Impulsive Maneuvers & Co-Planar Orbital Transfers | Cornell Astrodynamics

[00:00:00]so far we've been discussing perturbations and perturbing forces in very general terms we know that these can arise due to natural forces that are beyond the two-body model we know that forces can act on our orbiting body continuously or they can be one-time effects and as we go along we're going to be providing ourselves with additional descriptions of specific type of perturbing forces and additional tools for evaluating how they impact our two body orbits and cause those orbits to change in time but in addition to naturally occurring perturbations we can also think about forces due to our control of spacecraft and to do so just as with anything we need a model and what we're going to start with is a quick review of the impulsive burn model which is the simplest way of modeling orbital control the impulsive burn model unsurprisingly is based on the concept of impulses so recall that we can write the integral of a force between two times that is the integral of a force f acting on some particle p between times t1 and t2 is the same as the antiderivative of the derivative of the linear momentum of that particle with respect to an inertially non-accelerating point this is just a restatement of newton's second law which means that this integral is equal to the difference in the linear momentum of that particle between times t2 and t1 if we plug in our definition of linear momentum we can write that is the mass of the particle times its velocity at time t2 is equal to its mass times its velocity at time t1 plus this integrated force between times t1 and t2 and this integral is the quantity that we call the linear impulse which we will denote as f over bar sub p between times t1 and t2 if this force f sub p is nearly constant in time and if the time interval is very short then we can approximate this linear impulse as just f p delta t where delta t is t2 minus t1 so this will be our model for impulsive burns we will assume that we can apply a force to a spacecraft so quickly that our velocity changes effectively instantaneously while the position vector remains the same and so what that means is that you have an initial orbit with some position or orbital radius vector and some velocity vector we will add a burn to it and we will get the same position and a new velocity that's equal to the original velocity plus the impulse of that burn divided by the mass of the spacecraft since position and velocity fully define a two body orbit what we've done here is to create a new orbit that intersects the original orbit because the position vector hasn't had time to change in the application of our impulse obviously this is just a model it is impossible to create velocity changes in infinitesimal time actual burns by chemical propulsion systems take many seconds or minutes or even hours depending on the magnitude of the desired velocity change so this is a first cut model but using this we can get a very very long way towards the real answer and then to get all the way to the real answer you have to go into the realm of numerical integration and so we will provide ourselves with more advanced models when we talk about numerical integration and optimal control but as a first cut the impulsive burn model is incredibly powerful and we can learn a lot from it so what we'll do now is just to review some basic ideas about impulsive burn trajectories and refresh our memories on this concept burns can be either tangential or non-tangential a tangential burn happens at a flight path angle of zero so if you are on a circular orbit every burn is a tangential burn automatically but if you are on an elliptical orbit or an open orbit then the tangential burn can only happen at a turning point you will recall that we've previously described the relationships between velocity and semi-major axis via the vesviva relationship and also given ourselves tools for evaluating the eccentricity based on this we can say that at flight path angles of zero any change in the velocity vector will automatically increase the eccentricity and increasing the magnitude of velocity at a turning point will also increase the seven major axis whereas decreasing the velocity magnitude at a turning point will decrease the same major axis so when you burn at a turning point you are going to be either increasing or decreasing your semi-major axis and pretty much anything you do is going to end up increasing your eccentricity because eccentricity is defined such that it cannot go less than zero and so pretty much the only place you have to go is up there are some very very specific things that you could try to figure out to create a new circular orbit but if you're doing in-plane burns in the original parafocal plane of the orbit such that your new orbit is co-planar with the original orbit you are pretty much guaranteed to create an elliptical orbit out of your circular orbit as we've seen from our previous analysis a standard problem in orbital control and trajectory planning is transferring between two coplanar orbits with some caveats the most efficient way of doing so is by means of a home and transfer most efficient in this context means minimizing the net change in velocity that you have to apply so this delta v scalar is the sum of the absolute values of the changes in orbital velocity that are needed via two impulsive burns you need two impulsive burns because you are transferring in general between two non-intersecting orbits since a single impulse of burn always creates an intersecting orbit in order to get to a non-intersecting orbit you must burn twice the most efficient way of accomplishing this maneuver is by the application of two tangential burns so again if you're going between two circular orbits you can burn anywhere but if you're going between a circular and an elliptical orbit or between two elliptical orbits you're burning at two turning points and in the case of two elliptical orbits you are burning at the opposite turning points so for example you would start from periapse of one orbit and then burn at the apple apps of the second orbit with the setup what you're effectively doing is putting yourself on another elliptical orbit that patches between the two turning points of your initial and final orbits and the semi-major axis of this transfer orbit is given by half the sum of your initial orbital radius and your final orbital radius given the semi-major axis of the transfer orbit you can also figure out its period you are following this transfer orbit for exactly half of its period and therefore your transfer time is half of the orbital period of the transfer orbit and then from the vesviva equation and all the other tools that we've developed you can figure out the magnitudes of the changes in velocity that are required to accomplish this specific transfer we typically define four terms we have the initial velocity on your initial orbit we have the final velocity on our final orbit so these are essentially given by the specific problem statement that you're trying to solve and then you calculate the velocity on the transfer orbit which is just a function of the semi-major axis of the transfer orbit and this initially will be given by the initial orbital radius and similarly the final velocity on the transfer orbit before you change on to your final orbit so you have these four terms together they give you these two quantities you sum the absolute values of these two quantities and you get this delta v value this delta v value is important because it acts as a proxy for the amount of fuel mass that you have to consume we plug this into the ideal rocket equation and we get out a mass of fuel that is needed i said that the home and transfer was the most delta v efficient way to transfer between two co-planar orbits with some caveats and the caveat is that there exists very specific configurations of your initial and final orbits where you can actually squeeze out a little bit more efficiency by performing three burns instead of two burns and what that looks like is essentially stacking two homeon transfers on top of each other so you start from your initial orbit you do a tangential burn you put yourself on a transfer ellipse that takes you to an empty point in space you burn again and return back to your intended final orbit how does this gain us anything well to analyze this we can define two new parameters eta which is the ratio of the final orbit semi-major axis to the initial orbit semi-added axis and c which is the ratio of the radius of your transfer ellipse to the initial semi-major axis our t here is that radius from the central body to that empty point in space where you perform your second burn in the bioliptic transfer and then we can write the ratio of our net delta v scaled by the initial orbital velocity in the case of the homo transfer just as a function of eta and in the bioliptic case as a function of c and eta and you get these rather horrible looking expressions in the case of the bioptic transfer we can also figure out a limiting case where we take this radius of the transfer orbit to be infinite so this is you're essentially going out to infinity and then coming all the way back in so this is a limiting case it's not a physically practical case but it's convenient to parametrize this problem and then we can plot the ratios of the net delta v divided by the initial orbital velocity as functions of eta well remember eta parametrizes the basic problem because it is the ratio of the final semi-major axis to the initial semi-major axis when 8 is greater than 1 you're going outwards when eta is less than 1 you're going inwards and an interesting thing happens this black curve represents the home and transfer and you'll note that the y-axis here is on a logarithmic scale and the fact that it drops out completely here is just the fact that this home and transfer curve goes to zero which goes off of the logarithmic scale for eta of one when you have an eta of one when your final semi-major axis is equal to your initial semi-major axis there's nothing for the home and transfer to do and so there is zero delta v required in the case of a pile of the transfer however you could go out all the way to infinity and come back or go out to any distance and come back there's no reason to do so but the math supports non-zero values for this quantity for bioelectric transfer when ada is one the dash black curve here represents the bioelectric transfer performance for this limiting case of c equal to infinity which is equivalent to rt going to infinity and then the remaining color curves are the performance of the bio elliptic transfer for specific values of c remember what we're looking for is a metric of performance as parameterized by the required delta v we are looking for the minimal delta v needed and so we instantly see that there are two crossing points they're given by eta of 11.9 and a bit and it's reciprocal and between these two values the home and transfer is always going to be more efficient so you can instantaneously check for whether or not you need to bother trying to figure out a bioleptic transfer just by looking at your setup if the ratio of your final semi-major axis to your initial semi-major axis y is between 11.93876 and a few more decimal places and it's reciprocal then instantaneously you know that you don't need to bother with the biologic transfer because the home and transfer will be more efficient outside of this range if you're going inwards the biopic transfer is going to be more efficient on the other side when you're going outwards you have this interesting effect where these ada curves bounce off of the homing curve at values of eta equal to c so for example this c equals two curve shown here bounces off of the home and transfer curve at exactly an eight of two but this occurs in this 11.9 and one over 11.9 range so this curve will never be more efficient than the home and curve on the other hand this c equals 50 curve over here only goes above the homing curve at an a to a 50. and so in this range for it is between this 11.93 number and 50 a c of 50 bioelectric transfer will be more efficient and so on and so forth the maximum value for the homing curve occurs at 15.58 and a bit which is the curve being shown here in red right there so to summarize in this range between 1 over 11.9 and 11.9 the homan is always more efficient above 15.5817 the buy elliptic will be more efficient so long as c is greater than eta and below one over 11.9 the buy elliptic will always unconditionally be more efficient the last thing to know here is that you don't get anything for free when you switch the biologic transfer what you're trading is required delta v for time to be more efficient than the equivalent home and transfer the bile of the transfer has to take more time because you're going out to some distant point in space and coming back so in cases where you are incredibly fuel limited and can afford to wait and if your spacecraft has the required survivability then it's okay to go with a biologic transfer if you're trying to get somewhere fast then the home and transfer is your best bet and finally remember also that these are idealized models in reality you are never actually trying to create a completely coplanar change between two orbits real bodies in the solar system are not strictly co-planar even though they're quite near there are always going to be changes to other orbital parameters so as with everything this is a first cut model but it gives you an idea of the kind of delta v requirements that we will actually work out in the solution of the real problem