ASEN 5050 SPACEFLIGHT DYNAMICS Interplanetary Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 29: Interplanetary 1

Announcements HW 8 is out –Due Wednesday, Nov 12. –J2 effect –Using VOPs Reading: Chapter 12 Lecture 29: Interplanetary 2

Schedule from here out Lecture 29: Interplanetary 3 11/7: Interplanetary 2 11/10: Entry, Descent, and Landing 11/12: Low-Energy Mission Design 11/14: STK Lab 3 11/17: Low-Thrust Mission Design (Jon Herman) 11/19: Finite Burn Design 11/21: STK Lab 4 Fall Break 12/1: Constellation Design, GPS 12/3: Spacecraft Navigation 12/5: TBD 12/8: TBD 12/10: TBD 12/12: Final Review

Space News Orion’s EFT-1 Lecture 29: Interplanetary 4

Quiz #14 Lecture 29: Interplanetary 5 ✔ ✔

Quiz #14 Lecture 29: Interplanetary 6 ✔ ✖ ✔ ✔ ✖ ✔ N S Atm motion S/C motion (inertial) Perigee Point V ~ 8 km/s V atm ~ 0.48 km/s theta ~ 3.1 deg θ

Quiz #14 Problem 2 Lecture 19: Perturbations 7 Sun

Quiz #14 Problem 3 Lecture 19: Perturbations 8 Sun

Quiz #14 Lecture 29: Interplanetary 9

Quiz #14 Lecture 29: Interplanetary 10

ASEN 5050 SPACEFLIGHT DYNAMICS Interplanetary Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 29: Interplanetary 11

Interplanetary History Planets Moons Small bodies Lecture 29: Interplanetary 12 Today: tools, methods, algorithms!

Building an Interplanetary Transfer Simple: –Step 1. Build the transfer from Earth to the planet. –Step 2. Build the departure from the Earth onto the interplanetary transfer. –Step 3. Build the arrival at the destination. Added complexity: –Gravity assists –Solar sailing and/or electric propulsion –Low-energy transfers Lecture 29: Interplanetary 13

Patched Conics Use two-body orbits Lecture 29: Interplanetary 14

Patched Conics Gravitational forces during an Earth-Mars transfer Lecture 29: Interplanetary 15

Sphere of Influence Measured differently by different astrodynamicists. –“Hill Sphere” –Laplace derived an expression that matches real trajectories in the solar system very well. Laplace’s SOI: –Consider the acceleration of a spacecraft in the presence of the Earth and the Sun: Lecture 29: Interplanetary 16

Sphere of Influence Motion of the spacecraft relative to the Earth with the Sun as a 3 rd body: Motion of the spacecraft relative to the Sun with the Earth as a 3 rd body: Lecture 29: Interplanetary 17

Sphere of Influence Laplace suggested that the Sphere of Influence (SOI) be the surface where the ratio of the 3 rd body’s perturbation to the primary body’s acceleration is equal. Lecture 29: Interplanetary 18

Sphere of Influence Laplace suggested that the Sphere of Influence (SOI) be the surface where the ratio of the 3 rd body’s perturbation to the primary body’s acceleration is equal. Lecture 29: Interplanetary 19 Primary Earth Accel Primary Sun Accel 3 rd Body Sun Accel 3 rd Body Earth Accel

Sphere of Influence Laplace suggested that the Sphere of Influence (SOI) be the surface where the ratio of the 3 rd body’s perturbation to the primary body’s acceleration is equal. Lecture 29: Interplanetary 20 Primary Earth Accel Primary Sun Accel 3 rd Body Sun Accel 3 rd Body Earth Accel =

Sphere of Influence Find the surface that sets these ratios equal. Lecture 29: Interplanetary 21 After simplifications:

Sphere of Influence Find the surface that sets these ratios equal. Lecture 29: Interplanetary 22 Earth’s SOI: ~925,000 km Moon’s SOI: ~66,000 km

Patched Conics Use two-body orbits Lecture 29: Interplanetary 23

Interplanetary Transfer Use Lambert’s Problem Earth – Mars in 2018 Lecture 29: Interplanetary 24

Interplanetary Transfer Lambert’s Problem gives you: –the heliocentric velocity you require at the Earth departure –the heliocentric velocity you will have at Mars arrival Build hyperbolic orbits at Earth and Mars to connect to those. –“V-infinity” is the hyperbolic excess velocity at a planet. Lecture 29: Interplanetary 25

Earth Departure We have v-infinity at departure Compute specific energy of departure wrt Earth: Compute the velocity you need at some parking orbit: Lecture 29: Interplanetary 26

Earth Departure Lecture 29: Interplanetary 27 Departing from a circular orbit, say, 185 km:

Launch Target Lecture 29: Interplanetary 28

Launch Target Lecture 29: Interplanetary 29

Launch Targets C 3, RLA, DLA Lecture 29: Interplanetary 30 (In the frame of the V-inf vector!)

Launch Targets Lecture 29: Interplanetary 31

Mars Arrival Same as Earth departure, except you can arrive in several ways: –Enter orbit, usually a very elliptical orbit –Enter the atmosphere directly –Aerobraking. Aerocapture? Lecture 29: Interplanetary 32

Aerobraking Lecture 29: Interplanetary 33

Comparing Patched Conics to High- Fidelity Lecture 29: Interplanetary 34

Gravity Assists A mission designer can harness the gravity of other planets to reduce the energy needed to get somewhere. Galileo launched with just enough energy to get to Venus, but flew to Jupiter. Cassini launched with just enough energy to get to Venus (also), but flew to Saturn. New Horizons launched with a ridiculous amount of energy – and used a Jupiter gravity assist to get to Pluto even faster. Lecture 29: Interplanetary 35

Gravity Assists Gravity assist, like pretty much everything else, must obey the laws of physics. Conservation of energy, conservation of angular momentum, etc. Lecture 29: Interplanetary 36 So how did Pioneer 10 get such a huge kick of energy, passing by Jupiter?

Designing Gravity Assists Rule: Unless a spacecraft performs a maneuver or flies through the atmosphere, it departs the planet with the same amount of energy that it arrived with. Guideline: Make sure the spacecraft doesn’t impact the planet (or rings/moons) during the flyby, unless by design. Lecture 29: Interplanetary 37 Turning Angle

How do they work? Use Pioneer 10 as an example: Lecture 29: Interplanetary 38 INTO FLYBY OUT OF FLYBY

Gravity Assists We assume that the planet doesn’t move during the flyby (pretty fair assumption for initial designs). –The planet’s velocity doesn’t change. The gravity assist rotates the V-infinity vector to any orientation. –Check that you don’t hit the planet Lecture 29: Interplanetary 39

Gravity Assists We assume that the planet doesn’t move during the flyby (pretty fair assumption for initial designs). –The planet’s velocity doesn’t change. The gravity assist rotates the V-infinity vector to any orientation. –Check that you don’t hit the planet Lecture 29: Interplanetary 40

Designing a Gravity Assist Build a transfer from Earth to Mars (for example) –Defines at Mars Build a transfer from Mars to Jupiter (for example) –Defines at Mars Check to make sure you don’t break any laws of physics: Lecture 29: Interplanetary 41

Designing a Gravity Assist Another strategy: –Build a viable gravity assist that doesn’t necessarily connect with either the arrival or departure planets. –Adjust timing and geometry until the trajectory becomes continuous and feasible. Lecture 29: Interplanetary 42

Gravity Assists Lecture 29: Interplanetary 43 Please note! This illustration is a compact, beautiful representation of gravity assists. But know that the incoming and outgoing velocities do NOT need to be symmetric about the planet’s velocity! This is just for illustration. Please note! This illustration is a compact, beautiful representation of gravity assists. But know that the incoming and outgoing velocities do NOT need to be symmetric about the planet’s velocity! This is just for illustration.

Gravity Assists We can use them to increase or decrease a spacecraft’s energy. We can use them to add/remove out-of-plane components –Ulysses! We can use them for science Lecture 29: Interplanetary 44

Announcements HW 8 is out –Due Wednesday, Nov 12. –J2 effect –Using VOPs Reading: Chapter 12 Lecture 29: Interplanetary 45