 Hello and welcome. In this lecture, we will start our discussion on one of the most important as well as a critical mission of any space activity that is re-entry or return mission. The mission becomes critical because we are now recovering the object that we had sent to space and then we need to deal with the atmospheric effects, the gravitational effects as we had dealt with while creating the ascent mission. So, let us begin our discussion with the basic entry concept. So, let us first define what is the problem that we are talking about. Re-entry or return mission is typically the last part of a reusable launch vehicle mission in which an orbiter or a space capsule or any other similar object is brought back to earth in a controlled manner. It is also applicable for entry or landing on other planets and for that reason it is an extremely important aspect of any space mission. We also realize that this mission is complex from a mechanics perspective as large amount of energy which is imparted during the ascent or orbital phases is now required to be dissipated which is going to be a challenging task. Just to understand the toughness of the problem, let me just give a few statistics that is commonly applicable in such scenarios. So, if we assume that a spacecraft or a capsule is returning from a low earth orbit of 322 kilometer circular in nature, typically such orbits will have a velocity close to about 8000 meters per second and this represents a specific kinetic energy of the order of 32000 kilojoules per kg that is total kinetic energy per unit mass. It is not very clear from this number whether this is large or small. So, let us take one more example of an object which is returning from outer space mission on a hyperbolic trajectory. Typically, the minimum velocity would be of the order of 11000 meters per second and the corresponding specific energy is likely to be of the order of 60000 kilojoules per kg. So, that is practically double the amount. Even now it is not very apparent whether these numbers should be worried about or they are manageable. In order to understand this aspect, let us first explore the mechanisms which will dissipate this energy. Of course, we can use propulsion as we have used in the ascent mission to overcome the gravity. We can now use propulsion to generate a retarding acceleration so that the velocity will reduce continuously until it becomes zero at the landing or the touchdown point. But generally it is found that a more efficient way to dissipate this energy and without expending costly propulsion is to convert this into thermal energy through aerodynamic drag. So, you allow the drag to retard the vehicle and that energy which is consumed by the drag appears in the form of heating of the vehicle and if you can manage the heating, then it is possible to dissipate the energy in an efficient manner. So, supposing we were to do this complete the return mission, what would happen? So, let us take a hypothetical case that if we converted all the energy to heat, of course it will not happen like this, we would find that the vehicle actually would get vaporized while entering the atmosphere. Because the vaporizing heat for water is only 2300 kilojoules per kg, whereas even from the mission which is from a low earth orbit of 322 kilometers, the total energy content is of the order of 32000 kilojoules per kg. Similarly, the vaporizing heat for carbon is 60,460 kilojoules per kg. So, if you are coming from a low earth orbit, you might still survive in terms of carbon, but anything else would simply disappear. And the 60,000 kilojoules per kg is the minimum energy with which you will come when you are coming from a mission to outer planet. So, obviously you realize that using this mechanism is not going to be directly practically feasible unless you can manage the heat generation in an appropriate manner. Of course, we have examples of many meteors that do survive the atmosphere, which also contain similar material as the vehicles that we send out in the atmosphere and outer space. So, obviously it means that not all the heat is going to be impacting the objects because we realize that a significant part of this heat would get lost through radiation. So, there is a little bit of comfort in that idea. Further, most of the return missions and vehicles that are required to return make use of heat sheets that prevent a large amount of heat transfer to sensitive and critical parts of the vehicle so that the heat load can still be managed while entering the atmosphere, which obviously means that we now need a mechanism by which we control the way in which the heat is going to get dissipated or generated so that it remains within limits during the entry mission and that we can bring the spacecraft or a space capsule back in a safe manner. So, in order for us to manage the heat, it is directly clear that we are going to have to manage the aerodynamic drag and which obviously means that we have to manage the trajectory because the aerodynamic drag is going to depend on the velocity and the altitude together will decide the amount of drag force which is going to get generated and which brings us to the idea that the trajectory design for reentry mission is equally complex if not more in comparison to an ascent mission. However, in contrast to what we do as part of our ascent mission, in entry missions there are two mechanisms that are commonly employed. In one case, it is purely a ballistic object which means it does not have any aerodynamic lift that is similar to our ascent mission where we neglect the aerodynamic lift and we make use of only gravity to achieve the trajectory inclination along with the large amount of propulsion which generates the desired velocity and a large class of return vehicles follow this strategy of entry which is simpler to design and implement. Of course, with the advent of space shuttle and similar objects which are currently under development, there is also a possibility of using lift to manage the drag appropriately through a parameter called lift to drag ratio. So, you design a vehicle in such a manner which looks like an aircraft which what the space shuttle looks like and then appropriately design external geometry such that the lift to drag ratio is in this range of 0 to 3.5 and with that it is possible for us to manage the drag by generating an appropriate amount of lift as long as L by D is fixed the drag will be fixed the moment we fixed the lift and if we can control the lift through appropriate control surfaces we can control the drag. So, this is the philosophy of lifting entry we will discuss a little bit more about this in subsequent lectures and in addition to this there are large number of different guidance algorithms that help us to manage the lift and consequently the drag having different objectives including optimality criterion to achieve different kinds of entry missions with different constraints. Let us now move over to the description of the entry problem. So, objects when they enter the atmosphere encounter aerodynamic forces as shown in the picture alongside. So, we here make use of a spherical earth model and then we say that there is a concept of local horizon which is nothing but a tangent at the intersection point and represents the local horizon of the object with respect to which we now define an entry velocity what we call VRE and a flight path angle or what we call elevation angle phi RE which is in the opposite direction and generally it is a negative number. Apart from this we can clearly see that as it enters the atmosphere it will generate a drag force which is opposite to the velocity direction. It will have a gravitational force which will be pointed towards the radial direction and in general if it is a lifting vehicle it will also generate a lift a new force which will appear in this context is represented through CF or what is called a centrifugal force because the trajectory will not be a straight line but it will be a curved trajectory either circular or any other curvilinear form. Next you may generally find that a vehicle starts the entry into the earth's atmosphere the first point of content with the orbit of that object or the trajectory of that object is the point which is closest to the earth's surface which has a generic name called the periapsis in the context of earth a more common terminology of perigee is used but if you are talking about this problem in the context of another planet then a general name of periapsis is used to define a point which is closest to the surface of that planet so obviously that is the point which will enter the atmosphere first. The reason why this point is mentioned is that that is the point at which the object has the maximum velocity while in the orbit or on trajectory. So, the object will enter the atmosphere with a very large velocity and even if it is entering at a very high altitude so that the densities are small because the velocities are very large it will immediately generate a large amount of drag force of course you can see that if we are able to manage the heat load then this large drag force will be useful in generating a large deceleration and the velocity will quickly reduce and if the velocity reduces quickly then it is possible for us to manage the trajectory such that we are in a position to dissipate the heat by the time the vehicle is required to land or reach the destination. So, in this context what is commonly done is to design the entry trajectory from outer space in such a manner that the eccentricity of the trajectory is marginally greater than 1 so that after it passes once through the earth's atmosphere or planet's atmosphere its velocity becomes sufficiently small so that it can form immediately an orbit about that planet. Here let me make a mention of this parameter e or eccentricity which is an important parameter that defines the nature of the orbit of which you will learn more when you go through a course on orbital mechanics which I presume either you may have gone through or you might want to go through after this so I will not spend too much time on it but I will just make a mention that this is the parameter which is under the control for an entry object and can be designed suitably such that the moment it hits the atmosphere it will never go out of the atmosphere again as it would form an orbit and then it would keep cycling around the planet. Now once this happens there are two possibilities either we can use residual propulsion to directly make e square less than 0 so that the vehicle will have a trajectory that will directly bring it back to earth straight away instead of using the propulsion we allow the vehicle to orbit with part of a trajectory in the atmosphere so that in every cycle a certain amount of energy get dissipated slowly until the same condition of e square less than 0 is achieved after which the same act of vehicle coming back to earth straight away will happen. Let us try and understand this through a simple example so let us take the case of an object which is entering earth atmosphere the SOI expands as sphere of influence which is a common term used to say that the object is now under the influence of the gravitational field of a specific planet with a velocity of 2000 meters per second and at an altitude of 55.2 kilometer. Now if after one pass through atmosphere the speed reduces by 500 meters it is an estimate of amount of drag and the energy that it dissipates then determine after one pass what happens to the vehicle and the number of passes before it will fall back to earth so let us start with the basic orbital mechanics relation which I am reproducing here you can refer to these relations independently for greater clarity but it can be shown that the velocity at periapsis the VP which is the point closest to earth will be of the order of 11,310 meters per second for the case when it enters the earth sphere of influence at 2000 meters per second and with that the eccentricity as you can see is slightly greater than 1 so we are considering the case which we have discussed in the previous slide. Now we assume that in one pass it will reduce the velocity by 500 meters per second so at the periapsis we assume this to be an instantaneous act because the velocity is very large it will reduce the velocity from 11,310 meters per second to 10,810 meters per second and at this point the altitude is 6,954 sorry not altitude this is the angular momentum down we look at the energy and we find that after one pass the eccentricity becomes less than 1 so the e square is also lesser than 1 but not less than 0 with these parameters we find that it will take approximately 7 passes before e square will become less than 0 and then it will start falling back to earth as a ballistic object so in such situations you will find that following kind of trajectories are possible so on the right side when you arrive on a hyperbola you enter the atmosphere dissipate a certain amount of energy and immediately you end up forming an elliptic orbit and once you have formed the elliptic orbit either you can use the residual propulsion to directly reduce the velocity such that e square is less than 0 or you can allow multiple passes such that the trajectory becomes closer and closer until it reaches a point where e square again becomes less than 0 and then the vehicle falls back to earth you can clearly see that while this strategy that is on the right hand side is time efficient the strategy on the left hand side is energy efficient this is sometime also called aero assist or aerobraking strategy of pre-entry now let us describe the problem in a generic context where does this activity actually begin so contrary to what we do in the ascent mission where beyond 40 kilometers we assume that the atmosphere is absent which is primarily because of the fact that our thrust is a much larger force compared to drag in the entry mission there is no thrust and drag is the important force so obviously we need to estimate it with better accuracy and then we say that the drag is going to be considered to be significant when it becomes 1 percent of gravity which means if it becomes 0.01 g that is the point at which we will assume that the atmospheric entry has begun and this we find is going to happen at a higher altitude of about 75 kilometers because our velocities are very large and because of which our entry altitudes generally will be of the order of 80 to 120 kilometers so we now draw a boundary of what we call the sensible atmosphere and show you the same picture with respect to the local horizon and the entry elevation angle of phi re and the entry velocity we are now we have done enough modeling of two-dimensional motion in the atmosphere so we simply recall those equations but we will see that there are certain differences so as you can see we have the drag term which is now opposite to the velocity direction we have the gravitational component then we have m dv by dt the force term the thrust is missing and then we have one extra term m v square by r into sin phi this is the centrifugal force term because v square by r is the centrifugal acceleration as we are using a spherical coordinate system here it is worth noting that for small velocities if v is small then v square by r is a small term and perhaps can be ignored as has been done in the context of a center mission but in this case even though r is of the similar order the v is very large so this term also is reasonable and needs to be included in the trajectory model now this is the equation along the velocity direction we now have another equation perpendicular to the velocity direction where now you have the presence of a lift the gravitational component and then you have the normal inertia force and the component of centrifugal force in the normal direction of course similar to our ascension we also have dr by dt the kinematic equation which is same as dh by dt as v sin phi and then we have the ds by dt which is another kinematic equation as v cos phi so we realize that these equations are similar to those that we derived for modeling ascension except two quantities that thrust is missing and in its place we now have the centrifugal force term of course if we are going to make use of propulsion to retard the vehicle then we can add thrust to these equations without making any further changes and the centrifugal force term will always remain until you feel that velocities are sufficiently low for it to be ignored of course similar to ascension we immediately realize that there are no known general closed form solutions and some assumptions are going to be required to arrive at such solutions that we will look at in the next lecture. So, to summarize the entry or the re-entry missions involve dissipation of large energy through lose of drag it is also seen that basic motion is similar to ascent mission except that thrust is absent and centrifugal acceleration is important due to large velocities. Hi, so in this lecture we have looked at some of the basic concepts of an entry mission which requires that a vehicle on a mission in outer space is brought back to earth in a safe manner. We have also noted that an important requirement would be to manage the heat load which can be done indirectly through managing the drag acting on the vehicle which is responsible for heat generation. In the next lecture we will look at some basic solutions of the kinematic model that we have developed in this particular lecture for the return mission and understand the attributes of the solutions in different contexts. So, bye see you in the next lecture and thank you.