MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � � �� � Lecture 27 Variation of Parameters Chapter 10 Lagrange’s Variational Methods for Linear Equations Consider the equation dy1 �� � �� � � � d2y + y = sec t = dt = y2 = dy1 =0 1 y1 +0 dt2 ⇒ dy2 + y1 = sec t ⇒ dt y2 −10 y2 sec t dt which is equivalent to dy = Fy + gdt where �� � � � � y = yy12 and F = −0101 and g = sec0 t Now the Wronskian matrix W sin t − cos t dWW = satisfies = FW cos t sin t dt and the solution of the homogeneous equation is yh = Wc where c = c1 c2 We now seek a solution of the general equation dy = Fy + g of the form y = Wc(t)dt Substitute and obtain dW dc dc c + W = FWc + g which reduces to W = gdt dt dt Hence dc = W−1gdt which is solved by quadratures to obtain dc sin t cos t 0 dc1 dc2= or = 1 and = tan t dt − cos t sin t sec t dt dt Hence c1(t)= t + c1 and c2(t) = log(sec t + tan t)+ c2 so that the general solution is simply y = Wc(t)or y(t)= c1(t) sin t − c2(t) cos t 16.346 Astrodynamics Lecture 27Derivation of the Variational Equations O I dt⎡r d ⎤= ⎡ µ ⎤⎡ds r ⎤= Fs + η ⎢ ⎥ ⎢ ⎥⎢dt ⎢0 ⎣ v ⎦ ⎥ ⎢⎣ −⎥⎦⎢⎣ + ⇐⇒ where IO v ⎡ ⎤ 3 ⎥ ⎢ ⎥⎦⎥⎣ ad ⎦ �r(t, α) rs = s(t, α)= v(t, α) � ∂s dsTwo-Body Motion: = Fs Disturbed Motion: = Fs + η ∂t dt Seek solutions of the form s = s(t, α1,α2,α3,α4,α5,α6) T �0where, for example, α =[Ω iωaeλ = −nτ ] and η = ad Differentiate � ds ∂s ∂s dα ∂s dα = + = Fs + η = = η dt ∂t ∂ α dt ⇒∂ α dt Since ∂s ∂ α ∂ α ∂s = = I ∂ α ∂s ∂s ∂ α then ∂ α ∂s dα ∂ α �∂ α ∂ α ��0 ∂ α = η = = a∂s ∂ α dt ∂s ∂r ∂vad ∂v dso that � dα ∂ α = adt ∂v dx = f(ξ,η) � dx � f=� ξ fη dξ ξ = F (x, y) dξ F x F y dx = y = g(ξ,η) dy g g�� dη � η = G(x, y) � dη � � G G �� dy � � ξ η ydx � � x fξ fη �� FxFy �� dx � � fξ fη �� F⇒xF= = y = I dy gξ gη G x G y dy gξ gη G x G y � Variation of the Classical Elements �2 1� 2 T µ ∂a T da 2a2 µ − = v = v v = v v = =2v = = v ar a · ⇒a2 ∂v ⇒dt µ · d∂h ∂vdh h = r × v = Sr v = ⇒ = S = S I = = ∂v r∂v rS r ⇒= r × adt d16.346 Astrodynamics Lecture 27∂h h2 = hT h =⇒ 2h =2hT dh S r = ⇒ = ih · r a∂v × d = ih × r · ad = r idt θ · ad or, alternately, dh 1 h2 =(r × v) · (r × v)= rT rvT v − rT vrT v = ⇒ = rT (rvT dt − vrT )ah dh2 p = = a(1 − e 2de da dh) = ⇒ 2µae = µ(1 )dt − e 2µ dt − 2hdt de µ = ad × (r × v)+(ad × r) × v dt Variation of i and Ω From Page 84 in the textbook h = h ih = h(sin Ω sin i i x − cos Ω sin i i y+ cos i i z ) .6)dh dΩ di dhThen = h sin i i dt n − h i m + idt dt dt hwhere i n = cos Ω i x + sin Ω i y (2.5) i m= ih × i n = − sin Ω cos i i x+ cos Ω cos i i y+ sin i i z (2.8)Hence dΩ1 r sin θ = i sin n dt × r ad = ih ah i · h sin i · ddi 1 r cos θ = dt − i h m × r · ad = i a h h· dwhere θ = ω + f is the argument of latitude. Variation of the true anomaly f h2 ∂f ∂e 2h ∂h r(1 + e cos f)= = µ ⇒ re sin f = r cos f∂v ∂v − µ ∂v µ ∂f ∂e r v ∂h hFrom Eq. (3.29) re sin f = r h· v = ⇒ re cos f = T v −r sin f + · + r∂ ∂v µ ∂v µ Multiply the first by cos f , the second by sin f and add to obtain ∂f ∂h df h ∂f reh =(p cos f) rT (p + r) sin f to be used in = + a∂v −∂v dt r2 ∂v d (216.346 Astrodynamics Lecture 27Variation of ω i n = cos Ω i x + sin Ω i y = ⇒ cos θ = in · i r = cos Ω (ix · i r) + sin Ω (iy · i r)Then ∂θ ∂Ω ∂θ ∂Ω − sin θ =[− sinΩ(i x · i r) + cos Ω (i v ∂y· i r)] = ∂v ⇒ = ∂v −cos i∂vsince ix · i r = cos Ω cos θ − sin Ω sin θ cos i iy · i r = sin Ω cos θ + cos Ω sin θ cos i This gives the perturbative derivative of θ , i.e., the change in θ due to the change in i n from which the angle θ is measured. The total time rate of change of θ is the sum dθ∂θ ∂θ h dΩ = + ad = cos i dt ∂t ∂v r2 −dt Since θ = ω + f , then dω∂f dΩ = − adt d − cos i ∂v dt Gauss’ form of Lagrange’s variational equations in polar coordinates dΩ r sin θ = adt h sin idh di r cos θ = adt h dhdω 1 r sin θ cos i = [−p cos fadr +(p + r) sin fadθ] − adt he h sin i dhda 2a2 =� pe sin fadr + adθ dt h r de 1 = {p sin fadr +[(p + r)� cos f + re]adt h dθ}df h 1 = + [p cos fadr − (p + r) sin fadθ]dt r2 ehGauss’ form of the variational equations in tangential-normal coordinates dω 1 � � r � � r sin θ cos i = 2 sin fadt + …