Space Engineering I – Part I
Where are we?
Where are we?
Where we going?
Where are we relative to what?
Johannes Kepler
1571-1630
Issac Newton 1642-1727
 Newton's laws of motion and universal gravitation dominated
scientists' view of the physical universe for the next three centuries.
 Newton derived Kepler’s laws of planetary motion from his
mathematical description of gravity, removing the last doubts about
the validity of the heliocentric model of the cosmos.
Coordinate Systems
(X,Y,Z)
(R,q,l)
(R,j,l)
Origin?
 Center of Earth
 Sun or a Star
 Center of a planetary body
 Others….
Reference Axes
 Axis of rotation or revolution
 Earth spin axis
• Equatorial Plane
 Plane of the Earth’s orbit
around the Sun
• Ecliptic Plane
 Need to pick two axes and
then 3rd one is determined
Ecliptic and Equatorial Planes
Obliquity of the Ecliptic = 23.44 °
Vernal Equinox vector
- Earth to Sun on March 21st
- Planes intersect @ Equinox
Inertial
Coordinates
Relationship between Coordinate Frames
Orbital Elements
a - semi-major axis Ω - right ascension of ascending node
e - eccentricity
 - argument of perigee
i - inclination
 - true anomaly
Properties of Orbits
• a is the semimajor axis;
• b is the semiminor axis;
• rMAX = ra, rMIN = rp are the maximum and
minimum radius-vectors;
• c is the distance between the focus and
the center of the ellipse;
• e = c/a is eccentricity
b
 
a
2
= 1 - e2
• 2p is the latus rectum
(latus = side and rectum =
straight)
p — semilatus rectum or semiparameter
• A = ab is the area of the ellipse
p
= 1 + e cos q
r
Why do we need all this?
 Launch into desired orbit




Launch window, inclination
Ground coverage (ground track/swath)
LEO/GEO
Purpose of mission?
 Orbital Manoeuvers




Feasible trajectories
Minimize propulsion required
Station keeping
Tracking, Prediction
 Interplanetary Transfers



Hyperbolic orbits
Changing reference frames
Orbital insertion
 Rendezvous/Proximity Operations


Relative motion
Orbital dynamics
Bird’s Eye View
A 3D Situational Awareness Tool
for the Space Station
Review of Orbital Elements
e
120°
150°
90°
Eccentricity
(0.0 to 1.0)
Apogee
180°
v
True anomaly
(angle)
a
Semi-major
Perigee
0°
axis
e=0.8 vs e=0.0
e
a
v
defines ellipse shape
defines ellipse size
defines satellite angle from perigee
Inclination i
Intersection of the
equatorial and
orbital planes
Inclination
(above)
(angle)
i
(below)
Ascending
Node
Equatorial Plane
( defined by Earth’s equator )
Ascending Node is where a
satellite crosses the equatorial
plane moving south to north
Right Ascension of the ascending node
Ω and Argument of perigee ω
Ω = angle from
vernal equinox to
ascending node on
the equatorial plane
Perigee Direction
ω = angle from
ascending node to
perigee on the
orbital plane
ω
Ω
Ascending
Node
Vernal Equinox
The Six Orbital Elements
a
= Semi-major axis (usually in
kilometers or nautical miles)
e
= Eccentricity (of the elliptical
orbit)
v
= True anomaly The angle
between perigee and satellite in
the orbital plane at a specific time
i
= Inclination The angle between
the orbital and equatorial planes
Ω = Right Ascension (longitude)
of the ascending node The
angle from the Vernal Equinox
vector to the ascending node on
the equatorial plane
 = Argument of perigee
The
angle measured between the
ascending node and perigee
Shape, Size,
Orientation,
and Satellite
Location.
Computing Orbital Position
Mean Anomaly
Eccentric Anomaly
True Anomaly
Mean Motion
Two Line Orbital Elements
N ASA and NORAD Standard for specifying orbits of Earth-orbiting satellites
ISS (ZARYA)
1 25544U 98067A 08264.51782528 −.00002182 00000-0 -11606-4 0 2927
2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537
Ref: http://en.wikipedia.org/wiki/Two-line_element_set
Equations of Motion
Equations of Motion (2)
Equations of Motion (3)
Conservation of Energy
For a circular orbit, balancing the force of gravity and the centripetal acceleration
Since
or
Equations of Motion (4)
An orbit is a continually changing balance between potential and
kinetic energy
Potential Energy
Kinetic Energy
Using
For any Kepler orbit (elliptic, parabolic, hyperbolic or radial), this is the
Vis Viva equation
Orbital Period vs. Altitude
a3
P = 2 m
h = 160 n.mi
P = 90 minutes
“High” Earth Orbit
H = 3444 n.mi
P = 4 hours
Geosynchronous Orbit
h = 19,324 n.mi
P = 23 h 56 m 4 s
Orbital Velocity vs. Altitude
h = 160n.mi
V = 25,300 ft/s
“High” Earth Orbit
H = 3444 n.mi
V = 18,341 ft/s
Geosynchronous Orbit
h = 19,324 n.mi
V = 10,087 ft/s
Orbital Velocity vs. Altitude
(Elliptical Orbits)
h = 19,324 n.mi
V = 5,273 ft/s
h = 160 n.mi
V = 33,320 ft/s
Geosynchronous Transfer Orbit
a = 13,186 n.mi
e = 0.726
Possible Orbital Trajectories
• e=0 -- circle
• e<1 -- ellipse
• e=1 -- parabola
• e>1 -- hyperbola
e < 1 Orbit is ‘closed’ – recurring path (elliptical)
e > 1 Not an orbit – passing trajectory (hyperbolic)
Parabolic Trajectories
Total Energy = 0
Hyperbolic Trajectories
Total Energy > 0
As
State Vectors
Cartesian x, y, z, and 3D velocity
Integrating Multi-Body Dynamics
Solar and Sidereal Time
The Sun
Drifts east in the sky ~1° per day.
Rises 0.066 hours later each day.
(because the earth is orbiting)
The Earth…
Rotates 360° in 23.934 hours
(Celestial or “Sidereal” Day)
Rotates ~361° in 24.000 hours
(Noon to Noon or “Solar” Day)
Satellites orbits are aligned to the
Sidereal day – not the solar day
Ground Track
Ground tracks drift
westward as the Earth
rotates below.
360 deg / 24 hrs
= 15 deg/hr
Perturbations - J2000 Inertial Frame
Orbit Perturbations - Atmospheric Drag
Atmospheric density is a function
of latitude, solar heating, season,
land masses, etc.
Drag also depends on spacecraft
attitude
Effect is to lower the apogee of an elliptical orbit
Perigee remains relatively constant
Orbit Perturbations - Gravitational Potential
Spherical Term
Zonal Harmonics
where
m = Universal Gravitational Constant X Mass of Earth
r = Spacecraft Radius Vector from Center of Earth
ae = Earth Equatorial Radius
P() = Legendre Polynomial Functions
f = Spacecraft Latitude
l = Spacecraft Longitude
Jn = Zonal Harmonic Constants
Cn,m,Sn,m = Tesseral & Sectorial Harmonic Coefficients
Only the first 4X4 (n=4, m=4) elements are
used in Space Shuttle software.
J2 has 1/1000th the effect of the spherical
term;
Tesseral Harmonics Sectorial Harmonics
all other terms start at 1/1000th of J2’s
effect.
,
d
Orbit Perturbations - “The J2 Effect”
f
t
The Earth’s oblateness causes the
most significant perturbation of any of
the nonspherical terms.
i
h orbit
J2
l
S
h
Right Ascension of the Ascending
Node, Argument of Perigee and Time
since Perigee Passage are affected.
-1 0
100
-9
a
200
-7
-6.7
300
500
Typical
Shuttle
Orbits
-6
o
Nodal Regression is the most important
operationally.
Magnitude depends on orbit size (a),
shape (e) and inclination (i).
d
-8
-5
1000
-4
-3
Posigrade orbits’ nodes regress
Westward (0° < i < 90°)
Retrograde orbits’ nodes regress
Eastward (90° < i <180°)
N
h
-2
-1
0
10
20
30
0
3940 50 60 70 80 90
180 170 160 150 140 130 120 110 100
Inclination, Degrees
Nodal Regression
Orbital planes
rotate eastward
over time.
(above)
Ascending
Node
(below)
Nodal Regression can be used to
advantage (such as assuring
desired lighting conditions)
Coverage from GEO
TDRSS Comm Coverage
Geosynchronous Orbit
Sun-Synchronous Orbits
Relies on nodal regression to shift the ascending node ~1° per day.
Scans the same path under the same lighting conditions each day.
The number of orbits per 24 hours must be an even integer (usually 15).
Requires a slightly retrograde orbit (I = 97.56° for a 550km / 15-orbit SSO).
Each subsequent pass is 24° farther west (if 15 orbits per day).
Repeats the pattern on the 16th orbit
Used for reconnaissance (or terrain mapping – with a bit of drift).
Molniya - 12hr Period
‘Long loitering’ high latitude apogee. Once used
used for early warning by both USA and USSR
Changing Orbits - The Effects of Burns
Posigrade & Retrograde
Orbital
Direction
Initial O
rbit
h
Final O
rbit
V
V
T
y
p
i
c
a
l
S
h
u
t
t
l
e
O
r
b
i
t
s
:
1
f
t
/
s
e
c

V
y
i
e
l
d
s1
/
2
n
m
i

h
A posi-grade burn will RAISE orbital altitude.
A retro-grade burn will LOWER orbital altitude.
Note – max effect is at 180° from the burn point.
h
Changing Orbits - The Effect of Burns
Radial In & Radial Out
V= m
( r2
1
a
)
EXAMPLE: Radial In Burn at Perigee
Orbital
Direction
Resultant Velocity
Initial Velocity
Radial burns shift the argument of perigee without significantly
altering other orbital parameters
Initial O
rbit
Final O
rbit
Orbital Transfers - Changing Planes
V2
V
V1
•Burn point must be intersection
of two orbits (“nodal crossings”)
•Extremely expensive energywise:
For 160 nmi circular
orbits, a 1° of plane
change requires a V of
over 470 ft/sec.
Homann Transfer
•
Vc2
We want to move spacecraft from
LEO → GEO
GEO
•
Initial LEO orbit has radius r1 and
velocity Vc1
V c1 =
GM 
LEO
r1
•
Desired GEO orbit has radius r2
and velocity Vc2
•
•
At LEO (r1), Vc1 = 7,724 m/s
At GEO (r2), Vc2 = 3,074 m/s
•
Could accomplish this in many
ways
r1
Vc1
r2
Homann Transfer
•
Vc2
We want to move spacecraft from
LEO → GEO
GEO
•
Initial LEO orbit has radius r1 and
velocity Vc1
V c1 =
GM 
LEO
r
•
Desired GEO orbit has radius r2
and velocity Vc2
•
•
At LEO (r1), Vc1 = 7,724 m/s
At GEO (r2), Vc2 = 3,074 m/s
•
Could accomplish this in many
ways
r1
Vc1
r2
Homann Transfer
•
Vc2
We want to move spacecraft from
LEO → GEO
GEO
•
Initial LEO orbit has radius r1 and
velocity Vc1
V c1 =
GM 
LEO
r
•
Desired GEO orbit has radius r2
and velocity Vc2
•
•
At LEO (r1), Vc1 = 7,724 m/s
At GEO (r2), Vc2 = 3,074 m/s
•
Could accomplish this in many
ways
r1
Vc1
r2
Homann Transfer
•
Vc2
We want to move spacecraft from
LEO → GEO
GEO
•
Initial LEO orbit has radius r1 and
velocity Vc1
V c1 =
GM 
LEO
r
•
Desired GEO orbit has radius r2
and velocity Vc2
•
•
At LEO (r1), Vc1 = 7,724 m/s
At GEO (r2), Vc2 = 3,074 m/s
•
Hohmann Transfer Orbit
– Most Efficient Method
r1
Vc1
r2
Homann Transfer
Vc2
GEO
•
Impulsive V1 is applied to get on
geostationary transfer orbit (GTO)
at perigee:
GTO
LEO
 V1 =
2m
r1

2m
r1  r2

m
r1
r1
m = GM 
r2
•
Leave LEO (r1) with a total
velocity of V1
V1
Vc1
V1
Homann Transfer
Vc2
Apogee
GEO
•
Impulsive V1 is applied to get on
geostationary transfer orbit (GTO)
at perigee:
 V1 =
•
2m
r1

2m
r1  r2

GTO
LEO
m
r1
r1
Leave LEO (r1) with a total
velocity of V1
r2
•
V1
Vc1
Transfer orbit is elliptical shape
– Perigee located at r1
– Apogee located at r2
Perigee
V1
Homann Transfer
Vc2
V2
•
Arrive at GEO (apogee) with V2
•
When arriving at GEO, which is at
apogee of elliptical transfer orbit,
must apply some V2 in order to
circularize:
V2 =
•
•
m
r2

2m
r2

V2
GEO
GTO
LEO
r1
2m
r1  r2
This is exactly the V that should
be applied to circularize the orbit
at GEO (r2)
– Vc2 = V2 + V2
If this V is not applied,
spacecraft will continue on dashed
elliptical trajectory
V1
Vc1
r2
V1
Homann Transfer
•
Initial LEO orbit has radius r1 and
velocity Vc1
GM 
V c1 =
•
•
V2
V2
GEO
r
Desired GEO orbit has radius r2
and velocity Vc2
Impulsive V1 is applied to get on
geostationary transfer orbit (GTO)
at perigee:
 V1 =
•
Vc2
2m

r1
2m
r1  r2
m
r2

2m
r2

LEO
r1
m

r1
Coast to apogee and apply
impulsive V2:
V2 =
GTO
2m
r1  r2
V1
Vc1
r2
V1
3. Second
Hohmann burn
circularizes at
GEO
Super GTO
GEO
Target
Orbit
Initial orbit has greater
apogee than standard
GTO.
Plane change at much
higher altitude requires
far less ΔV.
PRO: Less overall ΔV
from higher inclination
launch sites.
CON: Takes longer to
establish the final orbit.
2. Plane change
plus initial
Hohmann burn
1. Launch to
‘Super GTO’
Orbital Rendezvous
(Shuttle with ISS)
Interplanetary Trajectories
(Patched Conic Approximation)
Gravity Assist
Farewell