nm0885: going on here er this gentleman is from [0.7] i think the Linguistics department om0886: [0.2] nm0885: right and he's he's doing study of th-, of the language that academics use and things like that [0.5] er so he's he's taping a random sampling of lecturers around the university [0.6] so if you're wondering why i'm replacing all technical terms with the names of frudg-, fruits and vegetables this time it's basically i'm having a go at him [0.7] so [laugh] [0.7] er [0.5] [0.7] well [2.0] what we are [0.4] at [0.6] is [0.2] that [0.2] formula [0.5] and we spent a lot of time yesterday [0.2] hacking our way up to this thing hacking that's a good word [0.5] er [laugh] [0.7] and [0.8] and we got it [0.3] and we're really happy about it [0.4] er [0.2] what is it [1.1] [laugh] [6.5] what is D-of-L-nu [1.5] sf0887: decrease of the radiance [1.3] nm0885: right [0.7] and some r-, if some radiation is passing through a slab of atmosphere a thin layer of atmosphere [0.4] er some of it's going to get lost [0.4] and the amount that gets lost is the change in radiance [0.2] D-L-nu [1. 0] okay [0.6] er [0.4] what happens to the radiance that gets lost [0.2] where does it go [4.6] there are a couple of different things that could happen to it [2.4] sf0888: could get scattered or absorbed [0.5] nm0885: right it can get scattered [0.4] or absorbed [2.3] okay [0.3] and so now the amount we worked out depends on a couple of different things [0.3] okay some of them are pretty obvious [0.3] y-, er you lose more if you have more going in [0.4] so it depends on the amount of radiance that's incident on our [0.2] little slab of atmosphere [0.7] it depends on [0.9] something called K [0. 6] what [0.3] does K tell us about [8.6] sf0889: [1.1] nm0885: er [1.0] scattering is one of the things that go that g-, that goes into it [2.0] sf0890: er is it the extinction coefficient [0.3] nm0885: yeah it it's it's the extinction coefficient which is the sum of the scattering coefficient plus the absorption coefficient [1.0] okay [0.2] and so what it's telling us about is it's telling us about the characteristics of the material [0.3] is this material an effective scatterer or an effective absorber [0.3] and we'll be getting into later on in the course what it is actually that makes the material a good scatterer or absorber [0.5] and then it depends on [0. 4] rho-secant-theta-D-Z what's that [0.6] that's the easy one [3.8] sf0891: pathlength [0.7] nm0885: pardon [0.2] sf0891: pathlength [0.4] nm0885: yep this one's the pathlength why do we have a rho in there [1.4] sf0892: [0.4] nm0885: yeah it's the density of the material that's doing the interaction [0. 5] so [0.2] if your total this thing up pathlength times the density that's the amount of material [0.3] per unit area [1.7] [0.8] okay [0.4] so that's for [0. 3] a very thin slab [0.3] and that means and because it's a very thin slab we can say well things like density [0.4] the amount of material [0.3] extinction coefficents all all of these things don't vary much as you go across the slab [0.4] i mean in principle [0.3] they can vary [0.3] so [2.1] if we have [0.8] a [0.2] finite [3.0] slab of material [2.8] if you have a finite slab of material [0.4] so [0.2] instead of just [0.3] a thin layer [1.2] D-Z here [0.3] we've actually got [0.3] a thick layer [0.4] which [0.3] ranges between [0.9] er [0. 4] you know [0.5] a finite slab of material [1.4] from Z-one [1.0] to Z-two [0. 8] so we've got [0.7] two heights involved and so if we say Z-one is the top where it's entering and Z-two is the bottom [0.5] where it's going out [1.1] er [0.2] [0.2] we can integrate [0.3] this expression [0.2] to add up [0.7] the changes in the er add up the amount of depletion in all the different layers [0.4] and give us the amount coming out the bottom [0. 3] so if we've got an initial radiance L-nought-sub-nu [1.3] in our direction here [0.7] passes through the slab [0.7] then [2.2] what comes out [0.5] is [1. 3] at the bottom is something less [0.4] and i'll call that L with a superscript- [0.4] D [0.5] so that's er the depleted beam so it's lost some stuff [0.7] and that's L-nu and it's [0.2] of course [0.4] coming out in the same direction [0.2] theta-phi [1.8] okay [0.7] so [0.4] if we have a finite slab of material from Z-one to Z-two [0.5] we [0.6] can [0.8] integrate [2.9] this [1.1] expression [2.2] er [3.8] to give [1.1] D [1.2] direct [2.7] compo-, [2.4] of the [0.2] emerging [1.9] beam [2.3] 'cause remember the emerging beam is also going to have contributions from emission [0.3] and contributions from scattered stuff which we're going to have to work on a little bit later [0.6] but er [0.4] but the direct part of it [1.3] L-D [0.5] and actually yeah i m-, i made a mistake that D stands for direct [4.2] we can get that by doing the integral [0.6] so [0.3] what we do [0.3] is we [0.5] take this expression [0.6] and er [0.7] you notice we've got [0.3] L-nu on both sides so we'll take this one over to the other side [0.4] so we'll write this as D-L-nu over L-nu [0.6] and that's [0.4] excuse me going to be equal to [0.4] minus- [1.5] K- [0.2] nu-E- [1.0] rho-secant- [0.2] theta-D-Z [0.9] okay now we integrate this [0.4] across [1.7] the layer [2.7] er [1.0] so [0.2] integral signs on both sides we're now going to sum up those layers [0.7] so [0.3] the integral on the right hand side we're integrating [0.3] with respect to Z [0.3] so we're integrating in height what are the limits of integration [2.2] what's the bottom one [0.5] sf0894: [0.9] nm0885: pardon [0.6] sf0894: [0.2] nm0885: Z-one [0.2] that's right [0.8] it's where the stuff's going in [0.3] and it goes up to Z-two where the stuff's coming out [0.4] now on the left hand side [0.3] we're integrating over radiance [0.4] D-L-nu [0.4] so what's the starting radiance where we start the integration [0.9] sm0895: L-nought-sub-nu [0.2] nm0885: yeah [0.2] L-nought- [0.5] sub-nu [0.4] and finishing up at L-D- [0.2] sub-nu [0.7] so [0.9] that's our integral [0.5] it's now a matter of doing it [0.7] now [0.9] left hand side [0.6] is [0.5] pretty simple what's the integral of D-L-over-L [1.5] sf0896: [0.4] nm0885: right log-L [1.0] so [0.3] on this side we get [0.8] log [0.3] of L-nu [1.7] and you're [0.2] ranging from L-nought to L-D- [0.4] sub-nu [0.3] and on this side [1.0] er what do we get there [6.6] this is a trick question so have a guess at it and y-, and y-, and you'll be wrong but but you won't be have to be embarrassed because it's [0.3] it's a trick question [4.0] sf0896: Z [1.7] nm0885: pardon sf0896: er [0.3] Z times all that [1.6] nm0885: Z times [0.2] that stuff okay [0.2] and that [0.2] is indeed the wrong answer [0.4] and [laugh] [0.3] the r-, r-, [0.7] but it's exactly what i was hoping you'd say [0.4] because that would be true [0.3] if all this stuff was constant [0.4] then you could just pull it out of the integral sign [0.5] but [0.4] these properties may well the secant-theta shouldn't vary [0.8] but the other stuff [0.3] might potentially vary as you go through the layer and certainly you know if you imagine a beam of sunlight going down through the atmosphere [0.4] er the density of just about everything [0.2] is going to be increasing as as you go down to the surface [0.5] and probably the only significant thing where that's not really true [0.4] is going to be something like ozone where it's really varying wildly [0.4] or clouds or something [0.3] so we can't do much [0.2] with that side [0.3] so what i'm going to do is i'm just going to leave that side as [0.2] an integral [2.0] from Z-one to Z-two K-nu-E-rho-secant- [0.5] theta- [1.1] D-Z [1.0] and er [1.6] i'm going [0.6] to give it [0.3] a name [2.9] and the name is going to be [0.6] optical thickness or optical depth [0.3] so i'm going to write that as minus- [0.7] delta lower case delta [0.5] and the er and we'd better put a subscript-nu on that [0.3] because this is going to be a function of frequency [3.3] so [1.2] where [0.8] delta-nu [0.2] is equal [0.5] by definition to this integral Z-one to Z-two [0.7] extinction coefficient [0.2] times density times [0.8] pathlength [0.5] secant-theta-D-Z [1.1] is the [1.2] optical [2.1] thickness [1.6] or [1.5] optical depth [3.0] these two terms mean the same thing and [0.4] people will tend to use them in-, interchangeably [1.5] [0.6] so what that is is that's a m-, [0.4] is that's going to be some measure [0.4] of how much [0.2] stuff is getting pulled out of the beamer or what or what fraction of it [1.0] and if we define the optical depth this way [1.5] then [0. 3] we can say that [0.3] our [1.7] direct beam [0.3] L-D-of-nu [0.4] is going to be equal to [1.3] the initial beam times [0.5] E-to-the-minus- [0.3] delta-nu [3.5] just rearranging that equation [3.6] and this [0.2] is a very important relationship i mean in some sense in in the theory of radiative transfer this was the first [0.4] really quantitative law [0.3] that was discovered [0.7] and er [1.4] and so it's named after its [0.2] discoverer [0.9] not what he was drinking at the time [0.4] Beer's law [1.4] er you will also find [0.3] this is sometimes called [1.3] Lambert's law [2.6] and it is also sometimes called [3. 0] Bouguer's law [2.6] and sometimes it's called the Beer-Lambert law and sometimes it's called the Lambert-Bouguer law and sometimes it's called the Beer-Lambert-Bouguer law [0.3] or whatever [0.3] er [0.3] i think it [0.2] probably has to do with which nation you are and which nationality this person [0.8] these people are [0.3] er [1.5] 'cause this 'cause this relationship was probably discovered independently several times [0.5] we're going to call it Beer's law [0.5] in this course just for simplicity [1.3] okay [0.2] and so what it says is that the amount of radiation that's coming through directly [0.4] is er [0.4] decreasing exponentially [0.7] in [0.5] the optical [0.2] thickness [0.3] so the fraction of energy [5.0] the fraction of [1.2] well radiance [3.5] that [0.8] emerges [1. 3] is [1.8] given [0.4] by [0.6] the [1.6] monochromatic [1.6] transmittance [4. 2] and it's monochromatic because we're always doing this as a function of wave number [0.2] or or sorry as a function of [0.5] er frequency but we can always add up over frequency [0.4] and that is defined as [0.2] tau [1.5] lower case Greek letter tau [0.6] and that by definition is going to be the amount that's coming out the bottom L-D [0.3] divided by the amount that's going in the top L- nu [0.6] and you can see immediately [0.2] that that's directly related to the optical depth [0.9] E-to-the-minus-delta-nu [3.4] so there's a couple of different ways that we can think about how much is going through we can think about a transmittance [0.4] wh-, so you may have a transmittance of point-five [0.3] which says half the stuff gets through [0.6] er you may have an optical depth here [0.4] and the [0.2] thing about the optical depth is that's going to be directly proportional [0.4] to things like pathlength and density and so on so if you double the density you double the optical depth [2.5] and er [1.6] yeah so that that's that's the amount [0.8] that gets through [0.6] okay so that's transmittance but of course remember [0.3] there are two other things [0. 3] two other ways that energy is coming out the bottom of the slab [0.5] what were they again [3.3] sf0897: scattering [0.5] nm0885: scattering [0.9] and [6.9] sf0898: emission [0.4] nm0885: emission thank you [0.5] it's one of these cases where [0.3] everyone's whispering the answer but no one's saying it loud enough [0.4] all right [0.2] so [0.4] so the slab can be emitting radiation [0.6] and also radiation that's entering the slab from other directions can be getting bounced into this direction [0.4] so let's let's write down some expressions for those ones as well [4.3] and we'll start by taking emission [4.0] so [2.4] energy [2.1] emitted [0.6] by [1.0] the slab [4.6] now once again [0.4] this is going to depend on [0.4] a couple of different factors so [0.7] this [1.3] depends [0.3] on [1.0] what's going to determine how much energy the slab is emitting [0.7] at some given wavelength [4.7] temperature [0.4] is going to be [0.4] a big one so this depends on [2.2] temperature [0.7] and what's the relationship that tells us how much emission we're going to get [0.2] at a given temperature [4.8] sm0899: Wein's law [0.6] nm0885: er Wein's law's going to tell us the wavelength where we get the most emission but what's going to give us the whole function at a as a function [0. 3] of any wavelength [0.6] sm0900: Stefan [0.4] sf0901: [0.3] nm0885: er [2.0] the Stefan-Boltzmann one is isn't the one i was looking for because that's giving you the total [0.8] i'm saying if you just have some arbitrary wavelength that's not at the peak what's the function that's going to tell you how much you get at that wavelength [8.2] we've done it in this course we've written it out on the board [0.8] i've shown plots of it on the projector [0.8] sf0902: Planck's law [0.3] nm0885: Planck's law that's right [1.2] so the Planck function [1.5] so [0.6] the energy emitted depends on the temperature [0.4] and that's given by the er Planck function [2.7] and er [3.0] well [0.8] what else is it going to depend on [1.4] so [0.2] if [0.3] if it's at a given temperature [0.7] what else is going to affect the amount of emission that we get [0.7] sf0903: [0.8] nm0885: sorry [0.5] sf0903: is it the amount of material [0.4] nm0885: the amount of material absolutely [4.7] the amount [0.2] of material [4. 2] does it depend on anything else [7.8] sf0904: density of material [1.0] nm0885: er [0.2] that's another w-, i would i'd say it's another way of saying amount [0.3] okay [1.6] sm0905: the heat capacity of the material [1.1] nm0885: er [5.4] probably i'm [0.4] i w-, i wouldn't write it in terms of heat capacity but it does depend on the type of material [0.4] different materials emit [0.4] with with er [1.0] wi-, with with different [0.4] efficiency [1.3] so it's it's also going to [0.5] er depend on the er [0.5] type [0.8] of [1.5] material [0.3] so some materials will be better emitters [0.2] than others [0. 3] and especially better emitters at given wavelengths and this'll depend on the molecular structure and things like that which w-, [0.3] which we'll see [0. 5] later on [0.8] er [3.6] something that you [2.3] well [0.9] i'll come back to that in a second [0.2] let's write it down [0.2] okay [0.3] so let's write the emitted radiance [0.6] and so i'm going to denote that as L with a superscript-capital-E [0.4] for the emitted part of the radiance subscript-nu 'cause we're at one given frequency [0.5] again [0.7] and oops [1.4] and let me not forget to write down the direction [0.2] theta-phi [0.7] so this is going to depend on [0.7] these various properties [0.4] so it's going to depend on the [0.5] Planck function [1.9] which is a function of temperature [0.4] now you'll notice i've written a B here [0.4] whereas before i wrote an E [0.5] when i wrote down the expression before [0.3] i wrote down [0.2] the formula for the [0.3] irradiance [0.2] emitted [0.5] so the total of what's coming out in all directions [0.4] but in this case we're actually thinking about just one direction from the slab [0.3] so we need the radiance [0.5] but it's easy to go between radiance and irradiance for a black body why why is that [2.7] sf0906: isotropically [0.8] nm0885: yeah that's right a black body emits isotropically so it emits the same in any direction [0.5] and that means that if we know the total [0.4] then [0. 4] we know what the amount going in any given direction is [0.2] and remember [0.6] and so [1.4] let's actually write this down where [0.8] B-nu-of-T [0.2] is [0.6] the [1.6] Planck [1.4] function [3.3] in [0.3] radiance units [4.3] you use the Planck Planck function in radiance units [0.4] and that means that er [1.4] if you'll recall and i don't necessarily expect you to remember this just at this moment [0.3] but you might want to remember it for the exam [0.6] is that er [0.2] the difference between [0.4] the radiance and the irradiance for an isotropic emitter is just a factor of pi [0.7] so [0.4] in one direction it's going to be E-nu- [0.2] of-T [0.3] divided by [0.3] pi [3.5] so [1.3] we've got the er [1.6] the temperature dependence in there because now we've got the Planck function [1.3] for the radiance [1.3] okay the next thing we want to do [0.6] is we want to put in something about [0.4] the type of material [1.2] and we'll write this in terms of the [0.8] mass absorption [0.4] coefficient [1.1] so K-A-of-nu is the mass absorption coefficient which we've already def-, [0.5] which we've already [0. 3] mentioned [0.4] why am i writing the [0.2] mass absorption coefficient [0.3] when i'm talking about emission [0.3] here [2.1] this is a question for somebody who's done the right physics course this is not something you're going to be able to figure out on the spot [2.8] unless your name is Kirchoff [0.8] and you're really clever [1.8] or you might be really clever but if your name isn't Kirchoff they still won't name the law after you [0.7] [laugh] [6.4] sf0907: is it because good emitters are good abs-, good absorbers [0.6] nm0885: that's [0.2] exactly right [0.7] and that that's Kirchoff's law [0.3] and what Kirchoff's law says and this is what what you have to wor-, work your way through in a physics course [0.5] is [0.2] Kirchoff's law [0.2] says [0.3] that if you're in thermodynamic [0.2] if you're in a local thermodynamic equilibrium [0.6] then [0.7] a property must be equally good at absorbing and emitting [0.5] and the reason for that is if you're in thermodynamic equilibrium [0.5] you're er you've got to be at the same temperature [0.7] in in different parts that are that are close together [0.9] and if you're at [0. 5] and if you're at the same temperature [0.4] but [0.5] one part of th-, th-, b-, but your substance is a good emitter [0.7] er but a very bad [0.4] absorber [0.2] or something like that then what that means [0.2] is that you'll end up with a flux of energy going from one part to the other [0.5] and one part of your substance will try and heat up relative to the other [0.5] and that's a contradiction with the thermodynamic equilibrium [0.6] so [0.2] if you're going to have an equilibrium and have nearby stuff at the same temperature [0.2] you've got to be equally good at absorbing and emitting [0.4] so that there's no net flow of energy from one place to another by radiation [0.8] so [0.8] anyway i mean th-, that that kind of detail doesn't matter [0.4] but [1.0] what matters for our purposes is [0.3] is that for any part of the atmosphere that we're interested in and this means not the [0.6] the far outer thermosphere or whatever [0.4] er [0.4] we're in thermodynamic equilibrium at least locally [0.5] and [0.6] the [0.3] absorption coefficient and the emission coefficient [0.2] are going to be [0.4] the same [0.2] thing [1.3] so er [2.7] K-A-of-nu [1.2] er [2.5] well gives [1. 2] the [1.3] emission [1.1] properties [4.1] of the material [1.8] for [1.9] substances [2.1] in [2.5] thermodynamic [1.5] equilibrium [4.1] and i'll just put in brackets here that this is er Kirchoff's law [8.6] so [0.2] it's the same for absorption [0.6] so K is the same for absorption and emission [0.9] okay [0.2] final bit we need is the amount of material how are we going to write the amount of material [1.3] this is an easy one 'cause we've done it before [6.1] sf0908: density [0.9] nm0885: sorry [0.5] sf0908: [0.4] nm0885: density times [4.3] sm0909: volume [2.3] nm0885: volume well [0.2] we're going to do it per unit area so density times pathlength [0.8] and that gives you the am-, the amou-, the amount of mass per unit area [0.4] and so that's going to be [0.2] rho-secant-theta- [0.2] D-Z [0. 4] we've already worked that one out [1.2] okay [0.3] so that's the emission [1. 0] the emitted part of it [0.3] and as long as we know the temperature [0.5] we know the properties of the material that we're dealing with and and i've sort of glossed over how this one comes about so we're going to have to go back to this later in the course [0.5] and if you know the pathlength [0.3] then you can work out how much radiance is going to be emitted [0.4] from a given [0.7] slab of atmosphere [0.3] and going off in a given direction [0.5] but of course [0.5] the emission's isotropic so here it's going to be the same in all the different directions anyway [1.1] okay [0.3] final point [0.3] that we need to get in this [0.5] is we need the scattered bit [1.1] so [0.3] our next [0.4] thing to talk about here [0.4] is [0.4] radiation [0.7] er [2.7] scattered [1.3] in [0.7] from [1.0] other [0.3] directions [10.6] so if we're sitting below our slab [1.5] here [0.4] and we're observing [0.5] a scattered radiance L-S-of-nu which is coming out in that direction [0.6] er that radiance could be getting into that direction [0.5] from [0.4] anywhere else [0.3] and it could be coming from above the slab it could be coming from below the slab [0.4] or whatever [1.0] so there could be a contribution [0.3] to this scattered irradiance [0.3] coming from [0.6] any direction so [1.8] by the way is this pen fading out too much can you still read it [1.2] okay give m-, give me a sh-, give me a shout if it's if it's getting bad [0.6] so [0.9] radiance [2.4] can be [0.8] scattered [1.9] can be scattered [1.9] into [2.2] direction [1.7] theta-phi [2. 1] from [0.9] any [1.1] other [0.4] direction [2.6] and so we'll write an arbitrary other [0.5] direction [0.5] as [0.3] a er [1.6] as a [0.8] theta- primed- [0.4] phi-primed [0.8] so from any direction theta-prime-phi-prime [0. 5] the energy could potentially [0.4] be scattered in into the direction that we're interested in [15.8] so [0.5] what's the scattered energy [0.3] going to depend on [21.6] sf0910: single scattering albedo [1.1] nm0885: right it's going to depend on the single scattering albedo [0.6] can you remember what that defines what does the single scattering albedo measure [4.5] sf0911: er [1.3] nm0885: er [0.7] sounds right [0.3] yeah it's it's going to measure [0.4] if the radiation interacts with the material [0.3] what fraction of it is going to be scattered [0.4] rather than absorbed so big albedo [0.2] omega-equals-one [0. 8] everything that interacts at all is going to be scattered [1.1] small albedo [0.2] omega-equals-zero [0.4] anything that interacts at all is going to be absorbed [0.8] so [4.5] so the er [2.7] scattered [1.2] radiation [1.6] depends [0.9] on [0.9] a single scattering albedo [0.2] omega-nu [1.0] now of course it de-, if it depends on the sing-, single scattering albedo [0.4] that's what happens to it [0.2] if it interacts in the first place [0.5] but er [0.9] what's going to determine [0.4] the probability that it interacts in the first place [1.2] what are we going to need to worry about there [1.8] sf0912: pathlength [0.5] nm0885: yeah [0.2] it's going to depend on [0.5] on the pathlengths or it's going to depend on the [0.4] amount of material [4.3] which of course you realize is going to be given by the density times the pathlength [0.9] er what else is it going to depend on [12.0] sf0913: size of [1.0] particle [1.5] nm0885: er [1.8] it is [0.4] actually [0.9] but i wasn't going to tell you that until next week [0.7] er [0.2] [laugh] [0.2] let's say for the moment it depends on again the properties of the material [0.5] and you're right it's going to turn out that size of the particles or size of the molecules is the p-, is the property that matters most [0.7] but it's going to depend on the [0.3] er [2.3] type [0.3] of [0.9] material or properties of the material [0.4] and so that's going to be given by [0.2] the er [4.8] well we we can write it [0.4] in terms of [0.8] the extinction coefficient [0.2] K-E-of-nu which is the probability [0.3] that it's going to interact with the radiation [1.6] or equivalently [0.3] we can write it as the [0.5] mass scattering coefficient K-S- nu [0.7] and the reason for that is because if we know the single scattering albedo that's the ratio between the two of these [0.4] so if [0.3] all we if we have any two of these three we can work out what the third one is [1.4] so it's going to depend on the type of material it's going to depend on [0.3] how much material [0.6] er [1.0] there's another pretty obvious one here [2.2] suppose that er [1.4] the sun goes very high in the sky and you now have more radiation [0.2] incident [1.4] what's going to happen to the scattering then [1.9] sm0914: [2.4] nm0885: yeah it's going to increase if you have more incident radiation you're going to get more scattered [0.3] into any [0.3] direction so it's going to depend on the [0.7] incident [0.4] radiance [3.3] and so that's going to be [1. 4] L-nu [1.4] in what direction [3.2] is the radiation coming in [0.2] that we want to worry about [1.6] someone's doing this which i think means any direction [0.6] yeah [0.3] so any direction so if we just pick an arbitrary direction [0.4] we can say [0.5] in a direction [0.2] theta-prime-phi-prime [0. 6] but [0.5] this brings us on to the last factor [0.3] which is really crucial here [0.6] is er [1.4] the contributions from different directions [0.3] theta- prime-phi-prime [0.4] because if radiation gets scattered off some particle [0. 2] a molecule or a [0.4] wa-, water drop an ice crystal whatever [0.3] it's not going to [0.4] you're not going to get the same amount going off in every direction the scattering is not [0.3] isotropic [0.3] the radiation will [0.2] interact with the particle [0.3] and it'll be sent off in different amounts in different directions [0.3] so the other thing [0.3] that we're going to have to know here [0.3] is we're going to have to know [0.2] that property of of of of the material we're going to have to know [0.4] which direction [0.4] the scattered radiation [0.2] is going to go in [0.2] in any [0.5] given interaction so [0.3] and [0.5] the [2. 1] direction [0.7] directional [0.6] dependence [4.6] of the scattering [4.0] and so we're going to need to define [0.4] an expression [0.4] for this [2.3] so [0.6] let's [0.6] define [2.8] the phase function [5.5] so the phase function [0.8] which we'll write with a er [0.7] capital-P [0.4] from direction [0.2] omega-prime [0.7] into [0.6] direction [1.0] omega [0.5] and i've used solid angle here so omega-prime would be theta-prime-phi-prime [0.4] and omega would be theta-phi [1.3] so what this is going to be [0.4] is this is going to be [1.6] the [1.1] probability [2.6] that [2.1] a photon [1.9] incident [7.2] in direction [1.6] omega-prime [2.3] is scattered [5.6] into [1.1] the direction [1.2] omega [1.5] so [0.9] some radiance [0.3] comes from or or is initially travelling in in direction [1.1] omega-prime [0.6] then [1.1] in this certain pathlength and given this certain nature of material it turns out that it [0.8] interacts with one of the particles there [0.3] now it could in principle get sent off in any direction [0.5] but the proportion that gets sent off [0.3] in the direction that we're interested in [0.4] omega [0.7] omega [0.4] is given by the phase function [1.7] so the phase function says whichever angle you come from [0.3] what are the odds that you're going to get scattered into the direction that we want [2.5] and er again [0.3] when we get on to talking about scattering in more detail [0.3] we're going to have to think about what this phase function looks like so what direction does [0.4] the radiance get scattered [3.3] okay so er [2.2] what this means then now [0.3] is that if we want to get [0.4] the radiance [2.3] which is scattered into our direction omega [0.4] what we're going to need to do [0.3] is we're going to need to integrate [0.3] over this phase function 'cause we're going to need to add up the contributions [0.3] from all the different possible directions [0.7] that get bounced into our direction [1.0] and [0.2] i'm fed up with this pen so i'm going to [1.5] throw it away [4.6] lovely [0.7] okay [1.5] so er [0.9] so [2.6] we must integrate [3.3] over [0.9] all [0.5] incident [1.4] directions [3. 2] omega-prime [1.2] to [2.3] to get [0.6] the [1.3] total [2.1] emerging [2.3] radiance [3.1] in direction [1.5] omega [4.0] so [0.5] what does this mean well [0.5] what this means is if we want to write [0.6] our [0.4] scattered radiance [0.4] so L-superscript-capital-S- [0.5] sub-nu [0.8] and er [1.1] forgot here let's [1.0] in the direction [0.2] omega [1.1] we can now [1.8] write this [0. 7] as er [3.6] what are we going to have well we've got to [0.2] get in [0.4] all our [0.8] little bits of stuff here [0.4] it's going to be in terms of the er [3.0] of the [1.7] well let's ca-, let's write it as the mass scattering coefficient [0.4] so i could also write this as [0.2] omega-nu times the extinction coefficient [0.5] let's write it as as the scattering coefficient [0.7] then [0.5] an integral [1. 0] over all possible directions [0.6] of er [6.0] of the [0.6] incident radiance [0.2] in that direction [0.9] so [0.4] L-nu- [1.2] omega-prime [1.5] and er [2.9] the [1.5] multiply by [0.3] the phase function [1.0] which is the [0.3] probability [0.6] that er [0.5] you're going to go from [0.3] omega-prime [0.4] into direction [0.2] omega [5.9] and let me [0.3] clear a little bit more space here [2.5] then we're going to need to write [0.6] the er [0.4] pathlength [0.7] which is part of the probability that you're going to get scattered [0.4] at all [0.3] so we've got a rho-secant-theta- [2.8] D-Z [4.0] and er [1.2] and then we're going to need to integrate over [0.3] all directions [0.2] omega-prime [0.7] and actually i mean it's a f-, a few things like like the D-Z we can take out of that integral [0.3] but the pa-, but the the s-, the secant-theta part you know the pathlength is going to depend on which direction [0.4] the radiance comes in [1.1] because if something's coming in at a different angle it's going to have a different probability of getting scattered if it's coming in at a low angle [0.4] it's going to get scattered [0.7] much more [1.8] so [0.8] what we'll do is we'll integrate this [0.3] over [1.9] the whole sphere [2.1] and that means we'll need to divide out [0.6] four-pi [0.4] we're integrating over the whole sphere all possible directions because radiation can [0.3] can get into the slab from above and below [0.6] why are we dividing out four-pi [12.2] in the context of solid angle and sphere what does four-pi represent [0.6] sf0915: whole [1.1] nm0885: yeah four-pi is is the ra-, is is the solid angle [0.2] of the whole sphere [0.3] so we're adding up contributions from all of this [0.3] and then we're dividing out the total solid angle [0.6] so 'cause remember [0.2] radiance is in units of per steradian [0.4] so we want to keep it as per steradian [1.7] okay [0.8] so [0.9] with that [0.6] hideous mess [0.5] we now have [0.4] all the contributions [0.4] to [0.7] the er [0.5] to the radiance [0. 6] so [2.3] the radiance [3.5] emerging [2.6] from a slab [1.3] is then going to be [1.9] what is it going to be it's going to be [1.0] L-nu [1.2] in some direction [1.5] theta-phi [0.2] and it's going to be the the sum [0.4] of all these bits [0.4] that we've put together [0.4] so it's going to be a sum of the [0.4] direct beam [0.2] L-D-nu- [0.2] theta-phi [0.9] plus [0.4] the er [1.3] emitted [0.2] that's right [0.2] L-E-nu [0.7] again what's emitted in our direction theta-phi although [0.6] emission of course is going to be [0.8] isotropic [0.8] and it's going to be [2.9] added to the scattering [0.7] L-S [0.6] of theta-phi [0.7] so [1.6] our [0.7] emission at the bottom of of the slab [0.4] is er [0.4] is given by these three terms [0.4] but remember [0.6] as as we were mentioning in the last lecture [0.3] these terms are not [0.2] all [0.4] always important [0.3] at certain wavelengths [0. 4] the radiation is going to be [0.3] dominated by er [0.3] by different [0.4] parts [0.4] of this for example with solar radiation [0.5] coming into a fra-, solar wavelengths [0.3] the emission by the atmosphere is going to be negligible the atmosphere's just too cold [0.5] to emit [0.4] those high energy photons [0.6] but on the other hand scattering is going to be [0.2] pretty important [1.1] and er [0.4] and of course there will be some depletion of the direct beam as well [1.7] okay [5.4] so that's kind of cool [0.2] that basically [0.4] covers the er [0.4] definition [0.2] section [0.3] of the course we now know what we're talking about [0.8] now for the rest of the course [0.2] we can talk about it [0.4] so what we're going to do during the rest of the course [0.3] is er [0.9] it's going to be divided into [0.4] two [0.4] big chunks [0.3] okay [0. 3] first big chunk [0.3] is we're going to do [0.3] scattering [0.9] and we're going to look at scattering in quite a bit of detail [0.2] we're going to look at what it is in the atmosphere that does the scattering [0.4] how those properties determine what the scattering does [0.5] and then [0.3] if you actually look up in the sky [0.3] how does [0.2] that knowledge of scattering actually explain the stuff [0.3] that we see [0.4] so we're going to do fairly obvious [0.2] or [0.8] reasonably obvious things like why is the sky blue [0.4] why is the er [1.0] why is the er well why why do rainbows form [0.4] we're going to get into slightly more subtle things like why do photographers always use polarizing filters when they want to look [0.3] want to photograph clouds [0.5] and [0.2] a few other things like that [1.0] and then after that we're going to do [0.4] emission and absorption and remember emission and absorption are actually sort of two sides of the same coin so we're going to do [0.3] that part together [0.2] and that means the direct beam [0.3] that depletion bit [0. 4] and the emission bit [0.2] we're going to handle them together [0.5] and er and there we're going to get into a little bit of the properties of the particles [0.3] so we're going to get into [0.3] a little bit of quantum mechanics we're not going to do it in an-, in any sort of great detail [0.2] but just enough that we can see why [0.3] some particles are better at certain things than some other particles [0.5] and er and then we're going to get [0.2] into at the end [0.4] how you actually do these calculations [0.3] if you happen to be interested in weather or climate or something [0.3] and you want to know [0.2] how to put those effects into [0.3] a model or into some kind of explanation [0.3] of how climate's going to change or how er [0.6] or how [0.4] you know is there going to be ice on the roads tomorrow morning this kind of thing we're going to get into [0.6] er how you'd actually do those calculations but we can only do that once we've actually gotten into detail of all [0.4] the processes [0.5] so we've just got five minutes left today [0.3] so [1.2] i want to er [0.4] point out [0.9] one slight subtlety [0.5] of what we've done [1.2] earlier on today [0.2] now you remember [0.4] optical depth [0.4] is a measure [0.3] of how much stuff you lose from the beam [1.2] but [0.7] optical depth [0.3] gets used a lot you'll hear that term a lot [0.2] but you want to be slightly careful with it because as we've seen [0.3] optical depth is composed of two bits it's composed of absorption [0.4] and scattering [0.5] and those can have very different [0.2] effects you know if if say you've got [0.5] er a water cloud in the atmosphere with an optical depth of [0.6] point-five or something like that [0.5] er [2.0] on a day like today which actually actually the o-, [0.3] what would you guess the optical depth was going to be [0.4] on a day like today [0.9] that's that's that's a good question [0.2] sli-, slightly subtle [7.1] let me give it to you as a multiple choice [0.7] ten-million [1.0] one [0.7] or [0.9] one-over-ten- million [1.9] ten-to-the-minus-seven [6.8] well [1.5] think of it this way [0.7] remember [0. 3] the transmittance the fraction of the radiation [0.3] that's getting through [0.6] okay [0.3] suppose the optical depth [0.2] were [0.8] ten-to-the-minus- seven what's E-to-the-minus ten-to-the-minus-seven is that going to be a [0.6] big number little number [9.1] let's draw a graph [0.3] so [0.5] E-to-the-minus- X as a function of X [3.7] what's it going to be at X-equals-zero [3.5] sf0916: one [1.1] nm0885: sorry [0.4] sf0916: one [0.5] nm0885: one [0.7] okay [0.2] going to be one [0.4] what's it going to be as X gets big [2.0] sm0917: [0.5] nm0885: sorry [2.4] sm0917: it goes small [0.6] nm0885: yeah [0.2] it's going to [1.3] fade away [2.4] er [0.8] where is it going to get to be [0.8] ten-and-one-halfish sort of thing how fast is it fading away [6.4] so if there's point-one-five [0.3] is that point there is that going to be closer to [1.4] ten-to-the-minus-seven or ten-million or to one or [0.5] which of those three [0.9] sf0918: one [1.3] nm0885: pardon sf0918: [0.5] nm0885: yeah it's go-, it's going to be kind of oneish 'cause [0.5] if if if [0. 4] if X is one then that's going to be E-to-the-minus-one which is going to be one-over-E [0.3] which is one-over-two-point-seven whatever [0.5] so [1.5] so basically [0.4] if you have [0.3] an optical depth which is really small sort of ten-to-the-minus-seven or something like that [0.3] then basically what you're saying is the transmission is pretty much one [0.4] all the light's getting through [0.8] so on a day like today a lot of the light is not getting through so the optical depth is not going to be that small [0.3] on the other hand if you've got an E that's something like [0.3] ten-million [0.4] then E is going to be very small and you're saying only a very very very small fraction of the radiation is getting through it's going to be very dark [0.4] and it's not that dark [1.3] er [0.5] so it's going to be something of order one [1.5] but [1.4] that's because the radiation is scattered [0.5] suppose we had an abs- , an absorbing case suppose we had a single scattering albedo in in our cloud today of zero [0.7] then what would the transmission look like [9.7] i'm asking that question in a co-, [0.2] kind kind of confusing way [1.1] the transmission what's directly getting through depe-, depends on [0.8] depends on the optical depth and that's not a problem [0.4] er the problem is relating [0.3] what you see when you look up at the sky [0.4] to the trans-, to the transmission because what we see when we look up at the sky [0.3] is actually [0.2] very little [0.2] direct beam [0.2] and it's almost all scattered radiation [0.5] so actually the optical depth today [0.4] is pretty big [1.6] but [0.4] because of the fact that the single scattering albedo [0.3] is pretty close to one [0.4] most of that light [0.3] that's getting taken out of the direct beam is getting bounced around and [0.5] some fraction of it [0.3] maybe half of it or something [0.3] is getting down to us [0.3] so we can still see where we're going [0.3] if we had a smoke cloud [0.4] like say from a forest fire or something like that and th-, did you guys see the Australian bush fires a couple of years ago [0.9] on T-V there's showing pictures of that [0.4] and basically it was black [0.6] under tho-, if you got close to those things and that was because [0.7] you had a cloud which in terms of optical depth was probably pretty similar to what we have today [0.3] but because it had a very low single scattering albedo [0.6] it was very dark [2.1] so [0.3] you got to watch out with optical depth optical depth tells you how much is taken out of the direct beam [0.3] but it doesn't tell you how much light you're actually going to see [0.8] on any given day that was actually the point that i wanted to make in a [0.3] slightly roundabout way [0.3] okay that's it for today see you next week