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A Complex Exponential Solution to the Unified Two-Body Problem Troy A. Henderson · John L. Junkins · Gianmarco Radice
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Abstract A complex exponential solution has been derived which unifies the elliptic and hyperbolic trajectories into a single set of equations and provides an exact, analytical solution to the unperturbed, Keplerian two-body problem. The formulation eliminates singularities associated with the elliptic and hyperbolic trajectories that arise from these orbits. Using this Complex Exponential solution formulation, a Variation of Parameters formulation for the perturbed two-body problem has been derived. In this paper, we present the analytical formulation of the complex exponential solution, numerical simulations, a comparison with classical solution methods, and highlight the benefits of this approach compared with the classical developments. Keywords Complex exponential · Unified · Two-body · Kepler · Elliptic · Hyperbolic
1 Introduction An alternative solution method for the universal two-body problem has been developed using complex exponential functions. This solution method provides an exact solution that can be efficiently implemented for numerical computation. While complex algebra is required for the derivation, the complex conjugate eigenstructure of the differential equations is exploited and results, as expected, in a purely real solution for Keplerian two-body motion. Previously presented as AAS 07-136 at the 17th AAS/AIAA Spaceflight Mechanics Meeting Sedona, Arizona, AAS 08-206 and AAS 08-230 at the 18th AAS/AIAA Spaceflight Mechanics Meeting Galveston, Texas T. Henderson Dept. of Aerospace Engineering, Texas A&M University, 744C H.R. Bright Building, College Station, TX 77843-3141 Tel: 979-862-1877 E-mail: troy
[email protected] J. Junkins Texas A&M Universiy E-mail:
[email protected] G. Radice University of Glasgow E-mail:
[email protected]
Click here to view linked References
2
This paper shows the derivation path which was developed using novel, rigorously linear differential equations. The results of the derivation are shown to be analogous to Battin’s development of the universal functions. Further applications of the solution method are developed through the corresponding exponential function form of the state transition matrix as well as a variation of parameters solution path for the perturbed two-body problem. The complex exponential solution, as currently formulated, exhibits a sigularity for the parabolic trajectory (and numerical singularities for the near-parabolic case) and thus is considered a “unified” solution. However, a series solution expansion for the near-parabolic trajectory eliminates or minimizes the effects of singularity and is discussed in Section 3. Thus, the series solution allows the formulation to be implemented as a branched universal solution.
2 Complex Exponential Solution Derivation 2.1 Derivation The motivation for the complex exponential solution arose from investigations into Battin’s universal solution[1] while the first author was enrolled in a graduate-level Celestial Mechanics course taught by the second author. The applications were developed while the first author was studying under the third author. As Battin’s text served as the major source of inspiration, we follow Battin’s derivation path and notation where possible. However, we are indebted to the historical contributions of Stumpff and Herrick. The Lagrange-Gibbs F and G functions serve as the starting point. As an unperturbed orbit plane may be defined through the two initial condition vectors, r(t0 ) and r˙ (t0 ), any orbit position vector and velocity, r(t) and r˙ (t), will lie in the orbit plane and can thus be expressed as a linear combination of the initial condition vectors. Therefore we have the well-known Lagrange-Gibbs F and G solution r(t) = Fr(t0 ) + G˙r(t0 )
(1)
˙ r(t0 ) ˙ 0 ) + G˙ r˙ (t) = Fr(t
(2)
Upon inserting these equations into the unperturbed orbit equation r¨ (t) =
−μ r r(t)3
(3)
an immediate property of the F and G functions is found to be ¨ = −μ F(t) F(t) r(t)3
(4)
¨ = −μ G(t) G(t) r(t)3
(5)
with the initial conditions F(t0 ) = 1
˙ 0) = 0 F(t
(6)
G(t0 ) = 0
˙ 0) = 1 G(t
(7)
Classically, the equations for F and G are solved for either the elliptic or the hyperbolic trajectory case. However, as we wish to unify the two trajectories, we choose at this
3
point (as did Battin) to employ one of the many regularization techniques, the Sundman Transformation[2] √ μ dt = rd χ (8) Using the chain rule, we obtain ddχ = dtd ddtχ and higher order derivatives in χ . It is then possible to derive the following linear–and most attractive–differential equations for the Lagrange-Gibbs F and G functions (which to the authors’ knowledge have been unpublished until now) dF d3 F +α =0 (9) dχ3 dχ dG d3 G +α =0 dχ3 dχ
(10)
where χ is the generalized eccentric anomaly as defined by the Sundman Transformation and 2 r˙2 (t0 ) 1 − α= = a r(t0 ) μ is the energy constant of the trajectory. The solutions of F and G are immediately obtained by solving the above third-order, linear differential equations using common, complex exponential functions. Note that the solutions for F and G have the same general form √ √ i F(χ ) = c0 − √ (c1 eiχ α + c2 e−iχ α ) α
(11)
However, care must be taken in imposing the boundary conditions; the differential equations have been regularized via the Sundman Transformation, Eqn. (8), and so the boundary conditions must be regularized as well. When evaluated at χ = 0, the regularized boundary conditions become F =1 G=0 (12) dF =0 dχ −1 d2F = dχ2 r0
dG r0 =√ dχ μ d2G σ0 =√ dχ2 μ
(13) (14)
√ where the subscript zero, (·)0 , implies evaluation at t = 0 and we define the constant σ μ = r · r˙ [3]. Applying the boundary conditions leads to the following (complex) expressions for the F and G functions √ √ 1 1 + (eiχ α + e−iχ α ) r0 α 2r0 α √ √ i r0 α − iσ0 iχ √α r0 α + iσ0 −iχ √α σ0 G(χ ) = √ − √ ( e − e ) √ √ α μ 2 αμ 2 αμ α √ √ √ μ i(eiχ α − e−iχ α ) ˙ χ) = √ F( 2rr0 α √ √ √ √ ((r0 α − iσ0 )eiχ α + (r0 α + iσ0 )e−iχ α ) ˙ √ G(χ ) = 2r α
F(χ ) = 1 −
(15) (16) (17) (18)
As can be readily seen, for a parabolic trajectory, α = 0, a singularity occurs in each of the F and G functions. This shows that the current form of the complex exponential solution
4
is not truly universal, but instead bonds the elliptic and hyperbolic orbits. A solution for the parabolic trajectory is treated in Section 3. We now follow Battin’s derivation to obtain the transformed Kepler equation. We begin with the following equations as derived by Battin[1] d2 σ + ασ = 0 dχ2
(19)
d3 r dr +α =0 3 dχ dχ
(20)
d 2t d4t +α 2 = 0 4 dχ dχ
(21)
Classically, Battin solved these equations in terms of the Stumpff-Herrick-Battin Universal functions of α and χ . However, they can easily be solved with complex exponential functions. The initial conditions are imposed at t = t0 , thus χ = 0, r = r0 , and σ = σ0 . The solutions for σ and r are
σ=
√ √ √ √ i i (r0 α − 1 − iσ0 α )eiχ α − (r0 α − 1 + iσ0 α )e−iχ α 2α 2α
(22)
r0 α − iσ0 − 1 iχ √α r0 α + iσ0 − 1 −iχ √α 1 +( )e +( )e α 2α 2α
(23)
r=
Alternatively, we can write the expression for r in a form based on (the previously computed) F and G functions as √ r = r · r = F 2 r02 + G2 r˙02 + 2FGσ0 (24) Then we integrate the Sundman Transformation, Eqn. (8), to get the unified Kepler equation, and not surprisingly, it is found to be the same as presented by Battin[1] √ α μ (t − t0 ) − χ + σ − σ0 = 0
(25)
or alternatively in complex exponential form √ √ √ i χ − α μ (t − t0 ) + √ (1 − r0 α + iσ0 α )eiχ α 2 α √ √ i − √ (1 − r0 α − iσ0 α )e−iχ α = 0 2 α
(26)
We note that this version of Kepler’s equation is transcendental and can be solved for the real values of χ by one of many available techniques. For the example shown in Section 4 of this paper, the authors chose to use the most familiar Newton-Raphson root finding method. This traditional method converged rapidly (within 6 iterations) to tight tolerances (ε = 10−10 ) as expected from its wide use. Other methods or improvements for speed and/or accuracy are certainly possible in solving the complex version of Kepler’s equation, but their discussion is outside of the scope of this paper.
5
2.2 Equivalence to Classical Formulations It can be easily shown that the complex exponential solution is homomorphic to Battin’s (classical) universal functions[1] for elliptic and hyperbolic trajectories, thus verifying the derivation of the complex exponential solution. From complex mathematics, Euler’s identity states that e−iθ = cos(θ ) − isin(θ ) (27) eiθ = cos(θ ) + isin(θ ) which holds for all values of θ . Using Euler’s identity in Eqn. (15) through Eqn. (18)–and collecting terms–gives us √ 1 F= (cos(χ α ) − 1) + 1 (28) r0 α √ √ r0 σ0 G = √ (1 − cos(χ α ) + √ sin(χ α )) (29) α μ α μ √ √ μ √ sin(χ α ) F˙ = (30) rr0 α √ √ σ0 r0 (31) G˙ = cos(χ α ) + √ sin(χ α ) r r α Mathematically, these equations are valid for all values of α . Notice that these expressions match the form Battin’s universal functions exactly for the elliptic trajectory. In order to exactly match Battin’s form for the hyperbolic trajectory, further complex trigonometry identities are required, cos(iθ ) = cosh(θ )
sin(iθ ) = isinh(θ )
(32)
It is then easily shown that for the hyperbolic trajectory that the complex exponential solution reduces exactly to Battin’s universal function form, as expected. Therefore, if one derives the elliptic case–and interprets it properly using trigonometric identities–the hyperbolic case comes along for free. In modern scripting languages such as MATLAB, there is no necessity to identify variables as complex quantities. As a result, with careful and general programming practices, the code generating an elliptic trajectory will also generate a hyperbolic trajectory simply by changing the initial conditions properly. It should be noted, and emphasized, that the choice of solution using exponential functions with complex arguments is not a unique one. This is due to the relationship between exponential functions with complex arguments and circular functions (i.e., sin and cos) through Euler’s identity. Instead of solving Eqns. (9) and (10) using exponential functions, the choice could have been made to use the sin and cos without loss. Traditionally, the sin and cos functions were reached using a power series solution by Stumpff, Herrick, and Battin, and not an exact solution path as presented here.
2.3 Implementation We note that the recursive Euler’s top-down method typically used for computing the infinite fraction form of the universal functions, while it converges well, is not required for the complex exponential solution. For the complex exponential method, the only transcendental function is Kepler’s equation containing the usual exponential function which can be specialized to the circular or hyperbolic sine and cosine for elliptic and hyperbolic cases, respectively.
6
A summary of the complex exponential solution equations, in order of solution, is now presented: r02 = x20 + y20 + z20
(33)
r˙02 = x˙20 + y˙20 + z˙20
(34)
α=
(35)
r · r˙ σ=√ μ
(36)
√ χ = α μ (t − t0 ) + σ − σ0
(37)
F(χ ) = 1 −
G(χ ) =
2 r˙0 2 − r0 μ
√ √ 1 1 + (eiχ α + e−iχ α ) r0 α 2r0 α
√ √ i r0 α − iσ0 iχ √α r0 α + iσ0 −iχ √α σ0 √ − ( e − e ) √ √ √ α μ 2 αμ 2 αμ α r=
F 2 r02 + G2 r˙02 + 2FGσ0 √
˙ χ) = F(
√
μ i(eiχ α − e−iχ √ 2rr0 α
√
α)
√ √ √ √ (r0 α − iσ0 )eiχ α + (r0 α + iσ0 )e−iχ α ˙ √ G(χ ) = 2r α
(38)
(39)
(40)
(41)
(42)
r(t) = Fr(t0 ) + G˙r(t0 )
(43)
˙ r(t0 ) ˙ 0 ) + G˙ r˙ (t) = Fr(t
(44)
The steps for orbit propagation are: given the initial conditions, form the magnitude of the position, Eq. (33), and velocity, Eq. (34), and then compute the energy constant, Eq. (35). These first three equations are only computed once as they are based solely on the initial conditions but must be stored throughout the computation of the entire trajectory as they will be used at every time step. Eq. (36) through Eq. (44) are computed sequentially for every time step and provide the position and velocity vector at every time step. Note that it is necessary to save the initial σ0 value as it is used in computation at every time step.
7
3 Parabolic Trajectories The complex exponential solution, as derived, contains a singularity for the parabolic orbit as α approaches zero. The equivalence between the complex exponential solution and classical solutions is not completely obvious for the parabolic case, but will be shown here. As α goes to zero, the modified form of Kepler’s equation becomes
χ = σ − σ0 Battin[1], using Barker’s method, computes the Lagrangian coefficients for a parabolic orbit to be χ2 χ F = 1− G = √ (2r0 + σ0 χ ) (45) 2r0 2 μ √ − μχ χ2 (46) G˙ = 1 − F˙ = rr0 2r By judiciously choosing a small range for α representing the near-parabolic orbit, and expanding about small α in a series expansion, the singularity can be altogether avoided. While this provides a branched solution approach, and the solution is no longer exact, it is a significant step toward making the complex exponential formulation universal. A Taylor series expansion for small α of Eqn. (15) through Eqn. (18) gives F = 1−
χ 2 1 α χ 2 (α χ 2 )2 + − . . .) ( − r0 2! 4! 6!
r0 χ α χ 2 (α χ 2 )2 σ 0 χ 2 1 α χ 2 (α χ 2 )2 G = √ (1 − + − . . .) + √ ( − + − . . .) μ 3! 5! μ 2! 4! 6! √ −χ μ α χ 2 (α χ 2 )2 + − . . .) (1 − F˙ = rr0 3! 5! r0 α χ 2 (α χ 2 )2 σ0 χ α χ 2 (α χ 2 )2 G˙ = (1 − + − . . .) + (1 − + − . . .) r 2! 4! r 3! 5!
(47)
(48) (49) (50)
For a parabolic orbit (or in the case that |α χ 2 | is zero to machine precision in computation), l’Hospital’s rule may be employed to take the limit of the F and G functions (Eqns. (15) through (18)). The Lagrange-Gibbs F and G functions become F = 1− F˙ =
χ2 2r0 √ − μχ rr0
χ G = √ (2r0 + σ0 χ ) 2 μ
(51)
r0 + χσ0 G˙ = r
(52)
These are exactly the leading terms in the Taylor series expansion above for α = 0. The functions agree exactly with Battin’s solution presented in Eqns. (45) and (46), except for the G˙ term. In order to show that the G˙ term derived in the Complex Exponential solution matches exactly with Battin’s solution, we must go back to Eqn. (20). For the parabolic case, α = 0 so that Eqn. (20) becomes d3 r =0 dχ3
(53)
8
The analytical solution to this equation is r = r0 + χ
dr χ 2 d2 r χ2 |0 + | = r + χσ + 0 0 0 dχ 2 dχ2 2
(54)
which when re-written and divided by r on both sides gives r0 χσ0 χ 2 = 1− − r r 2r
(55)
Plugging this result into Eqn. (52) gives r0 + χσ0 χσ0 χ 2 χσ0 χ2 G˙ = = 1− − + = 1− r r 2r r 2r
(56)
which matches exactly with Battin’s solution.
4 Application In this section, a practical application (besides the obvious integration of the equations of motion) of the complex exponential solution is introduced. In addition to orbit propagation previously derived, a variation of parameters method for the perturbed two-body problem is derived using the complex exponential solution.
4.1 Variation of Parameters for the Perturbed Two-Body Problem The Variation of Parameters method seeks an approach to vary the constants of integration (arising from the unperturbed, or unforced solution) so that the form of the homogenous solution remains valid for perturbed motion. Historically, this concept has proven to be extraordinarily powerful in solving multi-dimensional systems of nonlinear differential equations. For Keplerian two-body motion, the variation of parameters represents a continuous rectification of the osculating reference orbit, which is described by a set of parameters that when unperturbed, remain constant with respect to time[3]. These elements are generally chosen to be the six classical orbit elements, however the choice is not unique. In the presence of perturbative forces, these parameters become slowly time-varying to achieve osculation with the actual, perturbed motion.
4.2 Derivation of Variation of Parameters Equations The Variation of Parameters method (for solving the equations of orbit motion) seeks a system of first-order differential equations describing the rate of change of the time-varying elements, c, of the perturbed two-body problem such that the description of the perturbed motion instantaneously has the same algebraic form as the unperturbed motion. Defining ad as the perturbing acceleration, the perturbed two-body equation of motion is r¨ (t) =
−μ r + ad r3
(57)
The unperturbed (ad = 0) solution has the form r(t) = f (t, c)
(58)
9
r˙ (t) =
∂f (t, c) ∂t
(59)
−μ r ∂2 f (t, c) = 3 (60) 2 ∂t r The partial derivative is used without loss since the elements of c are constants for the unperturbed problem. In general, the chain rule implies ∂f ∂ f dc + (61) r˙ (t) = ∂t ∂ c dt r¨ (t) =
Comparing Eqn. (59) and Eqn. (61), the osculation constraint is derived ∂ f dc =0 ∂ c dt Physically, this constraint means that the osculating orbit, defined by c(t) and ∂ f (t,c) ∂t ,
“kiss” the actual motion so that the two-body velocity, Taking another derivative using the chain rule, 2 ∂2 f ∂ f dc r¨ (t) = 2 + ∂t ∂ t ∂ c dt
(62) dc(t) dt
must
is also the actual velocity.
(63)
Equating the unperturbed and perturbed motion equations, Eqn. (60) and Eqn. (63), the solvability condition is derived 2 ∂ f dc = ad (64) ∂ t ∂ c dt The constraints as derived for the Variation of Parameters formulation can be written compactly as ∂ f (t,c) ∂ r(t,c) dc dc dc 0 ∂c ∂ c (65) = = [L] = 2 2 ˙ r (t,c) ∂ f (t,c) ∂ a dt dt dt d ∂ t∂ c
∂ t∂ c
The 2N × 2N matrix [L] is known as the Lagrangian Matrix and is generally fully populated and may depend explicitly on time. By deriving the derivatives in the Lagrangian Matrix and inverting this matrix, an analytical expression for the first-order differential equations describing the rate of change for the osculating elements, dc dt , is obtained. Now we choose the vector c to be the initial position and initial velocity in an orbit (in a Cartesian frame) c = [x0 , y0 , z0 , vx0 , vy0 , vz0 ]T In order to derive the matrix [L], we now develop the partial derivatives of the current state with respect to the initial state as follows [L(t,t0 )] =
∂r ∂r ∂ r0 ∂ r˙ 0 ∂ r˙ ∂ r˙ ∂ r0 ∂ r˙ 0
(66)
The partial derivatives must now be taken with respect to the initial condition vectors. Note from Eqns. (15) through (18) that the F and G functions are, indeed, themselves functions of the initial conditions. It should be noted that we take the derivatives without any
10
reference to choice of coordinates or orbital parameters; up to this point we have only assigned the state vector. We choose to build the Lagrangian matrix by deriving each of the four 3 × 3 submatrices. ∂ r(t) ∂F ∂G + r˙ 0 (67) = F · I3×3 + r0 ∂ r(t0 ) ∂ r0 ∂ r0
∂ r(t) ∂F ∂G + r˙ 0 = G · I3×3 + r0 ∂ r˙ (t0 ) ∂ r˙ 0 ∂ r˙ 0 ∂ r˙ (t) ∂ F˙ ∂ G˙ = F˙ · I3×3 + r0 + r˙ 0 ∂ r(t0 ) ∂ r0 ∂ r0 ∂ r˙ (t) ∂ F˙ ∂ G˙ + r˙ 0 = G˙ · I3×3 + r0 ∂ r˙ (t0 ) ∂ r˙ 0 ∂ r˙ 0
(68)
(69)
(70)
Taking derivatives of the F and G functions with respect to r0 and (significantly) simplifying the following expressions are obtained
∂F = ∂ r0
˙ χ 1−F Fr + √ α 2α μ
˙ ∂χ ∂ α 1 − F ∂ r0 Fr + +√ ∂ r0 r0 ∂ r0 μ ∂ r0
˙ χ ˙ 02 G ∂ α ˙ ∂χ ˙ 0 ∂ r0 Frr (1 − F)r0 ∂ σ0 Frr Gr Gr ∂G = − + − +√ √ √ − ∂ r0 μ ∂ r0 μ ∂ r0 2α μ 2α μ α ∂ r0 μ ∂ r0 √ √ α μ χ μ ∂ F˙ ∂χ 1 1 F˙ ∂ α = + − 1−F − 1−F − ∂ r0 r r0 α ∂ r0 2r r0 α 2α ∂ r0 ˙ ˙ −F ∂ r0 F ∂ r + − r0 ∂ r0 r ∂ r0 √ G˙ ∂ r r0 α ∂ G˙ ∂ r0 α μ σ0 ∂χ 1 =− + − F −1+ G− √ ∂ r0 r ∂ r0 r r0 α ∂ r0 r α μ ∂ r0 2 √ ˙ 0 ∂ σ0 χ μ r σ0 ∂ α Fr G˙ 1 + 0 F −1+ − −√ − G− √ 2r r0 α 2α 2r α μ ∂ r0 μ ∂ r0
(71)
(72)
(73)
(74)
Note that the derivatives with respect to r˙ 0 can be seen to have the same form as Eqns. (71) through (74) with r0 replaced with r˙ 0 as the derivatives are still somewhat general. In Eqns. (71) through (74) there are scalar partial derivatives on the right hand side that must be taken in order to construct the full partial derivatives. Thus, we develop the following partial derivatives r0 T ∂ r0 = ∂ r0 r0 −2r0 T ∂α = ∂ r0 r03 r˙ T ∂ σ0 = √0 ∂ r0 μ
∂ r0 = 0T ∂ r˙ 0
(75)
−2˙rT0 ∂α = ∂ r˙ 0 μ
(76)
rT ∂ σ0 = √0 ∂ r˙ 0 μ
(77)
11
Now, the derivatives of the r and χ equations (Eqns. (23) and (26) respectively) must be taken. The derivatives with respect to r0 and r˙ 0 have the same form of ˙ 0 ∂ σ0 ˙ 0 ∂χ √ ∂r ∂ r0 Frr σ0 1 Frr = r0 α F − 1 + − √ − α μ G− √ + √ ∂ r0 r0 α ∂ r0 μ ∂ r0 α μ μ ∂ r0 (78) √ ˙ r0 (F − 1) Frr0 (σ0 − χ ) χ μ σ0 ∂α + + − G− √ √ α 2α μ 2 α μ ∂ r0
∂χ ∂ r0
˙ 0 α ∂ r0 1 Frr 1 + rα G˙ − r0 α F − 1 + = √ r0 α μ ∂ r0 1 ∂ σ0 −1 + r0 α F − 1 + r0 α ∂ r0 √ ˙ 0 (1 + 2r0 α ) ∂ α μ √ σ0 Frr + [t − t0 ] μ + G− √ + √ 2 α μ 2α μ ∂ r0 (1 + (F − 1) r0 α )(σ0 + χ ) r χ G˙ ∂ α + − 2α 2 ∂ r0
(79)
We now define the following scalar functions for brevity and clarity in the remaing portion of the derivation.
∂F ∂ r0 ∂α ∂χ = f r0 + fα + fχ ∂ r0 ∂ r0 ∂ r0 ∂ r0
(80)
∂G ∂ r0 ∂α ∂χ ∂ σ0 = gr0 + gα + gχ + gσ0 ∂ r0 ∂ r0 ∂ r0 ∂ r0 ∂ r0
(81)
∂ F˙ ∂ r0 ˜ ∂ α ∂χ ∂r = f˜r0 + fα + f˜χ + f˜r ∂ r0 ∂ r0 ∂ r0 ∂ r0 ∂ r0
(82)
∂ G˙ ∂ r0 ∂α ∂χ ∂ σ0 ∂r = g˜r0 + g˜α + g˜χ + g˜σ0 + g˜r ∂ r0 ∂ r0 ∂ r0 ∂ r0 ∂ r0 ∂ r0
(83)
∂r ∂ r0 ∂α ∂χ ∂ σ0 = rr0 + rα + rχ + rσ0 ∂ r0 ∂ r0 ∂ r0 ∂ r0 ∂ r0
(84)
∂χ ∂ r0 ∂α ∂ σ0 = χr0 + χα + χσ0 ∂ r0 ∂ r0 ∂ r0 ∂ r0
(85)
where the scalar values are the coefficients of the partial derivatives in Eqns. (71) through (74), (78), and (79), respectively. Finally, the terms are collected for each of the four submatrices forming the Lagrangian matrix. fr0 + χr0 f χ χσ0 f χ fα + χα f χ ∂ r(t) T −2 r + r r0 r˙ 0 T = F · I3×3 + √ 0 0 ∂ r(t0 ) r0 μ r03 (86) gr0 + χr0 gχ gσ0 + χσ0 gχ fα + χα f χ T T ˙ ˙ ˙ + −2 r + r r r √ 0 0 0 0 r0 μ r03
12
χσ f χ 2 ∂ r(t) fα + χα f χ r0 r˙ 0 T r0 r0 T − = G · I3×3 + √0 ∂ r˙ (t0 ) μ μ
gσ0 + χσ0 gχ 2 gα + χα gχ r˙ 0 r˙ 0 T + r˙ 0 r0 T − √ μ μ
(87)
∂ r˙ (t) = F˙ · I3×3 ∂ r(t0 ) ˜
fr0 + χr0 f˜χ + rr0 f˜r + χr0 rχ f˜r 2 + − 3 f˜α + χα f˜χ + f˜r rα + χα rχ f˜r r0 r0 T r0 r0
g˜r0 + χr0 g˜χ + rr0 g˜r + χr0 rχ g˜r 2 + − 3 g˜α + χα g˜χ + g˜r rα + χα rχ g˜r r0 r˙ 0 T r0 r0 χσ0 f˜χ + rσ0 f˜r + χσ0 f˜r rχ + r˙ 0 r0 T √ μ g˜σ0 + χσ0 g˜χ + g˜r rσ0 + χσ0 g˜r rχ + r˙ 0 r˙ 0 T √ μ (88)
χσ0 f˜χ + rσ0 f˜r + χσ0 f˜r rχ ∂ r˙ (t) r0 r0 T = G˙ · I3×3 + √ ∂ r˙ (t0 ) μ
2 ˜ fα + χα f˜χ + rα f˜r + χα f˜r rχ r0 r˙ 0 T − μ g˜σ0 + χσ0 g˜χ + g˜r rσ0 + χσ0 g˜r rχ + r˙ 0 r0 T √ μ
2 − g˜α + χα g˜χ + rα g˜r + χα g˜r rχ r˙ 0 r˙ 0 T μ
(89)
The Lagrangian matrix for this special choice of c has been shown to be symplectic for the two-body problem[1, 3]. By definition, the matrix Φ is symplectic if it satisfies the following condition
Φ T JΦ − J = 0
(90)
where J is equivalent to the matrix imaginary such that J 2 = −I, or J=
0n×n In×n −In×n 0n×n
(91)
The matrix Φ can be broken into four 3x3 submatrices which are equated to the Lagrangian matrix through Eqn. (66). As the derived Lagrangian matrix is symplectic, an analytical inverse is easily written as [L]−1 = Φ −1 = −J Φ T J =
T −Φ T Φ22 12 T ΦT −Φ21 11
(92)
13
The expression for Φ −1 contains four simpler matrices, Φi j , which are combinations of the identity matrix and the multiplicaion of two vectors. Thus, the transpose of each Φi j can be analytically written using the following matrix algebra properties (AB)T = BT AT
(A + B)T = AT + BT
(93)
Thus we can analytically invert Φ by writing the transpose of each submatrix fr0 + χr0 f χ χσ0 f χ fα + χα f χ T −2 r + r r˙ 0 r0 T √ 0 0 r0 μ r03 fα + χα f χ gr0 + χr0 gχ gσ0 + χσ0 gχ T r0 r˙ 0 + + −2 r˙ 0 r˙ 0 T √ r0 μ r03
−1 Φ11 = F · I3×3 +
−1 Φ12
χσ0 f χ 2 = G · I3×3 + √ r0 r0 T − fα + χα f χ r˙ 0 r0 T μ μ
gσ0 + χσ0 gχ 2 + r0 r˙ 0 T − gα + χα gχ r˙ 0 r˙ 0 T √ μ μ
−1 Φ21 = F˙ · I3×3 ˜
fr0 + χr0 f˜χ + rr0 f˜r + χr0 rχ f˜r 2 ˜ ˜ ˜ ˜ + − 3 fα + χα f χ + fr rα + χα rχ fr r0 r0 T r0 r0
g˜r0 + χr0 g˜χ + rr0 g˜r + χr0 rχ g˜r 2 + − 3 g˜α + χα g˜χ + g˜r rα + χα rχ g˜r r˙ 0 r0 T r0 r0 ˜ ˜ ˜ χσ0 f χ + rσ0 fr + χσ0 fr rχ + r0 r˙ 0 T √ μ g˜σ0 + χσ0 g˜χ + g˜r rσ0 + χσ0 g˜r rχ + r˙ 0 r˙ 0 T √ μ
−1 Φ22
(94)
(95)
(96)
χσ0 f˜χ + rσ0 f˜r + χσ0 f˜r rχ r0 r0 T √ μ
2 ˜ fα + χα f˜χ + rα f˜r + χα f˜r rχ r˙ 0 r0 T − μ g˜σ0 + χσ0 g˜χ + g˜r rσ0 + χσ0 g˜r rχ + r0 r˙ 0 T √ μ
2 − g˜α + χα g˜χ + rα g˜r + χα g˜r rχ r˙ 0 r˙ 0 T μ = G˙ · I3×3 +
(97)
With an analytical expression for the [L] matrix and it’s inverse, the variation of the orbital elements (parameters) formulation is complete.
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4.3 Numerical Validation of Variation of Parameters Equations The obvious first step in validating the derivation was to ensure that for ad = 0, the solution collapsed to the classical elliptic or hyperbolic trajectory. The elliptic and hyperbolic trajectories defined in Table 1 were numerically integrated in MATLAB using a fixed time-step, variable order (4th to 5th order) Runge-Kutta integrator. Cowell’s method of integrating the equations of motion was taken as “truth” and the complex exponential version was compared. Cowell’s method directly integrates the equations and perturbations as shown in Eqn. (57).
Table 1 Initial Values for Numerical Examples Parameter x y z x˙ y˙ z˙
Elliptic Value 7000km 0 100 0km/s 6.8 0.6
Hyperbolic Value 7000km 0 100 0km/s 11.2 0.6
The complex exponential variation of parameters formulation was simulated for unperturbed motion, ad = 0, to compare to the trajectories. For the elliptic trajectory, the magnitude of the radius exhibited an rms error of 10−15 km, which was the order of the tolerance used in the numerical integration, over the first five orbit periods with a Δ t = 10 seconds. For the hyperbolic trajectory, the magnitude of the radius exhibited an rms error of 10−15 km as well over 20,000 time steps. Thus, the variation of parameters solution collapsed to the classical solution in the unperturbed case. A second test was performed using a small perturbation to a spacecraft orbit. An elliptic trajectory, whose initial position and velocity are detailed in Table 1, was numerically integrated with the Earth’s oblateness taken into account. The disturbing acceleration due to the J2 effect is ⎛ 2 x ⎞ 1 − 5 rz r z 2 y ⎟ 3J2 μ req 2 ⎜ ⎜ ⎟ (98) aJ2 = − 2 ⎜ 1−5 r r ⎟ ⎝ 2r r z 2 z ⎠ 3−5 r r where J2 = 1082.63 × 10−6 . Using a time step of Δ t = 10seconds, the numerical integration and complex exponential variation of parameters formulation were compared over one year (365 days) on orbit. The radius of the orbit was found to have a maximum difference of 10−12 km and an rms error of 10−14 km over the year. Other sample elliptic trajectories exhibited similar results.
5 Conclusions This paper presents a novel derivation of unified equations for the elliptic and hyperbolic trajectories using regular complex exponential functions. The approach leads to an exact solution using complex argument exponential functions in lieu of the Stumpff-Herrick-Battin
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Universal functions, but the resulting solution is purely real functions that remove singularities typically associated with zero inclination and/or zero eccentricity. The solution is analytically shown to be equivalent to classical solutions and numerical tests show the same. Further applications are derived in the form of a state transition matrix and a variation of parameters formulation. These formulations are shown, numerically, to be equivalent to the solution produced by Cowell’s method.
References 1. Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education Series. American Institute of Aeronautics and Astronautics, Reston, VA (1999). 2. Sundman, K., “M`emoire sur le Probl`eme des trois corps,” Acta Mathematica, Vol. 36, pp. 105-179 (1912). 3. Schaub, H. and Junkins, J.L., Analytical Mechanics of Space Systems, AIAA Education Series. American Institute of Aeronautics and Astronautics, Reston, VA (2003).
