Mechanics and Mechanical Engineering Vol. 15, No. (011) 183 191 c Technical University of Lodz On The Generalized Hohmann Transfer with Plane Change Using Energy Concepts Part I Osman M. Kamel Astronomy and Space Science Dept. Faculty of Sciences, Cairo University, Egypt Adel S. Soliman Theoretical Physics Dept. National Research Center, Egypt assoliman16@hotmail.com Received (10 October 010) Revised (5 May 011) Accepted (8 August 011) We investigate in this article the optimized orbit transfer of a space vehicle, revolving initially around the primary, in a similar orbit to that of the Earth around the Sun, in an elliptic trajectory, to another similar elliptic orbit of an adequate outer planet. We assume the elements of the initial orbit to be that of the Earth, and the elements of the final orbit to be that of an outer adequate planet, Mars for instance. We assume the elements of the two impulse Hohmann generalized configuration (the case of elliptic, non coplanar orbits) to be a 1, e 1, a, e, a T, e T. From the very beginning, we should assign θ = α 1 + α, the total plane change required. α 1 is the plane change at the first instantaneous impulse at peri apse, which will be minimized, and α the plane change at the second instantaneous thrust at apo apse. Keywords: Orbital mechanics, elliptic Hohmann transfer with plane change, optimization 1. Introduction The Hohmann transfer is the minimum two impulse transfer between coplanar circular and elliptic orbits [1]. As for the derivations of the velocity change requirements V 1, V and transfer time, we can draw a graph which illustrates total energy/satellite mass as a function of orbit period P = πa3/ µ that means a plot of ( ) µ a versus π µ a 3/. Plotted results are extensively established [1], []. For classical Hohmann transfer if r r 1 15.58, r r 1, is not satisfied, then the Hohmann transfer is no longer optimal. For these conditions Bi elliptic transfers are always more economical in propellant than Hohmann transfer configurations [1]. The Hohmann
184 Kamel, OM and Soliman A transfer is a relatively simple maneuver, especially the classical model. It may be simplified or complicated easily, and we may encounter very difficult situations. This can be easily seen from the literature of orbit transfer [3]. There exist four feasible Hohmann configurations according to the coincidence of peri apse and apoapse of the three ellipses. We consider the first of them [4]. Radius or major axis change, in the process of orbit transfer may be coupled by a plane change for the circular or elliptic orbit transfer. This is an important practical procedure. The optimal two impulse transfer that satisfy these conditions is the Hohmann transfer, with split plane change. The first V 1 thrust not only produces a transfer ellipse but also induce a rotation of the orbital plane. At the second impulse, a second tilt is induced as well as the production of the final elliptic orbit. An engine firing in the out of plane direction is required for the change of plane. The point of firing becomes a point in the new orbit, and the burn point becomes the intersection of the current orbit and the desired orbit. Definitely, we should perform plane change in the smartest way, since it is fuel expensive, anyway you do them. Even without the examination of the specific equations, planning a space mission, reduces to a problem of geometry, timing, mechanics of orbital motion, and a lot of common sense.. Method and Results In this article we investigate the generalized Hohmann orbit transfer with split plane change. We take into account, the first configuration, where the apo apse of the transfer orbit coincides with the apo apse of the final orbit, and the peri apse of the initial and the transfer orbit are coincident [5], Fig. 1. V 1 produces at peri apse of initial orbit, a transfer ellipse as well as a plane change α 1. Similarly at apo apse, V rotates the orbit plane through an angle α = θ α 1, and designs the final elliptic orbit as shown in Fig., Fig. 3 and Fig. 4, represent the velocity vector triangular addition V 1, V respectively. Figure 1 Generalized Hohmann Transfer
On The Generalized Hohmann Transfer with Plane... 185 Plane Change at first Imp. Final OrbitOrbt Initial Orbit Plane Change at second Imp. Fig.: Generalized Hohmann transfer with plane change Figure V P tr. á 1 v1 ÄV 1 Fig. 3 V Pi Figure 3
186 Kamel, OM and Soliman A V Af vaf v ÄV è-á 1 V Atr vatr Fig.4 where Figure 4 µ (1 + e 1 ) V Pi = a 1 (1 e 1 ) µ (1 + e T ) V Ptr = a T (1 e T ) µ (1 e T ) V Atr = a T (1 + e T ) µ (1 e ) V Af = a (1 + e ) The increments of velocities at peri apse and apo apse of the elliptic transfer orbit is given by V1 = µ (1 + e 1) a 1 (1 e 1 ) + µ (1 + e T ) { µ (1 + e1 ) a 1 (1 e 1 ) a T (1 e T ) } { µ (1 + et ) a T (1 e T ) } cos α 1 (1) V = µ (1 e ) a (1 + e ) + µ (1 e T ) a T (1 + e T ) { } { } µ (1 e ) µ (1 et ) cos (θ α 1 ) () a (1 + e ) a T (1 + e T ) V T = V1 + V = V 1 + V (3)
On The Generalized Hohmann Transfer with Plane... 187 where θ is arbitrary and given by θ = α 1 + α and e T, a T are calculated from the formulae e T = [a (1 + e ) a 1 (1 e 1 )] [a (1 + e ) + a 1 (1 e 1 )] = b 4 b 1 b 4 + b 1 (4) a T = [a (1 + e ) + a 1 (1 e 1 )] = b 4 + b 1 (5) where b 1 = a 1 (1 e 1 ) b 4 = a (1 + e ) α 1 to be optimized by the condition of minimization V T α 1 = 0. When checking the second order conditions, we treat the functional V [y] as a function of ε i.e. V (ε). We set dv dε = 0 and consequently, we acquire the first order necessary condition for an extremal. For distinction between maximization and minimization we calculate the second derivative d V dt. We find the following second order necessary conditions: d V dt 0 for maximize of V, d V dt 0 for minimize of V. As for second order sufficient conditions: d V dt < 0 for maximize of V, d V dt > 0 for minimize of V, cf. [5] for definitions. For the first configuration Let A 1 = µ ( ) 1 + e1 a 1 1 e 1 C 1 = µ ( ) 1 e a 1 + e B 1 = µ ( ) 1 + et a T 1 e T D 1 = µ ( ) 1 et a T 1 + e T (6) whence V T = ( A 1 + B 1 A 1 B 1 cos α 1 ) 1/ (7) ( + C 1 + D 1 ) 1/ C 1 D 1 cos (θ α 1 ) By partial differentiation with respect to α 1 and equating to zero, and after rearrangements and clearing fractions, we find that A 1 B 1 sin α 1 A 1 + B 1 C 1 D 1 sin (θ α 1 ) = A 1 B 1 cos α 1 C 1 + D 1 C 1 D 1 cos (θ α 1 ) (8)
188 Kamel, OM and Soliman A From which we can deduce [ A 1 B 1 C 1 + D 1 ( C 1 D 1 bx + a )] (1 1 x x ) (9) = C 1 D 1 [A 1 + B 1 ] [ (a A 1 B 1 x b ) x + b abx ] 1 x where: x = cos α 1 1 x = sin α 1 a = sin θ b = cos θ (10) After some reductions, we find A 1 B 1 (C 1 + D 1 ) C 1 D 1 (A 1 + B 1 ) b ) + (C 1 D 1 A1 B 1 b ba 1 B 1 C1 D 1 x { A 1 B 1 (C 1 + D 1 ) + C 1 D 1 (A 1 + B 1 ) ( a b )} x { ( + ba 1 B 1 C1 D 1 + C 1 D 1 A1 B 1 a b )} x 3 = 1 x {aa 1 B 1 C1 D 1 abc 1 D 1 (A 1 + B 1 ) x ) + (4abC 1 D 1 A1 B 1 aa 1 B 1 C1 D 1 x } Let A 1 B 1 (C 1 + D 1 ) C 1 D 1 (A 1 + B 1 ) b = E 1 C 1 D 1 A1 B 1 b ba 1 B 1 C1 D 1 = E {A 1 B 1 (C 1 + D 1 ) + C 1 D 1 (A 1 + B 1 ) ( a b ) } = E 3 ( ba 1 B 1 C1 D 1 + C 1 D 1 A1 B 1 a b ) = E 4 aa 1 B 1 C1 D 1 = E 5 abc 1 D 1 (A 1 + B 1 ) = E 6 4abC 1 D 1 A1 B 1 aa 1 B 1 C1 D 1 = E 7 i.e. E 1 + E x + E 3 x + E 4 x 3 = 1 x ( E 5 + E 6 x + E 7 x ) (11) After squaring and some reductions, we may write ( E 1 E5 ) + (E1 E E 5 E 6 ) x + ( E 1 E 3 + E E 5 E 7 E6 + E5 ) x + (E 1 E 4 + E E 3 E 6 E 7 + E 5 E 6 ) x 3 (1) + ( E E 4 + E3 E7 + E 5 E 7 + E6 ) x 4 + (E 3 E 4 + E 6 E 7 ) x 5 + ( E 4 + E 7) x 6 = 0
On The Generalized Hohmann Transfer with Plane... 189 Set E 1 E 5 = Ψ 1 E 1 E E 5 E 6 = Ψ E 1 E 3 + E E 5 E 7 E 6 + E 5 = Ψ 3 E 1 E 4 + E E 3 E 6 E 7 + E 5 E 6 = Ψ 4 E E 4 + E 3 + E 5 E 7 + E 6 E 7 = Ψ 5 E 3 E 4 + E 6 E 7 = Ψ 6 ; E 4 + E 7 = Ψ 7 That means after some reductions, we get an equation of degree 6 in x = cos α 1 : ψ 1 + ψ x + ψ 3 x + ψ 4 x 3 + ψ 5 x 4 + ψ 6 x 5 + ψ 7 x 6 = 0 (13) where the E s appearing in the ψ s are functions of one parameter α 1 and could be expressed in terms of a 1, a, e 1, e [6], which are constants since they are the known elements of the terminal orbits. 3. Discussion The Hohmann transfer is an optimal two impulse transfer. We suppose that the first increment at peri apse V 1, not only produces a transfer elliptic orbit, but also rotates the orbital plane by an optimal angle α 1. At apo apse the second increment of velocity V will produce the trajectory of the final elliptic orbit and rotates the orbit plane by an angle α = Θ α 1. We have V T = V 1 + V. For the minimization of V T we have the condition V T / α 1 = 0. By expansions, rearrangements and clearing fractions, we acquire through a purely analytical method, except for the resolution of the algebraic sixth degree equation, the value of the optimized α 1 i.e. (α 1 ) opt, whence (α ) opt = Θ (α 1 ) opt. By substitution of (α 1 ) opt, we can easily evaluate ( V 1 ) opt and ( V ) opt from Eqs 1,, and ( V T ) Min = ( V 1 ) opt + ( V ) opt [6]. The sixth degree algebraic equation O(x 6 ) Eq.13 could be easily solved by a mathematica software program. In this article we laid the foundation of the theory. The approach is purely analytical except the solution of Eq. 13 which is numerical. No numerical illustrations are included, for the time being, for instance for the obtenition of numerical results for the elliptic Hohmann Earth outer planets, configurations. This will be done in the second part of the paper. We restrict ourselves here to the consideration of the first configuration. References [1] Prussing, J.E. and Conway, B.A.: Orbital Mechanics, Oxford University Press, 1993. [] Ehricke, K.A.: Space Flight, vol. : Dynamics, Van Nostrand, 196. [3] Lawden, D.F.: Optimal Transfers Between Coplanar Elliptical Orbits, J. Guidance, vol. 15, No. 3: Engineering notes, 1991. [4] Kamel, O.M. and Soliman, A.S.: On transfer between elliptical coplanar coaxial orbits, Bulletin of the Faculty of Sciences, Cairo University, vol. 67, 1999.
190 Kamel, OM and Soliman A [5] Chiang, A.C.: Elements of Dynamic Optimization, McGraw Hill, 199. [6] Chobotov, V.A.: Orbital Mechanics, nd ed. AIAA, Education Series, 1996. [7] Eagle, C.D.: [Internet : Windows 95/98/NT application (Hohmann exe)], 003. Appendix David Eagle [7], exposed an excellent clear topic about the optimization of the classical Hohmann transfer, for coplanar and non coplanar circular orbits invented in the year 195 by the German engineer Walter Hohmann. Eagle cited the following remarks concerning this transfer: 1. Two impulses 180 0 apart are required in the direction of motion, collinear with velocity vector at peri apse and apo apse of transfer orbit.. Velocity and not position of vehicle is changed instantaneously. 3. Both thrusts are posigrade, i.e. in direction of orbital motion. 4. The transfer time from first to second impulse is given by a τ = π 3 µ a = 1 (r i + r f ) Script i, f refer to initial circular orbit and final circular orbit respectively; a denotes semi major axis of transfer ellipse. He derived the following formulae for the purpose of his article: V 1 = V lc 1 + R 1 R 1 cos θ 1 V = V lc R + R 1 R 3 R R 3 cos θ V t = V 1 + V θ t = θ 1 + θ r f R 1 = R = r i r i R 3 = r i + r f r f r i + r f µ V lc = local circular velocity = ri θ 1 = plane change associated with first impulse θ = plane change associated with second impulse θ t = total plane change angle between initial and final orbit The necessary condition for optimization is: V t θ 1 = R 1 sin θ 1 1 + R 1 R 1 cos θ 1 + R R 3 sin θ R + R 1 R 3 R R 3 cos θ = 0
On The Generalized Hohmann Transfer with Plane... 191 Nomenclature: V 1 increment of velocity at peri apse impulse. V increment of velocity at apo apse impulse. V T = V 1 + V µ constant of gravitation. a 1 semi major axis of initial orbit. a semi major axis of final orbit. a T semi major axis of transfer orbit. e 1 eccentricity of initial orbit. e eccentricity of final orbit. e T eccentricity of transfer orbit. r 1 initial radius (classical Hohmann). r final radius (classical Hohmann). α 1 plane change at peri apse. α plane change at apo apse. Θ = α 1 + α total plane change.