Spacecraft Dynamics

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Spacecraft Dynamics
with your host.
Dr. Hy, the rocket scientist guy
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AERO 426, Lecture 5 Spacecraft Dynamics-
Questions Addressed
?How can we tell where our spacecraft is ?
?What are some simple ways to estimate the motion
of spacecraft in the vicinity of a NEA?
?How can we plan space trajectories and estimate
propulsion system requirements?
?Regarding available and future launch systems,
what are the implications for cost versus payload
size, weight, etc.?
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Suggested reading LW, Chap.5 intro or PM,
Sect. 3.3 (coordinate systems), LW, Sect. 6.1.1
- 6.1.3 or PM, Sect. 3.6 (Keplerian
orbits), LW, Sect. 6.3 (orbit
maneuvering), LW, Sect. 17.2 or PM, Sect.
4.2.1 and 4.3 (rocket propulsion and
motion), LW, Sect. 17.3 (types of
rockets), LW, Sect. 18.2 (launch system data)
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Whats our coordinates? Use Natures Gyros!
So, we have two axes that are fixed The
perpendicular to the orbit plane and the axis of
rotation of the Earth (which actually nutates
once every 26,000 years)
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Actually, in the Ecliptic coordinate systemWe
use the normal to the orbit plane (called
theEcliptic Pole) as the Z-axis
In the position of the vernal equinox, the
rotation axis vector is perpendicular to the
Sun-Earth vector and Northern Hemisphere spring
commences
X-axis
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Coordinate systems used in space applications
Coordinate Name Fixed with respect to Center Z-axis or Pole X-axis or Ref. Point Applications
Celestial (Inertial) Inertial space Earth or spacecraft Celestial Pole Vernal equinox Orbit analysis, astronomy, inertial motion
Earth-fixed Earth Earth Earth polecelestial pole Greenwich meridian Geolocation, apparent satellite motion
Spacecraft-fixed Spacecraft Defined by engineering drawings Spacecraft axis toward nadir Spacecraft axis in direction of velocity vector Position and orientation of spacecraft instruments
Ecliptic Inertial space Sun Ecliptic pole Vernal equinox Solar system orbits, lunar/solar ephemerides
Lunar The Moon Moon Lunar North pole Average center of Lunar Disk Locating lunar features
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Locating Events in Time
The Julian day or Julian day number (JDN) is the
integer number of days that have elapsed since
the initial epoch defined as noon Universal Time
(UT) Monday, January 1, 4713 BC in the Julian
calendar. The Julian date (JD) is a continuous
count of days and fractions elapsed since the
same initial epoch. The integral part gives the
Julian day number. The fractional part gives the
time of day since noon UT as a decimal fraction
of one day with 0.5 representing midnight UT.
Example A Julian date of 2454115.05486 means
that the date and Universal Time is Sunday 14
January 2007 at 131859.9. The decimal parts of
a Julian date 0.1 2.4 hours or 144 minutes or
8640 seconds 0.01 0.24 hours or 14.4 minutes
or 864 seconds 0.001 0.024 hours or 1.44
minutes or 86.4 seconds 0.0001 0.0024 hours or
0.144 minutes or 8.64 seconds 0.00001 0.00024
hours or 0.0144 minutes or 0.864 seconds. The
Julian day system was introduced by astronomers
to provide a single system of dates that could be
used when working with different calendars. Also,
the time separation between two events can be
determined with simple subtraction. To make
conversions, several handy web-sites are
available e.g., http//aa.usno.navy.mil/cgi-bin/a
a_jdconv.pl
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• Orbital Dynamics - Made Simple
• ?Most of the time (with many important
  exceptions) spacecraft orbital dynamics involves
  bodies that are either (1) very, very small
  relative to inter-body distances, or (2) are
  nearly spherically symmetric -- then
• ?Bodies behave (attract and are attracted) as if
  they are point masses.
• ?Motion can be described by keeping track of the
  centers of mass.
• ?Also, most of the time (with many important
  exceptions) spacecraft orbital dynamics is a
  two-body problem (the s/c and the Earth, or the
  s/c and the sun, or, etc.) - so we have two
  gravitationally attracting point masses, and
• ?Both bodies move in a plane (the same plane)
• ?Both trace out conic sections with one focus at
  the total center of mass.
• ?Each body moves periodically on its conic
  section, tracing and retracing the same curve
  forever.

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• ?Finally, most of the time (with many important
  exceptions), one of the bodies is much more
  massive than the other ( the Earth versus a s/c,
  or the sun versus the Earth, etc.). Then in
  addition to the above
• ? The smaller body moves on a conic section with
  a focus on the larger body's center of mass,
  which is also approximately the total center of
  mass.
• ? The motion of the smaller body does not depend
  on its mass.
• ? The smaller body's motion depends on the
  gravitational constant, G, and the larger body's
  mass only through the combination
• ? "The Gravitational Parameter"
• GM
• G 6.673 x 10-11 m3/ kg-s2
• M Mass of the larger body

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Euler Angle Description of the Orbit Plane
Orbit Plane
Equitorial Ecliptic plane
f
Periapsis
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Orbital Dynamics - Briefly Summarized
?
d
Parabola vmax (2?)1/2 vescape E 0
parabola
vf
v0
ellipse
rmax
hyperbola
Ellipse rmax rmin (vmax)2 / (2 ? -
(vmax)2) 0 ? E ? E0
Hyperbola vmax v0 1 (1 ?2)1/2 / ?,
rmin dv0 / vmax sin(?) ? /(1 ?2 )1/2 E ? 0
circle
For all orbits
e m / rmin E v2/2 - m / r ?? / 2a a
(rmax rmin)/ 2
Circle v vmax ?1/2 E E0 - ? /2
rmin
For bound orbits
vmax
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Location of a Body in its Orbit as a Function of
Time
b
r
E
F2
f
a
F1
ae
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Getting from Earth to a NEA - Patched Conics
Method
When S/C crosses asteroids activity sphere
boundary, subtract the asteroids velocity
relative to the sun.This gives initial conditions
for the asteroid-dominated portion of the
rendezvous
Sphere of Influence of the asteroid S/C
acceleration due to asteroid gt Perturbing
acceleration due to the Earth. SI radius given
by RSI ? RA-E (Masteroid /MEarth)2/5 (Masteroid
4.6X1010 kg MEarth5.9737X1024 kg ) (Masteroid
/MEarth)2/5 2.2626X10-6 Within SI and ref.
frame moving with the asteroid, S/C approx.
interacts only with the asteroid.
When outside the Earths activity sphere,
calculate only the S/C orbit around the Sun.
(which follows a conic section).
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?V (km/s) Topography
Mars
Sun
Low Mars orbit
4.1
Phobos
0.5
0.9
Phobos transfer
0.3
Deimos transfer
0.7
0.2
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Mars C3
Deimos
0.9
Mars transfer
0.6
Optional Aerobrake
Earth C3
0.7
Orbital location
GTO
0.7
2.5
1.6
1.6
GEO
3.8
1.7
LEO
L4/5
4.1
9.3 - 10
Lunar orbit
0.7
Earth
Moon
1.6
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Low Thrust Transfer Maneuvers
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Low Thrust Transfer Maneuvers - Continued
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Planar Circular Restricted 3-Body Problem(PCR3BP)

• Restricted Gravitational field is determined
  by two massive bodies (The primaries). The
  third body is too small to affect the primaries.
• Circular The primaries are in circular orbits
  about the total center of mass
• Planar All three bodies move in the same
  plane.
• Normalized Units
• Unit of mass m1m2
• Unit of length constant separation between
  m1and m2
• Unit of time Orbital period of m1and m2 is 2? (G
  1)
• The only parameter in the system is ?
  m1/(m1m2)

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Unit of distance L distance between m1 and m2
(km)Unit of Velocity V orbital velocity of m2
(km/s)Unit of time orbital period of the
primaries (s)
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Equations of Motion (In the rotating frame)
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Planar Circular Restricted Three Body Problem
(PCR3BP)
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Effective Potential The Open Realms and the
Forbidden Zone
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Five Cases of Possible Motions
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Types of Orbits in the Neck Region
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Tangled Trajectories in the Neck Region
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Structure of the Neck Region
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Global Orbit Structure Homoclinic/Heteroclinic
Chains
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Patched 3-Body Method The Interplanetary Super
Highway
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Patched 3-Body Method LL1 to EL2 in 40 days
with a single 14m/s ?V
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Patched 3-Body Method Space Mission Application
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Rotational Dynamics of Axisymmetric Rigid Bodies
Z axis inertia C
z
x
y
x and y axes moment of inertia A
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Rotational Dynamics of Axisymmetric Rigid Bodies
z
x
y
This does not happen when C gtA
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Propulsion Function Comments/ Typical Requirements
Launch and injection into LEO Really in the domain of "Launch Systems" - which we discuss separately below
DV for raising the orbit from LEO to a higher orbit 60 to 1500 m/s, Use kick motor
Acceleration to escape velocity from LEO parking orbit 3600 to 4000 m/s for injection into an interplanetary trajectory
Interplanetary trajectory - From Earth escape to in-mission parking orbit. Depends heavily on the trajectory design - Have a wide choice among min energy maneuvers, swing-by maneuvers, etc.
In-Mission Operations Orbit correction ?V Stationkeeping ?V "Formation Flying" ?V's 15 to 75 m/s per year, for Earth orbits Up to 45 to 55 m/s per year, Earth orbits Could be relevant to stand-off mode of operation.
Attitude control Acquisition of Sun, Earth, Star - for navigational and target acquisition purposes In-mission pointing control, 3-axis stabilization lt 5000 N-s total impulse, 1K to 10K pulses, 0.01 to 5.0 s pulse width 100K to 200K pulses, min impulse bit of 0.01N-s, 0.01 to 0.25s pulse width.
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Propulsion Systems - Key Parameters
Oxidizer
Fuel
F ? Thrust
? Ve
(dm/dt) Ve exhaust
velocity dm/dt
propellant and oxidizer mass
flow rate
dm/dt
Isp ? Specific Impulse F / (g dm/dt) --
depends on propulsion type
Nozzle
(liquid, solid, chemical, electric,
etc.) , energetics of chemical
reactions, etc.
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Key Propulsion Parameters Related to Important
Trajectory Parameters
Suppose we have a thruster burn event with
constant thrust (maybe to inject the spacecraft
into a higher orbit, etc.). Define m0 ? Total
mass of vehicle before burn event mp? Mass of
propellant ( oxidizer) used in burn event
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Trajectory Parameters/ Propulsion System
Relations
?V Total change in vehicle speed g
Isp ln (m0/( m0 - mp)) ?t Time elapsed
during burn event g Isp mp/ F

• Trajectory Requirements Needed DV
  and Dt

• Use above relations to estimate total mass of
  propellant
• Select propulsion system (F Isp) and design
  trajectory to minimize total propulsion system
• mass

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Determining Propulsion System Requirements - For
Transport of S/C to its Mission Station

• Lay out the entire trajectory and itemize the DV
  maneuvers.
• Start from the last DV maneuver and use the DV/
  mp equation to determine mp (where here, m0 - mp
  the final S/C mass), for several values of
  Isp
• From considerations of the Dt desired, or other
  practical constraints, determine any thrust level
  requirements. Now narrow the selection of
  propulsion systems to those consistent with
  required thrust levels.

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Now, carry out the above process for all the ?V
maneuvers, working back along the trajectory. Get
a range of values for mp and F.
Finally, obtain the total propulsion system
masses corresponding to different propulsion
system options. Select option with smallest
cost and/or launch weight.
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Launch Systems

• Key Parameters are
• Mass of payload that can be injected into LEO or
  GTO or GEO
• Fairing diameter and length

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Data for Systems with Fairing Diameters gt3.0 m
Launch System Upper Stage (if any) LEO (kg) GTO (kg) GEO (kg) Fairing Envelopes Fairing Envelopes
Launch System Upper Stage (if any) LEO (kg) GTO (kg) GEO (kg) Diam (m) Length (m)
ATLAS II Cent-2 6395 2680 570 4.2
SHUTTLE IUS TOS PAM-D 24,400 -- -- -- -- 5900 5900 1300 -- 2360 -- -- 4.6 18.3
TITAN III NUS PAM-D2 TRAN TOS 14,400 -- -- -- -- 1850 4310 5000 -- 1360 1360 -- 3.6 12.4 15.5 16.0
TITAN IV NUS Cent IUS 17,700 -- -- -- 5760 6350 -- 4540 2380 4.5 17.0 20.0 23.0,26.0
ARIANE 40 (France) 42P 42L 44P 44LP 44L H-10 H-10 H-10 H-10 H-10 H-10 EPS 4900 6100 7400 6900 8300 9600 18,000 at 550 km 1900 2600 3200 3000 3700 4200 6800 -- -- -- -- -- -- -- 3.6 8.6 to 12.4
H-2 (Japan) -- 10,500 4000 2200 3.7 10.0
LONG MARCH (China) CZ2E Star 63F 9265 3370 1500 3.8 7.5
PROTON (Russia) ENERGIA ZENIT 2 D1 D1e EUS, RCS Block D 20,000 90,000 13,740 -- 5500 -- 4300 -- 2200 18,000 4100 3.3 4.1 5.5 3.3 4.2-7.5 19-37 5.8-9
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Good Luck With Your Mission!
and watch out for those irate Romulans!
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