[MUSIC] Welcome to this second session. In this session, in this session we will learn how to interpret emergent spectra of exoplanets. First of all, we have to come back to a very useful concept in physics. Which is the concept of thermodynamic equilibrium. This is a concept I'm sure you are already familiar with, but we briefly review what it consists of. Basically, radiation and matter interact with each other and we need to understand how, if we want to retrieve physical information, from an observed planetary spectrum. Now in sufficiently dense environments collisions between particles will dominate our radiation processes. And these collision processes, will determine, and control the internal energy partitioning. Within matter. In practice what does it mean? For example it means that all the populations of the various energy levels. in matter be it in an atom or a molecule or a solid. All these populations of energy layouts can be. Computed in a relatively easy way. Knowing only one parameter, temperature. So using only the knowledge of temperature we can derive how the energy levels are populated. What is the ionization state of atoms. Or how the various electrons are populating the orbitals within an atom. This concept of thermodynamic equilibrium makes everything much simpler when trying to solve the equations describing the state of matter energization and For example the radiation field can be described in a very simple way in a way that is called black body radiation. Matter in thermodynamic equilibrium emits black body radiation and the radiation only depends on one thing again. Temperature. Of course complete TE is an idealization. In astrophysics, we often use the concept of local thermodynamic equilibrium, LTE. And this is a useful approximation that is very often valid. And that helps a lot, solving the equations of radiative transfer, for example. What is this approximation? So in LTE, matter is still in thermodynamic equilibrium, but small temperature gradients are allowed. And, in a similar, line the radiation field may depart, slightly from the one of a black body. So LTE applies to planetary atmospheres in general, except perhaps in the upper layers of the atmosphere, where the density is too low. [SOUND]. So black body radiation is the famous Planck curve that you see here on the right of the slide. And, this is always a useful approximation. The first order approximation to the spectrum of stars and planets in general. And, as you know, exoplanets span a very wide range in temperatures. The cool, cool, coldest bodies in the solar system may have a surface temperature of 30K or something like this. While the hottest exoplanets reach of temperatures of more than 2000 Kelvin. So what this means is that when you want to detect formal emission from this bodies, you will have to look at the proper wave lengths, the wave lengths when you expect the most of the flux will be edited. And this can vary a lot between the same for example which emits most of its lights indivisible as you can see in the plots that the same black boby will be shown by the yellow line. Compared to the earth, for example which would be the red lines in the, in, the red line in the plot, which emits most of its radiation in the mid-infrared. Okay, now, to better understand how radiation propagates and how emergent spectra are, are made, we have to do a little bit of radiative transfer here. So I'm showing here the Radiative transfer equation which describes how a beam of light with intensity I, sorry, will propagate through matter. So, this equation describes how the derivative of the intensity changes with position x. And the equation is relatively straight forward to understand. Basically what you have here on the right hand side is you've read that the change in intensity is the, the gains minus the losses at, at location x along the beam. The gains are described by the emissivity term. That you see on the right here, epsilon. While the last is the, the first term which is made of several other terms. The, the main aspect here is the, the, are the cross sections, the, the absorption and scattering. Cross sections that describe how photons are removed from the beam. So photons can, can be destroyed by absorption, they can also be scattered out of the beam, which is described by the scattering cross-section. You have to multiply these cross-sections by the density of particles, the n here. And then, all these multiplies the intensity I, to describe how, how much flux is removed from the beam at a position x. Another useful concept that we need here, to, to discuss, is the concept of optical depth. The definition of optical depth is shown in, in, in this slide. Basically the optical depth is the total absorption and scattering. It's the sum of all absorption and scattering sources. Along the, the beam. So you, you have a differential definition here. d tau is defined as the total absorption scattering at a location x times dx, a small distance. And so the total optical depth from point A to point B is just the integral of the total absorption and scattering between point A and point B. Now a very useful thing that we can do now is try to sol, solve the radiative transfer equation that we've seen in the previous slide by assuming that the emissivity is 0. This can happen, for example, of you're looking at the frequency of light where, you don't have any. The black body or any thermal emission. So in this case, the solution to the radiative transfer equation is quite easy. Because I remind you that we have a differential equation, where the increase or decrease in intensity is proportional to intensity itself. And as you know, this kind of differential equation has a relatively easy solution. Which is an exponential solution. And you can see here that this exponential solution has the form of so the emissivity at the location x is the density at zero times the exponential of minus the optical depth. Along the way from zero to x. So basically this gives us a way to interpret this concept of optical depth. We see immediately the beam weakens exponentially with optical depth. And so we can interpret the optical depth of one. Ad being the photons mean free path. The average distance a photon can travel before being absorbed or scattered and another thing that we can also derive from that. Is that when you are looking at the black body source. Yours, the photons that you received, the photons that has reached the observer. Cannot leave the body anywhere. They will have left the, the emitting body at an optical depth of about one, because otherwise if, if they originated deeper in the atmosphere of the body they wouldn't have made it to us, to the observer. They would have been absorbed before reaching the surface of the source. So now, now that we have understood these two concepts of radiative transfer and optical depth, we can move on to try to give some basic interpretation of emergent spectra, whether planetary or in stellar spectra. Let's first have a look at what happens if you have a body in local thermodynamic equilibrium which has an isothermal atmosphere. Well in this case we're actually in a complete thermodynamic equilibrium situation, which means that the emergent spectrum will just be a pure black body spectrum with no spectral features. If now we have another case where we still have an object in LTE, but with a temperature gradient, so a temperature decreasing outward. In this case, what we'll have is a black body spectrum with absorption lines. Why is that? Remember that we see, the photons we received. Left the body at optical depth of about one. Now if you compare two wavelengths, one wavelength opacity and another wavelength with low opacity. And you try to understand where the photons come from, at a high opacity wavelength, the photons will come from a higher level in the atmosphere. This higher level, because of the temperature gradient. We will be at a lower temperature. Lower temperature means lower black body flux. So we will have our relative low flux coming from there. If you now look at the wave length where the opacities are low, you will probe deeper into the atmosphere. Where, the temperature is higher, and the black body flux is higher as well. now, so, if you're in the core of a spectral line, you will have low flux and in the continuum, or the wings of a spectral line, you will have high flux. That's the description of an absorption line. And that explains why you would see absorption lines on top of a black bod-, for black body. If we now move to the opposite case where you have an object in LTE but with temperature inversion it its atmosphere. So the temperature now would be increasingly outward. Then, in this case, reasoning in the same way as before we would get emission lines on top of the black body emission. And finally a slightly different case that will be useful for us later when studying exoplanet atmospheres is the case of cold gas, absorbing light from the hot source that is, that is located behind the cold gas with respect to the line of sight of the observer. In this case, what, what happens, you would see the hot source spectrum but because the beam of light has to go, to pass through the cold gas, you would see absorption lines super imposed. On the, hot source spectrum. This happens for example, if you look at stars from the ground, looking through the earth atmosphere, you would see the, imprints of molecules that filter the light in the earth atmosphere. Now that we have acquired these tools to interpret spectra, we will in the next session, we will actually look at in more details of the chemical composition of the of exoplanets. Which will tell us more about the sources of obesities. Thank you. [MUSIC]