EN4 Dynamics and Vibrations Design Project
Contents
1. s laws Introduce the velocity of the satellite and moon as additional unknowns to write the equations of motion into the standard form required by MATLAB there will be a total of 12 unknowns Don t forget that the moon s gravity is weaker than the earth s Extend your MATLAB program to solve the new equations of motion Next you will need to determine initial conditions for this simulation The satellite will start from GTO perigee and its initial velocity can be computed using the results of step 5 You must also determine the initial position and velocity of the moon To find this find i the time f taken for the moon takes to travel from perigee to the impact location already determined in step 4 ii the time f taken for the satellite to travel from the appropriate point on the GTO orbit to the impact location determined in step 5 Then using your code that simulates the moon by itself start the moon from its perigee at t 0 and compute the position and velocity vectors at time t t The position and velocity of the moon at the end of this computation will be the initial conditions for the impact simulation Write an event function that will detect the time when the satellite hits the surface of the moon Run the code that simulates the satellite and moon together using initial conditions computed in step c and hence calculate the impact velocity i e the magnitude of the difference between the velocity of2. event indexevent i 1 for each value of i for the example above as there is only one type of event Create a plot that shows the satellite orbit and the lunar orbit on the same picture and shows the two potential impact points Neglect the effects of the moon s gravity on the satellite To do this e Solve the equation of motion for the satellite with appropriate initial conditions and plot it with figure hold on plot3 w_vals 1 w_vals 2 w_vals 3 e Solve the equation of motion for the moon with the appropriate initial conditions and plot it with plot3 w_vals 1 w_vals 2 w_vals 3 e Add the position of the moon at one of the impact points with e g rmoon wevent 1 1 3 plot3 rmoon 1 rmoon 2 rmoon 3 ro MarkerSize 11 MarkerFaceColor r and similarly for the second point It is helpful to color the two impact points differently so you can distinguish them It will also help you in later calculations if you can work out the direction of motion of the moon as it passes through each impact point 5 The satellite will be launched on a trajectory to Lumar orbit pine hit the moon by firing a rocket at some point around its orbit The rocket fires very quickly over a few minutes so its effect can simply be idealized as an impulse which changes the velocity of the satellite Trajectory The propulsion system will be used to simply increase the satellite s speed by some unknown amount AV without
3. but clear description of how results were calculated and correctly states results of calculations clearly written in good English 10 points Conclusions Concise summary of results of calculations 5 points References appropriate selection of source material properly cited 3 points Appendix Calculations all done correctly Calculations and MATLAB code described in sufficient detail that calculations can be checked and repeated Organized in a logical manner 17 points Presentation report layout typing quality of figures and tables 5 points Total 50 Points THE COVER PAGE OF YOUR REPORT MUST CONTAIN THE FOLLOWING STATEMENT This report has been prepared in accordance with the honor code of Brown University The report and MATLAB code are my own work 8 References Ariane V Users Manual http www arianespace com launch services ariane5 Ariane 5 User s Manual asp very large file 23MB Lunar orbit parameters http www usno navy mil USNO astronomical applications data services geo pos Ejecta plume simulations for Deep Impact mission http www astro cornell edu richardson ejector html NASA Deep Impact web site http www nasa gov mission_pages deepimpact main index html NASA LCROSS web site http Icross arc nasa gov APPENDIX A Data and Formulas A 1 The Geocentric Coordinate System coordinate frame which has a fixed orientation with respect to the nearby stars By convention the coordinate sy
4. giving two angles the right ascension of the moon the declination and the distance of the moon from the center of the earth These quantities are illustrated in the figure 1 The right ascension qa is specified in l hours minutes and seconds 24 hrs A i corresponds to 360 degrees 2 The declination is specified in degrees minutes and seconds Note that declination may be negative this means that degrees minutes and seconds are all negative be careful not to miss the negative sign 3 The radial coordinate p specifies the distance of the moon from the center of the earth Equatorial plane For the moon the angles are measured relative to the mean equator of the earth and the angle is measured relative to the Vernal Equinox direction HEALTH WARNING it is critical to use these tables correctly be particularly careful to identify the apogee and perigee correctly and to convert properly from degrees or hours minutes amp seconds to decimal if you get the lunar orbit wrong you won t be able to hit the moon The position vector of the moon in the fi jk basis can be calculated from these data using the following formula r p cos6 cos ai cos 6 sinaj sin k You can also identify the apogee and perigee of the moon s orbit from the data in the table The velocity vector of the moon at the perigee can be computed from Me tie v 2u Pa P Pp Pat Pp Pp where r is the position vector of the perigee
5. initial conditions for the moon or the satellite Also while testing this function you might get an error saying that wevent 1 1 3 is undefined or something like that This happens if MATLAB was unable to detect an event i e there was no point where the satellite got close to the impact point if this happens MATLAB does not assign a value to wevent and when you try to use the variable it gives the error If this happens use one of the handy swear words from the appendix Then simply comment out the last 3 lines of the function above add two lines that say d 1234 time_to_reach_min_point 4321 or substitute your favorite numbers and look at your predicted trajectories to find out what went wrong Once your function plots sensible trajectories the critical value of delta_V required to hit the moon can be found by plotting the minimum distance as a function of delta_V and identifying the point where the distance is minimum on your graph or by finding the min in your code this is sufficiently accurate for design purposes You will also need to calculate the time taken to reach the impact point You can do this using a simple loop which must be outside your mindisttomoon function for i 1 41 You can use more than 41 points if you like delta_V i enter formula here to vary delta V from 0 4 to 1 km s dist i traveltime i mindisttomoon delta_V i r0 v0 r_impact end plot dist delta_V distmin index m
6. orbit at which a satellite is furthest from the body it is orbiting Apogee the Apoapsis for a satellite orbiting the earth Apohelion the apoapsis for a satellite orbiting the sun GTO Geosynchronous Transfer Orbit LEO Low Earth Orbit Periapsis the point on an orbit at which a satellite is closest to the body it is orbiting Perigee the periapsis for a satellite orbiting the earth Perihelion the periapsis for a satellite orbiting the sun A7 A few useful swear words Bazdmeg Hungarian Dritt Norwegian Paska Finnish Do kurwy nedzy Polish Cach Welsh Drek Slovenian me cago en la hostia Spanish
7. p 9 are the radial coordinates of the apogee and perigee and m is a unit vector perpendicular to the plane of the orbit the sign of m must be chosen so that m k gt 0 don t forget this or the moon ends up moving backwards You can find the perigee in the table of positions by locating the point where the moon is closest to the earth You can find a unit vector normal to the moon s orbit by taking the cross product of the position vectors of two different points on the orbit make sure the two points are well separated don t forget to convert the cross product into a unit vector and be very careful to get the sign of the unit vector right if you get the sign wrong the moon orbits in the wrong direction and you won t be able to hit it DO NOT USE BOTH THE APOGEE AND PERIGEE FOR THE POINTS THAT DETERMINE THE NORMAL TO THE ORBIT can you see why A6 Useful data e Mean equatorial radius of the earth 6378 145km e Mean equatorial radius of the moon 1737 4km e Gravitational parameter u GM 3 986012 x 10 km s G gravitational constant M mass of earth e Ratio of moon s mass to earth s mass 0 012298 Orbit parameters for Ariane V GTO Inclination 7 degrees Altitude of perigee 250 km Altitude of apogee 35950 km Argument of perigee 178 degrees Longitude of first ascending node 180 degrees A6 List of geeky astronomical terms to impress your friends Apoapsis the point on an
8. the moon and the satellite at the instant of impact Note that everything has to work correctly for you to be able to impact the moon To debug you will find it helpful to animate the motion of the moon and satellite The following script will animate a trajectory assumed to be stored in a variable called ws with time values in times for i l length times CLE Clear the fram plot3 ws 1 ws 2 ws 3 SPlot th ntire trajectory hold on plot3 ws i 1 ws i 2 ws i 3 ro MarkerSize 10 MarkerFaceColor r pause 0 05 end You can modify this appropriately to show both the moon and the satellite 6 Report instructions PRELIMINARY REPORT Your preliminary report due Friday March 12 should consist of solutions to parts 1 3 of section 4 It should include e Free body diagrams for the satellite or the moon orbiting the earth e A derivation of the equation of motion for the satellite or moon orbiting the earth e MATLAB code that will simulate the motion of the satellite or moon around the earth e A graph of the total energy of the satellite as a function of time as the satellite completes at least 1 full orbit and a second graph of the total angular momentum of the satellite as a function of time for the same period of time e The predicted time for one complete orbit of the moon around the earth and a comparison of your prediction with the value determined from the lunar tables e Th
9. ATLAB scripts if you wish mi It is best not to make your report a blow by blow account of each individual calculation you did in the order that you did them In fact some calculations which you just ran to check your code need not be reported at all The purpose of your report is so that the reader can understand 1 why your results are important 2 What your conclusions are and 3 The reasoning or calculations that led to your conclusions The introduction should address 1 the mission calculations sections should address 3 and the conclusions should address 2 It is not easy to write good reports Often more effort goes into writing a report than the calculations themselves writing style grammar presentation and organization are all important If you get really good at report writing it will give your future career a huge boost Fortunately Brown is one of the best places in the world to learn how to write well 7 Final report grading rubric Grades will be based on the content of the final report as follows 1 2 3 i Abstract appropriate length accurately describes content of report clearly written 5 points Introduction Includes appropriate background information explains purpose of report gives outline of remainder of report clearly written in good English 5 points Mission calculations Calculations done correctly report organized so that important results can be found easily includes brief
10. EN4 Dynamics and Vibrations Design Project Orbital Design for a Lunar Impact Mission Synopsis NASA has identified a need for a low cost mission to launch a satellite that will impact the moon You will design the orbits for this mission and write a report summarizing your calculations and recommendations The project will give you some experience with realistic engineering analysis HEALTH WARNING i This is a challenging project don t put off starting on it to the last minute of course you would never do that but just in case and expect to get stuck The appendix lists useful swearwords in several languages 1 Background On July 4 2005 the highly successful Deep Impact mission crashed a spacecraft into Comet 9P Tempel 1 The ejecta plumes were monitored by space and earth based telescopes Various measurements such as spectroscopic analysis of the light flash the shape and size of the plume and the persistence of the gas and dust cloud left by the impact provided valuable scientific information concerning the composition of comets at least that s what the engineers told the government in order to get funding for the mission but the real truth is that engineers love crashing things and crashing a spacecraft is the most fun you can ever hope to have if all your friends are engineers A similar mission to impact the moon was proposed to NASA by Professor Schultz of the Planetary Geology department at Bro
11. alculations 2 Begin your calculations by writing a simple MATLAB script that will calculate the position vector of the satellite as a function of time in the Ariane GTO orbit The satellite orbit calculation done in class should be helpful in doing this calculation essentially you just need to extend this to 3D For this purpose Neglect the gravitation of the moon Assume that the earth is stationary Draw a free body diagram and obtain an equation of motion for the satellite Assume that at t 0 the satellite is at the perigee of the orbit with the correct initial velocity vector You can use the formulas in the Appendix to calculate the initial position and velocity from the data in the table in the preceding section don t forget to include the radius of the earth the altitude specified by Arianespace refers to altitude above the earth s surface e Write a MATLAB script that will integrate the equations of motion given the initial conditions Make the code plot the 3D trajectory of the satellite e Check your code by calculating and plotting the total energy and angular momentum of the satellite as functions of time and make sure they are approximately constant 3 Extend your program to simulate the motion of the moon by itself throughout this project you should just extend your code to do the new steps there is no need to write separate scripts for all the calculations To do this you will only need to call the ODE
12. ane The impact will be accomplished by firing a rocket that will increase the speed of the satellite without changing the direction of its velocity this is the most efficient way to use a rocket The increase in speed places the satellite on a trajectory that can be chosen to impact the moon As the orbital designer for this mission you have been charged with answering as many of the following questions as possible 1 Where are the two points where the lunar impact may take place 2 What are the dates and times that the moon will be at the impact points 3 What increase in satellite speed is required to hit the moon 4 At what point on the GTO orbit should the rocket be fired 5 How long does it take for the satellite to travel from the point where it leaves the GTO orbit to the moon 6 What is the resulting impact velocity with the moon All the necessary data to answer these questions is provided in this project description In addition a detailed set of instructions for doing the calculations is given in the next section In practice of course you would need to decide what data you need and then search for it you would also have to work out for yourself how to do the calculations 4 Detailed project instructions 1 Read the appendix to familiarize yourself with the terminology that is used to describe satellite orbits In particular make sure you are comfortable with the various coordinate systems used in astrodynamic c
13. changing its direction of motion The change in velocity required to hit the moon must be calculated For later calculations we must also determine how long it takes for the Lunarorbit satellite to reach the impact point Satellite orbit plane Potential impact point Shortest dist to impact point Potential impact point Initial satellite orbit To do this we will model the motion of the satellite as it moves from a pre selected point on the GTO orbit towards the impact point To begin with assume that the rocket is fired at the perigee of the satellite orbit This means that you know a the position vector and b the velocity of the satellite at the time the rocket is fired You can use these as initial conditions in your MATLAB code to calculate the subsequent trajectory of the satellite You don t need to account for the gravity of the moon in this calculation or to consider the motion of the moon the goal is simply to get the satellite from the perigee to the impact point It is helpful to start this calculation by creating a function that will a plot the trajectory of the satellite as it travels towards the moon and b calculate the closest approach distance of the satellite to the impact location as a function of the initial position and velocity of the satellite and the increase in velocity AV Your function which should be inside your main project function could look something like this This evaluates th
14. e function with delta_V 0 5 km sec just to test it out test_distance test_time mindisttomoon 0 5 r0 v0 r_impact function d time_to_reach_min_point mindisttomoon delta_V r0 v0 r_impact Function to calculate min distance to moon for satellite trajectory r0 and v0 are the position and velocity of the oe o9 o9 satellite just before the rocket is fired and delta_V is the increase in its velocity in km sec while r_impact is the position vector of the impact point computed in part 4 ae ol o9 time 8 24 60 60 8 days is enough to reach the moon w0 1 3 r0 Initial position of satellite w0 4 6 v0 delta_V v0 sqrt dot v0 v0 Initial velocity rtol 0 00000001 options odeset RelTol rtol Events min_dist Run the simulation to calculate the trajectory of the satellite t_vals w_vals tevent wevent ievent ode45 eom 0 time w0 options Plot the trajectory of the satellite on its way to the moon hold on plot3 w_vals 1 w_vals 2 w_vals 3 Find distance to impact point at the event oe r_impact is the position vector of the impact point computed in 4 rmin wevent 1 1 3 r_impact d sqrt dot rmin rmin min dist to moon time_to_reach_min_point tevent 1 Time to reach the min dist end Here eom is the equation of motion for the satellite which you coded in step 3 The Event function called min_dist which yo
15. e predicted distance of the moon from the earth at its perigee and apogee and a comparison with the values determined from the table FINAL REPORT Your report should be a formal presentation of your recommendations to management The main purpose of the report is to communicate clearly your findings A secondary purpose is to present enough details about your calculations that another engineer could check and repeat them Here is a possible outline for the report 1 Abstract a short summary of the content of the report In this project the conclusions are simple enough that you can report them in the abstract if you wish but that is often not possible 2 Introduction give a brief description of the mission explain the role of the report in mission planning outline the information contained in the report and summarize the main conclusions 3 Mission calculations this section should contain several sub sections that describe how you have addressed the questions listed in Section 3 In each case it is important to list any assumptions made in doing the calculations outline briefly how the calculations were done and present the results of the calculations 4 Conclusions this should clearly and concisely summarize your recommendations giving any quantitative information that you think is necessary for mission management References 6 Appendix here you can go nuts and include all the gory details of your calculations including M
16. ee The velocity vector of the satellite at the periapsis can be calculated from the following formula 1 2 Vp 2H a sin cosQ cos sin Qos 8 i cos cos Qcos 6 sin wsinQ j cos wsin Ok Tp Va rp where 44 GM is the gravitational parameter A 5 The Lunar orbit It is tricky to describe the orbit of the moon around the earth because the orbit is continuously changing with time For example the moon s orbital plane rotates Westward around the earth completing a full revolution in 18 6 years Similarly the line of apsides the line joining the perigee and the apogee of the orbit rotates with a period of 8 9 years This behavior is due partly to the gravitational effects of the sun and partly because the earth is not perfectly spherical and homogeneous so that it s center of gravity is not a fixed point Mission planning relies on detailed tables of lunar positions which can be found e g at http www usno navy mil USNO astronomical applications data services geo pos To use the tables select Astrometric Geocentric Right Ascension and Declination in the position menu and select Moon in the Celestial object box Set the Tabular Interval to hours and set the number of repetitions to 700 Then press compute data This will produce a table giving the position of the moon at hourly intervals over a 700 hour period should include a complete orbit The tables specify the position of the moon by
17. gle between the position vector of the moon and the normal is 90 degrees A formula for a unit vector normal to the satellite orbit is given in Section A2 e Add an Event function to your MATLAB code to locate the points where r n 0 Your event function could look something like this but think about how you could modify the code to identify the two impact points separately function event_val stopthecalc direction detect_impact_point t w Formula for unit vector normal to satellite orbit plane n enter your formula here Position vector of moon this assumes w 1 x w 2 y w 3 z r w 1 3 Detect when r n 0 event_val dot n 1r stopthecalc 0 direction 0 end e Run the code to determine the position vectors of the two potential impact points Also determine the dates and times that the moon will be in the correct position of course this will occur at many times during successive orbits you can create a table of possible impact dates and times to help mission management You should extract the time and location of the event from the ODE solver as follows options code to define the event function and tolerance t_vals w_vals tevent wevent indexevent ode45 etc etc The variable tevent will contain the time that the moon crosses each impact point and the variable wevent will contain the solution variable at these times while indexevent identifies the type of each
18. in dist Find the min value in the vector dist critical_delta_V_perigee delta_V index SPrint the magic delta V figure plot delta_V traveltime time_to_hit_from_perig traveltime index SPrint the travel time You will probably find that delta_V lies somewhere in the range 0 4 lt delta_V lt 1 km s 6 Repeat step 5 but this time start the satellite from the apogee of its orbit and try to hit the second ascending impact point For this purpose you will need to find the position and velocity vectors of the apogee of the satellite orbit This should be done by hand using conservation of angular momentum or energy and some elementary geometry For this case you can assume that delta_V lies somewhere in the range 1 lt delta_V lt 3 km s Note that you don t need to write another minimum distance function for this calculation all you need to do is to change the values of r0 v0 r_impact 7 Finally you should calculate the impact velocity with the moon more accurately For this purpose you will need to modify your script to account for the effects of the moon s gravity on the motion of the satellite it is not necessary to account for the effects of the satellite on the moon This means you must solve for the motion of the moon and the satellite at the same time a Derive equations of motion for the moon and the satellite choose a coordinate system draw a b c d e FBD and use Newton
19. on the orbit an onboard rocket will be fired to place the satellite on a trajectory to hit the moon Your primary Lunar orbit mission is to determine the point at which the rocket should be fired Initial satellite orbit Potential impact point A commercially available launch vehicle Ariane V will be used to place the satellite into its initial orbit Full details of this launch facility can be found in the huge Ariane V user manual Ariane V offers two orbits a Low Earth Orbit LEO which is not suitable for this mission or a more suitable Geosynchronous Transfer Orbit GTO Arianespace give details of the geometry of this orbit in their manual using the standard terminology for satellite orbits This terminology is described briefly in Appendix A and the parameters for the orbit can be found in Section A5 After launch the satellite will be allowed to complete several orbits of the earth to give mission controllers a chance to check on board systems and verify that the satellite is in the correct orbit It must then use an on board propulsion system to navigate to the moon and hit it Hitting the moon is complicated by the fact that the Ariane V GTO is not co planar with the moon s orbit A very powerful and expensive propulsion system is required to change the plane of a satellite s orbit so it is preferable to attempt to hit the moon at one of the two special points where the moon s orbit crosses the satellite s orbit pl
20. solver again with the appropriate initial conditions for the moon the equations of motion of the moon by itself are the same as those for the satellite Take the position vector and velocity vector of the moon at its perigee as the initial conditions You can calculate these using the procedure in Sect A5 Check your code by calculating the following e The time for one complete orbit of the moon around the earth you can check this against the tabulated data e The distance of the moon from the earth at its perigee and apogee 4 Note that the moon and the satellite do not Lunar orbit plane orbit the earth in the same plane This makes Normal to satellite orbit hitting the moon more difficult In principle Satellite orbit plane it is possible to change the plane of the Potential impact point satellite s orbit but this requires a very large and prohibitively expensive propulsion system It is better to try to hit the moon at one of the two special points where the moon s orbit crosses the plane of the satellite orbit You can use your MATLAB code to calculate the position vectors of these points For this purpose e Note that when the moon is in the plane of the satellite s orbit its position vector must satisfy r O where n is a unit vector normal to the plane of the satellite s orbit The figure shows ra z Potential impact point Lunar orbit Satellite orbit why this is the case at the special points the an
21. stem shown in the figure is used for this purpose For astrodynamical calculations it is important to use a Vernal equinox We choose three unit vectors i j k with k parallel to the Celestial North pole of the earth i e parallel to its axis of rotation and i parallel to the so called Vernal Equinox direction The Vernal Equinox direction i is illustrated in the figure At each equinox roughly March 12 and Sept 21 the sun lies in the equatorial plane of the Earth that is why the day and night have the same length The Vernal equinox direction is a unit vector pointing from the center of the earth towards the sun at the instant that the Vernal Equinox occurs in March The remaining unit vector j k xi Note that the i j vectors lie in the plane of the equator and eee eee lt j Orbit direction Earth at automnal equinox that the earth rotates with respect to the fi jk directions The origin for specifying position vectors is taken to be at the center of mass of the earth A 2 Specifying the geometry of a satellite orbit Orbit plane Descending tiode Line of nodes A A generic orbit is illustrated in the figure a Perigee The following are special points on the orbit Equatorial plane 1 The periapsis or perigee for an earth orbit is the point where the satellite Orbit dma is closest to the object it is orbiting 2 The apoapsis or apogee for an earth 3 orbit is the poin
22. t where the satellite is furthest from the object it is orbiting pozes 3 The line of nodes connects the two Ascending node points where the orbit intersects the i j plane These two special points are called nodes the ascending node is the point where the satellite crosses the i j plane with a positive velocity parallel to the k direction while the descending node is the other point The geometry of an orbit is described using the following numbers 1 2 3 4 The inclination of the orbit 0 is the angle between the plane of the orbit and the i j plane The altitude of the periapsis r is the distance from the surface of the planet to the periapsis The altitude of the apoapsis r is the distance from the surface of the planet to the apoapsis The longitude of the ascending node is the angle between the i direction and the ascending node The argument of the periapsis is the angle between the line of nodes and the position vector of the periapsis A table of the values of these parameters for the ARIANE GTO orbit can be found in Section A5 The position vector of the periapsis can be calculated in terms of these quantities as Ip p cos cos Q sin sin Qcos O i cos sinQ sin cosQcos 0 j sin wsin Ok It is also useful to note that n sinQsin Oi cos Qsin 0j cos Ok is a unit vector perpendicular to the plane of the orbit A 3 Calculating the velocity of the satellite at perig
23. tellite and the moon Your initial calculations should be submitted in a preliminary report see Sect 6 for details c Use your program to calculate critical mission parameters such as dates that the impact can take place the position of the impact the trajectory of the satellite necessary to hit the moon and the impact velocity d Write a formal report describing your calculations and presenting recommendations for the mission management COLLABORATION You may discuss your calculations with others but the report and MATLAB code that you submit for grading must be your own work DUE DATES Preliminary report due Tues March 15 Final report due Tues March 22 NOTE EXTENSION IN DUE DATES HELP Section and class hours on Wednesday March 9th will be used to provide help on this project Also we will not have a formal lecture on Thursday March 10 but people will be available in computer lab and in room 166 to help debug preliminary code Some additional weekend hrs office hrs will also be announced by email and on the course website 3 Overview of the mission Lunar orbit plane Sat i ce i a The mission has three stages ee Satelite oroit piana Cirit direction Potential impact poimi i The satellite will first be placed into an initial orbit around the earth see the figure ii The satellite will be permitted to orbit the earth a few times to check systems and to verify the orbit iii At a pre selected point
24. u must write and which must be nested inside the mindisttomoon function should identify the point where the trajectory is closest to the impact point This is a bit tricky because you need to find a way to use compute a value for the event_value variable in the event function that goes to zero at the point where the satellite passes closes to the moon Note that at this special point the position and velocity of the satellite must satisfy v 0 r Timpact where Fjmpact 18 the position vector of the impact point as calculated in step 4 r is the position vector of the satellite and v is the velocity of the satellite You can use r Timpact v as your event_value variable Note that you can only hit one of the two candidate impact points from perigee The moon should be descending i e crossing the equatorial plane from North to South at the correct impact point You will find it extremely helpful to test this function by plotting the trajectory of the satellite on its way to the impact point for several different values of delta_V try values between 0 4 and 1 Plot the trajectory on top of the plot of the moon orbit and impact points that you completed in step 4 If the satellite does not look as though it passes very close to one of the impact points for some choice of delta_V something is wrong and it s not worth continuing until you fix it The most likely error is that you have made a mistake in calculating the
25. wn The goal of this mission was to provide similar data for an impact into the silicate rich surface of the moon Scientific data that could be gathered from the mission could e Calibrations that can be used to interpret data from natural lunar impact flashes Comparison with the flash generated by Deep Impact The first direct measurements of luminous efficiency from a lunar impact Spectral emission data from the peak intensity Evaluation of crater scaling relations for asteroid impacts with the moon Some years ago a group of Senior Brown engineering students Bobby Legge Dan Manian Andrew Lind and Ross Barney completed a full conceptual design for this mission NASA followed these as well as recommendations from many other teams working on similar designs to complete the LCROSS lunar impact mission in the Fall of 2009 Detecting the presence of water on the moon was one of the major discoveries of the mission In this project you will complete a small part of this project by designing the trajectory of the spacecraft Of course NASA and other have written extensive special purpose software to do their orbit calculations see e g http www stk com but here you will write your own code using MATLAB 2 Summary of project requirements a Complete background reading and research to become familiar with some basic terminology of orbital mechanics and the lunar orbit b Write a MATLAB program that will calculate the orbits of the sa
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