Mechanics and Mechanical Engineering Vol. 15, No. 2 (2011) 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 [email protected] Received (10 October 2010) Revised (5 May 2011) Accepted (8 August 2011) 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 ¯nal orbit to be that of an outer adequate planet, Mars for instance. We assume the elements of the two impulse Hohmann generalized con¯guration (the case of elliptic, non coplanar orbits) to be a1, e1, a2, e2, aT , eT . From the very beginning, we should assign θ = ®1 + ®2, the total plane change required. ®1 is the plane change at the ¯rst instantaneous impulse at peri{apse, which will be minimized, and ®2 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 cir- cular and elliptic orbits [1]. As for the derivations of the velocity change require- ments ¢V1, ¢V2 and transfer time, we can draw a graph which illustrates total 3=2 energy/satellite mass as a function of orbit period P = 2¼ap that means a plot of ³ ´ ¹ ¡¹ p2¼ 3=2 2a versus ¹ a . Plotted results are extensively established [1], [2]. For classi- r2 cal Hohmann transfer if h15:58; r2ir1; is not satis¯ed, then the Hohmann transfer r1 is no longer optimal. For these conditions Bi{elliptic transfers are always more economical in propellant than Hohmann transfer con¯gurations [1]. The Hohmann 184 Kamel, OM and Soliman A transfer is a relatively simple maneuver, especially the classical model. It may be simpli¯ed or complicated easily, and we may encounter very di±cult situations. This can be easily seen from the literature of orbit transfer [3]. There exist four feasible Hohmann con¯gurations according to the coincidence of peri{apse and apo- apse of the three ellipses. We consider the ¯rst 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 ¯rst ¢V1 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 ¯nal elliptic orbit. An engine ¯ring in the out{of plane direction is required for the change of plane. The point of ¯ring becomes a point in the new orbit, and the burn point becomes the intersection of the current orbit and the desired orbit. De¯nitely, we should perform plane change in the smartest way, since it is fuel expensive, anyway you do them. Even without the examination of the speci¯c equations, planning a space mission, reduces to a problem of geometry, timing, mechanics of orbital motion, and a lot of common sense. 2. Method and Results In this article we investigate the generalized Hohmann orbit transfer with split { plane change. We take into account, the ¯rst con¯guration, where the apo{apse of the transfer orbit coincides with the apo{apse of the ¯nal orbit, and the peri{apse of the initial and the transfer orbit are coincident [5], Fig. 1. ¢V1 produces at peri{apse of initial orbit, a transfer ellipse as well as a plane change ®1. Similarly at apo{apse, ¢V2 rotates the orbit plane through an angle ®2 = θ¡®1 , and designs the ¯nal elliptic orbit as shown in Fig. 2, Fig. 3 and Fig. 4, represent the velocity vector triangular addition ¢V1 , ¢V2 respectively. Figure 1 Generalized Hohmann Transfer On The Generalized Hohmann Transfer with Plane ... 185 PlaneChange atfirst Imp. Final OrbitOrbt Initial Orbit PlaneChange atsecondImp. Fig.2:GeneralizedHohmanntransfer withplanechange Figure 2 VP tr. Dv1 ÄV1 aá11 VPi Fig.3 Figure 3 186 Kamel, OM and Soliman A VAf vAf Dv2 ÄV2 q-aè-á11 vAtr VAtr Fig.4 Figure 4 s ¹ (1 + e1) VPi = where a1 (1 ¡ e1) s ¹ (1 + eT ) VPtr = aT (1 ¡ eT ) s ¹ (1 ¡ eT ) VAtr = aT (1 + eT ) s ¹ (1 ¡ e2) VAf = a2 (1 + e2) The increments of velocities at peri{apse and apo{apse of the elliptic transfer orbit is given by 2 ¹ (1 + e1) ¹ (1 + eT ) ¢V1 = + a1 (1 ¡ e1) aT (1 ¡ eT ) s½ ¾ ½ ¾ ¹ (1 + e1) ¹ (1 + eT ) ¡2 cos ®1 (1) a1 (1 ¡ e1) aT (1 ¡ eT ) 2 ¹ (1 ¡ e2) ¹ (1 ¡ eT ) ¢V2 = + a2 (1 + e2) aT (1 + eT ) s½ ¾ ½ ¾ ¹ (1 ¡ e2) ¹ (1 ¡ eT ) ¡2 cos (θ ¡ ®1) (2) a2 (1 + e2) aT (1 + eT ) q q 2 2 ¢VT = ¢V1 + ¢V2 = ¢V1 + ¢V2 (3) On The Generalized Hohmann Transfer with Plane ... 187 where θ is arbitrary and given by θ = ®1 + ®2 and eT , aT are calculated from the formulae [a2 (1 + e2) ¡ a1 (1 ¡ e1)] b4 ¡ b1 eT = = (4) [a2 (1 + e2) + a1 (1 ¡ e1)] b4 + b1 [a (1 + e ) + a (1 ¡ e )] b + b a = 2 2 1 1 = 4 1 (5) T 2 2 where b1 = a1 (1 ¡ e1) b4 = a2 (1 + e2) @¢VT ®1 to be optimized by the condition of minimization = 0: @®1 When checking the second order conditions, we treat the functional V [y] as a function of " i.e. V ("). dV We set d" = 0 and consequently, we acquire the ¯rst order necessary condi- tion for an extremal. For distinction between maximization and minimization we d2V calculate the second derivative dt2 : We ¯nd the following second order necessary conditions: d2V dt2 · 0 { for maximize of V , d2V dt2 ¸ 0 { for minimize of V . As for second order su±cient conditions: d2V dt2 < 0 { for maximize of V , d2V dt2 > 0 { for minimize of V , cf. [5] for de¯nitions. For the ¯rst con¯guration Let µ ¶ µ ¶ ¹ 1 + e1 ¹ 1 + eT A1 = B1 = a1 1 ¡ e1 aT 1 ¡ eT (6) µ ¶ µ ¶ ¹ 1 ¡ e2 ¹ 1 ¡ eT C1 = D1 = a2 1 + e2 aT 1 + eT whence ³ p ´1=2 ¢VT = A1 + B1 ¡ 2 A1B1 cos ®1 (7) ³ p ´1=2 + C1 + D1 ¡ 2 C1D1 cos (θ ¡ ®1) By partial di®erentiation with respect to ®1 and equating to zero, and after rear- rangements and clearing fractions, we ¯nd that A B sin2 ® C D sin2 (θ ¡ ® ) 1 1p 1 = 1 1p 1 (8) A1 + B1 ¡ 2 A1B1 cos ®1 C1 + D1 ¡ 2 C1D1 cos (θ ¡ ®1) 188 Kamel, OM and Soliman A From which we can deduce h p ³ p ´i ¡ ¢ 2 2 A1B1 C1 + D1 ¡ 2 C1D1 bx + a 1 ¡ x 1 ¡ x (9) h p i h¡ ¢ p i 2 2 2 2 2 = C1D1 A1 + B1 ¡ 2 A1B1x a ¡ b x + b ¡ 2abx 1 ¡ x where: p 2 x = cos ®1 1 ¡ x = sin ®1 a = sin θ b = cos θ (10) After some reductions, we ¯nd 2 A1B1 (C1 + D1) ¡ C1D1 (A1 + B1) b ³ p p ´ 2 + 2C1D1 A1B1b ¡ 2bA1B1 C1D1 x © ¡ 2 2¢ª 2 ¡ A1B1 (C1 + D1) + C1D1 (A1 + B1) a ¡ b x n p p o ¡ 2 2¢ 3 + 2bA1B1 C1D1 + 2C1D1 A1B1 a ¡ b x p n p 2 = 1 ¡ x 2aA1B1 C1D1 ¡ 2abC1D1 (A1 + B1) x ³ p p ´ o 2 + 4abC1D1 A1B1 ¡ 2aA1B1 C1D1 x Let A B (C + D ) ¡ C D (A + B ) b2 = E 1 1 p1 1 1 1 p1 1 1 2 2C1D1 A1B1b ¡ 2bA1B1 C1D1 = E2 ¡ 2 2¢ ¡fA1B1 (C1 + D1) + C1D1 (A1 + B1) a ¡ b g = E3 p p ¡ 2 2¢ 2bA1B1 C1D1 + 2C1D1 A1B1 a ¡ b = E4 p 2aA1B1 C1D1 = E5 ¡2abC D (A + B ) = E 1 p1 1 1 6 p 4abC1D1 A1B1 ¡ 2aA1B1 C1D1 = E7 i.e. p ¡ ¢ 2 3 2 2 E1 + E2x + E3x + E4x = 1 ¡ x E5 + E6x + E7x (11) After squaring and some reductions, we may write ¡ 2 2¢ E1 ¡ E5 + (2E1E2 ¡ 2E5E6) x ¡ 2 2 2¢ 2 + 2E1E3 + E2 ¡ 2E5E7 ¡ E6 + E5 x 3 + (2E1E4 + 2E2E3 ¡ 2E6E7 + 2E5E6) x (12) ¡ 2 2 2¢ 4 + 2E2E4 + E3 ¡ E7 + 2E5E7 + E6 x 5 ¡ 2 2¢ 6 + (2E3E4 + 2E6E7) x + E4 + E7 x = 0 On The Generalized Hohmann Transfer with Plane ... 189 Set 2 2 E1 ¡ E5 = ª1 2E1E2 ¡ 2E5E6 = ª2 2 2 2 2E1E3 + E2 ¡ 2E5E7 ¡ E6 + E5 = ª3 2E1E4 + 2E2E3 ¡ 2E6E7 + 2E5E6 = ª4 2 2 2 2E2E4 + E3 + 2E5E7 + E6 ¡ E7 = ª5 2 2 2E3E4 + 2E6E7 = ª6; E4 + E7 = ª7 That means after some reductions, we get an equation of degree 6 in x = cos ®1 : 2 3 4 5 6 Ã1 + Ã2x + Ã3x + Ã4x + Ã5x + Ã6x + Ã7x = 0 (13) where the E`s appearing in the Ã`s are functions of one parameter ®1 and could be expressed in terms of a1, a2, e1, e2 [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 ¯rst increment at peri{apse ¢V1, not only produces a transfer elliptic orbit, but also rotates the orbital plane by an optimal angle ®1.