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Orbital Maneuvers

How to transfer the satellite from an orbit to another
coplanar one?

Standard Hohmann transfer coplanar maneuvers.

Bi-elliptic Hohmann transfer coplanar maneuvers.

General coplanar transfer between circular orbits.

Simple plane change maneuvers.

Combined Speed and Plane Change Maneuvers.

Effect of the launch site latitude on the initial orbit
of the satellite after its launch.
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Transferring from an orbit to another coplanar one
The total energy of the
satellite in an orbit:
The energy of orbit 2
is higher than orbit 1
Transferring satellite from orbit 1 to orbit 2 or
vice versa needs firing at an intersection point
If orbits are tangent (the intersection point is typically at the
perigee or apogee points)  Tangential orbit maneuver
If orbits are not tangent but intersected, firing is executed at one of
the intersection points  Non-tangential orbit maneuver
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Transferring from an orbit to another coplanar one
There is no perfect circular orbit
 Near circular orbit (e is very small)
 Apogee and Perigee points are existed but difficult to be allocated
If orbit 1 is assumed to be a perfect circular orbit, the selected point
to be a perigee or apogee for firing will assign the direction of orbit 2
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Case1:Transferring satellite from orbit 1
to orbit 2 needs forward-firing (V12), in
the same direction of satellite flight.
Case2:Transferring satellite from orbit 2
to orbit 1 needs retro-firing (V21), in the
opposite direction of satellite flight.
Case 1 represents Speed Up to Slow Down
Case 2 represents Slow Down to Speed Up
It is easy to apply the same methodology if orbit 1 is also elliptical.
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Standard Hohmann Transfer Coplanar Maneuvers (1925)
Transferring between two circular orbits
in same plane and have the same focus
Elliptical Transfer Orbit
Tangent to inner orbit at perigee (rP=r1)
Tangent to outer orbit at apogee (rA=r2)
2-impulse maneuvers are needed in
one-half of Hohmann transfer ellipse
Case 1: Transferring from Orbit 1 to Orbit 2:

where
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Standard Hohmann Transfer Coplanar Maneuvers (1925)
Transferring between two circular orbits
in same plane and have the same focus
Elliptical Transfer Orbit
Tangent to inner orbit at perigee (rP=r1)
Tangent to outer orbit at apogee (rA=r2)
2-impulse maneuvers are needed in
one-half of Hohmann transfer ellipse
Case 2: Transferring from Orbit 2 to Orbit 1:
amount of Case 1
Same
but in reverse directions
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Real Case: Either or both of the inner & the outer orbits are ellipses
There are two possibilities for the
Hohmann transfer elliptical orbit
Orbit 3
Orbit 3’
Knowns
Comparison for min. V
Ex.:
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Bi-elliptic Hohmann Transfer Coplanar Maneuvers
The strategy depends on performing three impulse maneuvers using
two transfer elliptical orbits
Share inner and outer orbits
in focus and line of apsides
Their shared apogee
beyond the outer orbit
is
For Bi-elliptic strategy
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Comparison: Standard Vs. Bi-elliptic Strategies
For Standard strategy:
For Bi-elliptic strategy:
Three regions:
Reg. 1: (rC/rA) < 11.94 (Standard)
Reg. 2: 11.94 < (rC/rA) < 15 ( same)
 Large (rB) favor the Bi-elliptical
 Small (rB) favor the Standard
Reg. 3: (rC/rA) > 15 (Bi-elliptic)
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Time of Flight for Hohmann Transfer Techniques
Referring to Kepler’s third law:
For standard Hohmann transfer strategy:
For Bi-elliptic Hohmann transfer strategy:
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Launchers
Launchers are classified as:
Rockets (ex.: Arian, SpaceX)
Space transportation system (STS)
For high altitude orbits, the satellite reaches an initial LEO circular
orbit at about 300 Km called (Parking orbit) by the launcher
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GEO using Standard Hohmann Coplanar Strategy
One shot: Two Maneuvers in one-half of Hohmann transfer elliptical orbit, or
called “GTO”, its perigee altitude at 300 Km and apogee altitude at 36000 Km
Vperigee : Rocket Injection or Perigee Kick Motor (PKM)
Vapogee : Apogee Kick Motor (AKM)
Steps: Multiple successive transfer orbits by multiple kicks at the apogee
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General Coplanar Transfer between Circular Orbits
 The conditions are:
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General Coplanar Transfer between Circular Orbits
The conditions are:
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General Coplanar Transfer between Circular Orbits
Givens:
r1, r2 for inner and outer orbits
e, P for the transfer ellipse
 Calculate (atransfer) and (h)
&
&
&
Due to conservation law of (h):
 Calculate (1) and (2)
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General Coplanar Transfer between Circular Orbits
The same methodology will be
followed if the satellite will
move from orbit 2 to orbit 1
Same amount of Vtotal but
V1 and V2 will be in the
opposite directions
Hohmann transfer technique is
a special case, where 1= 2=0:
&
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Simple Plane Change Maneuvers
Simple plane change means changing the plane of
the orbit without changing its shape and size
The Key Factor is to perform the Firing at the
intersection point of the initial and the final orbits
such that the orbit rotation is pivoting around the
line connecting the burn point to Earth’s center.
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Simple Plane Change Maneuvers
Simple plane change means changing the plane of
the orbit without changing its shape and size
1st application: Changing only the inclination (i)
of the elliptical orbit while all the other orbital
parameters (, a, e, , n) remains the same.
 i 
2 n a (1  e Cos ) Sin  
 2 
Vi 
1  e 2 Cos (   )
 : Argument of perigee , a: semi-major axis
 : True anomaly, e: Eccentricity, n: Mean motion
Special Case (1): Circular orbit (e=0, a=r, =0, =0)
 i  where
 Vi  2 Vinitial Sin  
Vinitial  V final V Asc. / Des.  n r 
 2 
Firing at Ascending or Descending node

r
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Simple Plane Change Maneuvers
1st application: Changing only the inclination (i) of the elliptical orbit while all
the other orbital parameters (, a, e, , n) remains the same.
 i 
2 n a (1  e Cos ) Sin  
 2 
Vi 
1  e 2 Cos (   )
Special Case (2):
Elliptical orbit such that:
Perigee is located as Descending node
Apogee is located as Ascending node
 =180, apogee= 180
Best Firing point for plane change
maneuver is at Apogee point (Lowest
speed) to minimize the required Vi
It is easy to prove that:
 i 
Vi  2 Vapogee Sin   where
 2 
 2
1 

Vinitial  V final  Vapogee  

 rapogee a 


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Combined Speed and Plane Change Maneuvers
Both the speed and plane change maneuvers will be done at the same firing point.
The best Firing point for plane change maneuver is the Apogee (Lowest speed) to
minimize the required Vi
Method (1):
Plane change followed by Speed change
 i 
V  2V1 Sin 
  V2  V1
 2 
The initial orbit and the final orbit may be both Circular or both Elliptical or one
of them Circular whereas the other Elliptical. So, V1 and V2 at apogee will be as:
Vapogee 

rapogee
 2
1
(Circular ) & Vapogee   
  ( Elliptical )
 rapogee a 


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Combined Speed and Plane Change Maneuvers
Both the speed and plane change maneuvers will be done at the same firing point.
The best Firing point for plane change maneuver is the Apogee (Lowest speed) to
minimize the required Vi
Method (2):
Speed change followed by Plane change
 i 
V  V2  V1  2V2 Sin  
 2 
The initial orbit and the final orbit may be both Circular or both Elliptical or one
of them Circular whereas the other Elliptical. So, V1 and V2 at apogee will be as:
Vapogee 

rapogee
 2
1
(Circular ) & Vapogee   
  ( Elliptical )
 rapogee a 


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Combined Speed and Plane Change Maneuvers
Both the speed and plane change maneuvers will be done at the same firing point.
The best Firing point for plane change maneuver is the Apogee (Lowest speed) to
minimize the required Vi
Method (3):
Speed change accompanied by Plane
change (Most Efficient)
V  V12  V22  2V1V2 Cos i
The initial orbit and the final orbit may be both Circular or both Elliptical or one
of them Circular whereas the other Elliptical. So, V1 and V2 at apogee will be as:
Vapogee 

rapogee
 2
1
(Circular ) & Vapogee   
  ( Elliptical )
 rapogee a 


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Combined Speed and Plane Change Maneuvers
Both the speed and plane change maneuvers will be done at the same firing point.
The best Firing point for plane change maneuver is the Apogee (Lowest speed) to
minimize the required Vi
Method (3):
Speed change accompanied by Plane
change (Most Efficient)
V  V12  V22  2V1V2 Cos i
For V1 = V2 , the previous Equation reduces again to the simple plane change Equation
(change only the inclination at apogee while preserving other orbital parameters):
 i 
Vi  2 Vapogee Sin  
 2 
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Another Application for Simple Plane Change Maneuvers
2nd application:
For Circular orbits: changing “Right ascension
of ascending node ()” of the orbital plane.
Firing at North or South Pole
where: ( ) is the plane change angle.
This type of maneuvers is used to change
( ) even if the orbit is not polar (i  90).
Ex.: The Sun-synchronous orbit needs
d/dt=0.9856/day
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Mass of Consumed Propellant in Hohmann Maneuvers
Satellite Fuel Consumption
maneuvers (Vtotal):
in Hohmann transfer coplanar
(Sec.)
ISP represents fuel weight consumed per second to achieve a certain thrust.
As ISP of the satellite propulsion system increases, the consumption rate of
fuel weight and consequently, the required amount of fuel, decreases.
The reduction rate of the satellite mass with time (dm/dt) equals the rate
of the exhaust mass flow out from the nozzle ( ).
Sea-level gravitational acceleration go=9.81 m/sec2
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Launch Site
(isatellite) in the initial orbit (immediately after launch) depends on , Az
of the trajectory path
Napier’s rule solves the Right spherical triangle: Cos(i)=Cos() Sin(Az)
Satellite launch is preferred to be in prograde orbit (easterly direction)
0<i<90º
 Cos(i) = +ve
,
-90º90º
 Cos() = +ve
 0<AZ<180º
To have min. (isatellite)
for fixed ():
Cos(i) =max.
Sin(Az) =max.
Az =90º
min. isatellite = 
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Launch Site
For GEO satellites (the required inclination i  0):
In case of selecting a launch site with a latitude as near as possible to
the equator  The min. isatellite (after Launch) =site = very small value
(For example: Kourou site, site = 5N, which launches Arian Rockets)
By executing a small South maneuver,
 the satellite reaches the required
inclination of the GEO (i  0)
 little amount of the fuel consumption
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Satellite Injection in Space
The launcher stacks the satellites on top of
each other with the use of a tandem adaptor
(acceleration force)
Clamp-bands
Spacecraft is ejected by compressed springs
The timing is automatically adjusted
between ejections to avoid collision
Many launchers spin up the satellites before ejection using a spin table
equipped by roller bearing to provide gyroscopic stability
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Rocket Engine Architecture
Fuel & Oxidizer propellant tanks
Turbo-pump engine pumps the propellants
to the thrust chamber
Injectors diffuse the liquids to a fine mist
Ignition occurs just diffused fuel & oxidizer
mix together, yielding the desired rocket
thrust
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Thank you
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