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Orbital • Planetary launch and entry overview • Energy and velocity in • Elliptical orbit parameters • • Coplanar orbital transfers • Noncoplanar transfers • Time in orbit • Interplanetary

© 2020 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Orbital Mechanics MARYLAND 1 ENAE 791 - Launch and Entry Vehicle Design Orbital Mechanics: 500 years in 40 min. • Newton’s Law of Universal Gravitation Gm m F = 1 2 r2 • Newton’s First Law meets vector algebra

⇥F = m⇥a

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 2 ENAE 791 - Launch and Entry Vehicle Design Relative Motion Between Two Bodies

2 d r1⃗ m1m2 r21⃗ m2 m = G ⇥r2 1 dt2 2 r21⃗ r21⃗

⇥ m1m2 m1m2 F12 = G r21⃗ = G (r2⃗ − r1⃗ ) ⇥r1 3 3 F⇥21 r21⃗ r21⃗ 2 d r2⃗ m1m2 m m2 = G (r1⃗ − r2⃗ ) 1 dt2 3 r12⃗

F⇥12 = due to body 1 on body 2

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Gravitational Motion

Let µ = G(m1 + m2) d2~r ~r + µ = ~0 dt2 r3 “Equation of Orbit” - Orbital motion is simple harmonic motion

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Orbital Angular Momentum

⃗ h is angular momentum vector (constant) =⇒ ⃗r and ⃗v are in a constant plane

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 5 ENAE 791 - Launch and Entry Vehicle Design Fun and Games with Algebra

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U N I V E R S I T Y O F Orbital Mechanics MARYLAND 6 ENAE 791 - Launch and Entry Vehicle Design More Algebra, More Fun

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Orientation of the Orbit

~e eccentricity vector, in orbital plane ⌘ ~e points in the direction of periapsis

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 8 ENAE 791 - Launch and Entry Vehicle Design Position in Orbit

θ = : angular travel from perigee passage

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 9 ENAE 791 - Launch and Entry Vehicle Design Relating Velocity and Orbital Elements

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 10 ENAE 791 - Launch and Entry Vehicle Design Vis-Viva Equation 1 e2 p a(1 e2)= ⌘ 2 v2 r µ 1 2 v2 a = r µ ✓ ◆ 2 1 v2 = µ <--Vis-Viva Equation r a ✓ ◆ v2 µ µ = 2 r 2a

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 11 ENAE 791 - Launch and Entry Vehicle Design Energy in Orbit • 1 K.E. v2 K.E. = mν 2 ⇒ = 2 m 2 • mµ P.E. µ P.E. = − ⇒ = − r m r • Total Energy v2 µ µ Const. = − = − <--Vis-Viva Equation 2 r 2a

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 12 ENAE 791 - Launch and Entry Vehicle Design Suborbital Tourism - Spaceship Two

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 13 ENAE 791 - Launch and Entry Vehicle Design How Close are we to ? • Energy for 100 km vertical climb

µ µ km2 MJ + = 0.965 2 = 0.965 rE + 100 km rE sec kg

• Energy for 200 km

µ µ km2 MJ + = 32.2 2 = 32.2 2(rE + 200 km) rE sec kg • Energy difference is a factor of 33!

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design Implications of Vis-Viva • Circular orbit (r=a) µ v = circular ! r • Parabolic escape orbit (a tends to infinity) 2µ v = escape ! r • Relationship between circular and parabolic

vescape = √2vcircular

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 15 ENAE 791 - Launch and Entry Vehicle Design Some Useful Constants • Gravitation constant µ = GM – Earth: 398,604 km3/sec2 – Moon: 4667.9 km3/sec2 – Mars: 42,970 km3/sec2 – : 1.327x1011 km3/sec2 • Planetary radii

– rEarth = 6378 km

– rMoon = 1738 km

– rMars = 3393 km

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 16 ENAE 791 - Launch and Entry Vehicle Design Classical Parameters of Elliptical Orbits

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 17 ENAE 791 - Launch and Entry Vehicle Design Basic Orbital Parameters • Semi-latus rectum (or parameter) 2 p = a(1 − e ) • Radial distance as function of orbital position p r = 1+e cos θ • Periapse and apoapse distances

rp = a(1 e) ra = a(1 + e) − • Angular momentum ⃗ h = ⃗r × ⃗v h = √µp

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 18 ENAE 791 - Launch and Entry Vehicle Design The Classical Orbital Elements

Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 19 ENAE 791 - Launch and Entry Vehicle Design The Hohmann Transfer

r2 v2 r1 vapogee

v1 vperigee

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design First Maneuver Velocities • Initial vehicle velocity µ v1 = !r1 • Needed final velocity µ 2r2 vperigee = !r1 !r1 + r2 • Required ΔV µ 2r2 ∆v1 = 1 !r1 "!r1 + r2 − #

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 21 ENAE 791 - Launch and Entry Vehicle Design Second Maneuver Velocities • Initial vehicle velocity

µ 2r1 vapogee = !r2 !r1 + r2 • Needed final velocity µ v2 = !r2 • Required ΔV µ 2r1 ∆v2 = 1 !r2 " − !r1 + r2 #

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Limitations on Launch Inclinations

Equator

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 23 ENAE 791 - Launch and Entry Vehicle Design Differences in Inclination

Line of Nodes

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 24 ENAE 791 - Launch and Entry Vehicle Design Choosing the Wrong of Apsides

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 25 ENAE 791 - Launch and Entry Vehicle Design Simple Plane Change

v1 Δv2 vapogee v vperigee 2

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 26 ENAE 791 - Launch and Entry Vehicle Design Optimal Plane Change

v v perigee 1 Δv2 vapogee Δv1 v2

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 27 ENAE 791 - Launch and Entry Vehicle Design First Maneuver with Plane Change Δi1 • Initial vehicle velocity µ v = 1 r 1 • Needed final velocity µ 2r v = 2 p r r + r 1 1 2 • Required ΔV v = v2 + v2 2v v cos i 1 1 p 1 p 1 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 28 ENAE 791 - Launch and Entry Vehicle Design Second Maneuver with Plane Change Δi2

• Initial vehicle velocity µ 2r v = 1 a r r + r 2 1 2 • Needed final velocity µ v = 2 r 2 • Required ΔV v = v2 + v2 2v v cos i 2 2 a 2 a 2 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 29 ENAE 791 - Launch and Entry Vehicle Design Sample Plane Change Maneuver

Optimum initial plane change = 2.20° U N I V E R S I T Y O F Orbital Mechanics MARYLAND 30 ENAE 791 - Launch and Entry Vehicle Design Calculating Time in Orbit

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 31 ENAE 791 - Launch and Entry Vehicle Design Time in Orbit • Period of an orbit a3 P = 2⇥ µ • (average angular velocity) µ n = a3 • Time since pericenter passage M = nt = E e sin E ➥M= U N I V E R S I T Y O F Orbital Mechanics MARYLAND 32 ENAE 791 - Launch and Entry Vehicle Design Dealing with the • Relationship to orbit r = a (1 e cos E) • Relationship to true anomaly 1 + e E tan = tan 2 1 e 2 • Calculating M from time interval: iterate

Ei+1 = nt + e sin Ei until it converges U N I V E R S I T Y O F Orbital Mechanics MARYLAND 33 ENAE 791 - Launch and Entry Vehicle Design Example: Time in Orbit • Hohmann transfer from LEO to GEO

– h1=300 km --> r1=6378+300=6678 km

– r2=42240 km • Time of transit (1/2 ) 1 a = (r + r ) = 24, 459 km 2 1 2

P a3 ttransit = = ⇥ = 19, 034 sec = 5h17m14s 2 µ

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 34 ENAE 791 - Launch and Entry Vehicle Design Example: Time-based Position Find the spacecraf position 3 hours afer perigee

µ 4 rad n = = 1.650x10 a3 sec r e = 1 p = 0.7270 a Ej+1 = nt + e sin Ej =1.783 + 0.7270 sin Ej E=0; 1.783; 2.494; 2.222; 2.361; 2.294; 2.328; 2.311; 2.320; 2.316; 2.318; 2.317; 2.317; 2.317

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 35 ENAE 791 - Launch and Entry Vehicle Design Example: Time-based Position (cont.) E =2.317 r = a(1 e cos E) = 12, 387 km 1+e E tan = tan = = 160 deg 2 1 e 2 ⇥ Have to be sure to get the position in the proper quadrant - since the time is less than 1/2 the period, the spacecraf has yet to reach apogee --> 0°<θ<180°

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 36 ENAE 791 - Launch and Entry Vehicle Design Velocity Components in Orbit p r = 1+e cos dr d p p( e sin d ) v = = = dt r dt dt 1+e cos (1 + e cos )2 ⇥ pe sin d v = r (1 + e cos )2 dt p r2 d e sin 1+e cos = v = dt r r p ⇤h = r v ⇤ ⇥ ⇤ U N I V E R S I T Y O F Orbital Mechanics MARYLAND 37 ENAE 791 - Launch and Entry Vehicle Design Velocity Components in Orbit (cont.)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 38 ENAE 791 - Launch and Entry Vehicle Design Patched Conics • Simple approximation to multi-body motion (e.g., traveling from Earth orbit through solar orbit into Martian orbit) • Treats multibody problem as “hand-offs” between gravitating bodies --> reduces analysis to sequential two- body problems • Caveat Emptor: Tere are a number of formal methods to perform patched conic analysis. Te approach presented here is a very simple, convenient, and not altogether accurate method for performing this calculation. Results will be accurate to a few percent, which is adequate at this level of design analysis. U N I V E R S I T Y O F Orbital Mechanics MARYLAND 39 ENAE 791 - Launch and Entry Vehicle Design Example: Insertion

• v2 is velocity of moon around Earth • Moon overtakes spacecraf with velocity of (v2-vapogee) • Tis is the velocity of the spacecraf relative to the moon while it is effectively “infinitely” far away (before lunar accelerates it) = “hyperbolic excess velocity”

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 40 ENAE 791 - Launch and Entry Vehicle Design Planetary Approach Analysis

• Spacecraf has vh hyperbolic excess velocity, which fixes total energy of approach orbit v2 µ µ v2 = = h 2 r 2a 2 • Vis-viva provides velocity of approach 2µ v = v2 + ! h r • Choose such that approach is tangent to desired final orbit at periapse 2 2µ µ ∆v = vh + ! r − ! r

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 41 ENAE 791 - Launch and Entry Vehicle Design Patched Conic - Lunar Approach • Lunar orbital velocity around the Earth µ 398, 604 km vm = = =1.018 !rm !384, 400 sec • Apogee velocity of Earth transfer orbit from initial 400 km

2r1 6778 km va = vm =1.018 =0.134 !r1 + rm !6778 + 384, 400 sec • Velocity difference between spacecraf “infinitely” far away and moon (hyperbolic excess velocity) km vh = vm va = vm =1.018 0.134 = 0.884 − − sec U N I V E R S I T Y O F Orbital Mechanics MARYLAND 42 ENAE 791 - Launch and Entry Vehicle Design Patched Conic - Lunar Orbit Insertion • Te spacecraf is now in a hyperbolic . Te velocity it will have at the perilune point tangent to the desired 100 km low lunar orbit is

2µ 2(4667.9) km v = v2 + m = 0.8842 + =2.398 pm h r 1878 sec r LLO r • Te required delta-V to slow down into low lunar orbit is 4667.9 km v = vpm vcm =2.398 =0.822 r 1878 sec

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 43 ENAE 791 - Launch and Entry Vehicle Design ΔV Requirements for Lunar Missions

To: Low Earth Lunar Low Lunar Lunar Lunar Orbit Transfer Orbit Descent Landing From: Orbit Orbit Low Earth 3.107 Orbit km/sec

Lunar 3.107 0.837 3.140 Transfer km/sec km/sec km/sec Orbit Low Lunar 0.837 0.022 Orbit km/sec km/sec

Lunar 0.022 2.684 Descent km/sec km/sec Orbit Lunar 2.890 2.312 Landing km/sec km/sec

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 44 ENAE 791 - Launch and Entry Vehicle Design LOI ΔV Based on Landing Site

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 45 ENAE 791 - Launch and Entry Vehicle Design LOI ΔV Including Loiter Effects

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 46 ENAE 791 - Launch and Entry Vehicle Design Interplanetary Types

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 47 ENAE 791 - Launch and Entry Vehicle Design Interplanetary “Pork Chop” Plots • Summarize a number of critical parameters – Date of departure – Date of arrival – Hyperbolic energy (“C3”) – Transfer geometry • determines available C3 based on window, payload • Calculated using Lambert’s Teorem U N I V E R S I T Y O F Orbital Mechanics MARYLAND 48 ENAE 791 - Launch and Entry Vehicle Design C3 for Earth-Mars Transfer 1990-2045

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 49 ENAE 791 - Launch and Entry Vehicle Design Earth-Mars Transfer 2033

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 50 ENAE 791 - Launch and Entry Vehicle Design Earth-Mars Transfer 2037

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 51 ENAE 791 - Launch and Entry Vehicle Design Interplanetary Delta-V

Hyperbolic excess velocity Vh 2 ⌘ C3 = Vh 2 Vreq = Vesc + C3 V = Vp2 + C V esc 3 c 2033 Window:p V =3.55 km/sec 2037 Window: V =3.859 km/sec V in departure from 300 km LEO U N I V E R S I T Y O F Orbital Mechanics MARYLAND 52 ENAE 791 - Launch and Entry Vehicle Design Earth-Moon Potential Field

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 53 ENAE 791 - Launch and Entry Vehicle Design The Earth-Moon System

Note: Earth and Moon are in scale with size of L4 orbits

Geostationary Orbit Moon Earth L3 L1 L2

Photograph of Earth and Moon taken by Mars Odyssey April 19, 2001 from a distance of 3,564,000 km L5

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 54 ENAE 791 - Launch and Entry Vehicle Design Free-Body Diagram with Spherical

v !

g r

✓

U N I V E R S I T Y O F Orbital Mechanics MARYLAND 55 ENAE 791 - Launch and Entry Vehicle Design Orbital Planar State Equations

Inertial angular velocity v ! ! =˙ ✓˙ Sum of normal to velocity vector g g cos = !v r Sum of accelerations ✓ perpendicular to velocity vector g sin =˙v U N I V E R S I T Y O F Orbital Mechanics MARYLAND 56 ENAE 791 - Launch and Entry Vehicle Design Orbital Planar State Equations (2) r˙ = v sin r✓˙ = v cos v ! =˙ ✓˙ =˙ cos r v g cos = ˙ cos v r v2⇣ ⌘ g cos =˙v r ✓ ◆ v2 1 g cos =˙v rg ✓ ◆ U N I V E R S I T Y O F Orbital Mechanics MARYLAND 57 ENAE 791 - Launch and Entry Vehicle Design Canonical Orbital Planar State Equations 1 v2 ˙ = 1 g cos v v2 ✓ c ◆ v˙ = g sin r˙ = v sin v ✓˙ = cos r Coupled first-order ODEs r 2 g = g o o r U N I V E R S I T Y O F ⇣ ⌘ Orbital Mechanics MARYLAND 58 ENAE 791 - Launch and Entry Vehicle Design Numerical Integration - 4th Order R-K Given a series of equations y¯˙ = f¯(t, x¯)

k¯1 =t f¯(tn, y¯n) t k¯ k¯ =t f¯ t + , y¯ + 1 2 n 2 n 2 ✓ ◆ t k¯ k¯ =t f¯ t + , y¯ + 2 3 n 2 n 2 ✓ ◆ k¯4 =t f¯ tn +t, y¯n + k¯3 k¯ k¯ k¯ k¯ y ¯ =¯y + 1 + 2 + 3 + 4 +O(t5) n+1 n 6 3 3 6 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 59 ENAE 791 - Launch and Entry Vehicle Design References for This Lecture • Wernher von Braun, Te Mars Project University of Illinois Press, 1962 • William Tyrrell Tomson, Introduction to Space Dynamics Dover Publications, 1986 • Francis J. Hale, Introduction to Space Prentice- Hall, 1994 • William E. Wiesel, Spaceflight Dynamics MacGraw- Hill, 1997 • J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 60 ENAE 791 - Launch and Entry Vehicle Design