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.where R, called the disturbing function, can be written as R = Gm �x 1 1 1 1 Gm ρ r Gm ρ − r · ρ d � = Gm �rρ cos α = cos = ρ3 d − α νx ρ3 � ρ �d −���� ρ � �−� Since 2 2 2 � �� � ρ d ν d (r − ρ) · (r − ρ) r− 2rρ cos α + ρ= = =1 2νx+x 2 ρ = =(1 2νx+x 2)−1 2 ρ2 ρ · ρ ρ2 − ⇒d−Generating Function for Legendre Polynomials L(x, ν)=(1 − 2νx + x 2)−1 ∞2= k� Pk(ν)x k=0 The Disturbing Function and Its Gradient Gm � 1+ � ∞r kR = Pρk(cos α)ρ k=2 � �� d2r µ m �∞� r �k+ r = G [Pdt2 r3 ρ2 ρ k +1(cos α) iρ − Pk (cos α) ir]k=1 16.346 Astrodynamics Lecture 29 � � �� Lecture 29 The Disturbing Function & Legendre Polynomials #8.4 The Disturbing Function Three-body equations of motion d2r1 = Gm2 (r2 − r1)+ Gm3 (r3 − r1)dt2 r3 r3 12 13 d2r2 m1 m3= G 3 (r1 − r2)+ G 3 (r3 − r2)dt2 r21 r23 d2r3 m1 m2= G (r1 − r3)+ G (r2 − r3)dt2 r3 r3 31 32 Ignore the third equation. Subtract the first equation from the second and define r = r2 − r1 ρ = r3 − r1 d = r − ρ µ = G(m1 + m2) m = m3 Then, the equation of relative motion may be written as d2rT + µ rT = −Gm 1 dT +1 ρ T = Gm ∂ 1 d − 1 r · ρ = ∂R dt2 r3 d3 ρ3 ∂r ρ3 ∂rProperties of Legendre Polynomials Ro� drigues’ formula1 dn P n(ν)= F [−n, n +1;1; 12(1 − ν)] = �� (nn0� ν2 P2− 1)n(ν)=1 n! dνPHyp1(ν)= ν ergeometric function P2 2(ν)= 1 2(3ν − 1) nP � n(ν) � − (2n 1)νP �� − n 1(ν)+(n � −12(5ν3 − 1)P n(ν)=0−2P3(ν)= 4 − 3ν) Recursion P2 81 ��formula 4(ν)= 1(35ν − 30ν +3) �� P m(ν)P n(ν) dν =0 �Orthogonality property −1 (n − 1)Pn (ν) − (2n − 1)νPn (ν)+ nP (ν)=0−1 n−2Laplace’s Sphere of Influence #8.5 Motion of m2 relative to m1 (planet) Motion of m2 relative to m3 (sun) d2r G(m1 + m2) 11 d2d G(m2 + m ) Gm d + ρ = 311 = r Gmdt− d r ρ 2 � ��r3 −3 � d3 ρ3 dt2 −d3 −1 r−ρ3 p � � 3 � aap d d 21 21 a23 a23 Primary accelerations �� �� � � �� �� �� � p G(m1 + m ) a = − 2G(m1 + m2) = pG(mr − i a = 1 + m2)21 r3 r2 r21 r2 p G(m2 + m3) G(m2 + m3) G(m2 + ma3) 23 = − d = ip a = d3 −d2d23 r2− 2rρ cos α + ρ2 Disturbing accelerations 1 1 Gmr2ad Gm23 = −Gm� r 3 − 1 1 ρ 3 � = 2� i1 (2 i2 ρ − i r � = x 2ρ r− i ) ρ r ρ rrad 21 � 11 = −Gm3 d + ρ �= �∂R3 �T Gm= 3 � ∞x k[Pk +1(ν) i P (ν) i ] d3 ρ3 ∂r ρ2 ρ−k rk=1 From Eq. (8.72) Gm3 Gm≈ x[P2 (ν) i −3� �� � ρ2 ρ P1 (ν) i r]= x(3ν iρ2 ρ − i r) a d Gm1 x 2 Gm Gmi − i = 1 � Gm= | | x4 2 1 2 4 1 23 2 ρ r 2x cos α +1= 1 2νx + x r r2 −r2 � − ≈r2 d Gm3 Gm3 Gma2 3 2 21 = x|3ν iρ − i r| = x � 9νρ2 ρ− 6ν cos α +1= x 2 ρ2 � 1+3ν16.346 Astrodynamics Lecture 29Determine the ratios ad 2 ρ2 23 Gm1 r 2rνρ + m11 2p −≈ × = (1 2νx + x )a r2G2 23 (m2 + m3) m2 + m3 ×x× − 1 ad Gm � ≈21 3 r2 m= 3 x �� 2 � x 2 p x 1+3ν2 1+3νa2 21 ≈ρ� ×G(m1 + m2) m1 + m2 ×� ×Set the ratios equal ad d 21 a23 m mp= 3p = a21 a23 ⇒ m1 + m2 × �1+3ν2 × x 5 = 1 (1 2νx + x 2)m2 + m3 × −5 m1(m1 + m2) 1x = m3(m2 + m3) ×√1+3ν2 Finally, 1 ≤ (1+3ν2) 1 10< 1.15 and for m2  m1 and m2  m3 r≈ � m2 x = 51 ρ m3 �� � 2 m5 Radius of Sphere of Influence in miles = P mS × aP in miles where mP is the mass of the Planet, aP is the semimajor axis of the Planet and mS is the mass of the Sun. 16.346 Astrodynamics Lecture