EXTROVERT
Space Propulsion 03
Basic Orbits and Mission Analysis
1
EXTROVERT
Space Propulsion 03
Newton's Law of Gravitation
To find the velocity increment required for various missions, we
must calculate trajectories and orbits. This is done using Newton's
Law of Gravitation:
Fr 
-GMm
r2
Here the lhs is the "radial force" of attraction due to gravitation,
between two bodies; the big one of mass M,
and the little one of mass m.
The universal gravitational constant G is 6.670 E-11 Nm2/kg2.
EXTROVERT
Space Propulsion 03
Keplerian Orbits
Johannes Kepler (1609: first law, 1619: third law)
Kepler’s First Law: The orbit of each planet is an ellipse, with the sun at
one focus.
Second Law: The line joining the planet to the sun sweeps out equal areas
in equal times.
Third Law: The square of the period of a planet is proportional to the cube of
its mean distance from the sun
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Space Propulsion 03
Kepler's Laws applied to satellite of mass m, orbiting a much
bigger object of mass M. i.e, m << M):
phy-061062.blogspot.com/
1. The satellite travels in an
elliptical path around its
center of attraction, which is
located at one of focii of the
ellipse. The orbit must lie in a
plane containing the center
of attraction.
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Space Propulsion 03
Second Law: The line joining the planet to the sun sweeps out
equal areas in equal times.
Implication: The radius vector from the center of attraction sweeps equal areas
of the orbit per unit time. As the satellite moves away, its speed decreases. As it
nears the center of attraction, its speed increases.
Found on the web at a
secondary site. Primary
developer of this application is
unknown
EXTROVERT
Space Propulsion 03
Third Law: The square of the period of a planet is proportional to the cube of its
mean distance from the sun
Implication: The ratio of the squares of the orbital periods of any two satellites
about the same body equals the ratio of the cubes of the semi-major axes of
the respective orbits.
2
3
1
 22

a1
a23
EXTROVERT
Space Propulsion 03
Satellite Equations of Motion
Newton’s Second Law
Law of Gravitation
 F  mr
F
 GMm
r2

m
r2
F=magnitude of force due to gravity (N)
17
G=Universal constant of gravity = 6.67 x10
Nkm
Kg 2
M=mass of Earth (kg)
M =satellite mass
R =distance from center of Earth to the satellite (km)
 = GM for Earth = 398,600.5 km3 sec 2
Equation for satellite acceleration (from the above two Newton laws)
r

r
3
r 0
(2-body equation of motion)
The solution to this is the polar equation of a conic section.
EXTROVERT
Space Propulsion 03
Conic Section Orbits / Trajectories
Why Conic Sections are called
conic sections:
Total specific mechanical Energy of an orbit:
(mechanical energy per unit mass)
V2

2


r


2a
where a is the semi major axis and V is magnitude of velocity.
Conic
Energy 
Circle
Semi-major
axis a
e
<0
= radius
0
Ellipse
<0
>0
0<e<1
Parabola
0
Hyperbola
>0
Source: http://www.bartleby.com/61/imagepages/A4conics.html
Courtesy Houghton-Miflin Co.

<0
1
>1
EXTROVERT
Space Propulsion 03
A Few Commonly-Used Results on Orbits
Circular orbit around Earth:
Escape Velocity from Earth:

 R  631.3481
 7.905366 E  
r
r
 r 
2
R
892.8611
Vescape 
 11.17988 E 
r
r
r
Vc 
Speed at any point in an elliptical orbit: v
v apogee  g0r02
1  e   v
1  e  min
Time for one orbit is:
r in km is radius of orbit
V in km/s; r in km
 2 1
 g0r02   
r a
v perigee  g0r02
1  e   v
1  e  max
2 a3

r0 g0
Specific angular momentum: h
(no plane change)
 r xV is constant for the 2-body problem, per Kepler
EXTROVERT
Space Propulsion 03
Orbit Maneuvers
V  Vneed  Vcurrent
where V is the velocity at that point in the corresponding orbit.
The assumption made usually in simplifying orbit maneuvers is that we are dealing with a 2-body
problem. For transfers between earth orbit and say, Mars orbit, we assume initially an Earth-vehicle
problem. When the vehicle leaves the vicinity of Earth, we assume a Sun-vehicle problem, and finally
a Mars-vehicle problem
Hohmann transfer orbit:
VB
Transfer orbit’s ellipse is tangent to initial and final
orbits, at the transfer orbit’s pengee and apogee
respectively
 Most efficient transfer between two circular orbits.
rB
Two-burn transfer between circular orbits at altitudes :
rA & rB
rA
VA
EXTROVERT
Hohmann Transfer
V for a two-burn transfer between circular orbits at altitudes .
VTotal  VA  VB
Example:
rA & rB
1
1
1
1 


 2
1 2  1 2  2
1 2  1 2  
       
     
    
r
a
r
r
a
 A
 rB   
tx 
tx 
 B
 A



For Earth, with delta-v in km/s,
atx 
Space Propulsion 03
  631.3481
rA  rB 
2
rA  6567 km
rB  42160km
 VA  2.46km / s; VB  1.49km / s
 VTotal  3.95km / s
When the radii of the two orbits are close, the delta-v’s are approximately equal.
EXTROVERT
Space Propulsion 03
One-Tangent Burn
This is a quicker orbit transfer than the Hohman transfer ellipse, but is more costly in delta-v.
VB
VA
EXTROVERT
Space Propulsion 03
Orbit Plane Changes
Simple plane change through angle q at an orbital velocity of Vi.
q 
v  2v i Sin  
2
Plane changes are often combined with orbit-change burns. Usual practice is to do a little
of the plane change at the initial burn (at the lower orbit where velocity is higher) and most
of the plane change at the final burn at the higher altitude where the velocity is lower. Plane
change combined with Hohman transfer.
v 


v i 2  v f 2  2v i v f Cos q 
EXTROVERT
Space Propulsion 03
Launch from Earth
Note that the objective is to achieve the required tangential velocity for orbit – the need to gain height
above Earth is mainly dictated by the need to get out of the atmosphere and minimize drag losses.
As the vehicle trajectory begins to slant over, gravity pulls the trajectory down. At low speeds, this
requires a substantial pitch angle between the instantaneous velocity vector and the thrust direction
to counter gravity.
Gravity Turn
1. Begin ascent with short vertical rise, during which it starts usually and completes a roll from
launch-pad azimuth to desired flight azimuth.
2. Pitch-over (pitch 
vector from vertical)
0
because angular change of axis occurs faster than change of velocity
3. Adjust altitude to keep 
0
in trajectory through aerodynamic load region.
EXTROVERT
Space Propulsion 03
Gravity turn
(F  D )
dt
m
gldt
glCos g dt
V(t)
dg
V(t+dt)
g
Local horizontal
If the angle of the trajectory with respect to the local horizontal is gamma (g) then,
this angle changes as
ge cos gdt
 dg 
V  higherorderterms
(rad.)
or
g cos g
dg
 e
dt
V
(rad/s)
EXTROVERT
Space Propulsion 03
Controlled Angle of Attack – Linear Tangent Steering
Angle of attack or the tangent of the thrust angle with respect to the local horizontal, are controlled
as functions of time to minimize losses.
Angle of attack can be increased as dynamic pressure decreases beyond the “max q” point,
subject to limitations on vehicle bending moments. Tangent of the thrust angle usually starts
at a high value and is reduced according to a linear or other specified schedule,
i.e., the slope B is usually negative. .
Tanq  A  Bt
EXTROVERT
Space Propulsion 03
Launch Trajectories
EXTROVERT
Space Propulsion 03
EXTROVERT
Space Propulsion 03
Acceleration During Launch
EXTROVERT
Space Propulsion 03
Interplanetary Trajectories
Earth-Mars Hohmann scheme
1.
1.
2.
Escape velocity from Earth – to an orbit around the Sun, with zero velocity with
respect to Earth.
Impulsive burn to get into transfer orbit.
UMars
Impulsive burn at Mars orbit to match velocity with Mars.
uHoffman  uesc  uT  uMars
Change in orbital energy per unit mass (w.r.t. Earth)
 Hoffman   esc
2
uT 


2
2
uT 


2
UT
EXTROVERT
Data on Solar System Destinations
Mass of Earth = 5.975 E24 kg.
Space Propulsion 03
Mean Diameter = 12742. 46km
Body
Relative
Mass
Rel. Mean
Dia.
Orbit
radius,
Millions
of km
Eccentricity
Period, Earth
yrs
Sun
332,488.
109.15
Mercury
0.0543
0.38
57.9
0.2056
0.241
Venus
0.8136
0.967
108.1
0.0068
0.615
Earth
1.0000
1.000
149.5
0.0167
1.000
Mars
0.1069
0.523
227.8
0.0934
1.881
Jupiter
318.35
10.97
777.8
0.0484
11.862
Saturn
95.3
9.03
1426.1
0.0557
29.458
Uranus
14.58
3.72
2869.1
0.0472
84.015
Neptune
17.26
3.38
4495.6
0.0086
164.788
Plato
<0.1?
0.45
5898.9
0.2485
247.697
Moon
0.0123
0.273
0.3844
0.0549
27.322 days
EXTROVERT
Space Propulsion 03
Earth-Mars Oberth Scheme
Escape from Earth while carrying enough propellant to produce the transfer velocity is
wasteful in work against gravity. Can be avoided by achieving the right velocity magnitude
and direction in the first maneuver.
The launch velocity is larger at Earth, directly placing the vehicle in the Hohman transfer
ellipse. Thus
u 2Oberth GMe
Oberth 

2
Re
u 2Oberth
2

u 2esc
2

uT
2
2
uOberth  u 2esc  uT 2
 uesc  uT
uOberth  uOberth  uMars
Complications: 1) Atmospheric drag
2) Orbit desired @ Mars
EXTROVERT
Space Propulsion 03
Faster Trajectories for Human-Carrying Missions
Four impulsive burns:
1. Escape from Earth towards Mars with an excess over the
Hohman transfer velocity.
2. Increment to get captured into Mars orbit: Most trajectories put
the vehicle in Mars orbit, moving slower around the Sun than
Mars – need a burn to get captured.
3. Opposite of the above to escape Mars into a transfer trajectory
towards Earth
4. Burn to decelerate to a safe Earth re-entry velocity.
Months
Total delta-v,
fps
5
120,000
8
55,000
12
25,000
15
19,000
EXTROVERT
Space Propulsion 03
Values for Sample Missions: Garrison & Stuhler
V m/s
Earth- LEO (270km above Earth’s surface)
7600
LEO – GEO (42,227 km radius)
4200
LEO – Earth escape
3200
LEO – Lunar Orbit (7 days)
3900
LEO – Mars orbit (0.7 yrs)
5700
LEO – Mars orbit (40 days)
85000
LEO – Neptune Orbit (50 yr)
13,400
LEO – Neptune Orbit (29.9 yr)
8,700
LEO – Solar escape
70,000
LEO – 1000 AU (50yr)
LEO – Alpha Centauri (50yr)
142,000
30 Million
1 Astronautical Unit = mean distance from Earth to Sun = 149,558,000km
Impulsive burns assumed in all the above.
EXTROVERT
Space Propulsion 03
Spiral Transfer: Low, Continuous Thrust
Delta v required is greater than that for Hohman transfer, but a small engine can be used. Also,
possibility of using solar energy as the heat source. Thrust applied along the circumferential directio
is then given by
 d 2q
 dr  dq  
T  m  r 2  2  
 
 dt
dt
dt




or
Tr d  2 dq 
 r
m dt  dt 
1/ 2

d  3 d 2
 

2

d 
d

where
r

r0
 

with initial conditions at
GM
r0
3
t
 0
dq
d
0
1
d
d
 1
Tr02
T


GMm mg
Solution shows that the delta v required with a very small thrust-to-EarthWeight ratio is about 2.4
times the impulsive-burn delta v. As thrust-to-weight approaches 1, this comes down close to 1.
EXTROVERT
Space Propulsion 03
Spiral Transfer: Earth-Moon
“SMART-1 to continue
alternating between long
and short thrust arcs until
15 Nov lunar capture: orbital
period 4 days, 19 hrs;
apogee altitude has
increased from 35,880 km to
200,000 km; lunar
commissioning, science
operations begin in mid-Jan
2005”
http://sci.esa.int/
EXTROVERT
Gravity Assist Maneuvers
http://science.howstuffworks.com/framed.htm?
parent=question102.htm&url=http://www.jpl.nasa.gov
/basics/bsf4-1.html
Vehicle approaches another planet with velocity
greater than escape velocity for that planet.
Planet’s gravity adds momentum to the vehicle
in desired direction.
With respect to the planet, vehicle gains speed on
approach and decelerates as it moves away.
But when the planet’s velocity is considered, there
is net gain in angular momentum with respect
to the Sun.
Space Propulsion 03
EXTROVERT
Space Propulsion 03
EXTROVERT
Space Propulsion 03
Orbital Elements
a: semi-major axis.
e: eccentricity .
 1  


e     v 2   r  r  v v 
r

   
i: inclination of the angular momentum vector from the Z-axis. (Z is usually Earth’s axis of rotation.
W: right ascension of the ascending node: Angle from the vernal equinox to the “ascending node.”
“node” : one of two points where the satellite orbit intersects the X-Y (equatorial) plane.
Ascending node is the point where the satellite passes through the equatorial plane moving
from south to north.
Right ascension is measured in degrees of right-handed rotation about the pole (Z axis).
w:
argument of perigee. The angle from the ascending node to the eccentricity vector,
measured in degrees in the direction of the satellite’s position. The eccentricity vector
points from the center of the Earth to the perigee with magnitude equal to the eccentricity
of the orbit, in degrees.
: “true anomaly”. Angle from the eccentricity vector to satellite’s position vector, measured in
the direction of the satellite motion. Can also use the time since perigee passage, in degrees.