ORBITAL MECHANICS: HOW OBJECTS MOVE IN SPACE

1. ORBITAL MECHANICS: HOW OBJECTS MOVE IN SPACE FROM KEPLER FIRST LAW: A SATELLITE REVOLVES IN AN ELLIPTICAL ORBIT AROUND A CENTER OF ATTRACTION POSITIONED AT ONE FOCI OF THE ELLIPSE. SECOND LAW: THE RATE OF TRAVEL ALONG THE ORBIT IS DIRECTLY PROPORTIONAL TO THE AREA OF SWEEP IN THE ELLIPSE. THIRD LAW: PERIOD OF THE ORBIT SQUARED IS PROPORTIONAL TO THE MEAN DISTANCE TO CENTER CUBED. ORBIT PLANE APOGEE CENTER OF ATTRACTION (FOCI) AREA OF SWEEP PERIGEE SATELLITE EARTH EXAMPLE

2. ORBITAL MECHANICS: WHY OBJECTS MOVE THE WAY THEY DO NEWTONIAN THEORY Fg = G M m / r2 Fc = m V2 / r A SATELLITE MAINTAINS ITS ORBIT WHEN Fc = Fg G = UNIVERSAL GRAVITY CONSTANT M = MASS OF EARTH m = MASS OF SATELLITE r = DISTANCE EARTH CENTER TO SATELLITE IN NEAR CIRCULAR ORBITS THE ORBITAL VELOCITY IS ABOUT CONSTANT. IN HIGHLY ELLIPTICAL ORBITS THE SATELLITE SPEEDS UP TO MAX VELOCITY AT PERIGEE AND SLOWS DOWN TO MIN VELOCITY AT APOGEE. V = ORBITAL VELOCITY ORBITAL PATH Fc = CENTRIPETAL FORCE DUE TO REVOLUTION ALTITUDE Fg = GRAVITATION FORCE

3. Newton’s Laws • A body remains at rest or in constant motion unless acted upon by external forces • The time rate of change of an object’s momentum is equal to the applied force • For every action there is an equal and opposite reaction • The force of gravity between two bodies is proportional to the product of their masses and inversely proportional to the square of the distances between them.

4. Acceleration, Time, Distance • F = ma • Vf = V0 + at • s = V0t + at2 2

5. Vector Addition V1+ V3 = V2 Vector addition is done by adding the two head to tail vectors to equal the tail to tail and head to head vector V3 V1 V2 Law of Sines a / sin A = b / sin B = c / sin C Law of Cosines a2 = b2 + c2 - 2bc Cos A b2 = a2 + c2 - 2ac Cos B c2 = a2 + b2 - 2ab Cos C C a b A B c

6. Trigonometric Functions y r q x sine = opposite/ hypotenuse = y/r cosine = adjacent / hypotenuse = x/r tangent = opposite / adjacent = y/x sin 0 = 0 cos 0 = 1 sin 90 = 1 cos 90 = 0 tan 0 = 0 tan 90 = + infinity -1< sin or cos <+1 - infinity < tangent < + infinity y x r2 = x2 + y2 sin2q +cos2 q = 1

7. Rocket Engines • Liquid Propellant • Mono propellant • Catalysts • Bi-propellant • Solid Propellant • Grain Patterns • Hybrid • Nuclear • Electric Performance Energy Safety Simplicity Expanding Gases Thrust Termination Restart

8. Specific Heat cp dQ w dT >1 c = k = cv • Specific Heat: the amount of heat that enters or leaves a unit mass while the substance changes one degree in temperature. • c = Btu per lbsm - degree Rankine • cp : specific heat at constant pressure • cv : specific heat at constant volume • k : ratio of specific heats

9. Specific Impulse 1 - ( ) ( )( ) Pek-1 Pc [ ] Tc mg k k - 1 k F Isp = . W Isp = 9.797 Pe : Nozzle Exit Pressure (psi) <14.7 psi Pc : Combustion Chamber Pressure (psi) 6,000 psi Tc : Combustion Chamber Temperature (degrees Rankine) 5,000o R mg : average molecular weight of combustion products (lb/mole) 2H2+O 2 2H2O 18 lbs/mole = mg

10. Launch Velocity Losses • Gravity losses • Pitch over to get correct velocity vector alignment for orbital insertion • Drag from atmosphere • Not instantaneous application of velocity Losses are between 15 and 17 % of DV

11. minitial mfinal MR = F Isp = . W . W Ve F = + Ae ( Pe - Pa) g Rocket Formulas Rocket Equation DV = Ispx g x ln MR Mass Ratio Specific Impulse Thrust

12. Three Stage Booster Burn Time (sec) Weights (lbs.) Stage Weight Isp (sec) MR Structure Propellants 1st Stage 35,000 365,000 100 280 400,000 599,000 2.56 2nd Stage 10,000 125,000 120 290 135,000 199,000 2.69 3rd Stage 4,000 50,000 80 250 54,000 4.57 Payload 10,000 All three stages DV1 = (280)(32.2)ln (2.56) = 8,475 ft/sec DV2 = (290)(32.2)ln (2.69) = 9,238 ft/sec DV3 = (250)(32.2)ln (4.57) = 12,232 ft/sec Vl = 29,945 ft/sec Vposigrade = 29,535 ft/sec Vretrograde = 30,183 ft/sec Can place payload in posigrade orbit, but not in retrograde orbit,

13. ORBIT FORMULAS a r’+ r = 2a if r= r’ then at that point 2r = 2a r = a . a2 - c2 . . b = ELLIPTICAL & CIRCULAR ORBITS } a 2a = rA + rP } } b c rA = eccentricity e rP = Apogee Perigee a c rP c = a - a - rP e = r r’ a b rP e = 1 - a c 1 - 2rP e = rA + rP rA - rP e = rA + rP

14. CONSTANTS FOR ORBIT m H2 PHYSICAL m GEOMETRIC E = Specific Energy H = Specific Momentum e a b c = Universal gravitational attraction mr m2 r n e E < 0 , = 0 for a circle V e E < 0 , 0 < < 1 for an ellipse m1 Trajectory m2 e = 1 for a parabola m e > 1 for a hyperbola = r 1 + 2EH2 m 1 + cos n Polar coordinates for any conic section pages 32, 33, 34 “Handout” 2 m H2 e k = e = 2EH2 1 + 2

15. a3 4 p 2 Orbital Period CIRCULAR ORBIT ELLIPTICAL ORBIT 2 p a 2 p r P = Period = V V m a = mean distance from focus = = semi major axis V = r 4 p r 2 2 P2 = P2 = m m r KEPLER’S THIRD LAW 4 p r 3 2 P2 = P2 = (2.805 x 1015)a3 m sec2 units = ft3

16. for circular orbits a = r therefore EARTH SATELLITES Eccentricity = e rp = a - c ra = a + c rp + ra = 2a Major Axis = 2a Minor Axis = 2b c a e = E = specific energy H = specific angular momentum m c2 = a2 - b2 E = V2 M (1) = m 2 2a r m m H = Vr cos V = r for elliptical orbits from (1) 2 - V = r a

17. Cartesian Coordinates Abscissa = x Ordinate = y (x,y) Polar Coordinates Radius Vector = r Vectorial Angle = q (r,q) Coordinate Systems +y +x -x -y r q

18. Description of Orbit • Right Ascension • Measured eastward from the vernal equinox • In Spring when the sun’s center crosses the equatorial plane once thought to be aligned with the first point of the constellation Aries • Inclination • Argument of Perigee • Two of the following • Eccentricity • Perigee • Apogee

19. Orbit Calculations b b a a x2 y2 a2 b2 + = 1 c e = a Ellipse is the curve traced by a point moving in a plane such that the sum of its distances from the foci is constant. ra rp r’ r c r + r’ = 2a a2 = b2 + c2

20. INCLINATION 270o FUNCTION OF LAUNCH AZIMUTH AND LAUNCH SITE LATITUDE cos i (inclination) = cos (latitude) sin (azimuth) N North = 0 degrees Azimuth AZIMUTH azimuth 90o cos i + cos (lat) sin (az) East West lat. sin 90o = 1 sin 0o = 0 sin 180o = 0 sin 270o = -1 180o South launch azimuth from 180o to360o = retrograde orbit launch azimuth from 0o to 180o = posigrade orbit S

21. Argument of Perigee (w) Perigee Celestial Equator Inclination Right Ascension W Orbit Trace Celestial Sphere

22. ORBITAL MECHANICS: GROUND TRACES INCLINED ORBIT SAT ORBIT 4 ORBIT 3 EQUATORIAL ORBIT ORBIT 2 ORBIT 1 MULTIPLE ORBITS EARTH MOTION BENEATH SATELLITE ANGLE OF INCLINATION (0 DEG. FOR EQUATORIAL) • GROUND TRACES • THE POINTS ON THE EARTH’S SURFACE OVER WHICH A SATELLITE PASSES AS IT TRAVELS ALONG ITS ORBIT • PRINCIPLE : GROUND TRACE IS THE RESULT OF THE ORBITAL PLANE BEING FIXED AND THE EARTH ROTATING UNDERNEATH IT • AMPLITUDE OF GROUND TRACE (LATITUDE RANGE) IS EQUAL TO THE ORBITAL INCLINATION • MOVEMENT OF GROUND TRACE IS DICTATED BY THE SATELLITE ALTITUDE AND THE CORRESPONDING TIME FOR IT TO COMPLETE ONE ORBIT

23. ORBITAL MECHANICS: SPECIFIC ORBITS AND APPLICATIONS • POLAR (100- 700 NM AT 80 - 100 DEG. INCLINATION) • SATELLITE PASSES THROUGH THE EARTH'S SHADOW AND PERMITS VIEWING OF THE ENTIRE EARTH’S SURFACE EACH DAY WITH A SINGLE SATELLITE • SUN SYNCHRONOUS (80 - 800 NM AT 95 - 105 DEG INCLINATION) • PROCESSION OF ORBITAL PLANE SYNCHRONIZED WITH THE EARTH’S ROTATION SO SATELLITE IS ALWAYS IN VIEW OF THE SUN • PERMITS OBSERVATION OF POINTS ON THE EARTH AT THE SAME TIME EACH DAY • SEMISYNCHRONOUS (10,898 NM AT 55 DEG INCLINATION) • 12 HR PERIODS PERMITTING IDENTICAL GROUNDTRACES EACH DAY • HIGHLY INCLINED ELLIPTICAL (FIXED PERIGEE POSITION) • SATELLITE SPENDS A GREAT DEAL OF TIME NEAR THE APOGEE COVERING ONE HEMISPHERE • CLASSICALLY CALLED “MOLNIYA ORBIT” BECAUSE OF ITS HEAVY USE BY THE RUSSIANS FOR NORTHERN HEMISPHERE COVERAGE • GEOSYNCHRONOUS (GEO) (CIRCULAR, 19,300 NM AT 0 DEG INCLINATION) • 24 HR PERIOD PERMITS SATELLITE POSITIONING OVER ONE POINT ON EARTH. • ORBITAL PERIOD SYNCHRONIZED WITH THE EARTH’S ROTATION (NO OTHER ORBIT HAS THIS FEATURE)

24. radians S/r q = qf qf radians Vtangential = r w ft/sec wavg = sec tf to wf wo radians atangential = r a ft/sec2 aavg = sec2 tf to wf = wo + a t wo t+ a t2 q = 2 Linear and Angular Motion ANGULAR MOTION LINEAR MOTION Distance S = r q ft Velocity Acceleration (1 radian = 57.3 degrees)

25. s r r Q = s radians Q

26. CONSERVATION OF: ENERGY MOMENTUM m r V2 2 Angular Momentum = mr2w constant = F - = Specific Energy# mr2w = constant = mH H = V r cos f Specific Angular Momentum Vr f ^ V #Specific means per 1b mass

27. CONSTANTS FOR ORBIT m H2 PHYSICAL m GEOMETRIC E = Specific Energy H = Specific Momentum e a b c = Universal gravitational attraction mr m2 r n e E < 0 , = 0 for a circle V e E < 0 , 0 < < 1 for an ellipse m1 Trajectory m2 e = 1 for a parabola m e > 1 for a hyperbola = r 1 + 2EH2 m 1 + cos n 2 m H2 e k = e = 2EH2 1 + 2

28. ESCAPE VELOCITY m m ft3 sec2 ft 2 (2) 14.075x1015 sec r 20.9 x 106 ft E = 0 V2 2 r E1 = V2 0 = 2 r m Vescape = = 36,700 =