ARO309 - Astronautics and
Spacecraft Design
Winter 2014
Try Lam
CalPoly Pomona Aerospace Engineering
Introductions
Class Materials at http://www.trylam.com/2014w_aro309/
•Course: ARO 309: Astronautics and Spacecraft Design (3 units)
•Description: Space mission and trajectory design. Kepler’s laws. Orbits,
hyperbolic escape trajectories, interplanetary transfers, gravity assists. Special
orbits including geostationary, Molniya, sunsynchronous. [Kepler's equation,
orbit determination, attitude dynamics and control.]
•Prerequisite: C or better in ME215 (dynamics)
•Section 01: 5:30 PM – 6:45 PM MW (15900) Room 17-1211
Section 02: 7:00 PM – 8:15 PM MW (15901) Room 17-1211
•Holidays: 1/20
•Text Book: H. Curtis, Orbital Mechanics for Engineering Students,
Butterworth-Heinemann (preference: 2nd Edition)
•Grades: 10% Homework, 25% Midterm, 25% Final, 40% Quizzes (4 x 10%
each)
Introductions
• Things you should know (or willing to learn) to be
successful in this class
– Basic Math
– Dynamics
– Basic programing/scripting
What are we studying?
What are we studying?
Earth Orbiters
Pork Chop Plot
High Thrust Interplanetary Transfer
Low-Thrust Interplanetary Transfer
Low-Thrust Europa End Game
Low-Thrust Europa End Game
Low-Thrust Europa End Game
Orbit Stability
Stable for > 100 days
Enceladus
Orbit
Juno
Other Missions
Other Missions
Lecture 01 and 02:
Two-Body Dynamics: Conics
Chapter 2
Equations of Motion
where r = R2 - R1
Equations of Motion
• Fundamental Equations of Motion for 2-Body
Motion
˙r˙ = - m3 r
r
é ˙x˙ù
é xù
˙r˙ = - m3 r = ê y˙˙ú = - m3 ê yú
ê ú
r
r ê ú
êë ˙z˙úû
êë zúû
˙˙ - R
˙˙
where m = G( M + m) and ˙r˙ = R
2
1
Conic Equation
From 2-body equation to conic equation
˙r˙ = -
m
r
3
r
h =r ´v
h2 / m
p
r=
=
1+ ecosq 1+ ecosq
˙r˙ ´ h = - m3 r ´ h
r
Angular Momentum
Other Useful Equations
h =r ´v
and
h = rvcosg
h m
v^ = = (1+ ecos q )
r h
vr =
m
h
esinq
h2 / m
rp =
1+ e
vr
esin q
tan g = =
v^ 1+ ecosq
Energy
v2 m
e = K + P = - = constant
2 r
e = ep =
v 2p
2
-
m
rp
=-
m 2 /h2
2
2
1
e
( )
NOTE: ε = 0 (parabolic), ε > 0 (escape), ε < 0 (capture: elliptical and circular)
Conics
a(1 - e2 )
p
r=
=
1+ ecosq 1+ ecosq
Circular Orbits
e = 0 and e < 0
p
h2
r=
= p=
1+ 0× cosq
m
vcircular =
e=-
m 2 /h2
2
=-
m
2r
h = rv
m
r
Tcircular =
2pr
m
r
= 2p
r3
m
Elliptical Orbits
0 < e <1 and e < 0
Elliptical Orbits
p
p
r=
Þ ra =
1+ ecosq
1- e
since 2a = rp + ra
Þ ra = a(1+ e)
and
e=-
Telliptical
and
p
Þ a=
1 - e2
rp = a(1 - e)
m 2 /h2
2
p
rp =
1+ e
NOTE : b = a 1- e2
m
(1 - e ) = - 2a
2
2pab
2pab
a3
=
=
= 2p
h
m
ma(1 - e2 )
Elliptical Orbits
ra - rp
e=
ra + rp
h = ma(1 - e2 )
va = h /ra
and
v p = h /rp
vr
esin q
tan g = =
v^ 1+ ecosq
Parabolic Orbits
• Parabolic orbits are borderline case between
an open hyperbolic and a closed elliptical orbit
e =1 and e = 0
v2 m
2m
e = 0 = - Þ vparabolic =
= vescape
2 r
r
vr
esin q
1 × sinq
tan g = =
=
v^ 1+ ecosq 1+1 × cosq
g parabolic = q /2
NOTE: as v  180°, then r  ∞
Hyperbolic Orbits
e >1 and e > 0
e2 -1
sin q ¥ =
e
æ 1ö
b = cos ç ÷
è eø
-1
r =-
a(1 - e2 )
1+ ecosq
Hyperbolic Orbits
e=
Hyperbolic
excess speed
m
v2 m m
e= - =
2 r 2a
2a
m
v¥ =
a
=
m
h
esinq ¥ =
v2 m v¥2 m v¥2
e= - = - =
2 r
2 r¥ 2
C3 = v¥2
m
h
e2 -1
2m
2
v =v +
= v¥2 + vescape
r
2
2
¥
Properties of Conics
0<e<1
Conic Properties
Vis-Viva Equation
v2 m
m
e= - =2 r
2a
æ2 1ö
v = mç - ÷
è r aø
2
Mean Motion
2p
m
n=
=
T
a3
Vis-viva equation
Perifocal Frame
“natural frame” for an orbit centered at
the focus with x-axis to periapsis and zaxis toward the angular momentum
vector
r = x pˆ + y qˆ
and
ˆ = h /h
w
r = r cosq pˆ + r sin q qˆ
v = r˙ = x˙ pˆ + y˙ qˆ
(
)
+( r˙ sin q + r q˙ cosq ) qˆ
v = r˙ cos q - r q˙ sin q pˆ
Perifocal Frame
FROM
THEN
r˙ = vr =
m
h
esinq
and
m
˙
r q = v^ = (1+ ecos q )
h
˙x = - m sin q and y˙ = m (e+ cosq )
h
h
v = r˙ = x˙ pˆ + y˙ qˆ
v=
pˆ
h =r ´v = x
x˙
( -sin q ) pˆ + (e+ cosq ) qˆ]
[
h
m
ˆ
qˆ w
ˆ
y 0 = x y˙ - y x˙ w
y˙ 0
(
)
and
(
)
h = x y˙ - y x˙
Lagrange Coefficients
• Future estimated state as a function of current state
r = x pˆ + y qˆ
v = x˙ pˆ + y˙ qˆ
Where
Solving unit
vector based on
initial conditions
r = f r0 + g v0
v = f˙ r0 + g˙ v0
and
r0 × v0
vr 0 =
r0
Lagrange Coefficients
• Steps finding state at a future Δθ using Lagrange Coefficients
1. Find r0 and v0 from the given position and velocity vector
2. Find vr0 (last slide)
3. Find the constant angular momentum, h
h = r0v^ 0 = r0 v02 - vr20
4. Find r (last slide)
5. Find f, g, fdot, gdot
6. Find r and v
Lagrange Coefficients
• Example (from book)
Lagrange Coefficients
• Example (from book)
Lagrange Coefficients
• Example (from book)
Lagrange Coefficients
• Example (from book)
Since Vr0 is < 0 we know that S/C is approaching periapsis (so 180°<θ<360°)
ALSO
CR3BP
• Circular Restricted Three Body Problem (CR3BP)
p1 =1- p 2
m1 x1 + m2 x2 = 0
x1 + r12 = x2
x1 = -p 2 r12
x2 = p1r12
CR3BP
Kinematics (LHS):
r1 = ( x + p 2 r12 )iˆ + yˆj + zkˆ
m˙r˙ = F1 + F2
r2 = ( x - p1r12 )iˆ + yˆj + zkˆ
r = xiˆ + yˆj + zkˆ
CR3BP
Kinematics (RHS):
m˙r˙ = F1 + F2
F1 = -
m1m
3
1
r
r1
and
F2 = -
m2 m
3
2
r
r2
CR3BP
Lyapunov
Orbit
CR3BP Plots are
in the rotating
frame
Tadpole Orbit
DRO
Horseshoe Orbit
CR3BP: Equilibrium Points
x˙˙ = y˙˙ = ˙z˙ = x˙ = y˙ = z˙ = 0
Equilibrium points or
Libration points or
Lagrange points
L4
z=0
L3
L1
L2
Jacobi Constant
L5