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 28 Effect of Atmospheric Drag on Satellite Orbits #10.6 Terminology Disturbing acceleration a = −cρv2 dit Ballistic coefficient c Atmospheric density r)= ρ r q  ae ρ(0 exp −− = ρ0 exp[−ν(1 −cos E)] whereν =H HRadius of orbit r = a(1 −e cos E) Pericenter radius of orbit q = a(1 −e) Density at pericenter radius ρ0 Scale height of atmosphere H The Variational Equation de 1 r de 2 = 2(e + cos f)adt − sin fadn  = ⇒ = (e + cos f)adt v a dt v dta(1 −e2) r = a(1 − e cos E)= 1+ e cos f cos E = − e e + cos f cos f cos E = 1 − e cos E 1+ e cos f 2 2 1 2 2 1+ e cos E v = µ = n a r − a 1 − e cos E Hence de  1+ e cos Ep = −2cρv(e + cos f)= −2c × ρ cos dt 0 exp[−ν(1 − cos E)] × na E 1 − e cos E ×r so that de pna 1+ e cos E = dt −2ν cρcos E0 e−ν e cos E r  1 − e cos E  Series Representation Expand in a power series See Appendix C 1+ e cos E =1+2e cos E +2e 2cos2 E +2e 3cos3 E +1 − e cos E ··· 1+ e cos E cos E = cos E(1 + e cos E + 1 e 2 2cos2 E + 1 e 3 cos3 E + )1 − e cos E 2···This can be converted to a Fourier Cosine Series A0 + A1 cos E + A2 cos 2E + A3 cos 3E + ··· using Euler’s pattern 16.346 Astrodynamics Lecture 28cos2 E = 12(cos 2E +1) A0 = 1e2(1 + 3e 28) cos3 E = 1(cos 3E 4+ 3 cos E) A1 =1+ 3 e 2 8+ 15 e 4 Prob. 5–11 64cos4 E = 18(cos 4E + 4 cos 2E +3) A12 = e22(1 + 1e 2) cos5 E = 1 E E 16(cos 5 + 5 cos 3 + 10 cos E) A3 = 1 e 2(1 + 15 e 28 16)A more meaningfull result is obtained by averaging over a complete orbit de 1 2π de 1 2πr de = dM = dE dt 2π  0 dt 2π  0 a dt to obtain de ∞= −2−ν cρ0pne AkIk(ν)dt k=0 where 1 πIk(ν)= π  e ν cos E cos kE dE 0 is the modified Bessel function of the first kind of order k. Because of the relation to Bessel functions through the identity Ik(ν)= i−kJk(iν) it is sometimes referred to as a Bessel Function with Imaginary Argument. It can also be expressed as the series expansion  ∞( 1 ν2 )k+2jIk(ν)= j!(k + j)!j=0 Calculating Modified Bessel Functions Since ν is generally large, we can use the asymptotic expansion to calculate Ik(ν): √k2 ν 4− 12 (4k2 − 12)(4k2 − 32) e−2πν Ik(ν) ∼− + 1! 8ν 2! (8ν)2 (4k2 − 12)(4k2 − 32)(4k2 − 52)− +3! (8ν)3 ··· as obtained by Carl Gustav Jacob Jacobi in 1849. Although the series will eventually diverge as the number of terms increases, it can be used for numerical computation by employing only those terms whose magnitude decreases as we take more and more terms. The order of magnitude of the error at any stage is equal to the magnitude of the first term omitted. Note: Leonard Euler was using divergent series for computation in the middle 1770’s but the formal theory of asymptotic series was developed much later by Henri Poincar´e in 1886. 16.346 Astrodynamics Lecture 28A series of the form aa2 0 + 1 a+ + ··· x x2 where the ai ’s are independent of x, is said to represent the function f(x) asymptotically for large x whenever a a lim x n af(x) x→∞ − an0 + 1 + 2 + + =0 x x2 ···xn for n =0, 1, 2, 3,.... The series will  usually diverge.  We can use the recursion formula −ν ν2k e Ik+1(ν)= e− Ik−1(ν) − e−ν I (ν) νkto calculate higher-order values of e−ν Ik(ν) starting with e−ν I0(ν) and e−ν I1(ν). Also we can use the continued fraction e−ν I1(ν)1 ν = 2e−ν I0(ν)( 1 ν)2 1+ 2 ( 1 ν)2 2+ 2 ( 1 ν)2 3+ 2 ( 1 ν)2 4+ 2 5+ ... to calculate e−ν Iν1(ν) from e− I0(ν). A sensible receipt would be First: Calculate e−ν I0(ν) from the asymptotic series e−ν √12 12 32 12 32 52 2πν I0(ν) ·∼ 1+ + +· ·+ 1! 8ν 2! (8ν)2 3! (8ν)3 ···Next: Calculate e−ν I1(ν) using the continued fraction. Higher order functions can then be obtained using the recursion formula. 16.346 Astrodynamics Lecture