 So, let us see the variation of these four parameters this is something that you will need in calculation. So, let us see what it is. So, let us look at P and T with geometric altitude. So, I have just hidden and I want you to go layer by layer. So, we will just remove one one layer. So, first we look at the troposphere, the troposphere okay which is the bottom most portion. We notice the red line shows that the pressure is reducing from around 288 it reduces linearly up to a value of I think 212 Kelvin linearly. The pressure does not change linearly, but it reduces okay. It is a nonlinear variation because DP has got that rho and rho is changing with altitude. If we will remove one more layer and we come to the troposphere and the troposphere and tropopause, we find that in tropopause the temperature is remaining constant okay. However, pressure is still reducing, but the rate of change of reduction is reduced like it is still reducing, but at a lower rate. If we go one step further, now we find the temperature is increasing. However, pressure is still reducing, but at a very very low rate hardly any change, but there is change. Then we go to stratopause which is a very small zone. Again temperature remains constant pressure still reduces and above this we have what is called as the mesosphere, mesopause, thermosphere, but we are not bothered about that too much okay because we will not take that part. When you do spacecraft dynamics or you do introduction to spacecrafts, we will look at that right now. So what is happening is the bottom layer is the gradient layer where temperature is reducing linearly. So therefore the reduction in temperature up to 11 kilometer, this is something which you should try to remember. This is called as basic aerospace engineering knowledge. From sea level to 88.16 degree Kelvin or 15 degrees centigrade up to around 11 kilometers, it reduces linearly at the rate of 6 and a half degrees per kilometer, temperature and with that you can get the values of other parameters. In the isothermal layer which is from 11 to 25 kilometers generally, temperature remains constant okay. In the next layer from 25 to 47 it increases by 3 degrees Kelvin per kilometer. Above that up to 53 kilometers again it remains constant. Beyond that we do not care because we will hardly look at aircrafts which fly beyond 50 kilometers okay. So this is not important for us right now, but I will just reveal that again there is a reduction, again there is a constant, again there is an increase up to 105. So you are not expected to remember all the values but at least I would expect it will remember two things that up to 11 it reduces, it remains constant to 25 and beyond 25 up to 47 it becomes 3 degrees per after that I do not expect you to remember okay. Alright so this is something yes we will see it, we have that see there is one special section on measurement of pressure and velocity. There I will talk about instruments used to measure. Today also I will talk about it okay. Now you have some homework to do or self study. It is very easy for you to actually derive these expressions looking at standard textbooks. Many of you who are from aerospace background may have done this as part of your gate preparation or as part of your undergraduate studies. Those who have not done you can look up any textbook and do it, it is very straight forward because temperature is varying at some particular rate you can get the relationship between the pressure ratios and the density ratios. So G0 is 9.876 the gravity, R is the universal gas constant 287 joules per kg degree Kelvin, T is the temperature in Kelvin and S2 minus H1 is the temperature height difference. So from some height to some height the pressure ratio is this. So if you want to remember I will tell you what to remember this value will become P1 by P2 will be equal to T1 by T2 to the power 5.658 okay and for density it will become sorry for density it is 4.658 and for pressure it is 5.658 that is what you can remember. Yeah no it can be anything see here the value of H H2 and H1 both of them are from the mean sea level. This formula is actually meant for the geometric altitude okay. So H2 minus H1 is a difference it does not matter if the as long as you use the same reference does not matter it is delta H. So if you want if you know the pressure at 1 kilometer and you want at 2 kilometer you can use T2 by T1 power 5.258 and get the value of P2 by P1 got it okay. But this is only true in the isothermal region that is the region on the top from 11 to yeah this one is what is applicable in the gradient layer. So I am sorry the value which I gave you 5.258 is for this region actually G0 by LR and G0 by LR plus 1. So yeah so the rho by rho 1 is going to be T by T1 to the power minus of 5.258 and P by P1 will be T2 by 2 to the power 4.25 it is very simple G0 is 9.807 is 6.5 degree per kilometer and R is so just calculate if you have a calculator you will get a number that number you just remember okay. And you get this simply by 2 magical equations one which we have already derived the standard equation of hydrostatics and the other one is a universal law which says that P is equal to rho RT as long as the gas does not dissociate. So using these two you can get these two expressions alright let us go ahead viscosity this is a very important parameter okay sometimes in calculations when you do aircraft performance you need to know the viscosity of the air. So why is viscosity important where does it play a role yes but through which parameter does it show up through Raylons number and that is where you need viscosity. So viscosity calculation there are many formulae available but there is one very elegant formula given by Sutherland. I like it because it is very easy to code this formula very easy to enter this data. So it just says that mu at any altitude is equal to mu 0 into T by T0 power 1.5 into T0 plus 110 and T plus 110.4 okay. So if you want to know how this has come there is a link given below which you can click and you can read it yourself. Now we look at the other three altitudes. Now the pressure altitude is basically an altitude depending on a standard pressure. So if you look at the ISA International Standard atmosphere the pressure at 0 is 101 325 Newton per meter square that corresponds to 29.92 inches of mercury. So with respect to that pressure at datum value when you read the height that is called as a pressure altitude. In other words if an aircraft is flying at a height x and in the ISA the or I should say the other way round the aircraft is flying at a height h and the ambient air pressure at that condition is y. You look up in the ISA at what height do you have y that height is the pressure altitude simple. So I may be at sea level and the ambient pressure may be equal to pressure at 1 kilometer in ISA. So although I am at sea level my pressure altitude is 1 kilometer because the ambient air pressure at the height which I am flying corresponds to a height of 1 kilometer under ISA. This is called as a pressure altitude. Do you understand this? It is only a concept but why is it given? Why do we want to talk this kind of a complicated thing? The answer is the pilot does not know where the sea level is. The pilot only knows what the ambient air pressure is through an instrument and assume there are errors removed etc. Assume that the pilot knows the true pressure. So when the pilot talks to the ATC that is the question that you asked how do you have a reconciliation? So the pilot says I am operating at a pressure altitude of 3 kilometers. So that is for uniformity. So the theoretical location where at 15 degrees Celsius the altimeter reading will be so and so mercury that is the altitude. In other words look at ISA, find the pressure comes out of the envelope and that is the value. We also have density altitude. Now can you guess what is density altitude? Can you guess? Perfect. So if I am at some altitude at which density of air is 1.2256 kg per meter cube I am flying at sea level density altitude. I could be on average but my density altitude is 1.220 sorry is 0. Here the reference is the density under ISA. Now the first one pressure altitude is useful for the pilot. Who uses this altitude? Density altitude who is talking about it? The airport people because pilots cannot measure density of air. They do not have a small container and a mass spectrometer. You collect air, weigh it and then divide or they fly the aircraft. So they have a pressure instrument. But the people who are at the airport they want to communicate to the pilot that the density altitude is 2 kilometers. That means this is a very hot, very you know very less dense place. So your landing distances will change. So the density altitude is mostly used by operators, airport people etc. So what is it? Pressure altitude corrected for non-standard temperature. So for that there is a chart. So just to tell you how it is read. So what you do is you say okay I am flying at some particular pressure altitude. Let us say I am flying at this pressure altitude okay 9000 feet. And the temperature outside is not the temperature which is at 9000 feet under ISA. The pressure temperature outside is say 38 degrees. So what I will do is I will say okay the OAT. Now the pilots can measure OAT right because they can have a thermometer kind of probe. So the pilot will say okay 38 degrees is where I am. I go to 9000 feet okay and then I can find out what is the approximate density altitude by going along this line. So I will show you how this is read. There is a video which shows you. I will talk about that. And then we also have temperature altitude which is temperature based. If your temperature when you fly is 15 degrees centigrade you are at sea level temperature altitude okay. So let us see now. Let us see how this correction is carried out okay. Now look at the cold weather. If you are flying in a cold weather the true altitude may be lower than what is indicated okay. You may get a false feeling that I am at high altitude. If the temperature is colder than ISA you will have this problem. So what will happen? You will think you are at high altitude but actually you are at low you will go and hit an obstacle. Very dangerous. So therefore the pilots have to correct the altimeter readings for a non-standard temperature. So this is how they do okay. They use a chart or nowadays they use a software. So this is a typical airport chart. I will talk about it sometime later. So it shows here that the elevation of the airport is 5885 feet. That means this is in feet right. So you find out the altitude of the runway. Then identify the correct and corrected altitude which is 9000. This information comes also from the chart. It is written here 9000. So this will tell you the height above the airport. That is 3100 feet. I will just pause it here to explain to you. If some people are confused I will just tell you what it is. This is basically a standard chart. This is for Denver airport for example. Now I conduct a course on air traffic management or it is a part of the course on air transportation. There we go into more detail on how to read these charts, how to plan the routing. I will spend little bit time to tell you what it is. This number tells you the elevation 5885 that is the altitude at which the airport is located. And this one tells you at what height you have to come in when you come in to operate. So you should be flying at 9000 and the airport is at 5900. You just round it up to 100 feet. So you get 3100. That means the pilot knows I am at least I am roughly above the ground at 3100 feet. Now you also look at what temperature you are. So you have temperature minus 30 degree centigrade. You have height 3100 between these two. So then you are going to read out. You are going to just draw a line between 3100, 3000 and 4000 and minus 30. So it is between 570 and 760. Just do the correction. So it is 589 feet. So which means the error is 589 feet for this operating condition. And with that, so this is the corrected altitude. In other words, because you are operating at a lower temperature minus 30 degree centigrade you have to be, so this is the minimum procedure turn altitude, minimum PTA it is called. And this is the final approach fixed crossing altitude. These are all details which we can probably skip right now. So this is how it is done. You have a chart where you have a reported temperature on the y-axis and on the x-axis you have the height above the airport corrected between the height at which you are allowed to come in and the height of the airport rounded up to 100 feet. So much above ground level and so much temperature it gives you what is the error. And these charts are either digitally available or available in the form of hard copies which are used by the palliars to interpolate. Precisely and that affects the measurement. That is the reason. So what are the important points to be kept in mind? Whenever you refer to the tables, ISA tables you should refer geometric altitude. Whenever you refer pressure, density and temperature altitudes you must use the geometrical altitudes. Secondly, there could be a situation when the aircraft is flying at a pressure altitude of 2 kilometers, density altitude of 3 kilometers and temperature altitude of 4 kilometers all at the same time. Do you agree it is possible? So it is possible because the references for them are the three parameters. So this is very important. You can be flying at many different altitudes and of course these altitudes may not be same as geometric altitudes because the temperature pressure and density do not follow ISA. They have their own whims and fancies. So you have to be careful.