GN/MAE155 1

Orbital Mechanics Overview 3

MAE 155

G. Nacouzi

GN/MAE155 2

Orbital Mechanics Overview 2

• Interplanetary Travel Overview– Coordinate system– Simplifications

• Patched Conic Approximation– Simplified Example– General Approach

• Gravity Assist: Brief Overview• Project Workshop

GN/MAE155 3

Interplanetary Travel: Coordinate System

• Use Heliocentric coordinate system,i.e., Sun centered– Use plane of earth’s orbit around

sun as fundamental plane, called ecliptic plane

– Principal direction, I, is in vernal equinox

• Define 1 Astronomical Unit (AU) = semimajor axis of earth orbit (a) = 149.6 E6 km (Pluto, a ~ 39.6 AU)

Heliocentric-ecliptic coordinate system for interplanetary transfer: Origin-center of Sun; fundamental plane- eclipticplane, principal direction - vernal equinox

GN/MAE155 4

Principal Forces in Interplanetary Travel

• In addition to forces previously discussed, we now need to account for Sun & target planet gravity

• Simplify to account only for gravity forces, ignore solar pressure & other perturbances in mission planning => forces considered - Gravity effects of Sun, Earth, Target Planet on S/C

GN/MAE155 5

Principal Forces in Interplanetary Travel

• A very useful approach in solving the interplanetary trajectory problem is to consider the influence on the S/C of one central body at a time => Familiar 2 body problem– Consider departure from earth:

• 1) Earth influence on vehicle during departure

• 2) Sun influence=> Sun centered transfer orbit

• 3) Target planet gravity influence, arrival orbit

GN/MAE155 6

Interplanetary Trajectory Model: Patched Conic

• Divide interplanetary trajectory into 3 different regions => basis of the patched-conic approximation– Region 1: Sun centered transfer (Sun gravity

dominates) - solved first– Region 2: Earth departure (Earth gravity

dominates) - solved second (direct or from parking orbit)

– Region 3: Arrival at target planet (planet gravity dominates) - solved third

GN/MAE155 7

Interplanetary Trajectory Model: Patched Conic

• Two body assumption requires calculating the gravitational sphere of influence, Rsoi, of each planet involved Rsoi (radius) = a(planet) x (Mplanet/Msun)^0.4

Earth Rsoi = 1E6 km

Rsoi

gravity

Sphere of Influence

GN/MAE155 8

Interplanetary Trajectory Model: Patched Conic

• Sun centered transfer solved first since solution provides information to solve other 2 regions

• Consider simplified example of Earth to Venus– Assume circular, coplanar orbits (constant

velocity and no plane change needed)– Hohmann transfer used:

• Apoapsis of transfer ellipse = radius of Earth Orbit

• Periapsis of transfer ellipse = radius of Venus Orbit

GN/MAE155 9

Simple Example• Required velocities

calculated using fundamental orbital equations discussed earlier– ra= 149.6 km;

Va = 27.3 km/s

– rp= 108.2 km; Vp = 37.7 km/s

– Time of transfer ~146 days

Earth @Launch

Venus @arrival

GN/MAE155 10

Simple Example• Example calculations:

Hyperbolic Excess Velocity, VHE ,S/C velocity wrt Earth

VHE = VS/S - VE/S

where,

VS/S ~ vel of S/C wrt Sun, VE/S ~ Vel Earth wrt Sun

VHE = 27.29 - 29.77 = -2.48 km/s

Similarly, @ target planet, hyperbolic excess vel, VHP needs to be

accounted for

VHP = VS/S - VP/S = 37.7 - 35 = 2.7 km/s; VP/S ~ Vplanet wrt Sun

Note: C3 = VHE2, Capability measure of LV

GN/MAE155 11

Patched Conic Procedure

• 1) Select a launch date based on launch opportunity analysis

• 2) Design transfer ellipse from earth to tgt planet

• 3) Design departure trajectory (hyperbolic)

• 4) Design approach trajectory (hyperbolic)

Reference: C. Brown, ‘Elements of SC Design’

GN/MAE155 12

Patched Conic Procedure• 1) Launch opportunity

To minimize required launch energy, Earth is placed (@ launch) directly opposed to tgt planet @ arrival– Calc. TOF, ~ 1/2 period of

transfer orbit

– Calculate lead angle = Earth angular Vel (e)x TOF

– Phase angle, r = 2 pi - lead angle Wait time = r - current /(target - e)

Earth

Mars

r

Synodic period~ period between launch opport., S = 2pi/(e - target)S = 2yrs for Mars

GN/MAE155 13

Patched Conic Procedure• 2) Develop transfer

ellipse from Earth to Target Planet (heliocentric) accounting for plane change as necessary– Note that the transfer

ellipse is on a plane that intersects the Sun & Earth at launch, & the target planet at arrival.

Plane change usually made at departure to combine with injection and use LV energy instead of S/C

GN/MAE155 14

Patched Conic Procedure• 3) Design Departure trajectory to escape Earth SOI, the departure

must be hyperbolic

where, Rpark~ parking orbit, VHE~hyperbolic excess velocity

• 4) Design approach trajectory to target planet

where Vpark is the orbital velocity in the parking orbit and V is the SC velocity at arrival. Vretro is the delta V to get into orbit

Vearth

2

Rpark

he he

Vhe 2

2

V 2

2 V 2

R

park

Vretro

Vpark V

GN/MAE155 15

Patched Conic Procedure

GN/MAE155 16

Gravity Assist Description

• Use of planet gravity field to rotate S/C velocity vector and change the magnitude of the velocity wrt Sun. No SC energy is expended

Reference: Elements of SC Design, Brown

GN/MAE155 17

Gravity Assist Description

• The relative velocity of the SC can be increased or decreased depending on the approach trajectory

GN/MAE155 18

Examples and Discussion