ASEN 5050 SPACEFLIGHT DYNAMICS Conversions, f & g, Orbit Transfers Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 9: Conversions, Orbit Transfers 1 Announcements • Homework #3 is due right now – You must write your own code. – For this HW, please turn in your code (preferably in one text/Word/PDF document) – After this assignment, you may use Vallado’s code, but if you do you must give him credit for work done using his code. If you don’t, it’s plagiarism. • Homework #4 is due Friday 9/26 at 9:00 am – You’ll also have to turn in your code for this one. • No Quiz over the weekend! Enjoy your weekend. • I’ll be at the career fair Monday, so I’m delaying Monday’s office hours to 2:00. • Reading: Chapter 6 (SIX, we jumped a few) Lecture 9: Conversions, Orbit Transfers 2 Concept Quiz 7 ✔ ✔ ✔ ✔ ✔ Lecture 9: Conversions, Orbit Transfers 3 Concept Quiz 7 Lecture 9: Conversions, Orbit Transfers 4 Concept Quiz 7 The class is 50-50 split on this! Talk it over with your neighbor and convince him/her why you’re right. Lecture 9: Conversions, Orbit Transfers 5 Concept Quiz 7 Lecture 9: Conversions, Orbit Transfers 6 Quizzes • Speaking of quizzes, I had a bug in my grade book and only the first two quiz scores were shown. As of now, Quiz 1-6 should be shown. Lecture 9: Conversions, Orbit Transfers 7 Space News • Sunday: MAVEN arrives at Mars! • MOI: this Sunday at 19:37 Mountain • LASP is holding a viewing party Lecture 9: Conversions, Orbit Transfers 8 Space News Lecture 9: Conversions, Orbit Transfers 9 Space News Lecture 9: Conversions, Orbit Transfers 10 Space News • Then Wednesday: MOM arrives at Mars! • MOI: Wednesday at 21:00 Mountain – It will enter occultation at 21:04 – MOI will end at 21:24 – We’ll know if it’s successful around 21:30 Lecture 9: Conversions, Orbit Transfers 11 Space News Lecture 9: Conversions, Orbit Transfers 12 Challenge #4 • If you were to plot the position and velocity of a satellite over time using VNC (Velocity-NormalConormal) coordinates, what would you find? – Say, an elliptical orbit V Lecture 9: Conversions, Orbit Transfers C 13 ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate Transformations Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 9: Conversions, Orbit Transfers 14 Principal Axis Rotations Lecture 9: Conversions, Orbit Transfers 15 Principal Axis Rotations rX ¢ = rX cosq + rY sinq Y¢ rY ¢ = -rX sinq + rY cosq rZ ¢ = rZ é cosq ê rX ¢Y ¢Z ¢ = ê -sin q ê 0 ë sin q cosq 0 écosq ê ROT 2(q ) = ê 0 êë sin q Lecture 9: Conversions, Orbit Transfers 0 ù ú 0 ú rXYZ = ROT 3(q )rXYZ 1 úû 0 -sin q ù ú 1 0 ú 0 cosq úû Y q rX rXYZ q é1 0 ê ROT1(q ) = ê0 cosq êë0 -sin q Equation 3-15 X¢ rY ¢ rX ¢ rY X 0 ù ú sinq ú cos q úû 16 r and v From Orbital Elements Express the position and velocity in the perifocal system ( xˆ goes through periapse, zˆ in the direction of h , y perpendicular to xˆ and zˆ , in the orbit plane.) Y rPQW v PQW Qˆ é p cosn ù ê1 + e cosn ú ê ú ˆ = ê p sinn ú = r cosn Pˆ + r sinn Q ê1 + e cosn ú ê ú ê ú 0 ê ú ë ûPQW ér cosn - rn sinn ù = êr sinn + rn cosn ú ê ú 0 ëê ûú PQW Lecture 9: Conversions, Orbit Transfers rn = r= y = rsin n n m p(1+ ecos n ) h m = = (1+ ecos n ) r p p rn esin n m = (esin n ), 1+ ecos n p rPQW thus X x = r cosn Pˆ é ê ê ê =ê ê ê ê ê ë ù ú sin n ú p ú m ú (e + cos n) ú p ú 0 ú ú û v PQW m 17 PQW r and v From Orbital Elements Now we simply need to rotate into the geocentric equatorial system. The order of the rotations does matter Y Qˆ é IJK ù rIJK = ROT 3 ( -W) ROT1 ( -i ) ROT 3 ( -w) rPQW = ê ú rPQW PQW ë û é IJK ù v IJK = ROT 3 ( -W) ROT1 ( -i ) ROT 3 ( -w) v PQW = ê ú v PQW ë PQW û rPQW n é cos W cosw - sin W sin w cosi -cos W sin w - sin W cosw cosi sin W sini é IJK ù ê ê ú = ê sin W cosw + cos W sin w cosi -sin W sin w + cos W cosw cosi -cos W sini ë PQW û ê sin w sini cosw sini cosi ë Pˆ ù ú ú ú û Algorithm 10 in book. Ex. 2-6 Lecture 9: Conversions, Orbit Transfers 18 X f and g Series r = xPˆ + yQˆ v = x˙Pˆ + y˙ Qˆ Lecture 9: Conversions, Orbit Transfers r = fro + gv o v = fro + gv o 19 f and g Series Start by crossing the position vector into the initial velocity vector: r ´ vo = ( fro + gvo )´ vo = f (ro ´ vo ) + g(v o ´ vo ) The second term is zero, and the other terms are normal to the plane: xy˙ o - x˙ o y = fh xy˙ o - x˙ o y f = h Differentiating this last equation: ˙f = x˙y˙ o - x˙ o y˙ h Lecture 9: Conversions, Orbit Transfers 20 f and g Series Now cross the initial position vector into the position vector: ro ´ r = ro ´ ( fro + gv o ) = f (ro ´ ro ) + g(ro ´ vo ) The first term is zero, and the other terms are normal to the plane: x o y - xy o = gh x o y - xy o g= h Differentiating this last equation: x o y˙ - x˙y o g˙ = h Lecture 9: Conversions, Orbit Transfers 21 f and g Series Look at the cross-product: h = r ´ v = ( fro + gv o )´ ( fro + gv o ) = ff (ro ´ ro ) + fg(ro ´ v o ) + fg(v o ´ ro ) + gg(v o ´ v o ) = fgh - fgh Which can only be true if: fg - fg =1 x = r cosn x=Lecture 9: Conversions, Orbit Transfers m p sin n A good test! y = r sin n y= m p (e + cos n ) 22 f and g Series Which gives: Lecture 9: Conversions, Orbit Transfers 23 f and g Series So, to summarize, given an initial position and velocity, we can calculate a future position and velocity given the change in the true anomaly Dn: Which you can test using Example 2-4 in the textbook. Lecture 9: Conversions, Orbit Transfers 24 f and g Series: State Transition Matrix We can re-express our f and g series representation: r = fro + gv o v = f˙r + g˙ v o o in terms of a state-variable relationship: ér ù X =ê ú ëv û é ro ù éF Xo = ê ú X = FX o = ê ëv o û ëF˙ é f 0 0ù ê ú where F = ê 0 f 0 ú, etc., êë 0 0 f úû and F is the State Transition Matrix Lecture 9: Conversions, Orbit Transfers Gù Xo ú G˙ û 25 f and g Series • What uses do these functions have? – Given two states, find the time of flight between them. – Given two states, find an orbit that connects them. • Big fan of this application. – Using an iterative technique, such as Newton Raphson, can determine a future state given a current state and a transfer time or transfer angle. Lecture 9: Conversions, Orbit Transfers 26 ASEN 5050 SPACEFLIGHT DYNAMICS Orbital Maneuvers Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 9: Conversions, Orbit Transfers 27 Orbital Maneuvers • Orbital maneuvers are used to do many things: – Change a satellite’s orbit • Size • Shape • Orientation – – – – – Change the phase of a satellite in its orbit Rendezvous and/or proximity operations Avoid collisions (debris) Change the satellite’s groundtrack Etc. Lecture 9: Conversions, Orbit Transfers 28 Terminology • Coplanar maneuvers: no change to the orbit plane; the maneuvers can only change a, e, w. • Impulsive maneuvers: instantaneous change in velocity: ΔV – Requires an infinitely powerful engine • Finite maneuvers: maneuvers that require a duration of time to achieve • Ballistic: the trajectory of an object under the effects of only external forces (no maneuver firings). Lecture 9: Conversions, Orbit Transfers 29 Orbital Maneuvers Tangential Burns: in velocity/anti-velocity vector direction - Do not change velocity orientation, just magnitude - Do not change flight path angle Lecture 9: Conversions, Orbit Transfers 30 Orbital Maneuvers Nontangential: plane changes, orbit rotations Lecture 9: Conversions, Orbit Transfers 31 Orbital Maneuvers Hohmann Transfer – Walter Hohmann (1880-1945) showed minimum energy transfer between two orbits used two tangential burns. Lecture 9: Conversions, Orbit Transfers 32 Hohmann Transfer (math, compute DV and DT) Lecture 9: Conversions, Orbit Transfers 33 Hohmann Transfer atrans rinitial + rfinal = 2 v initial = v final = m rinitial m rfinal Dv a = v transA - v initial Dv total = Dv a + Dv b t trans 3 Ptrans atrans = =p 2 m v transA = v transB = 2m rinitial 2m rfinal - m atrans m atrans Dv b = v final - v transB (Algorithm 36, Example 6-1) Can also be done using elliptical orbits, but must start at apogee or perigee to be a minimum energy transfer. Lecture 9: Conversions, Orbit Transfers 34 Hohmann Transfer • We just argued that the Hohmann Transfer is (usually) the most energy-efficient orbital transfer. • Why? – Consider Elliptical—Elliptical transfer – Tangential Burns – Energy efficiency considerations 2 V m E= 2 R dE =V Þ dV Lecture 9: Conversions, Orbit Transfers DE » V × DV V is highest at perigee DV 2 DE = V × DV + 2 35 Energy Changes Lecture 9: Conversions, Orbit Transfers 36 Orbital Maneuvers Bi-elliptic Transfer – Uses two Hohmann transfers. Can save Dv in some cases. rb must be greater than rfinal, but can otherwise be optimized. Lecture 9: Conversions, Orbit Transfers 37 Bi-elliptic Transfer atrans 1 rinitial + rb = 2 v initial = v trans 1 B = v trans 2 C = atrans 2 m v trans 1 A = rinitial 2m rb - 2m rfinal m v trans 2 B = atrans 1 - m v final = atrans 2 Dv a = v trans 1 A - v initial Dv c = v final - v trans 2 C t trans = p 3 atrans 1 m rb + rfinal = 2 +p 2m rinitial 2m rb - - m atrans 1 m atrans 2 m rfinal Dv b = v trans 2 B - v trans 1 B Dv total = Dv a + Dv b + Dv c 3 atrans 2 (Algorithm 37, Example 6-2) m Much longer flight times for bi-elliptic transfer, but sometimes less energy. Lecture 9: Conversions, Orbit Transfers 38 Hohmann vs Bi-elliptic Lecture 9: Conversions, Orbit Transfers 39 One-Tangent Burns Lecture 9: Conversions, Orbit Transfers 40 Orbit Transfer Comparison Lecture 9: Conversions, Orbit Transfers 41 Changing Orbital Elements • • • • • • Δa Hohmann Transfer Δe Hohmann Transfer Δi Plane Change ΔΩ Plane Change Δω Coplanar Transfer Δν Phasing/Rendezvous (later discussion) Lecture 9: Conversions, Orbit Transfers 42 Changing Inclination • Δi Plane Change • Inclination-Only Change vs. Free Inclination Change Lecture 9: Conversions, Orbit Transfers 43 Changing Inclination • Let’s start with circular orbits Vf V0 Lecture 9: Conversions, Orbit Transfers 44 Changing Inclination • Let’s start with circular orbits Vf V0 Lecture 9: Conversions, Orbit Transfers 45 Changing Inclination • Let’s start with circular orbits Are these vectors the same length? Vf Δi V0 What’s the ΔV? Is this more expensive in a low orbit or a high orbit? Lecture 9: Conversions, Orbit Transfers 46 Changing Inclination • More general inclination-only maneuvers Where do you perform the maneuver? How do V0 and Vf compare? What about the FPA? Line of Nodes Lecture 9: Conversions, Orbit Transfers 47 Changing Inclination • More general inclination-only maneuvers Lecture 9: Conversions, Orbit Transfers 48 Changing The Node Lecture 9: Conversions, Orbit Transfers 49 Changing Argument of Perigee Lecture 9: Conversions, Orbit Transfers 50