[MUSIC] After the [UNKNOWN] descriptions of the transit now we will move on the quantitative aspects. So, technically speaking there is a couple of parameters that is required for transit. There is the impact parameters. How far, its called B. How far you transit from the center of the star. There is also the duration. The duration is accounted when the planet is inside, fully inside, the star. It's T. There is also the contrast. The depth of the transit. Which is here specified as delta. In addition to that, we have a specific time scale like the ingress. Ingress is the very specific time when the planet enter into the transit. It's a very special moment. Which is just here, between the fine, between the time where the transit is outside the disk of the star to move inside. That is exactly the same when the planet gets out of the transit, which is called in this case, the ingress. So they maybe different, depends on the eccentricity, but most of the time the difference is very tiny. So we will consider here that this parameter, which is called tau, will be the same. And that's, this time. So, you can then define a couple of the parameters. When you have the specific timing, but people are more useful, I mean, used to, is the, called the mid-transit time. Which is the time as defined at the time you have lose, you, you have, by half, half of the flux. So corresponds to the time of the transit plus the tiny fractions of the ingress and that is described here. The T tot is the full transit time [UNKNOWN], and then if you subtract the ingress you back. The time that I described before which is a time when the planet is inside the disk of the star. So there is a lot of equations that I want to go through that. But there is a couple of simplification we can use to come up with simple numbers to get some figure in mind. So first of all we assume that the ratio between the size of the planet and the star would qualify it as k. It's a k number. We'll follow this number through the presentations. So we assume the planet is small. If it's not the case,we move to the binary stars, which is another descriptions, very much complicated. So we assume it's a small mass object which is fine with a planet because Jupiter is 1000 time smaller mass than the sun. So we also assume that we're dealing with systems where the size of the planet will be smaller from the size of the star. But we also assume that the size of the star is smaller than the object to be considered. So, if you really have an orbiting planet very, very close to the stars the equation I show you doesn't really match. There is a correction to the equation we have to find out. So, we also consider that the impact parameter is not too big. I mean, you don't have what's called a grazing transit. When the planet is just, about getting into the transit. And we also consider there very close to 0 transicities. So it a lot of simplifications and practically most of the case you have a transit this or these configurations. So I am going to give you a series of good numbers to work with. All this number are good for the earth, if you have a transit of the earth for example. So from this number you can come up with a couple of equations which is just approximations. The first one is, there is a typical time scale which is easy to compute. It's a size of the star, the period of the systems, and then the [UNKNOWN] axis. We can put number into that. And you end up with 13 hours for, a star like the sun, which has the density of the sun. And for a period like the Earth. So the Earth, would need 13 hours to transit the sun. Then you can play around with the number. If you decrease the period then the duration will decrease. So this T0 connect to the real T which is the time when the planet is inside the disk of the stars. So then there is a correction which depends how far the transit is made from the center of the star, is impacted by a meter. So obviously when the impact parameter is, is larger than let's say half of the stars. Then the duration will become smaller. You have the longest transit when the impact parameter is 0. It's practically when you cross the star at the equator. Then you compute also the egress time. So engress time is also connected to this T0, I described before. But then, you need to balance with the contrast ratio between the star and the planet. So, if the star and the planet has a contrast ratio 1 10th, like Jupiter and then, and, and the sun. Then you can describe, you can decrease by ten, the, the time scale of the ingress of the egress. So we give you a bit the feeling how you compute the ingress by this number. And then we also need to collect the impact parameters. So, another interesting aspect here, if, if you pay attention to the T and k here in the equations. You realize the stellar radius disappears and instead of the stellar radius, we have the equations. We have the size of the planet divided the semi-major axis. So to give you a scale factor, the ratio of the size of a planet against this major axis gives you an idea of the ingress or egress timescale. It's called a dimensionless number. Which is a very bottom number you can extract from the latter, from the transit detections. So now we have been to the transit in time, now we will move to the transit in shape. How would you describe the shape of the transit. So simi, similarly there is a complex equation to that and we'll trying to have a simple approach to this. So what you see in the diagram is the full transit situation. When you have an orbiting planet. You have on one side, the occutations, when the player goes behind, and you have obviously the transit. Transit is bigger than the occutations. We come back on this and explain this on why. So when you zoom in, the interesting number to, to get here. Which is also, also an intrinsic, feature, which is the global shape of, of the flux. You do have this kind of sine wave. Looks like a sine wave. Which is actually what's called the phase function. So because you see different phase of the planet. Namely, the dark side, and the bright side. This will affect the total flux you will get from the system. When I mean total flux, is the flux from the star plus the planet. So that's why you have the sine shape. It's also something that comes into the descriptions of a, of a transit. So, in term of number, we very, kind of conventions on this we, we used to express the flux against the flux of the star. So the flux has been measured is the flux, the real flux, F t, against the flux of the star. So that's the real number. So, when you compare the flux from the planet against the flux from the stars, we also used to do this following assumption. Which is the ratio of the size of the star or the planet, multiplied by the intensity of of the flux from the planet and the stars. So it's assuming that you can cor, you can have the completes matching between the intensity size. And then, and then the flux, which is the case in most of the cases. Then, the equation is this one. This one give you a full descriptions of the transit. Practically what you do have here is, you have first the flux from the planets. This famous sine wave. The phase functions, so it is, this is a flux corrected from the flux from the star, remember. So you left with additional flux, which is the flux to the planet. And this one depends of the ratio between the size of the planet and the stars, and the ratio of the intensity of the planet and in the stars. So this is a famous sine. That's why there is a t factor, a time scale. Because this depends on what you see from the planet. If you see a dark or a bright star. And then you have two options, you can be in transit,. In this case, it is the ratio of the size between the star and the planet, the k2. So it's practically the area, this is the shadow area of the planet and on the stars multiplied by parameters. Which we describe after which give you the shape of the transit for exactly you time the shape. So I give you the scale with the k2. And then the shape of it comes from different aspects. And then, there's the time when you are out of transits which is why you have a 0. And then, there's a time when there is your quotations. So a quotation is completely different because in this case you do see practically only the star. So that's why you have to subtract the planet, practically, times the parameter. Which is the shape exactly of how you enter into the, the occultation. Which is alpha-occultation similarly to the, to the, to the planet. So the interesting aspect here, if you pay attention to these two parameters,. They are both the same. It's k, k square, I p over I stars. Then you will subtract, practically, the planet from the time of the occultations. So at the time of the full occultations, you will be seeing only the star. It will be one because you will have subtracted it. When you are slightly off on coming into the occultation then you have to take into account the shape of it. That's why this line is exactly the flux on the stars. So you have to see the, the diagram as the flux on the star, plus something. Which is a sine wave. Which is the fact that this is a phase function from the planet, and we'll come to that describe that produced this. And then, you have an eclipse and then occultations. And occultation is the only moment you have on the, the emission from the star. It's a bit funny to see that this way because it's, the planet is so small. But then, practically the planet effect a little bit what you see from the systems. And if you want to make sure you have only the stars. The only way to do that is to observe the system only when you are in interpretations. So, coming back then on the shape, particularly on the systems, we'll spend a little bit of time on this alpha here. That describe the exact shape of the transit. So the alpha is given as I said via, via let's say a dimension factor that says whole, whol, how deep is the transit. So this is the delta I mentioned before this is a k square minus a little bit of something. Because, you have something because you have also the emission from the, from the planet, remember. So, the full description is, this one. This is the, the, practically from the, from the transit, and the occultation is only the one that is emitted from the, from the planet. So for the sake of the argumentation here the amount of flux coming from the planet is something. But by compare reason with the strength, and, and, and of the transit, how deep is the transit is really a marginal factor. So we will completely neglect it. So practically, we do assume, then, that the transit def is only the ratio between the star and the planet to the square, because it is the area. And then the occultations is the same ratio. And then multiply by the amount of intensity that you have from the planet. So, speaking about the detailed shape, it depends of many parameter. One of them is the impact parameters. How far you will impact from the equator, from the center of the star. So you see a couple of different descriptions here. That tells you how you can play with different impact parameters. It depends directly on the exact angle of the systems. That's why you have the impact parameters. That is related to the cosine. And then, you also have, depending of the color, you look at the systems. You have to take into account of the shape of the star. The star is, is a sphere. [INAUDIBLE] the disk. So, when you enter the star, you will see the atmosphere of the star which will not be the same that we are in the middle of the transit. So that's also the parameters that you have to take into account. It's called a line darkening effecting. And the line darkening would be stronger in the blue. And if you look at the bottom of the figure you see that there is a strong shape compared to the red part of the transit. Which is almost like a square. So you have to take into account this, what is exact wavelength you are detecting. And all this together comes into the shape. And, you get a perception when you look at the Venus transit, remember I showed you before. This is this is when he is just getting out of the sun. And you see that the sun is not the same color if you look slightly to the edge or slightly to the center. So this is exactly the descriptions I mentioned here. Which is a [UNKNOWN]. You don't see it exactly the same kind of atmosphere. Because you see it in a very grazing way. So this will affect the global shape you have into the system. So altogether and that all to be too much an equation here, this, this two parameters. So in darkening the impact parameters will in fact affect of shape of the system. And that's where we have this alpha of transit into the equation here. And all this together comes into. On the top of that will be scaling factor which is the depth of the transit. So that's it to the descriptions we have in the systems. So in the next chapter, we will try to understand whole from the description of the transit. We will get into physical parameters of the system. The real size, the real descriptions of the stars and of the planet. [MUSIC]