Transcript Orbital-Mechanics

ORBITAL MECHANICS: HOW
OBJECTS MOVE IN SPACE
FROM KEPLER
FIRST LAW: A SATELLITE REVOLVES
IN AN ELLIPTICAL ORBIT AROUND A
CENTER OF ATTRACTION
POSITIONED AT ONE FOCI OF THE
ELLIPSE.
SECOND LAW: THE RATE OF TRAVEL
ALONG THE ORBIT IS DIRECTLY
PROPORTIONAL TO THE AREA OF
SWEEP IN THE ELLIPSE.
ORBIT PLANE
APOGEE
CENTER OF
ATTRACTION (FOCI)
PERIGEE
SATELLITE
THIRD LAW: PERIOD OF THE ORBIT
SQUARED IS PROPORTIONAL TO THE
MEAN DISTANCE TO CENTER CUBED.
EARTH EXAMPLE
ORBITAL MECHANICS: WHY
OBJECTS MOVE THE WAY THEY DO
NEWTONIAN THEORY
Fg = G M m / r2
V = ORBITAL VELOCITY
ORBITAL
PATH
Fc = m V2 / r
Fc = CENTRIPETAL FORCE
DUE TO REVOLUTION
ALTITUDE
Fg = GRAVITATION FORCE
A SATELLITE MAINTAINS ITS
ORBIT WHEN Fc = Fg
G = UNIVERSAL GRAVITY CONSTANT
M = MASS OF EARTH
m = MASS OF SATELLITE
r = DISTANCE EARTH CENTER TO SATELLITE
IN NEAR CIRCULAR ORBITS THE ORBITAL
VELOCITY IS ABOUT CONSTANT.
IN HIGHLY ELLIPTICAL ORBITS THE SATELLITE
SPEEDS UP TO MAX VELOCITY AT PERIGEE AND
SLOWS DOWN TO MIN VELOCITY AT APOGEE.
Newton’s Laws
• A body remains at rest or in constant motion
unless acted upon by external forces
• The time rate of change of an object’s momentum
is equal to the applied force
• For every action there is an equal and opposite
reaction
• The force of gravity between two bodies is
proportional to the product of their masses and
inversely proportional to the square of the
distances between them.
Acceleration, Time, Distance
• F = ma
• Vf = V0 + at
2
• s = V0t + at2
Vector Addition
Vector addition is done by adding the two head to tail vectors to equal the tail to tail and
head to head vector
V3
V1
V1+ V3 = V2
V2
Law of Sines
a / sin A = b / sin B = c / sin C
Law of Cosines
a2 = b2 + c2 - 2bc Cos A
b2 = a2 + c2 - 2ac Cos B
c2 = a2 + b2 - 2ab Cos C
C
a
b
A
c
B
Trigonometric Functions
sine = opposite/ hypotenuse = y/r
cosine = adjacent / hypotenuse = x/r
tangent = opposite / adjacent = y/x
-1< sin or cos <+1
- infinity < tangent < + infinity
y
r
q
y
sin 0 = 0
cos 0 = 1
sin 90 = 1
cos 90 = 0
tan 0 = 0
tan 90 = + infinity
x
x
r2 = x2 + y2
sin2 q + cos2 q = 1
Rocket Engines
• Liquid Propellant
– Mono propellant
• Catalysts
– Bi-propellant
• Solid Propellant
– Grain Patterns
• Hybrid
• Nuclear
• Electric
Performance
Energy
Safety
Simplicity
Expanding Gases
Thrust Termination
Restart
Specific Heat
• Specific Heat : the amount of heat that enters or leaves a unit mass
while the substance changes one degree in temperature.
– c = Btu per lbsm - degree Rankine
– cp : specific heat at constant pressure
– cv : specific heat at constant volume
– k : ratio of specific heats
dQ
c = w dT
cp
k= c
v
>1
Specific Impulse
Isp = 9.797
(
k
k-1
Pe
Pc
Tc
mg
k-1
)( )[ 1 - ( ) ]
k
Pe : Nozzle Exit Pressure (psi)
<14.7 psi
Pc : Combustion Chamber Pressure (psi)
6,000 psi
Tc : Combustion Chamber Temperature (degrees Rankine)
5,000o R
mg : average molecular weight of combustion products (lb/mole)
2H2+O 2
2H2O
Isp =
18 lbs/mole = mg
F
.
W
Launch Velocity Losses
• Gravity losses
• Pitch over to get correct velocity vector alignment
for orbital insertion
• Drag from atmosphere
• Not instantaneous application of velocity
Losses are between 15 and 17 % of DV
Rocket Formulas
Rocket Equation
DV = Isp x g x ln MR
Mass Ratio
MR =
minitial
mfinal
Specific Impulse
F
Isp = .
W
.
W
Ve + Ae ( Pe - Pa)
F=
g
Thrust
Three Stage Booster
Structure
Propellants
Burn
Time
(sec)
35,000
365,000
100
Weights (lbs.)
All three
stages
1st Stage
2nd Stage
3rd Stage
Payload
10,000
4,000
10,000
125,000
50,000
120
80
Isp
(sec)
Stage
Weight
280
400,000
599,000
135,000
199,000
54,000
290
250
DV1 = (280)(32.2)ln (2.56) = 8,475 ft/sec
DV2 = (290)(32.2)ln (2.69) = 9,238 ft/sec
DV3 = (250)(32.2)ln (4.57) = 12,232 ft/sec
Vposigrade = 29,535 ft/sec
Vretrograde = 30,183 ft/sec
MR
2.56
2.69
4.57
Vl = 29,945 ft/sec
Can place payload in posigrade orbit,
but not in retrograde orbit,
ORBIT FORMULAS
ELLIPTICAL & CIRCULAR ORBITS
a
e
=
c
= eccentricity
a
}
2a = rA + rP
}b
rA
Apogee
rP
c
c = a - rP
e=
e=
e=
e=
a - rP
1-
r’
a
rP
a
1 - 2rP
rA + rP
rA - rP
rA + rP
a
r’ + r = 2a
if r = r’ then at that point
2r = 2a
r=a
a2 - c2
.. . b =
r
b
c
Perigee
CONSTANTS FOR ORBIT
PHYSICAL
GEOMETRIC
E = Specific Energy
H = Specific Momentum
m = Universal
gravitational attraction
mr
e
a
b
c
m2
n
E<0
V
,
E<0 , 0<
m1
e
= 0 for a circle
e
< 1 for an ellipse
e
m
H2
r
e
=
1+
1+
ke
m
2
cos n
m
=
2
1 + 2EH
> 1 for a hyperbola
Polar coordinates for any
conic section
pages 32, 33, 34 “Handout”
H2
=
e
2EH2
= 1 for a parabola
m2
Orbital Period
CIRCULAR ORBIT
ELLIPTICAL ORBIT
2pa
V
2pr
V
P
V =
m
r
a = mean distance from focus =
= semi major axis
P2
4 p2r2
m
r
Period =
=
P2 = 4 p
2
r3
=
P2
P2
=
= (2.805 x
2
4 p a3
m
1015)a3
m
units
2
= sec ft3
KEPLER’S
THIRD
LAW
EARTH SATELLITES
Eccentricity =
e
Major Axis =
2a
Minor Axis =
2b
rp = a - c
ra = a +
c
rp + ra = 2a
c
e =
a
E = specific energy
H = specific angular momentum
E = V2
2
M
r
=
m
2a
(1)
for circular
orbits a = r
H = Vr cos
V
=
2m - m
r
a
c2 = a2 - b2
for elliptical
orbits from (1)
therefore
V=
m
r
Coordinate Systems
• Cartesian Coordinates
– Abscissa = x
– Ordinate = y
– (x,y)
• Polar Coordinates
– Radius Vector = r
– Vectorial Angle = q
– (r,q)
+y
r
+x
-x
-y
q
Description of Orbit
• Right Ascension
– Measured eastward from the vernal equinox
• In Spring when the sun’s center crosses the equatorial plane once
thought to be aligned with the first point of the constellation Aries
• Inclination
• Argument of Perigee
• Two of the following
– Eccentricity
– Perigee
– Apogee
Orbit Calculations
Ellipse is the curve traced by a point
moving in a plane such that the sum
of its distances from the foci is
constant.
r
ra
p
b
r’
r
c
a
b
a
x2 y2
+
= 1
a2 b2
r + r’ = 2a
e=
c
a
a2 = b2 + c2
INCLINATION
FUNCTION OF LAUNCH AZIMUTH AND LAUNCH SITE LATITUDE
cos i (inclination) = cos (latitude) sin (azimuth)
N
North = 0 degrees Azimuth
azimuth
270o
90o
East
West
180o
cos i + cos (lat) sin (az)
sin 90o = 1
sin 0o = 0
sin 180o = 0
sin 270o = -1
South
S
launch azimuth from 180o to360o = retrograde orbit
launch azimuth from 0o to 180o = posigrade orbit
Celestial Sphere
Argument of
Perigee (w)
Perigee
Celestial
Equator
Inclination
Right Ascension
Orbit Trace
W
ORBITAL MECHANICS: GROUND TRACES
INCLINED ORBIT
SAT
ORBIT 4
ORBIT 3
ORBIT 2
ORBIT 1
EQUATORIAL ORBIT
MULTIPLE ORBITS
EARTH MOTION BENEATH SATELLITE
ANGLE OF INCLINATION
(0 DEG. FOR EQUATORIAL)
GROUND TRACES
THE POINTS ON THE EARTH’S SURFACE OVER WHICH A SATELLITE PASSES AS IT
TRAVELS ALONG ITS ORBIT
PRINCIPLE : GROUND TRACE IS THE RESULT OF THE ORBITAL PLANE BEING FIXED AND
THE EARTH ROTATING UNDERNEATH IT
AMPLITUDE OF GROUND TRACE (LATITUDE RANGE) IS EQUAL TO THE ORBITAL INCLINATION
MOVEMENT OF GROUND TRACE IS DICTATED BY THE SATELLITE ALTITUDE AND THE CORRESPONDING
TIME FOR IT TO COMPLETE ONE ORBIT
ORBITAL MECHANICS: SPECIFIC
ORBITS AND APPLICATIONS
•
POLAR (100- 700 NM AT 80 - 100 DEG. INCLINATION)
– SATELLITE PASSES THROUGH THE EARTH'S SHADOW AND PERMITS VIEWING OF THE
ENTIRE EARTH’S SURFACE EACH DAY WITH A SINGLE SATELLITE
•
SUN SYNCHRONOUS (80 - 800 NM AT 95 - 105 DEG INCLINATION)
– PROCESSION OF ORBITAL PLANE SYNCHRONIZED WITH THE EARTH’S ROTATION SO
SATELLITE IS ALWAYS IN VIEW OF THE SUN
– PERMITS OBSERVATION OF POINTS ON THE EARTH AT THE SAME TIME EACH DAY
•
SEMISYNCHRONOUS (10,898 NM AT 55 DEG INCLINATION)
– 12 HR PERIODS PERMITTING IDENTICAL GROUNDTRACES EACH DAY
•
HIGHLY INCLINED ELLIPTICAL (FIXED PERIGEE POSITION)
– SATELLITE SPENDS A GREAT DEAL OF TIME NEAR THE APOGEE COVERING ONE
HEMISPHERE
– CLASSICALLY CALLED “MOLNIYA ORBIT” BECAUSE OF ITS HEAVY USE BY THE
RUSSIANS FOR NORTHERN HEMISPHERE COVERAGE
•
GEOSYNCHRONOUS (GEO) (CIRCULAR, 19,300 NM AT 0 DEG INCLINATION)
– 24 HR PERIOD PERMITS SATELLITE POSITIONING OVER ONE POINT ON EARTH.
– ORBITAL PERIOD SYNCHRONIZED WITH THE EARTH’S ROTATION (NO OTHER ORBIT
HAS THIS FEATURE)
Linear and Angular Motion
ANGULAR MOTION
Distance
Velocity
Acceleration
q=
wavg
aavg =
S/r
qf
=
tf
wf
S = r q ft
radians
qf
radians
to
sec
wo
radians
tf
LINEAR MOTION
to
sec2
Vtangential = r w ft/sec
atangential = r a ft/sec2
wf = wo + a t
q = wo t + a t2
2
(1 radian = 57.3 degrees)
r
Q
s
s Q= r
radians
CONSERVATION OF:
ENERGY
V2
2
-
m
r
=
MOMENTUM
constant = F
Specific Energy#
Angular Momentum = mr2w
mr2w = constant = mH
H = V r cos f
Vr
#Specific means per 1b mass
^
f
V
Specific
Angular
Momentum
CONSTANTS FOR ORBIT
PHYSICAL
GEOMETRIC
E = Specific Energy
H = Specific Momentum
m = Universal gravitational attraction
e
a
b
c
mr
m2
n
E<0
V
,
E<0 , 0<
m1
e
= 0 for a circle
e
< 1 for an ellipse
e
m
H2
r
e
=
1+
1+
ke
m
2
cos n
H2
m
=
e
2EH2
=
2
1 + 2EH
m2
= 1 for a parabola
> 1 for a hyperbola
ESCAPE VELOCITY
E=0
V2
E1 =
2
V2
0=
2
Vescape=
m
r
m
r
2 m
r
=
(2)
14.075x1015
20.9 x 106 ft
sec2
= 36,700 ftsec