Astrodynamics (AERO0024)

Astrodynamics (AERO0024) 6. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L)

Course Outline THEMATIC UNIT 1: ORBITAL DYNAMICS Lecture 02: The Two-Body Problem Lecture 03: The Orbit in Space and Time Lecture 04: Non-Keplerian Motion THEMATIC UNIT 2: ORBIT CONTROL Lecture 05: Orbital Maneuvers Lecture 06: Interplanetary Trajectories 2

Motivation 3

6. Interplanetary Trajectories 6.1 Patched conic method 6.2 Lambert s problem 6.3 Gravity assist 4

6. Interplanetary Trajectories 6.1 Patched conic method 6.1.1 Problem statement 6.1.2 Sphere of influence 6.1.3 Planetary departure 6.1.4 Hohmann transfer 6.1.5 Planetary arrival 6.1.6 Sensitivity analysis and launch windows 6.1.7 Examples 5

? Hint #1: design the Earth-Mars transfer using known concepts Hint #2: division into simpler problems Hint #3: patched conic method 6

What Transfer Orbit? Constraints? Dep. V 1 Arr. Arr. V 2 Dep. Dep. Arr. Motion in the heliocentric reference frame 7

Planetary Departure? Constraints? v Earth / Sun v? V 1 β Motion in the planetary reference frame 8

Planetary Arrival? Similar Reasoning Transfer ellipse SOI SOI Arrival hyperbola Departure hyperbola 9

Patched Conic Method Three conics to patch: 1. Outbound hyperbola (departure) 2. The Hohmann transfer ellipse (interplanetary travel) 3. Inbound hyperbola (arrival) 6.1.1 Problem statement 10

Patched Conic Method Approximate method that analyzes a mission as a sequence of 2-body problems, with one body always being the spacecraft. If the spacecraft is close enough to one celestial body, the gravitational forces due to other planets can be neglected. The region inside of which the approximation is valid is called the sphere of influence (SOI) of the celestial body. If the spacecraft is not inside the SOI of a planet, it is considered to be in orbit around the sun. 6.1.1 Problem statement 11

Patched Conic Method Very useful for preliminary mission design (delta-v requirements and flight times). But actual mission design and execution employ the most accurate possible numerical integration techniques. 6.1.1 Problem statement 12

Sphere of Influence (SOI)? Let s assume that a spacecraft is within the Earth s SOI if the gravitational force due to Earth is larger than the gravitational force due to the sun. Gm m r > Gm m E sat S sat 2 2 E, sat rs, sat r < 5 E, sat 2.5 10 km The moon lies outside the SOI and is in orbit about the sun like an asteroid! 6.1.2 Sphere of influence 13

Sphere of Influence (SOI) Third body: sun or planet Spacecraft m 2 d m j r ρ µ d ρ && r + r = Gm 3 j + 3 3 r d ρ O m 1 Central body: sun or planet Disturbing function (L04) 6.1.2 Sphere of influence 14

If the Spacecraft Orbits the Planet && r ( m ) G m + r r p v sv sp pv + r 3 pv = Gms + 3 3 r pv rsv rsp p:planet v: vehicle s:sun && r A = P pv p s Primary gravitational acceleration due to the planet Perturbation acceleration due to the sun 6.1.2 Sphere of influence 15

If the Spacecraft Orbits the Sun && r G ( m + m ) + r = Gm r + s v pv sp sv 3 sv p 3 3 r sv rpv rsp r p:planet v: vehicle s:sun && r A = P sv s p Primary gravitational acceleration due to the sun Perturbation acceleration due to the planet 6.1.2 Sphere of influence 16

SOI: Correct Definition due to Laplace It is the surface along which the magnitudes of the acceleration satisfy: P A p s = P A s p Measure of the planet s influence on the orbit of the vehicle relative to the sun Measure of the deviation of the vehicle s orbit from the Keplerian orbit arising from the planet acting by itself r SOI 2 m 5 p ms r sp 6.1.2 Sphere of influence 17

SOI: Correct Definition due to Laplace P A p s > P A s If the spacecraft is inside the SOI of the planet. p A A > p The previous (incorrect) definition was 1 s 6.1.2 Sphere of influence 18

SOI Radii Planet SOI Radius (km) SOI radius (body radii) Mercury 1.13x10 5 45 Venus 6.17x10 5 100 OK! Earth 9.24x10 5 145 Mars 5.74x10 5 170 Jupiter 4.83x10 7 677 Neptune 8.67x10 7 3886 6.1.2 Sphere of influence 19

Validity of the Patched Conic Method The Earth s SOI is 145 Earth radii. This is extremely large compared to the size of the Earth: The velocity relative to the planet on an escape hyperbola is considered to be the hyperbolic excess velocity vector. vsoi v This is extremely small with respect to 1AU: During the elliptic transfer, the spacecraft is considered to be under the influence of the Sun s gravity only. In other words, it follows an unperturbed Keplerian orbit around the Sun. 6.1.2 Sphere of influence 20

Outbound Hyperbola The spacecraft necessarily escapes using a hyperbolic trajectory relative to the planet. Hyperbolic excess speed When this velocity vector is added to the planet s heliocentric velocity, the result is the spacecraft s heliocentric velocity on the interplanetary elliptic transfer orbit at the SOI in the solar system. Lecture 02: v = v + v v + v 2 2 2 2 2 esc SOI esc Is v SOI the velocity on the transfer orbit? 6.1.3 Planetary departure 21

Magnitude of V SOI The velocity v D of the spacecraft relative to the sun is imposed by the Hohmann transfer (i.e., velocity on the transfer orbit). H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 6.1.3 Planetary departure 22

Magnitude of V SOI By subtracting the known value of the velocity v 1 of the planet relative to the sun, one obtains the hyperbolic excess speed on the Earth escape hyperbola. µ 2R sun 2 vsoi = vd v1 = 1 v R ( ) 1 R1 + R 2 Imposed Known Lecture05 6.1.3 Planetary departure 23

Direction of V SOI What should be the direction of v SOI? For a Hohmann transfer, it should be parallel to v 1. 6.1.3 Planetary departure 24

Parking Orbit A spacecraft is ordinary launched into an interplanetary trajectory from a circular parking orbit. Its radius equals the periapse radius r p of the departure hyperbola. H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 6.1.3 Planetary departure 25

V Magnitude and Location a Lecture 02: known v = = 2 h µ e µ a 2 1 1 r p = known µ h = 2 h µ (1 + e) e v 2 1 e = 1+ h = r v + p r v p µ 2 µ 2 2 rp h v = v p v = µ c r 1 β = cos 1 p rp e 6.1.3 Planetary departure 26

Planetary Departure: Graphically Departure to outer or inner planet? H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 6.1.3 Planetary departure 27

Circular, Coplanar Orbits for Most Planets Planet Inclination of the orbit to the ecliptic plane Eccentricity Mercury 7.00º 0.206 Venus 3.39º 0.007 Earth 0.00º 0.017 Mars 1.85º 0.094 Jupiter 1.30º 0.049 Saturn 2.48º 0.056 Uranus 0.77º 0.046 Neptune 1.77º 0.011 Pluto 17.16º 0.244 6.1.4 Hohmann transfer 28

Governing Equations v D µ 2R sun 2 v1 = R ( ) 1 R1 + R2 1 v Dv1 v 2 µ sun 2R1 va = 1 R R + R ( ) 2 1 2 v 2 v A Signs? v 2 - v A, v D - v 1 >0 for transfer to an outer planet v 2 - v A, v D - v 1 <0 for transfer to an inner planet 6.1.4 Hohmann transfer 29

Schematically Transfer to outer planet Transfer to inner planet 6.1.4 Hohmann transfer 30

Arrival at an Outer Planet For an outer planet, the spacecraft s heliocentric approach velocity v A is smaller in magnitude than that of the planet v 2. v + v = v 2 A v > v 2 A v 2 v and have opposite signs. 6.1.5 Planetary arrival 31

Arrival at an Outer Planet The spacecraft crosses the forward portion of the SOI 6.1.5 Planetary arrival 32

Enter into an Elliptic Orbit If the intent is to go into orbit around the planet, then must be chosen so that the v burn at periapse will occur at the correct altitude above the planet. = r 1+ p 2µ r v p 2 h µ (1 + e) 2 µ µ (1 + e) v = vp, hyp vp, capture = = v + r r r r 2 p p p p 6.1.5 Planetary arrival 33

Planetary Flyby Otherwise, the specacraft will simply continue past periapse on a flyby trajectory exiting the SOI with the same relative speed v it entered but with the velocity vector rotated through the turn angle δ. e = 1+ r v p µ 2 δ = 2sin 1 1 e 6.1.5 Planetary arrival 34

Sensitivity Analysis: Departure The maneuver occurs well within the SOI, which is just a point on the scale of the solar system. One may therefore ask what effects small errors in position and velocity (r p and v p ) at the maneuver point have on the trajectory (target radius R 2 of the heliocentric Hohmann transfer ellipse). 2µ 1 v + R2 2 δ r 1 p rp δ v δ µ p = + 2 R2 R1v D vdv rp rp vd vp 1 2 µ sun 6.1.6 Sensitivity analysis and launch windows 35

Sensitivity Analysis: Earth-Mars, 300km Orbit µ = 1.327 10 km / s, µ = 398600 km / s sun 11 3 2 3 2 1 R = 149.6 10 km, R = 227.9 10 km, r = 6678 km 6 6 1 2 v = 32.73 km / s, v = 2.943 km / s D 2 δ R δ r 2 p δ v = 3.127 + 6.708 R r v p A 0.01% variation in the burnout speed v p changes the target radius by 0.067% or 153000 km. A 0.01% variation in burnout radius r p (670 m!) produces an error over 70000 km. p p p 6.1.6 Sensitivity analysis and launch windows 36

Sensitivity Analysis: Launch Errors Ariane V Trajectory correction maneuvers are clearly mandatory. 6.1.6 Sensitivity analysis and launch windows 37

Sensitivity Analysis: Arrival The heliocentric velocity of Mars in its orbit is roughly 24km/s. If an orbit injection were planned to occur at a 500 km periapsis height, a spacecraft arriving even 10s late at Mars would likely enter the atmosphere. 6.1.6 Sensitivity analysis and launch windows 38

Cassini-Huygens

Existence of Launch Windows Phasing maneuvers are not practical due to the large periods of the heliocentric orbits. The planet should arrive at the apse line of the transfer ellipse at the same time the spacecraft does. 6.1.6 Sensitivity analysis and launch windows 40

Rendez-vous Opportunities θ = θ + θ 1 10 1 = θ + 2 20 2 φ = θ θ 2 1 n t n t φ π = φ + 2 ( n n ) Tsyn 0 0 2 1 φ = φ + n n t ( ) 0 2 1 T syn = 2π n n 1 2 T syn = TT 1 2 T T 1 2 Synodic period 6.1.6 Sensitivity analysis and launch windows 41

Transfer Time Lecture 02 t 12 π 1 2 = µ sun R + R 2 3/ 2 φ = π n t 0 2 12 6.1.6 Sensitivity analysis and launch windows 42

Earth-Mars Example T syn 365.26 687.99 = = 365.26 687.99 777.9 days It takes 2.13 years for a given configuration of Mars relative to the Earth to occur again. t 12 φ 0 = s = = 7 2.2362 10 258.8 days 44 o The total time for a manned Mars mission is 258.8 + 453.8 + 258.8 = 971.4 days = 2.66 years 6.1.6 Sensitivity analysis and launch windows 43

Earth-Jupiter Example: Hohmann Galileo s original mission was designed to use a direct Hohmann transfer, but following the loss of Challenger Galileo's intended Centaur booster rocket was no longer allowed to fly on Shuttles. Using a lesspowerful solid booster rocket instead, Galileo used gravity assists instead. 6.1.7 Examples 44

Earth-Jupiter Example: Hohmann Velocity when leaving Earth's SOI: v v v D µ 2R E sun 2 1 = = = R ( ) 1 R1 + R 2 1 8.792km/s Velocity relative to Jupiter at Jupiter's SOI: J µ sun 2R v2 va = v = = R ( ) 2 R1 R + 2 1 1 5.643km/s Transfer time: 2.732 years 6.1.7 Examples 45

Earth-Jupiter Example: Departure Velocity on a circular parking orbit (300km): v c µ R + h E = = E 7.726km/s 2 2µ v = v + 7.726 km/s = 6.298 km/s r p e 2 rpv = 1+ = 2.295 µ 6.1.7 Examples 46

Earth-Jupiter Example: Arrival Final orbit is circular with radius=6r 2 2 µ µ (1 + e) v = v + = 24.95 17.18 = 7.77km/s r r p p J e=1.108 6.1.7 Examples 47

Hohmann Transfer: Other Planets Planet Mercury Venus Mars Jupiter Saturn Pluto v departure (km/s) 7.5 2.5 2.9 8.8 10.3 11.8 Transfer time (days) 105 146 259 998 2222 16482 Assumption of circular, co-planar orbits and tangential burns 6.1.7 Examples 48

Venus Express: A Hohmann-Like Transfer C 3 = 7.8 km 2 /s 2 Time: 154 days Real data Why? C 3 = 6.25 km 2 /s 2 Time: 146 days Hohmann 6.1.7 Examples 49

6.1.7 Examples

6. Interplanetary Trajectories 6.2 Lambert s problem 51

Motivation Section 6.1 discussed Hohmann interplanetary transfers, which are optimal with respect to fuel consumption. Why should we consider nontangential burns (i.e., non- Hohmann transfer)? L05 6.2 Lambert s problem 52

Non-Hohmann Trajectories Solution using Lambert s theorem (Lecture 05): If two position vectors and the time of flight are known, then the orbit can be fully determined. 6.2 Lambert s problem 53

Venus Express Example 6.2 Lambert s problem 54

Porkchop Plot: Visual Design Tool C3 contours C3 contours Arrival date Departure date In porkchop plots, orbits are considered to be non-coplanar and elliptic. 6.2 Lambert s problem 55

6.2 Lambert s problem

Type II transfer for cargo: the spacecraft travels more than a 180 true anomaly Type I transfer for piloted: the spacecraft travels less than a 180 true anomaly

6.2 Lambert s problem

6.2 Lambert s problem

6. Interplanetary Trajectories 6.3 Gravity assist 6.3.1 Basic principle 6.3.2 Real-life examples 61

V Budget: Earth Departure SOYUZ Planet Mercury Venus Mars Jupiter Saturn Pluto C 3 (km 2 /s 2 ) [56.25] 6.25 8.41 77.44 106.09 [139.24] Assumption of circular, co-planar orbits and tangential burns 62

V Budget: Arrival at the Planet A spacecraft traveling to an inner planet is accelerated by the Sun's gravity to a speed notably greater than the orbital speed of that destination planet. If the spacecraft is to be inserted into orbit about that inner planet, then there must be a mechanism to slow the spacecraft. Likewise, a spacecraft traveling to an outer planet is decelerated by the Sun's gravity to a speed far less than the orbital speed of that outer planet. Thus there must be a mechanism to accelerate the spacecraft. 6.3.1 Basic principle 63

Prohibitive V Budget? Use Gravity Assist Also known as planetary flyby trajectory, slingshot maneuver and swingby trajectory. Useful in interplanetary missions to obtain a velocity change without expending propellant. This free velocity change is provided by the gravitational field of the flyby planet and can be used to lower the v cost of a mission. 6.3.1 Basic principle 64

What Do We Gain? Spacecraft outbound velocity SOI Spacecraft inbound velocity V out = V in 6.3.1 Basic principle 65

Gravity Assist in the Heliocentric Frame Resultant V out SOI Resultant V in Planet s sun relative velocity v = v v, out, in 6.3.1 Basic principle 66

A Gravity Assist Looks Like an Elastic Collision Inertial frame Frame attached to the train Frame attached to the train Inertial frame 6.3.1 Basic principle 67

Leading-Side Planetary Flyby A leading-side flyby results in a decrease in the spacecraft s heliocentric speed (e.g., Mariner 10 and Messenger). 6.3.1 Basic principle 68

Trailing-Side Planetary Flyby A trailing-side flyby results in an increase in the spacecraft s heliocentric speed (e.g., Voyager and Cassini-Huygens). 6.3.1 Basic principle 69

What Are the Limitations? Launch windows may be rare (e.g., Voyager). Presence of an atmosphere (the closer the spacecraft can get, the more boost it gets). Encounter different planets with different (possibly harsh) environments. What about flight time? 6.3.1 Basic principle 70

V V E J G A See Lecture 1 71

Messenger 6.3.2 Real-life examples 72

6.3.2 Real-life examples

Hohmann Transfer vs. Gravity Assist Gravity assist Planet C3 (km 2 /s 2 ) Transfer time (days) Real mission C3 (km 2 /s 2 ) Transfer time (days) Mercury [56.25] 105 Messenger 16.4 2400 Cassini Saturn 106.09 2222 16.6 2500 Huygens Remark: the comparison between the transfer times is difficult, because it depends on the target orbit. The transfer time for gravity assist mission is the time elapsed between departure at the Earth and first arrival at the planet. 6.3.2 Real-life examples 74

New Horizons: Mission to Pluto 6.3.2 Real-life examples 75

Direct Transfer Was Also Envisioned 6.3.2 Real-life examples 76

Launch Window Combining GA and Direct 6.3.2 Real-life examples 77

6.3.2 Real-life examples

6. Interplanetary Trajectories 6.1 PATCHED CONIC METHOD 6.1.1 Problem statement 6.1.2 Sphere of influence 6.1.3 Planetary departure 6.1.4 Hohmann transfer 6.1.5 Planetary arrival 6.1.6 Sensitivity analysis and launch windows 6.1.7 Examples 6.2 LAMBERT S PROBLEM 6.3 GRAVITY ASSIST 6.3.1 Basic principle 6.3.2 Real-life examples 79

Even More Complex Trajectories 80

Astrodynamics (AERO0024) 6. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L)