Astronautics
••
••
••
••
••
••
••
Introduction
Introduction to
to space
space systems
systems concepts
concepts
Space
Space environment
environment
Orbital
Orbital mechanics
mechanics
Attitude
Attitude dynamics
dynamics and
and control
control
Propulsion
Propulsion and
and launch
launch vehicles
vehicles
Space
Space law
law and
and policy
policy
Space
Space industry
industry
Sample Space Applications
Communications
Communications
Weather
Weather
Materials
Materials processing
processing
Astronomy
Astronomy
Space
Space Exploration
Exploration
Transportation
Transportation
Navigation
Navigation
Surveillance
Surveillance
Search
Search &
& rescue
rescue
Weapons
Weapons
Solar
Solar Power
Power
Colonization
Colonization
Relay
Relay
Science
Science
Tourism
Tourism
Mapping
Mapping
Basic Elements of Space Missions
•• Subject
Subject -- the
the “thing”
“thing” which
which interacts
interacts with
with or
or isis sensed
sensed by
by the
the
payload
payload
•• Space
Space segment
segment -- spacecraft
spacecraft comprised
comprised of
of payload
payload and
and
spacecraft
spacecraft bus
bus
•• Launch
Launch segment
segment -- launch
launch facilities,
facilities, launch
launch vehicle,
vehicle, upper
upper
stage.
stage. Constrains
Constrains spacecraft
spacecraft size.
size.
•• Orbit/constellation
Orbit/constellation -- spacecraft’s
spacecraft’s trajectory
trajectory or
or path
path through
through
space
space
•• CC33 Architecture
Architecture -- command,
command, control
control and
and communications
communications
•• Ground
Ground segment
segment -- fixed
fixed and
and mobile
mobile ground
ground stations
stations
necessary
necessary for
for TT&C
TT&C
•• Mission
Mission operations
operations -- the
the people,
people, policies
policies and
and procedures
procedures
occupying
occupying the
the ground
ground (and
(and possibly
possibly space)
space) segments
segments
Bus
Payload
Mission
Operations
Space segment
Orbit and
Constellation
Ground
Segment
Reference: Larson
& Wertz, Space
Mission Analysis
and Design
Command, Control and
Communications
Architecture
Launch
Segment
Subject
Payload and Bus Subsystems
Basic requirements
• Payload must be
pointed
• Payload must be
operable
• Data must be
transmitted to users
• Orbit must be
maintained
• Payload must be
“held together”
• Energy must be
provided
Spacecraft
Payload
Sensors
Cameras
Antennas
Radar
Bus
Power
Solar arrays
Batteries
PMAD
TT&C
Power switching
Encoder/decoder
Processors
ADCS
Sensors
Actuators
Processors
Propulsion
Orbit injection
Stationkeeping
Attitude control
Thermal Control
Coatings
Insulation
Active control
Structure
Primary structure
Deployment
mechanisms
Deep Space 1
Deep Space 1 Deployed
Kepler
Kepler Sunside
A-Train Formation
StarDust
Giotto
In the LV
Lightband
MESSENGER Vibration Test
Thoughts on Space
Some comments overhead at the Officer’s Club
• It’s a really big place with no air.
• There’s nothing out there, is there?
• How many g’s is that satellite pulling
when the ground track makes those
turns?
• Why can’t I have my spy satellite
permanently positioned over Moscow?
Useful Characteristics of Space
•• Global
Global perspective
perspective or
or “There’s
“There’s nothin’
nothin’ there
there
to
to block
block your
your view”
view”
•• Above
Above the
the atmosphere
atmosphere or
or “There’s
“There’s no
no air
air to
to
mess
mess up
up your
your view”
view”
•• Gravity-free
or “In
“In free-fall,
free-fall, you
you
Gravity-free environment
environment or
don’t
don’t notice
notice the
the gravity”
gravity”
•• Abundant
Abundant resources
resources or
or “Eventually,
“Eventually, we
we will
will
mine
mine the
the asteroids,
asteroids, collect
collect more
more solar
solar power,
power,
colonize
colonize the
the moon,
moon, .. .. .”
.”
Global Perspective
Amount
Amount of
of Earth
Earth that
that can
can
be
be seen
seen by
by aa satellite
satellite isis
much
much greater
greater than
than can
can be
be
seen
seen by
by an
an Earth-bound
Earth-bound
observer.
observer.
Low-Earth
Low-Earth orbit
orbit isis closer
closer
than
than you
you think.
think.
Space isn't remote at all. It's only
an hour's drive away if your car
could go straight upwards.
— Fred Hoyle
Instantaneous Access Area
H
λ
Re
IAA
IAA= K A (1 − cos λ )
K A = 2.55604187 × 108 km 2
Re
cos λ =
Re + H
Example: Space shuttle
A friend of mine once sent me a post card
with a picture of the entire planet Earth
taken from space. On the back it said,
“Wish you were here.”
— Steven Wright
Re = 6378 km
H = 300 km
cos λ = 0.9551 ⇒ λ = 17.24o
IAA = 11,476,628 km 2
Above the Atmosphere
•• This
This characteristic
characteristic has
has several
several
applications
applications
–– Improved
Improved astronomical
astronomical observations
observations
–– “Vacuum”
“Vacuum” for
for manufacturing
manufacturing processes
processes
–– Little
Little or
or no
no drag
drag to
to affect
affect vehicle
vehicle motion
motion
•• However,
However, there
there really
really isis “air”
“air” in
in space
space
–– Ionosphere
Ionosphere affects
affects communications
communications signals
signals
–– “Pressure”
“Pressure” can
can contaminate
contaminate some
some processes
processes
–– Drag
Drag causes
causes satellites
satellites to
to speed
speed up
up (!)
(!) and
and
orbits
orbits to
to decay,
decay, affecting
affecting lifetime
lifetime of
of LEO
LEO
satellites
satellites
Vacuum Effects
•• While
While space
space isis not
not aa perfect
perfect vacuum,
vacuum, itit isis
better
better than
than Earth-based
Earth-based facilities
facilities
-7
-5
–– 200
200 km
km altitude:
altitude: pressure
pressure =
= 10
10-7 torr
torr =
= 10
10-5 Pa
Pa
-7
–– Goddard
vacuum
chambers:
pressure
=
10
Goddard vacuum chambers: pressure = 10-7 torr
torr
•• Outgassing
Outgassing
–– affects
affects structural
structural
characteristics
characteristics
–– possibility
possibility of
of
vapor
vapor condensation
condensation
Wake Shield Facility
(shuttle experiment)
Atmospheric Drag
•• Can
Can be
be modeled
modeled same
same as
as with
with “normal”
“normal” atmospheric
atmospheric flight
flight
D = C D Aρ V
1
2
ρ ≈ ρ SL e
2
−h / H
•• Key
Key parameter
parameter isis the
the “ballistic
“ballistic coefficient”:
coefficient”:
m /(C D A)
•• Larger
Larger ballistic
ballistic coefficient
coefficient (small
(small massive
massive satellite)
satellite) implies
implies slower
slower
orbital
orbital decay
decay
•• Smaller
Smaller ballistic
ballistic coefficient
coefficient implies
implies faster
faster orbital
orbital decay
decay
•• Energy
Energy loss
loss per
per orbit
orbit isis
≈ 2πrD
Better not take a dog on the space shuttle, because if he sticks his head out
when you're coming home his face might burn up. — Jack Handey
Weightlessness
This illustration from Jules
Verne’s Round the Moon
shows the effects of
“weightlessness” on the
passengers of The Gun
Club’s “bullet” capsule that
was fired from a large gun
in Florida.
The passengers only
experienced this at the halfway point between the
Earth and the Moon.
Physically accurate?
Zero-Gravity?
This plot shows
how gravity drops
off as altitude
increases.
Note that at LEO,
the gravitational
acceleration is
about 90% of that
at Earth’s surface.
Microgravity
••
••
Weightlessness,
Weightlessness, free
free fall,
fall, or
or zero-g
zero-g
Particles
Particles don’t
don’t settle
settle out
out of
of solution,
solution, bubbles
bubbles don’t
don’t rise,
rise,
convection
convection doesn’t
doesn’t occur
occur
•• Microgravity
Microgravity effects
effects in
in LEO
LEO can
can be
be reduced
reduced to
to 10
10-1-1 gg (1
(1 µg)
µg)
On Earth, gravity-driven buoyant convection causes a candle flame to be
teardrop-shaped (a) and carries soot to the flame's tip, making it yellow. In
microgravity, where convective flows are absent, the flame is spherical, sootfree, and blue (b).
A Brief History of Orbital Mechanics
Aristotle
(384-322
Aristotle
(384-322 BC)
BC)
Ptolemy
(87-150
Ptolemy
(87-150 AD)
AD)
Nicolaus
Nicolaus Copernicus
Copernicus(1473-1543)
(1473-1543)
Tycho
(1546-1601)
Tycho Brahe
Brahe
(1546-1601)
Johannes
(1571-1630)
Johannes Kepler
Kepler
(1571-1630)
Galileo
(1564-1642)
Galileo Galilei
Galilei
(1564-1642)
Sir
(1643-1727)
Sir Isaac
Isaac Newton
Newton
(1643-1727)
Kepler’s Laws
I.I. The
The orbit
orbit of
of each
each planet
planet isis an
an ellipse
ellipse with
with the
the
Sun
Sun at
at one
one focus.
focus.
II.
II. The
The line
line joining
joining the
the planet
planet to
to the
the Sun
Sun sweeps
sweeps
out
out equal
equal areas
areas in
in equal
equal times.
times.
III.
III. The
The square
square of
of the
the period
period of
of aa planet’s
planet’s orbit
orbit
isis proportional
proportional to
to the
the cube
cube of
of its
its mean
mean
distance
distance to
to the
the sun.
sun.
Kepler’s First Two Laws
I. The orbit of each planet is an ellipse with the Sun at
one focus.
II. The line joining the planet to the Sun sweeps out
equal areas in equal times.
Kepler’s Third Law
III.
III. The
The square
square of
of the
the period
period of
of aa planet’s
planet’s orbit
orbit isis
proportional
proportional to
to the
the cube
cube of
of its
its mean
mean distance
distance to
to the
the sun.
sun.
T = 2π
a3
µ
Here T is the period, a is the semimajor axis of the ellipse,
and µ is the gravitational parameter (depends on mass of
central body)
µ ⊕ = GM ⊕ = 3.98601×10 km s
5
3 −2
µ sun = GM sun = 1.32715 × 1011 km 3s − 2
Earth Satellite Orbit Periods
Orbit
Orbit Altitude
Altitude (km)
(km)
LEO
300
LEO
300
LEO
400
LEO
400
MEO
3000
MEO
3000
GPS
20232
GPS
20232
GEO
35786
GEO
35786
Period
Period (min)
(min)
90.52
90.52
92.56
92.56
150.64
150.64
720
720
1436.07
1436.07
You should be able to do these calculations! Don’t forget to add the
radius of the Earth to the altitude to get the orbit radius.
Canonical Units
•• For
For circular
circular orbits,
orbits, we
we commonly
commonly let
let the
the orbit
orbit
radius
radius be
be the
the distance
distance unit
unit (or
(or DU)
DU) and
and select
select
the
the time
time unit
unit so
so that
that the
the period
period of
of the
the orbit
orbit isis
2π
2π TUs
TUs
a3
T = 2π
µ
•• Clearly
Clearly these
these choices
choices lead
lead to
to aa value
value of
of the
the
gravitational
gravitational parameter
parameter of
of
µµ == 11 DU
DU33/TU
/TU22
•• For
For the
the Earth’s
Earth’s orbit
orbit around
around the
the Sun,
Sun,
11 DU
DU == 11 AU
AU ≈≈ 90,000,000
90,000,000 miles
miles
•• What
What isis the
the duration
duration of
of 11 TU?
TU?
Examples
•• The
The planet
planet Saturn
Saturn has
has an
an orbit
orbit radius
radius of
of
approximately
approximately 9.5
9.5 AU.
AU. What
What isis its
its period
period in
in
years?
years?
•• The
The GPS
GPS satellites
satellites orbit
orbit the
the Earth
Earth at
at an
an altitude
altitude
of
of 20,200
20,200 km.
km. What
What isis the
the period
period of
of the
the orbit
orbit
of
of aa GPS
GPS satellite
satellite in
in minutes?
minutes? …hours?
…hours?
•• The
The Hubble
Hubble Space
Space Telescope
Telescope orbits
orbits the
the Earth
Earth
at
at an
an altitude
altitude of
of 612
612 km.
km. What
What isis the
the period
period
of
of the
the orbit
orbit in
in minutes?
minutes? …hours?
…hours?
Newton’s Laws
•• Kepler’s
Kepler’s Laws
Laws were
were based
based on
on observation
observation
data:
data: “curve
“curve fits”
fits”
•• Newton
Newton established
established the
the theory
theory
–– Universal
Universal Gravitational
Gravitational Law
Law
GMm
Fg = − 2
r
–– Second
Second Law
Law
m
r
M
r
r&
&
F = mr
AOE
AOE 4134
4134 Astromechanics
Astromechanics covers
covers the
the
formulation
formulation and
and solution
solution of
of this
this problem
problem
Elliptical Orbits
•• Planets,
Planets, comets,
comets, and
and asteroids
asteroids orbit
orbit the
the Sun
Sun
in
in ellipses
ellipses
•• Moons
Moons orbit
orbit the
the planets
planets in
in ellipses
ellipses
•• Artificial
Artificial satellites
satellites orbit
orbit the
the Earth
Earth in
in ellipses
ellipses
•• To
To understand
understand orbits,
orbits, you
you need
need to
to
understand
understand ellipses
ellipses (and
(and other
other conic
conic
sections)
sections)
•• But
But first,
first, let’s
let’s study
study circular
circular orbits
orbits
–– aa circle
circle isis aa special
special case
case of
of an
an ellipse
ellipse
Circular Orbits
•• The
The speed
speed of
of aa satellite
satellite
in
in aa circular
circular orbit
orbit depends
depends
on
on the
the radius
radius
µ
vc =
r
v
a=r
•• IfIf an
an orbiting
orbiting object
object at
at aa particular
particular
radius
then itit isis in
in an
an elliptical
elliptical
radius has
has aa speed
speed << vvcc,, then
orbit
orbit with
with lower
lower energy
energy
•• IfIf an
an orbiting
orbiting object
object at
at radius
radius rr has
has aa speed
speed >>
vvcc,, then
then itit isis in
in aa higher-energy
higher-energy orbit
orbit which
which
may
may be
be elliptical,
elliptical, parabolic,
parabolic, or
or hyperbolic
hyperbolic
Examples
•• The
The Hubble
Hubble Space
Space Telescope
Telescope orbits
orbits the
the Earth
Earth
at
at an
an altitude
altitude of
of 612
612 km.
km. What
What isis its
its speed?
speed?
•• The
The Space
Space Station
Station orbits
orbits the
the Earth
Earth at
at an
an
altitude
altitude of
of 380
380 km.
km. What
What isis its
its speed?
speed?
•• The
The Moon
Moon orbits
orbits the
the Earth
Earth with
with aa period
period of
of
approximately
approximately 28
28 days.
days. What
What isis its
its orbit
orbit
radius?
radius? What
What isis its
its speed?
speed?
The Energy of an Orbit
•• Orbital
Orbital energy
energy isis the
the sum
sum of
of the
the kinetic
kinetic energy,
energy, mv
mv22/2,
/2,
and
and the
the potential
potential energy,
energy, -µm/r
-µm/r
•• Customarily,
Customarily, we
we use
use the
the specific
specific mechanical
mechanical energy,
energy, EE
(i.e.,
(i.e., the
the energy
energy per
per unit
unit mass
mass of
of satellite)
satellite)
v2 µ
E= −
2 r
⇔
−µ
E=
2a
•• From
From this
this definition
definition of
of energy,
energy, we
we can
can develop
develop the
the
following
following facts
facts
EE << 00
EE == 00
EE >> 00
⇔
⇔
⇔
⇔
⇔
⇔
orbit
orbit isis elliptical
elliptical or
or circular
circular
orbit
orbit isis parabolic
parabolic
orbit
orbit isis hyperbolic
hyperbolic
Examples
•• A
A satellite
satellite isis observed
observed to
to have
have altitude
altitude 500
500 km
km
and
and velocity
velocity 77 km/s.
km/s. What
What type
type of
of conic
conic
section
section isis its
its trajectory?
trajectory?
•• A
A satellite
satellite isis observed
observed to
to have
have altitude
altitude 380
380 km
km
and
and velocity
velocity 11
11 km/s.
km/s. What
What type
type of
of conic
conic
section
section isis its
its trajectory?
trajectory?
•• What
What type
type of
of trajectory
trajectory do
do comets
comets follow?
follow?
Asteroids?
Asteroids?
Conic Sections
Any
Any conic
conic section
section can
can
be
be visualized
visualized as
as the
the
intersection
intersection of
of aa plane
plane
and
and aa cone.
cone.
The
The intersection
intersection can
can be
be
aa circle,
circle, an
an ellipse,
ellipse, aa
parabola,
parabola, or
or aa
hyperbola.
hyperbola.
Properties of Conic Sections
Conic
Conic sections
sections are
are characterized
characterized by
by
eccentricity,
ee (shape)
eccentricity,
(shape) and
and
semimajor
semimajor axis,
axis, aa (size)
(size)
Conic
Conic
Circle
Circle
Ellipse
Ellipse
Parabola
Parabola
Hyperbola
Hyperbola
ee
ee == 00
00 << ee << 11
ee == 11
ee >> 11
aa
aa >> 00
aa >> 00
aa == ∞
∞
aa << 00
Properties of Ellipses
b
a
r
ae
vacant
focus
2a-p
„
focus
p=a(1-e2)
a
a(1+e)
a(1-e)
Facts About Elliptical Orbits
•• Periapsis
Periapsis isis the
the closest
closest point
point of
of the
the orbit
orbit to
to the
the
central
central body
body
rrpp == a(1-e)
a(1-e)
•• Apoapsis
Apoapsis isis the
the farthest
farthest point
point of
of the
the orbit
orbit from
from
the
the central
central body
body
rraa == a(1+e)
a(1+e)
•• Velocity
Velocity at
at any
any point
point isis
1/2
vv == (2E+2µ/r)
(2E+2µ/r)1/2
•• Escape
Escape velocity
velocity at
at any
any point
point isis
1/2
1/2
vvesc
=
(2µ/r)
=
(2µ/r)
esc
Examples
•• A
A satellite
satellite isis observed
observed to
to have
have altitude
altitude 500
500 km
km
and
and velocity
velocity 77 km/s.
km/s. What
What isis its
its energy?
energy?
What
What isis its
its velocity
velocity at
at some
some later
later time
time when
when its
its
altitude
altitude isis 550
550 km?
km?
•• A
A satellite
satellite isis in
in an
an Earth
Earth orbit
orbit with
with semimajor
semimajor
axis
axis of
of 10,000
10,000 km
km and
and an
an eccentricity
eccentricity of
of 0.1.
0.1.
What
What isis the
the speed
speed at
at periapsis?
periapsis? …at
…at apoapsis?
apoapsis?
•• A
A Mars
Mars probe
probe isis in
in an
an Earth
Earth parking
parking orbit
orbit with
with
altitude
altitude 400
400 km.
km. What
What isis its
its velocity?
velocity? What
What isis
its
its escape
escape velocity?
velocity? What
What change
change in
in velocity
velocity isis
required
required to
to achieve
achieve escape
escape velocity?
velocity?
Orbital Elements
^
K
Equatorial plane
ν
ω
^
I
Ω
i
ne
a
l
p
l
Orbita
^
n
^
J
Orbit is defined by 6 orbital elements (oe’s):
eccentricity, e;
semimajor axis, a;
inclination, i;
right ascension of ascending node, Ω;
argument of periapsis, ω;
and true anomaly, ν
Orbital Elements(continued)
• Semimajor axis a determines the size of the ellipse
• Eccentricity e determines the shape of the ellipse
• Two-body problem
– a, e, i, Ω, and ω are constant
– 6th orbital element is the angular measure of satellite motion
in the orbit – 2 angles are commonly used:
• True anomaly, ν
• Mean anomaly, M
• In reality, these elements are subject to various
perturbations
–
–
–
–
Earth oblateness (J2)
atmospheric drag
solar radiation pressure
gravitational attraction of other bodies
Ground Track
This plot is for a satellite in a nearly circular orbit
ISS (ZARYA)
90
60
latitude
30
0
−30
−60
−90
0
60
120
180
longitude
240
300
360
Ground Track
This plot is for a satellite in a highly elliptical orbit
1997065B
90
60
latitude
30
0
−30
−60
−90
0
60
120
180
longitude
240
300
360
Algorithm for SSP, Ground Track
••
••
••
••
••
••
Compute
Compute position
position vector
vector in
in ECI
ECI
Determine
at epoch,
epoch, θθg0
Determine Greenwich
Greenwich Sidereal
Sidereal Time
Time θθgg at
g0
Latitude
Latitude isis δδss == sin
sin-1-1(r
(r33/r)
/r)
Longitude
Longitude isis LLss == tan
tan-1-1(r
(r22/r/r11)-)- θθg0
g0
Propagate
Propagate position
position vector
vector in
in “the
“the usual
usual way”
way”
+ω ⊕(t-t
Propagate
Propagate GST
GST using
using θθgg =
= θθg0
(t-t00))
g0 +ω⊕
where
the angular
angular velocity
velocity of
of the
the Earth
Earth
where ω
ω⊕⊕ isis the
•• Notes:
Notes:
http://www.aoe.vt.edu/~chall/courses/aoe4134/sidereal.pdf
http://www.aoe.vt.edu/~chall/courses/aoe4134/sidereal.pdf
http://aa.usno.navy.mil/data/docs/WebMICA_2.html
http://aa.usno.navy.mil/data/docs/WebMICA_2.html
http://tycho.usno.navy.mil/sidereal.html
http://tycho.usno.navy.mil/sidereal.html
Elevation Angle
Elevation angle, ε, is
measured up from
horizon to target
Minimum elevation
angle is typically based
on the performance of
an antenna or sensor
IAA is determined by
same formula, but the
Earth central angle, λ,
is determined from the
geometry shown
R⊕
λ0
λ
ε D
η
ρ
The angle η is called the nadir angle
The angle ρ is called the apparent
Earth radius
The range from satellite to target is
denoted D
Geometry of Earth-Viewing
R⊕
λ0
λ
ε D
η
ρ
•• Given
Given altitude
altitude H,
H, we
we can
can state
state
sin
= RR⊕⊕ // (R
(R⊕⊕+H)
+H)
sin ρρ =
= cos
cos λλ00 =
ρ+
ρ+ λλ00 =
= 90°
90°
•• For
For aa target
target with
with known
known position
position vector,
vector, λλ isis
easily
easily computed
computed
cos
cos δδtt cos
cos ∆L
∆L ++ sin
sin δδss sin
sin δδtt
cos λλ == cos
cos δδss cos
•• Then
Then tan
tan ηη == sin
sin ρρ sin
sin λλ // (1(1- sin
sin ρρ cos
cos λ)
λ)
•• And
sin λλ // sin
sin ηη
And ηη ++ λλ ++ εε == 90°
90° and
and D
D=
= RR⊕⊕ sin
Earth Oblateness Perturbations
•• Earth
Earth isis non-spherical,
non-spherical, and
and to
to first
first
approximation
approximation isis an
an oblate
oblate spheroid
spheroid
•• The
The primary
primary effects
effects are
are on
on Ω
Ω and
and ω:
ω:
m
2
3
J
nR
2
e
& =
Ω
cos i
2
2 2
2a (1 − e )
r
− 3 J 2 nR
ω& = 2
(4 − 5 sin 2 i )
4a (1 − e )
2
e
2 2
M
The Oblateness Coefficient
J 2 = −1.0826 × 10
−3
Main Applications of J2 Effects
•• Sun-synchronous
Sun-synchronous orbits:
orbits:
The
The rate
rate of
of change
change of
of Ω
Ω can
can be
be chosen
chosen so
so that
that
the
the orbital
orbital plane
plane maintains
maintains the
the same
same
orientation
orientation with
with respect
respect to
to the
the sun
sun throughout
throughout
the
the year
year
•• Critical
Critical inclination
inclination orbits:
orbits:
The
The rate
rate of
of change
change of
of ω
ω can
can be
be made
made zero
zero by
by
selecting
selecting ii ≈≈ 63.4°
63.4°
Sun-Synchronous Orbit
Orbit rotates to
maintain same
angle with sun
a
Equinox
a
Summer
a
Sun
Equinox
a
Winter
Basic Space Propulsion Concepts
•• Three
Three functions
functions of
of space
space propulsion
propulsion
––
––
place
place payload
payload in
in orbit
orbit (launch
(launch vehicles)
vehicles)
transfer
transfer payload
payload from
from one
one orbit
orbit to
to another
another (upper
(upper
stages)
stages)
–– control
control spacecraft
spacecraft position
position and
and pointing
pointing direction
direction
(thrusters)
(thrusters)
•• Rockets
Rockets provide
provide aa change
change in
in momentum
momentum according
according
to
to the
the impulse-momentum
impulse-momentum form
form of
of Newton’s
Newton’s 2nd
2nd
law:
law:
r
r
r r
r
F(t 2 − t1 ) = p 2 − p1 ⇔ F∆t = ∆p
–– usually
usually we
we talk
talk about
about the
the change
change in
inrvelocity
velocity provided
provided by
by
aa rocket
∆V
rocket ⎯
⎯ its
its “Delta
“Delta Vee”
Vee”
–– generally
generally involves
involves change
change in
in magnitude
magnitude and
and direction
direction
Thrust, Mass Flow Rate,
& Exhaust Velocity
•• The
The simplest
simplest model
model of
of thrust
thrust generated
generated by
by aa rocket
rocket isis
F = −m& Ve
m& isis the
the rate
rate at
at which
which propellant
propellant isis ejected
ejected (negative)
(negative)
Ve isis the
the exhaust
exhaust velocity
velocity relative
relative to
to the
the rocket
rocket
•• An
An important
important performance
performance measure
measure isis the
the Specific
Specific
Impulse
Impulse
I sp = Ve / g
I sp has
has units
units of
of seconds
seconds
–– isis the
the number
number of
of pounds
pounds of
of thrust
thrust for
for every
every pound
pound of
of
propellant
propellant burned
burned in
in one
one second
second
•• Using
Using I sp the
the thrust
thrust generated
generated by
by aa rocket
rocket isis
F = − m& gI sp
•• The
The gg that
that isis used
used isis ALWAYS
ALWAYS the
the standard
standard mean
mean
acceleration
acceleration due
due to
to gravity
gravity at
at the
the Earth’s
Earth’s surface
surface
Typical Functional Requirements
•• Launch
Launch from
from Earth
Earth to
to LEO:
LEO:
∆V ≈ 9.5 km/s
–– initial
initial velocity
velocity isis ≈≈ 00
–– LEO
LEO circular
circular orbit
orbit velocity
velocity isis vvcc=√(µ/r)
=√(µ/r) ≈≈ 7.7
7.7 km/s
km/s
–– add
add aa bit
bit for
for losses
losses due
due to
to drag
drag
•• LEO-to-GEO
LEO-to-GEO orbit
orbit transfer:
transfer:
–– use
use Hohmann
Hohmann transfer
transfer
–– perigee
perigee burn:
burn: ∆v
∆v11≈≈ 2.4257
2.4257 km/s
km/s
–– apogee
1.4667 km/s
km/s
apogee burn:
burn: ∆v
∆v22≈≈ 1.4667
∆V ≈ 4 km/s
Hohmann Transfer
•Initial
•Initial LEO
LEO orbit
orbit has
has radius
radius
rr11,, velocity
velocity vvc1c1
•Desired
•Desired GEO
GEO orbit
orbit has
has
radius
velocity vvc2c2
radius rr22,, velocity
•Impulsive
•Impulsive ∆v
∆v isis applied
applied to
to
get
get on
on geostationary
geostationary
transfer
transfer orbit
orbit (GTO)
(GTO) at
at
perigee
perigee
∆v1 =
2µ
r1
−
2µ
r1 + r2
−
µ
µ
r2
−
2µ
r2
v2
∆v2
GTO
LEO
r2
r1
r1
•Coast
•Coast to
to apogee
apogee and
and apply
apply
another
another impulsive
impulsive ∆v
∆v
∆v2 =
vc2
− r12+µr2
∆v1
vc1
v1
GEO
Other Typical Requirements
•• Orbit
Orbit maintenance
maintenance for
for GEO
GEO satellites
satellites
–– East-West
East-West stationkeeping
stationkeeping
–– North-South
North-South stationkeeping
stationkeeping
33 to
to 66 m/s
m/s per
per year
year
45
45 to
to 55
55 m/s
m/s per
per year
year
•• Reentry
Reentry for
for LEO
LEO satellites
satellites 120
120 to
to 150
150 m/s
m/s
•• Attitude
Attitude control
control typically
typically requires
requires 3-10%
3-10% of
of
the
the propellant
propellant mass
mass