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Solving the two body Lambert problemLambert’s problem is concerned with the determination of an orbit that passes between two positionswithin a specified time-of-flight. This classic astrodynamic problem is also known as the orbital twopointboundary value problem (TPBVP).The time to traverse a trajectory depends only upon the length of the semimajor axis a of the transfertrajectory, the sum r i r fof the distances of the initial and final positions relative to a central body, andthe length c of the chord joining these two positions. This relationship can be stated as follows:From the following form of Kepler’s equationtof tof r r , c,aifwe can write3at t0 E esinE3at E E esinE sin E 0 0where E is the eccentric anomaly associated with radius r, E0is the eccentric anomaly at r 0, and t 0when r r0.At this point we need to introduce the following trigonometric sun and difference identities: a sinsin 2sin cos2 2 a cos cos 2sin sin2 2 a coscos 2cos cos2 2If we let E and E0 and substitute the first trig identity into the second equation above, we havethe following equation:With the two substitutions given by3a E E0 E E0t E E0 2sin ecos 2 2page 16
EE0 ecos cos2 2EE0 sin sin2 2the time equation becomesFrom the elliptic relationships given by3a t 2sin cos 2 2 r a 1ecosEx a cos E ey a sin E 1e2and some more manipulation, we have the following equations: r r c r r c s 2a 2a 2a a0 0cos1 1 1 r r c r r c s c 2a 2a 2a a0 0sin 1 1 1This part of the derivation makes use of the following three relationships: rrcos cos 12 2 202E E0 E E0 e sin sin sin 1 cos2 2 2 22 2 2 2x x0 y y0c sinsin 2 2 2a 2a 2aWith the use of the half angle formulas given bysins sins c2 2a2 2aand several additional substitutions, we have the time-of-flight form of Lambert’s theorempage 17
Page 1 and 2: A Computer Program for Trans-Lunar
Page 3: The software allows the user to spe
Page 6 and 7: ===================================
Page 8 and 9: sma (km) eccentricity inclination (
Page 10 and 11: Z coordinate (ER)vx (kps) vy (kps)
Page 12 and 13: Z coordinate (MR)The following is a
Page 14 and 15: vx (kps) vy (kps) vz (kps) vmag (kp
Page 18: 3at sin sin A discussion a
Page 21 and 22: where rmmbremam r rmb emr, t m3 3m
Page 23 and 24: The software solves the B-plane tar
Page 25 and 26: prca0cosihˆ0where rpand i are the
Page 27 and 28: yˆpˆxˆFinally, the components of
Page 29 and 30: Algorithm resources“Lunar Traject
Page 31 and 32: theta = orientation of the b-plane

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