Lecture 2: The Two-Body Problem (Cont.)

Lecture 2 The Two Body Problem Continued The Eccentricity Vector or The Laplace Vector µe = v × h − µ r r Explicit Form of the Velocity Vector #3.1 Using the expansion of the triple vector product a × (b × c) = (a · c)b − (a · b)c we have h × µe = h × (v × h) − µ h × r = h2 v − (h · v)h − µh × ir = h2 v − µh ih × ir r since h and v are perpendicular. Therefore: h × µe =⇒ v= µ i × (e + ir ) h h or hv = ih × (e ie + ir ) = e ih × ie + ih × ir = e ip + iθ µ Then since ip = sin f ir + cos f iθ we have hv = e sin f ir + (1 + e cos f ) iθ µ which is the basic relation for representing the velocity vector in the Hodograph Plane. See Page 1 of Lecture 4 Conservation of Energy p p hv hv · = v · v = 2(1 + e cos f ) + e2 − 1 = 2 × − (1 − e2 ) = p µ µ µ r 2 1 − r a  which can be written in either of two separate forms each having its own name: Energy Integral Vis-Viva Integral 1 µ 1 2 µ = c3 v − =− 2 2 r 2a 2 1 − v2 = µ r a The constant c3 is used by Forest Ray Moulton, a Professor at the University of Chicago in his 1902 book “An Introduction to Celestial Mechanics” — the first book on the subject written by an American. 16.346 Astrodynamics Lecture 2 Conic Sections Ellipse or Hyperbola in rectangular coordinates ( e = 1 ) y 2 = r2 − x2 = (p − ex)2 − x2 = (1 − e2 )[a2 − (x + ea)2 ] y2 (x + ea)2 + =1 a2 (1 − e2 ) a2 Semiminor Axis:   b2 = a2 (1 − e2 ) = |a| p Ellipse Fig. 3.1 from An Introduction to the Mathematics and Methods of Astrodynamics. Courtesy of AIAA. Used with permission. Hyperbola Fig. 3.2 from An Introduction to the Mathematics and Methods of Astrodynamics. Courtesy of AIAA. Used with permission. Parabola in rectangular coordinates ( e = 1) y 2 = r2 − x2 = (p − x)2 − x2 =⇒ Parabola 16.346 Astrodynamics Lecture 2 y 2 = 2p( 12 p − x) Fig. 3.3 from An Introduction to the Mathematics and Methods of Astrodynamics. Courtesy of AIAA. Used with permission. Alternate Forms of the Equation of Orbit #4.1 Origin at focus r + ex = p Origin at center r + ex = a With x now measured from the center which is at a distance ae from the focus, then r + ex = p r + e(x − ae) = p = a(1 − e2 ) r + ex = a Origin at pericenter r + ex = q With x now measured from pericenter which is at a distance of a from the center and a distance of q = a(1 − e) from the focus, then r + ex = p r + e(x + q) = p = q(1 + e) r + ex = q These are useful to derive other properties of conic sections: • Focus-Directrix Property: r = p − ex: Page 144 p  PF = r = e − x = e × PN e or PF =e PN • Focal-Radii Property: r = a − ex: Page 145 P F 2 = (x − ea)2 + y 2 P F ∗2 = (x + ea)2 + y 2 P F ∗2 = P F 2 + 4aex so that = r2 + 4aex ∗  PF = = (a − ex)2 + 4aex = (a + ex)2 a + ex −(a + ex) ellipse hyperbola a>0 a < 0, x < 0 Thus, P F ∗ + P F = 2a P F ∗ − P F = −2a • Euler’s Universal Form: ellipse hyperbola From r = q − ex: Page 143 y 2 = r2 − (q + x)2 = (q − ex)2 − (q + x)2 Then 16.346 Astrodynamics y 2 = −(1 + e)[2qx + (1 − e)x2 ] Lecture 2 Basic Two-Body Relations d2 r µ + 3r = 0 2 dt r Vector Equations of Motion r× Angular Momentum Vector dv ×h dt Eccentricity Vector µe · r Equation of Orbit dv =0 dt =⇒ =⇒ h × µe Velocity Vector =⇒ p≡ h2 µ dv µ = − 3r dt r r × v = constant ≡ h 1 v × h − ir = constant ≡ e µ r= =⇒ p h2 /µ = 1 + e cos f 1 + e cos f v= 1 h × (e + ir ) p p Orbital Parameter Dynamics Definition: or p = a(1 − e2 ) Geometric Definition: Total Energy or Semimajor Axis or Mean Distance a 1 2 µ µ v − = constant ≡ − 2 r 2a 2 2 (x + ea) y Geometric Definition: =1 + 2 2 a a (1 − e2 ) Dynamics Definition:  dθ 2 µ d2 r − r + =0 dt2 dt r2 Eqs. of Motion in Polar Coord. d  2 dθ  r =0 dt dt Kepler’s Laws dA 1 dθ h = r2 = constant = dt 2 dt 2 Second Law First Law r= Third Law 16.346 Astrodynamics p 1 + e cos f h πab = P 2 or =⇒ r = p − ex a3 µ = constant = 2 P 4π 2 Lecture 2