, you know, that environmental scientists tend to believe more in human caused climate change. And so I wanted to explore the reason behind that. And the reason is actually, I mean, there are more complicated models of climate science, which we're actually going to talk about on this episode, but even the simplest models are fairly accurate and tell us some very telling things. Yeah.
00:02:52
Speaker
And real quick, do you mind, before we dive into it, I want to actually mention the sources that you used in preparing this episode.
00:02:59
Speaker
Oh yeah, a lot of it is from this resource from harvard.edu. And this resource has some flaws with it. For example, they use mm when it means micrometers and it doesn't explain everything exactly well. There's a lot of you have to figure out on your own. We're actually going to put out a paper in about a week after this episode airs that explains this a lot better. And eventually we might actually be doing a poster, but you can find that at tinyurl.com slash J
Understanding Climate Models
00:03:27
Speaker
Z Q D R 4 0 And we'll probably put we'll put a link up to that on our Twitter as well and our Facebook cool So yeah, so there's this episode for those who are interested is based on that paper so if you listen to this episode and then if you want to hear more go to that paper or Read them at the same time. Well, actually you shouldn't do it at the same time. That's hard
00:03:51
Speaker
Yeah, why do you do that? You can, you know, compose a concerto. Yeah, why not? Why not? Looks like in this episode, we have quite a few things we're going to do. We're going to talk about our story so far, specifically with respect to climate science. We're going to talk about how global average temperatures have changed in this last century. We're going to talk about some models. We're then going to go to talk about the data and the tools and how we know what we know.
00:04:18
Speaker
And then we're going to go over some properties of radiative energy as a whole. Radiative energy meaning energy that's carried by photons or other particles. This gets real meaty. Yeah, we're going to really get into the meat of physics here. Oh yeah, this is all very concrete stuff. And then we're going to be talking about some properties of gases and how they absorb energy.
00:04:39
Speaker
And then we're going to talk about some models of global warming. Yep. So both simple models as well as improved models with more information and more mathematics and just how how our knowledge about greenhouse gases changes. As well as the difference between the types of models and what is useful for what and why we need more advanced models and why we don't. Yes. So this will be a thoroughly meaty episode.
00:05:05
Speaker
So according to the resource from Harvard, I'm sorry, it was harvard.edu. So it looks like just in this last century alone, the average global temperatures of the Earth have changed by 0.5 degrees Kelvin and they have changed positively. We've seen a gain increase of 0.5 K. And while that might not seem like that large of a change, you got to remember that the difference between now and our last ice age was only 4 Kelvin or so.
00:05:33
Speaker
Yeah, that's insane. That's insane. And again, just knowing what happened in the last ice age. Yeah, like slabs of ice that were over in New York that were about a mile thick. Yes, yeah, absolutely. And also... One to five Kelvin of climate change, positive climate change is expected in the century to come, depending on how quickly we act.
00:05:57
Speaker
Now, real quick, so is that the conservative estimate? Because I've heard 1.5, I've also heard 2, and I've heard more than 2. Oh yeah, 1
Challenges in Climate Model Accuracy
00:06:05
Speaker
to 5 is kind of the range. Oh, I read that as 1.5. So, how do we know this?
00:06:11
Speaker
Well, I mean, there's models that we're going to go into later, but the models all kind of treat the Earth as basically a substance or not even a substance, just like an object that has energy and can absorb and reflect energy. And it treats the atmosphere, it treats the atmosphere as something that can absorb and reflect energy and treats Earth as something that basically reflects ambient energy. There's also models of the Earth which actually take momentum of
00:06:38
Speaker
like airflow into consideration and all kinds of things. But we got to consider how we know that models are accurate. And we know that is accurate if it generates data, like present data from past data, or if it generates a trend from a small sample of the data. Right. So basically what we're saying is we have a model.
00:07:01
Speaker
And according to this model, it'll show a temperature trend. We have recorded temperature data for the last, what is it, 100 years, 150 years? Something like that. And then if this model, we can do a, is it a retrofit, like a reverse fit kind of thing? Yeah. Like we could see, we could like put in the atmosphere a hundred years ago and if it generates the trends that actually have existed throughout the last century, then we know that it might be an accurate model.
00:07:29
Speaker
However, there's also the problem of overfitting a model, which is, uh, where you, a model just fits the past very, very well, but has no core relation to the future. And that usually happens when you put in a lot of variables into a function, like not, not a lot, not necessarily just a lot of variables, but a lot of, a lot of constants that are hard to change or that mean specific things. And these can even be embedded in the math
The Utility of Climate Models
00:07:55
Speaker
sometimes.
00:07:55
Speaker
So models can certainly be a tricky thing with complex systems. That's for sure But I mean it's kind of the same thing It's like globes like you know any globe that we have is is like a perfect sphere Well, we know the earth is not a perfect sphere. The earth is what is a spheroid oblate oblate spheroid not only that but you know as you increase the resolution There's all kinds of rocky things and you know, like it's every single model that we have is just a tool for us They're absolutely
00:08:21
Speaker
Yeah, perfect model would be a replica of the earth and the sun and every star and it would just be a completely separate universe that we'd somehow interact with. Yeah, which is a model would completely go out of it would stop working immediately though because Our model our universe would start to be based on that model. Yes
00:08:39
Speaker
So a very valid scientific question is how useful are these models? And I've heard a lot of critiques on models like if you're watching Fox News or something. They always have people on there who love to rag on the models. There's a reason why these models haven't been thrown away. These models aren't useless. They are extremely good. They are extremely useful. And yeah, we're going to get into the nitty gritty of them and talk about what they're useful for as well as the shortcomings of them.
00:09:03
Speaker
And also the reason why these models that we're going to be discussing today are very useful is because the more the models have been refined, the more predictive power they have gotten. But not only that, it's not like we invent new models and then we come to some radically different conclusion. It's not like 50% of models don't show global Kool-Aid and the other 50% show global warming. They all are in consensus with one another.
00:09:28
Speaker
And from the simplest model to the most complex model, we see the same thing. You mentioned a potential problem with this called the telephone game. Can you elaborate on that in the outline?
00:09:43
Speaker
Oh, yeah. I mean, that's just what I put in the notes. But what I meant by that is if a model disagrees with one other model, which disagrees with one further model, then you can have so many degrees of separation between the models that you're using a justification to justify justification that's justified itself by justification. But that's why a scientific consensus where studies support one another is important. It's almost like Moss. The stronger it is, the more interwebbed it is.
00:10:11
Speaker
Now at this, there's a URL here that we have in the website. This is the one that you gave earlier, is that right? Yeah, and this has a lot of this information on it. And you could actually use it almost as a reference to this episode until we have the, we're gonna put a reference to, once we finish the paper in about a week or so, we're gonna put a link to it at the beginning of the episode. We do that now with Anchor.
00:10:36
Speaker
So yeah, um, so just take a tiny URL.com slash J Z Q D R four zero with a grain of salt.
00:10:47
Speaker
That website has a discussion of the models that we just previewed, correct? Oh yeah, very in-depth in a lot of formulas and stuff. I can imagine a lot of our listeners would like to go more in-depth on the models, even our listeners who may be skeptical of climate science, at least for a resource to see what information we have on the models and their shortcomings. Oh yeah, I encourage any skepticism and I encourage it to be met with scientific inquiry.
00:11:10
Speaker
Absolutely, and also it's it is worth noting. We said this earlier, but this website that we gave you it has further resources linked to it However, you're gonna find a few things that aren't quite refined such as mm Which is usually millimeters instead of micrometers. So watch out for that Yeah, and like little little things like that.
Historical Temperature Analysis
00:11:31
Speaker
It doesn't explain certain elements of the aspects of the tool Extremely well, but what are you gonna do?
00:11:39
Speaker
So let's talk about the data and the tools that helped us to get that data for what we know about our climate and our Earth's history.
00:11:48
Speaker
Yeah, and what's interesting is that apparently there is actually a natural fluctuation in the Earth's temperature that's occurred since the 1700s. And the global temperature has actually been occasionally higher than it is now. But the thing is, it hasn't increased as quickly as it has now. But given the graphs, it is sometimes hard to see human-caused
00:12:10
Speaker
It's hard to see a distinctive human caused trend just visually and that's why we need tools to inspect I mean there are certain graphs that show it pretty distinctively, but I'm just saying that if you look at the temperature itself you could You could come to some very reasonable conclusions
00:12:28
Speaker
Yeah, and actually you just said something before. I have heard the same. I've heard this, you know, on certain shows, on TV, on certain news outlets where they talk about it's been hotter before and life thrived, you know, where you had dinosaurs, you know, like as far as certain parts of the northern hemisphere where they wouldn't be able to live now. However, the rate of change is the kicker here.
00:12:51
Speaker
And if you want to know a really, really good resource that really illustrates the rate of change is there is an XKCD comic strip that does this very, very well where it lays out what Earth has been like for the last 20,000 years.
00:13:06
Speaker
Yeah, and you can find that at xkcd.com slash 1732. Yes. One criticism, especially from my friends, has been, oh, this is just a comic. You can make a comic of whatever you want. Right. Well, this comic actually has its references in it as well. If you look down the right side of the comic strip, there are four
00:13:26
Speaker
references, including the 2013 IPCC report as well. Did they really say that because it's a comic, it's just because of that, even though it has resources? That's like saying that you can't do accounting on anything but claytatslabs, because that's what I grew up with.
00:13:46
Speaker
Yeah, sure sure so but yeah this this shows a fantastic illustration of the rate of change where it starts off at 20,000 BCE and we see I believe it is a global average temperature at negative 4 degrees Celsius to what it is right now and
00:14:05
Speaker
And then it shows a real gradual increase as the ice age ends. And if you keep scrolling down, it's just a very, very, very gradual change here. Very gradual change. And then toward the very, very end, you have human involvement. And suddenly it ramps up. I don't want to say exponentially. That may not be the correct mathematical term here. It's a sudden jerk upward very, very, very fast in terms of average global temperatures.
00:14:31
Speaker
I mean, there's a lot of measurements of like, and people might be like, how do you know, how do we know how hot the earth was 15,000 years ago? We have ice core measurements. We have tree core measurements. We have coral data, coral data. We have so many things. That's why we talk about a scientific consensus. It's not like we have one study or one type of study that says this. It's many.
00:14:53
Speaker
Yeah. I think in a lot of the papers, you'll read about the term proxy data where you don't have an actual thermometer, but oftentimes you'll have data collected from, for instance, tree rings, and they will correlate the data with what we do have in our temperature record. And if you see a significant correlation, then it's assumed that that correlation exists backward in time as well.
Radiative Energy and Climate Science
00:15:18
Speaker
Yeah. I mean, you could think of it as a picture of a blurry thermometer. Yeah, pretty much.
00:15:28
Speaker
So we're going to talk about radiative energy a lot because the sun radiates energy onto the earth and the earth radiates energy outward when it's hot. And it's measured in watts per square meter, which is power per area. And measuring the radiative flux for each wavelength and summing it up gets the total radiative flux.
00:15:49
Speaker
And the radio reflux is different at each wavelength because okay, like let's say we get a piece of iron, right? And we hit up the first color it turns is red, right? Yes. Because it's emitting in an iron is almost like a black body. So actually, we'll just talk about black bodies to to typify this or exemplify this something like that.
00:16:10
Speaker
So it turns red first because it's generating a lot of heat, but also a few red photons which are low wavelength photons. And then it turns yellow because it starts getting yellow and green wavelengths which combine with red to make yellow in our eyes. And then as it gets more and more wavelengths, it gets whiter and whiter.
00:16:32
Speaker
So that's for a black body, but a black body is a perfect object that can absorb every wavelength and emit every wavelength. But we know from experimental data that if an object can absorb a wavelength, then it can emit it just as efficiently as it absorbs it.
00:16:49
Speaker
So there's something that I'm going to talk about called the flux distribution for a black body. The flux distribution for a black body was discovered by Max Planck in 1900 by the equation. This is a bit of a wordy equation here. You don't have to take notes on this. This is something that you can just Google search. But the equation that he found was 2 times pi times Planck's constant times c squared. Which is the speed of light squared. Correct. Speed of light squared.
00:17:15
Speaker
All of that divided by L, which we just mentioned above, I don't think we actually said it. L is the wavelength to the fifth power, times E raised to Planck's length times the speed of light, all divided by Boltzmann's constant times the temperature times the wavelength. And that's all over E.
00:17:34
Speaker
Yeah, I fully get and then there's a minus one. Did y'all get that? Did everyone get that? Okay. Okay. We're good. Just Google search it just you know, anyways, yeah, so there's an equation here that exists and and the peaks of this This shows that peaks at wavelengths are inversely proportional to the temperature.
00:17:50
Speaker
Yeah. So if it's a really, really high temperature, then it peaks at a very short or very, very energetic wavelength. Correct. Yeah. All of that math, just to say that single point. If the object absorbs a certain percentage of radiation at some frequency, then it will also radiate at that ratio. So like if you absorb like 50% of red light, then what that means is that when you're heated up, you will emit 50% as much red light as a black body would emit.
00:18:17
Speaker
This can allow us to generate the emission spectrum from the absorption spectrum. Literally, multiply the absorption spectrum by the blackbody radiation spectrum component-wise. Like over the frequency. So you have the first graph, which is the blackbody radiation spectrum, which goes something like...
00:18:34
Speaker
over time and then like you might have the absorption spectrum which might be actually very like abrupt like no nothing nothing nothing and then one for a little bit then a half for a little bit then the absorption spectrum especially with the atmosphere is something that's gonna be very messy and if you but if you multiply the two then you get you know the emission spectrum what it emits when you heat it up and that's essential for climate science because the energy emitted by the atmosphere and the earth is what it's heated and
00:19:02
Speaker
Yeah, actually that's the main point of this so that now that we've talked about how you can know the emission spectrum or the heat or the energy that objects radiate if you know that based on their absorption spectrum and vice versa. If you know the absorption spectrum, then you know the emission spectrum. So now let's talk about two very specific bodies that relate to climate science and that is the Earth and the Sun. What do we know about the Earth and the Sun?
00:19:27
Speaker
Not much. I'm just kidding. Solar radiation peaks in the visible range, which is not surprising because we're designed to look at things that reflect solar energy, which is about 400 to 700 nanometers. And it's maximum in the green, which is 500 nanometers. And about half is at infrared frequency, which is less than 700 nanometers. And a very small fraction is in UV.
00:19:53
Speaker
So this is the composition of the sun's energy. It just tells us what we know is like about half is like in the infrared, which is very closely related to heat. And the reason why infrared is closely related to heat is because gases, it was we'll see in a second, absorb infrared radiation by either rotating or vibrating. Yeah, we'll talk about that in a second.
00:20:16
Speaker
Yeah, actually that's something I found very, very interesting is I had no idea, like, you know, we talk about emissions and absorption. I didn't know what happened at an individual molecule level. I guess there is, there's all kinds of interactions, you know, including vibrations where I think of like, you know, a spring, like if a molecule were to spring, it would squish and expand, right? Oh, yeah. Not just in one direction as a spring does, but in multiple directions. But then there's rotational energy where you've got something like spinning around, like I think of rolling down a hill, you know, and you get all kinds of heat there.
00:20:45
Speaker
Or somebody's playing tennis and they put a spin on it. Yeah, exactly. So there's all kinds of things related to heat at the individual molecular level. Fascinating stuff, I find actually. Now we know that the Earth is not sufficiently hot to emit significant amounts of radiation in the visible range. We know that because it's not like a lump of coal that glows red in a fire.
00:21:11
Speaker
Yeah, if you looked at, I mean, sometimes the earth glows red like at volcanoes, but that's pretty much it. Yeah. Yeah. That's not saying the earth. Yeah. No, not the whole. Yeah. So that's a very dramatic explanation of why it doesn't like how we could see that it doesn't emit significant amounts in. Yeah. Now by the laws of thermodynamics, everything radiates. So the earth certainly does radiate. It's just not, I mean, as long as it has any motion whatsoever, it radiates.
00:21:37
Speaker
Um, any, any heat whatsoever, I believe. And I think emotion might be related through some kind of, I think Hawking studied that, uh, that black bodies that are traveling through space have a slightly different temperature. I'm not sure. Yeah. Interesting stuff. Yeah. Learn more about that in our black hole episode, where we literally talk about the thermodynamics of black holes, fascinating stuff as well. This is not nearly as complicated, not nearly as complicated. No, not nearly. Yeah.
00:22:01
Speaker
Although the models themselves are more nitty-gritty as we will see, I will have to say that for them, but yeah, the math itself is very basic. Now, let's talk about the thermal equilibrium of the Earth and the surrounding space.
00:22:14
Speaker
Oh yeah, so basically if the earth is at a constant temperature, thermal equilibrium is when the amount of radiative energy going into the earth from the sun and stars and everything is equal to the amount of radiative energy, which is the almost microscopic amount of visible radiation as well as very large amounts of infrared radiation emitted by the earth are equal.
00:22:41
Speaker
The amount of energy that is reflected is very important. It's a property called albedo, and it's reflected back into space by clouds, snow, ice, whatever. And it's equal to 0.28 for the Earth, which means that 28% of the light that reaches Earth is radiated away.
Role of Gases in Climate Models
00:23:00
Speaker
In the IR range, though, it's almost 100%.
00:23:03
Speaker
Interesting. Now, this is something that we'll talk about more and more with, again, as you said, with Earth, the albedo, or rather the amount that is readied back into space is A equals 0.28. For other planets, it's completely different.
00:23:15
Speaker
Yeah. And what's interesting is that the mean temperature of Earth, because of the albedo of the Earth, and this is a very simple equation. This doesn't relate to temperature within the Earth, but it's 255 Kelvin, which is about negative 18 degrees Celsius, or I think negative 25 Fahrenheit or something like that. Okay. I don't know. I'm not doing the math in my head right now. Yeah, me neither. So what about something like Venus?
00:23:42
Speaker
Now Venus, which has a famously hot atmosphere that can melt, I believe, lead? Yeah, it can melt lead. So even though it can melt lead, it has an average temperature of negative 41 degrees Celsius, or 232 Kelvin. You might wonder, how can this be? And the reason why is because of all the clouds on Venus. They reflect about 75% of the light that reaches them.
00:24:06
Speaker
Wow. Interesting. So here's the thing though. So if it's reflecting that much light, if it reflects 75%, it's closer to the sun than the earth, then yet it's still hot enough to melt lead. Yeah. And the reason why is because of the difference between the ground temperature and the atmospheric temperature. And that's represented by a separate model, as we'll see in a second.
00:24:29
Speaker
Okay. And I think it's interesting because, I mean, like we said, it's like 700 degrees Celsius or like 500 or something like that on the surface of Venus. It's hot enough to melt lead. Wow. But its average temperature is negative 41 degrees Celsius as seen by an observer in space. Wow. Interesting. So higher albedo still would mean like if it were a lower albedo, it would be even hotter.
00:24:51
Speaker
Oh yeah. Well, not necessarily on the surface though, because lower albedo might mean that the composition of the clouds are different, and clouds are actually one of the biggest varying factors in climate models. So now that we've talked a little bit about radiative effects as well as the earth and the sun, it's time to talk about something else, namely gases. Yeah, and why are gases important? Because the atmosphere is made of gas.
00:25:21
Speaker
Exactly, exactly. So, a few things about gases. First of all, the internal energy in a gas is quantized in a series of electronic, vibrational, and rotational states, at least according to our current understanding. Increase in internal energy corresponded to a change in state. Now, this is where it gets... Right away, and that's a current... Gabriel mentioned current understanding, and I wanted to expand on that. That's a current understanding according to even the standard models.
00:25:45
Speaker
Yes, yeah, so that's pretty solid that yeah when I say that it's not like it's gonna change you know It could but according to the standard model. That's that is basically as rigorous as it currently gets that again This is this is again. You learn something new every day. This is something that I was unaware of
00:26:02
Speaker
Each of these three quantized mode of a gas corresponds to a different amount of energy. For example, in order to increase the electronic energy of the gas, am I saying that right?
00:26:16
Speaker
Oh yeah, the electronic internal energy in a gas. You would do that through applying UV or ultraviolet radiation to it. And this is actually interesting because one of the early experiments in quantum physics was charging gold foil using UV light and showing that it can't be done even with a ton of red light because of the... We go into this more in detail, I believe. Do we go into detail on that in the Black Hole episodes?
00:26:42
Speaker
I don't, uh, we should. If we didn't, we should have. Listen to the whole thing to just check. Just, yeah, listen to everything, everything we've done and come back to us. How's that sound? But finish the episode, no. Yeah, yeah, yeah, exactly. Yeah. Um, yeah, then, uh, for like, uh, vibrational changes in energy, you need near infrared energy, which is like very, very low red to like,
00:27:02
Speaker
Like 700 nanometers to like 20 micrometers. Correct, yep. And that's just again the, should we call it, lack of a better term, the springiness of the individual molecule? Yeah, the buzziness, springiness. How much they vibrate. Sure. And then lastly, you have the rotational energy. If you want to change the rotational energy of molecule, then you'll have to apply something greater than 20 micrometers. So that's far infrared. Yeah, far infrared. I believe that in some, I'd really tell microwaves work actually. Okay.
00:27:30
Speaker
Okay. I mean, you could like blast even radio waves, like lower frequency radio waves and then change the rotation of molecules. Yeah. Interesting. And then it says here that visible changes, there's nearly no changes in visible light of them? What's interesting is that the visible spectrum doesn't cause many changes in gas states. It doesn't change UV, I mean, it doesn't change electronic internal energy, nor does it change vibrational or rotational significantly.
00:28:00
Speaker
Couple of considerations here. So if the charge is distributed symmetrically, then it cannot acquire vibrational states since the vibrational states affect the dipole movement, which affects the structure of the molecule. Let me say that again. So you have different atoms and you've got different gases. And the valence electrons are arranged differently in each of them. There are some of them like the noble gases where you've got like eight
00:28:27
Speaker
valence electrons, right? And there's an issue here. If they're distributed evenly, then you're really not going to be able to add a whole lot of vibrational energy to it. Is that right?
00:28:36
Speaker
Yeah, well, let's take nitrogen for an example. Nitrogen is two nitrogen molecules connected together. It's charged and distributed perfectly evenly, like perfectly evenly, like exactly evenly. And because of that, basically it can't bend or anything like that. And since it can't bend, there's nothing to vibrate really.
00:28:58
Speaker
Imagine having a molecule in front of you and it's just made of little balls connected by very, very tight springs. The more you can like, if you did nitrogen, if you tried to toying nitrogen, there would only be one vibration and it would be throughout the whole thing. But if you did something like carbon dioxide, you could bend it back and forth and because you could bend it back and forth, it wobbles.
00:29:24
Speaker
Okay. And so just the kind of wobbliness is correspondent to vibrational energy. So basically, again, you're going to have different shapes based on the arrangement of electrons. And some of these shapes have more vibrational energy than others, than others. Yeah. Or more receptive to vibrational energy. Correct. Yes. And that, yeah, that includes carbon dioxide.
00:29:45
Speaker
Yeah, and molecules that can acquire a charge asymmetry by stretching or flexing. So actually, let me do carbon dioxide as like a counterexample. So carbon dioxide is two carbons on either side, two oxygens on the other side, one carbon in the middle. So it has like the negative charges on the outside and the positive in the middle. So it's perfectly symmetrical. So in that state, it can't acquire any vibrational energy. However, if it gets knocked out of that state, like for example, if the
00:30:14
Speaker
carbon gets closer to one of the oxygens just perchance or it bends in the middle, kind of like a joint, then it can acquire more vibrational energy in that state. And so those are greenhouse gases. So hydrocarbons, O3, N2O, H2O, carbon dioxide. Yeah, and although water is a greenhouse gas, it's a very, very short-lived one.
00:30:41
Speaker
as we'll see that affects how potent of a greenhouse gas something is. Yeah, as you said earlier, molecules that cannot acquire charge asymmetry by flexing or stretching, things like N2, O2, or H2 are not, therefore, greenhouse gases.
00:30:57
Speaker
And the efficiency of absorption in our atmosphere is actually about 100% in the UV due to the electronic transitions of oxygen and O3 in the stratosphere. And it's about 100% in infrared because of greenhouse gases. But between 8 and 13 micrometers, near the peak of terrestrial emission, there's like a weak window between 8 and 13 micrometers, but like a peak for O3 at 9 point. So basically, it's a complicated atmosphere is what we're saying.
00:31:26
Speaker
And you need models that reflect this.
Building Climate Models
00:31:28
Speaker
But we can go do some simple models too. Yes. So with all that said, this is a part that if you've made it this far on the podcast, let me just say congratulations. You've been listening well or just playing it in the background. For this next part, we're going to talk about a simple model.
00:31:47
Speaker
So a simple model, we start by pretending that the atmosphere is like a shell above the surface of the earth that can absorb energy. Yeah. I mean, it already is above the earth, but what we're talking about is like, like separate from the earth. Yeah. Like, like, uh, imagine like a ring. Yeah. Imagine like you got rid of the white part of the shell of an orange and all you're left with is that orange on the outside and the fruit on the inside. That's kind of the model we have. Sure. Okay.
00:32:14
Speaker
Cool. So yeah, in this simple model, it implies something or it implores something that we call Kirchhoff's law. As an electrical engineer, we use Kirchhoff's law, but it's used in many, many things about the flow of energy. Kirchhoff's law basically states that the electricity that goes, the current that goes into a node is going to be equal to the current that leaves that node. They have to be the same. It's sort of a conservation law, if you will.
00:32:42
Speaker
Yeah, and it applies here because we're saying that the shell that the atmosphere is pumps energy outwards and inwards, outwards towards space and inwards towards the Earth. Correct, correct. And they have to be, yeah, what goes out has to be what comes in and vice versa. And we also have a sun that is generating, that is giving a constant amount of solar radiation.
00:33:03
Speaker
So in this simple model, if we were to calculate the incoming solar radiation, we would simply have to add the energy that's radiated, the energy that's radiated from the atmosphere into space, that's the outer layer, plus the energy that's radiated from the Earth into the atmosphere. Again, in the simple model, the atmosphere is above. So it's basically, it's Perkov's law. What comes in from the Earth is equal to what is absorbed outward.
00:33:30
Speaker
And we know the amount that is radiated from Earth into the atmosphere and from atmosphere into space, because as we mentioned before, the amount that's absorbed is, the fraction absorbed is the fraction radiated. And we use Kirchhoff's law also to show that the temperature of the, the amount of energy absorbed by the atmosphere from the Earth's surface temperature is equal to the total amount of energy being irradiated away from the atmosphere, both into space and into Earth.
00:33:56
Speaker
And we'll have more details on that in the paper to come. But basically we get this formula that the temperature of the Earth is equal to the mean solar flux times one minus the albedo, all divided by four times sigma, which is a constant that relates the fourth power of temperature to a black body's energy. So we're assuming that the atmosphere is a black body. This is called the gray atmosphere model.
00:34:26
Speaker
So it's over four sigma times one minus F, which is the amount of the fraction absorbed by the clouds divided by two, all to the one quarter. So as you can see, if we change the fraction that is absorbed, and we might have an app where you can play around with this actually, it'll change the temperature pretty significantly, which explains why Venus, even though it's colder than the Earth, is hotter at the surface.
00:34:51
Speaker
That'd be a very cool app, actually. Yeah, and so we showed that the global surface temperature is about 288 Kelvin. Which is what, Fahrenheit? Like, I think it's like 40 or 50 degrees. Okay, yeah, we can do a quick... Yeah, it's like 58 degrees. Cool, very good. Right on.
00:35:07
Speaker
Let's talk about, obviously, there were some simplifications made in that model. We have improved models as well. Yeah. For example, when we were calculating the previous thing, the amount of energy absorbed by the atmosphere of the Earth's surface temperature and stuff like that, we could have looked at how it affects every single different spectrum differently and then sum it up together. That's something that we did not do. We just assumed the Earth was black and then the atmosphere was black.
00:35:34
Speaker
Yeah, yeah, basically we're saying is that the model can be approved by not considering the earth and atmosphere as black bodies and still improve further with something called GCMs or general circulation models. Basically those provide 3D equations for energy, mass and momentum. Man, that gets real math-y. For those of you guys who like spaghetti of math equations, this is right up your alley. You'll love this.
00:35:59
Speaker
Oh, yeah. But what's interesting, though, is that this model that we're talking about here, you can see in the link that we linked to earlier, that it actually predicts a slightly more global warming than we've had in the last century that
Impact of Greenhouse Gases
00:36:14
Speaker
actually happened. But it's still pretty fairly accurate, even this small, almost insignificant model.
00:36:19
Speaker
Yeah, yeah. So the simple models are still very, very good. And they only get more and more specific. It's like we're just getting more refined models. Yeah. Yeah. And like with the GCMs, the general circulation models. Yes. You know, I have it on good authority that some of our listeners might be like us, fairly, fairly nerdy, and they like the nitty gritty details. Should we tell them some more of the points that are in the more advanced models?
00:36:42
Speaker
Yeah, well basically it's one of the more nitty-gritty points that it has in that thing is optical depth, which is how much flux has absorbed and scattered over time. You could take the different layers of the Earth's atmosphere, you know, the stratosphere, ionosphere, troposphere, and I think thermosphere. You could take all of those into consideration. Like, there's so much stuff you could do. Like, the world is your oyster when it comes to this stuff. Yeah, yeah, yeah, yeah.
00:37:07
Speaker
But what's interesting too is that although there are disagreements in the predictive surface warmings resulted from a given increase in greenhouse gases, all GCMs tend to show a linear relationship between the initial radiative forcing and the initial ultimate perturbation at the surface temperatures. Let's break that down. What is radiative forcing? That is just the radiative perturbation associated with an increase in a greenhouse gas.
00:37:30
Speaker
So it's saying if we add one pound of carbon dioxide to the atmosphere, how much radiative energy increase are we going to see at the surface? And we relate things to carbon dioxide because it's kind of a baseline. And also, I think it's most prevalent greenhouse gas. But I mean, methane, over a period of 20 years, is 62 times as potent as carbon dioxide as a greenhouse gas. Over 500 years, though, because it has sort of lifelines, only eight times as potent.
00:37:59
Speaker
But if we want to look at a really potent greenhouse gas, check out sulfur hexafluoride. Oh, wow. What do you have for that one? Let's see. Lifetime of 3.2 millennia. In over 20 years, it's 16,500 times as bad as carbon dioxide. 24,900 in 100 years. In over 500 years, 36,500 times as potent as carbon dioxide. Wow, that is insane.
00:38:22
Speaker
So basically what we see then is over a hundred year time horizon reducing use of sodium hexafluoride emissions by one kilogram is as effective from a greenhouse perspective as reducing carbon dioxide emissions by 24,900 kilograms. That's important for designing regulatory goals.
00:38:40
Speaker
And another important, I mean, also we have to, we can't just, if there's like only point, if there's only one milligram of sodium hexafluoride released into the atmosphere per year versus a billion pounds of carbon dioxide, we still have to pay more attention to carbon dioxide.
Conclusion: Human Responsibility for Climate Change
00:38:53
Speaker
So that's another, that's not the actual stats as you might've guessed.
00:38:56
Speaker
but it's something to take into consideration. And that's how it can help us design better factories. And once we have regulatory goal points when it comes to greenhouse gases, we can even design factories in terms of radiative forcing, taxum in terms of how much radiative forcing they generate.
00:39:17
Speaker
Like there's so much stuff we can do with this data. And I think that's a big takeaway from this. Oh, and these are the models which tell us that we are generating enough carbon dioxide to create a significant effect. As we've seen, global warming is happening and we are mostly to blame. Actually, I believe that if I'm not mistaken, humanity is entirely responsible for the current warming as we should actually be in a cooling trend.
00:39:44
Speaker
Oh, I did not know that. We have learned about how radiative forcing relates to climate science, how gases interact with light, and what we might be able to do about it. I'm Sophia. And I'm Gabriel. And this has been Breaking Math. Just a reminder, we are on patreon.com slash Breaking Math, and facebook.com slash Breaking Math podcasts, where you can see new releases, things like that. Facebook, BreakingMathPodcast.com. What else?
00:40:10
Speaker
So this has been a very exciting series for me. We have interacted with some real, real top level researchers. I'm not sure I'm at liberty to say who we've been chatting with. We'll say that just in case it doesn't pan out. Absolutely. Yeah. This topic is a huge topic the world over with the Australian wildfires that have been exacerbated by climate change to California fires.
00:40:33
Speaker
foreign fires, flooding in low-lying areas, all kinds of things. I mean, even Syria can be attributed in a large part to the rapidity of climate change. Correct. If it changes the availability of natural resources, then it'll have an impact on things like
00:40:52
Speaker
economics because I mean if you want to see how if you want to watch something I would check out by CGP Grey on YouTube it's the rules for rulers and it talks about her resource distribution can lead to either democracy or tyranny correct correct absolutely
00:41:10
Speaker
So yeah, and on our Twitter recently in the last six months, I have followed probably two dozen climate scientists, including people like Dr. Catherine Hayhoe, lead author of the US climate assessment from 2018, also Dr. Michael Mann, as well as a slew of others as well. So send us your climate change related questions and questions for any of these doctors. And I mean, you know, we'll see if we can find answers to them.
00:41:40
Speaker
I'm Sophia. And I'm Gabriel. And you're listening to Blakey Ma- I'm faking math.