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9 Orbital Maneuvers
Philosophy is such an impertinently litigious lady that a man had as good be engaged
in lawsuits as to have to do with her.
Isaac Newton in a letter to his friend Edmund Halley, June 20, 1687
10.1 Introduction
10.1.1 Orbital Energy
Spacecraft is not inserted in an orbit to stay forever! A spacecraft may need
to change its orbit once or more during its life time due to many reasons. A
launch vehicle may insert a geostationary (GEO) satellite into an initial low
Earth orbit (LEO) which is much lower than the final operational orbit. Then,
the satellite should transfer from the initial orbit to its final orbit. Another
need may arise if a surveillance satellite has to change its orbit in order to
track a new target. Interplanetary missions usually require many orbit
transfers until the spacecraft is inserted into the operational orbit or to use
the same spacecraft to accomplish more than one mission. At the satellite
end of life (EOF), the satellite may be kicked out of its orbit whether to
reenter the Earth’s atmosphere or to rest in a graveyard orbit.
Any analysis of orbital maneuvers, i.e., the transfer of a satellite from one
orbit to another by means of a change in velocity, logically begins with the
energy as
2 1
𝑉 2 = 𝜇( − )
𝑟 𝑎
( 10-1)
Sir Isaac Newton (1643-1727). English
Physicist, Astronomer and Mathematician
who described universal gravitation and
the three laws of motion, laying the
groundwork for classical mechanics,
which dominated the scientific view of the
physical Universe for the next three
centuries and is the basis for modern
engineering.
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Where V is the magnitude of the orbital velocity at some point, r the
magnitude of the radius from the focus to that point, a the semimajor axis
of the orbit, and μ the gravitational constant of the attracting body. Fig.
illustrates r, V, and a .
Equation can be rearranged as
𝑉2 𝜇
𝜇
− =−
2
𝑟
2𝑎
( 10-2 )
Where it is evident that
Kinetic energy + potential energy = total energy
Satellite mass
satellite mass
satellite mass
Note that total energy/satellite mass is dependent only on a. as a increases,
energy increases.
Apogee
V
2a
r
Perigee
Figure 10-1. Conservation of energy relates r, V and a.
10.2 Basic Orbital Maneuvers
Orbital maneuvers are based on the principle that an orbit is uniquely
determined by the position and vector at any point. Conversely, changing
the velocity vector at any point instantly transforms the trajectory to a new
one corresponding to the new velocity vector. Any conic orbit can be
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transformed into another conic orbit by changing the spacecraft velocity
vector.
2
Figure 10-2. Basic orbital maneuver.
10.2.1 Delta–V Budget
Orbital transfers are usually achieved using the propulsion system onboard
the spacecraft. Since the propellant mass on board is limited, it is very
crucial for mission planning to estimate the propellant required for every
transfer. The overall need for propulsion is usually expressed in terms of
spacecraft total velocity change, or DV (Delta-V) budget. We assume the
propulsion is applied impulsively, i.e. the velocity change will be acquired
instantaneously. This assumption is reasonably valid for high-thrust
propulsion.
V+DV
V
V
(3)
(2)
Figure 10-3. Delta-V Budget.
(1)
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From rocket theory,
Figure 10-4. Delta-V Budget.
𝐹 = 𝑚𝜈̇ 𝑒 = 𝑚̇𝐼𝑠𝑝 𝑔0
𝑀
𝑑𝑉
𝑑𝑀
=−
𝜈
𝑑𝑡
𝑑𝑡 𝑒
𝑑𝑉
𝑑𝑀
=−
𝜈𝑒
𝑀
( 10-3 )
𝑀
𝑀
∆𝑉 = −𝜈𝑒 ln ( 𝑀𝑓 ) = 𝜈𝑒 ln(𝑀 𝑖 )
𝑖
𝑓
M
M
∆V = -νe ln (Mf ) = νe ln(Mi
i
f
Where 𝐼𝑠𝑝 =specific impulse = thrust/rate of fuel consumption
𝑀𝑖 = spacecraft initial mass
𝑀𝑓 = spacecraft final mass
𝑀𝑝 =propellant mass used
𝑔0 =9.81m/s²
𝑀𝑝
∆𝑉 = 𝐼𝑠𝑝 𝑔0 ln(1 + 𝑀 )
𝑓
𝑀𝑝 = 𝑀𝑓 [𝑒𝑥𝑝 (𝐼
∆𝑉
𝑠𝑝 𝑔0
) − 1]
−∆𝑉
= 𝑀𝑖 [1 − 𝑒𝑥𝑝 (𝐼
𝑠𝑝 𝑔0
)]
( 10-4)
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10.3 Satellite Launch
High-altitudes (above 200 km) may be achieved through two burns separated
by coasting phase. The first burn is nearly vertical and places the satellite into an
elliptic orbit with apogee at the final orbit radius. The satellite then coast (no
burn) until it reaches the apogee. A second burn can be used to insert the
satellite into its final LEO orbit.
Figure 10-5.Satellite launch.
10.3.1 Single Coplanar Maneuver
A single burn can transfer a circular orbit into an elliptic orbit
𝜇
2𝜇 𝜇
𝑉𝑐𝑖𝑟𝑐 = √ , 𝑉𝑝 = √ −
𝑅
𝑅 𝑎
( 10-5)
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Figure 10-6.Single coplanar maneuver.
10.3.2 General Coplanar Maneuver
A single burn can transfer a satellite from some initial orbit to another
intersecting coplanar orbit by changing the velocity from V1 on the initial
orbit to V2 on the final orbit. The velocity on the final orbit will be the vector
sum of the initial velocity vector and the velocity change vector.
Figure 10-7.General coplanar maneuver.
∆𝑉 = 𝑉1 2 + 𝑉2 2 − 𝑉1 𝑉2 cos 𝛼
( 10-6 )
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10.3.3 Hohmann Transfer
The Hohmann’s transfer is the minimum two-impulse transfer between
coplanar circular orbits. It can be used to transfer a satellite between two
nonintersecting orbits (Walters Hohmann 1925). As illustrated in Fig. , the
Hohmann transfer is quite a simple maneuver. This maneuver employs an
intermediate elliptic orbit which is tangent to both initial and final orbits at
their apsides. To accomplish the transfer, two burns are needed. The first
burn will insert the S/C into the transfer orbit, where it will coast from
periapsis to apoapsis. At apoapsis, the second burn is applied to insert S/C
into final orbit.
A tangential ΔV1 is applied to the circular orbit velocity. The magnitude of
ΔV1 is determined by the requirement that the apogee radius of the
resulting transfer ellipse must equal the radius of the final circular orbit.
When the satellite reaches apogee of the transfer orbit, another ΔV must be
added or the satellite will remain in the transfer ellipse. This ΔV is the
difference between the apogee velocity in the transfer orbit and the circular
orbit velocity in the final orbit. After ΔV2 has been applied, the satellite is in
the final orbit, and the transfer has been completed.
2𝜇 𝜇
𝜇
∆𝑉1 = 𝑉𝑝,𝑡 − 𝑉1 = √ − − √
𝑟1 𝑎
𝑟1
(10-7)
𝜇
2𝜇 𝜇
−√ −
𝑟2
𝑟2 𝑎
(10-8)
∆𝑉2 = 𝑉2 − 𝑉𝑎,𝑡 = √
𝑇𝑂𝐹 =
1
𝑎3
𝑃𝑡 = 𝜋√
2
𝜇
(10-9)
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𝑟𝑝,𝑡 = 𝑟1 , 𝑟𝑎,𝑡 = 𝑟2
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(10-10)
Figure 10-8.Hohmann transfer.
10.3.4 General Coplanar Transfer between Circular Orbits
Transfer between circular coplanar orbits only requires that the transfer
orbit intersect or at least be tangent to both of the circular orbits. It is
obvious that the periapsis radius of the transfer orbit must be equal to or
less than the radius of the inner orbit and the apoapsis radius must be equal
to or exceed the radius of the outer orbit if the transfer orbit is to touch
both circular orbits. This condition can be expressed mathematically as
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Figure 10-9.General coplanar transfer between circular orbits.
10.3.5 Out-of-Plane Orbit Change
A velocity change which lies in the plane of the orbit can change its size or
shape, or rotate the line of apsides. To change the orientation of the orbit
plane in space, DV should have a component which is perpendicular to the
orbital plane.
Figure 10-10.Out-of-plane orbit change.
10.3.6 Simple Plane Change
Orbital maneuvers are characterized by a change in orbital velocity. If a
velocity increment ΔV, which is a vector, is added to a satellite velocity, V1,
also a vector, then a new satellite velocity V2, results. A finite ΔV which
does not change the speed and flight-path angle of the satellite, will only
change the plane of the orbit. Thus, the maneuver is called simple plane
change.
A plane change maneuver rotates the orbit plane in inertial space while
holding the size and shape, a and e fixed. During a circular orbit plane
change, through an angle θ, the ΔV is applied so as to rotate the velocity, V
through the angle θ. The resulting velocity triangle is isosceles, as observed
in Fig. from the right triangle,
∆𝑉 = 2𝑉 sin
𝜃
2
(10-11)
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|𝑉𝑖 | = |𝑉𝑓 | = 𝑉
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(10-12)
An example of a simple plane change would be changing an inclined orbit to
an equatorial orbit by changing the orbit plane through an angle, q. To
equatorialize an orbit (i.e. make its i = 0), ΔV must be applied at one of the
nodes. Large plane changes are prohibitively expensive in terms of ΔV.
V1
ΔV
P
V2
Orbit 1
Orbit 2
0
Figure 10-11.Transfer of orbits from 1 to 2; vector difference ΔV=V2-V1.
V
ΔV
θ
V
Figure 10-12.Velocity vector triangle for circular orbit plane change.
10.3.7 Combined Maneuver
Frequently, the spacecraft orbit needs to be raised as well as titled. Two
orbital transfers may then be applied:
-A simple coplanar maneuver to raise the orbit (change radius), then
-A simple plane change to tilt the orbit (change inclination)
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Alternatively, these two maneuvers can be combined in one maneuver to
perform both tasks in one burn which is more economic (require less fuel)
and faster.
Figure 10-13.Combined maneuver.
𝜃
∆𝑉 = ∆𝑉1 + ∆𝑉2 = 𝑉𝑓 (1 + 2 sin ) − 𝑉𝑖
2
(10-13)
∆𝑉𝑐 = √𝑉𝑖 2 + 𝑉𝑓 2 − 2𝑉𝑖 𝑉𝑓 cos 𝜃
(10-14)
Problems
1. Given two circular orbits:
Initial
Final
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r1=6660km(h1=282km)
i=30 deg
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rf=133200km
i=0 (equatorial)
Calculate the component and total ΔVs for the following transfer techniques
from the initial orbit to the final orbit:
a) Plane change and then Hohmann transfer:
Vc1
Descending
node
ΔV2
VATR
Equator
30˚
Vc1
Apogee
ΔV1
b) Hohmann transfer and then plane change:
υ
ΔV2
VATR
Equator
30˚
Vc1
ΔV3
30˚
Ω
ΔV1
Vcf
c) Hohmann transfer with plane change at apogee in a vectorial combination
(two impulses):
υ
30˚
Vc1
ΔV1
ΔV2
VATR
Ω
Vcf
d) Bi-elliptic transfer with vectorial plane change at rt=266400km (three
impulses):
υ
30˚
Vc1
ΔV1
ΔV3
ΔV2
VATR1
30˚
Ω
VATR2
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ΔV3
υ
e) Hohmann transfer with optimally split-plane changes (two impulses)
2. The sketch illustrates three circular orbits about the Earth. The radii, as show,
are 9, 16 and 25 Earth radii. Determine the characteristic velocity in meters per
second. (ΔVT=sum of ΔV) for a double Hohmann transfer from the inner orbit
to the outer orbit (A--B--C--D). Calculate ΔVT in meters per second for a
single Hohmann transfer (A--D). Finally, determine ΔVT for an
intermediate bi-elliptic transfer (A--B--E).
3. Given an elliptical orbit whose apogee radius rA=9r0 and perigee radius
rP=3r0 (where r0 is the radius of the assumed spherical Earth), compute
the velocity requirements for two modes of transfer from the surface of
the Earth to the ellipse. The first mode is an impulsive launch into a
bitangential transfer ellipse that is tangent to the Earth’s surface and to
the target ellipse at perigee of the target ellipse. At this point, the
vehicle impulsively achieves the target orbit. The second mode is via a
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bitangential transfer ellipse that is tangent to the Earth’s surface and the
target ellipse at its apogee.
a) Calculate four velocity increments in meters per second.
b) Determine the most economical mode.
4.
A satellite is in a circular polar orbit. If, at the ascending node, the
velocity vector is rotated counterclockwise 90 deg, what is the new orbit
inclination? If the rotation is clockwise 90 deg, what is the new i? if the
same rotations occur after the satellite has moved 60 deg and 90 deg
from the ascending node, what are the new inclinations?
5. Given a set of injection conditions corresponding to the sketch, determine the
true anomaly of the injection point as a function only of γ (and perhaps
constants), and determine the eccentricity of the resulting orbit as a function
only of γ (and perhaps constants).
6. A space vehicle at the South Pole is instantaneously launched, ΔV1, in a
horizontal direction into a parabolic orbit. When the vehicle crosses the
equator, point 2, a velocity increment ΔV2 is applied that instantaneously
places the vehicle into a polar, circular orbit. Assuming a spherical Earth radius
r0=6371km, determine the magnitudes of ΔV1, V2, Vc2, and ΔV2 in meters per
second, and determine the values of γ2 and α in degrees.
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7. An astronaut is heading east in a circular equatorial orbit about the Earth at an
altitude h=3r0. At 0˚ longitude, he applies a velocity increment ΔV1, which
places him in a polar orbit whose perigee grazes the Earth’s surface 180 deg
away in central angle on the equator.
a) What is the magnitude of ΔV1?
b) What is the angle between ΔV1 and the original circular orbit velocity?
c) What is the retro velocity increment ΔV2 at perigee that will reduce his
total velocity to zero (soft-land)?
8. A satellite is in a polar orbit (orbit 1 on the sketch) about a spherical Earth with
no atmosphere. Its perigee and apogee are in the equatorial plane. The perigee
altitude is 400 n.mi.; the apogee altitude is 2000 n.mi. Transfer from orbit 2 to
orbit 1 can occur in several ways. Determine the total ΔV for transfer via
circular orbit 3 from apogee to apogee. Determine the total ΔV for transfer via
circular orbit 4 from perigee to perigee. Determine the single ΔV at point X to
accomplish the transfer. Would the ΔV at point Y be identical in magnitude? In
direction?
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At an arbitrary point, 1, in an initial orbit i, a velocity increment ΔV is added
in the radial direction. A final orbit f is thus achieved. Compare the angular
moment h and the semilatus recta p in the two orbits. Determine the radius
in the final orbit at the point that is 180 deg in central angle away from
point 1.
Boris, a Russian cosmonaut, is in a circular equatorial orbit of radius
r=1.44r0 about the moon (see sketch). He decides to pay a surprise visit to
his American friends camped at the North Pole by transferring with ΔV1 into
a polar elliptical orbit whose pericenter is at the camp. When Bpris reaches
the camp, he retrofires with ΔV2 to reduce his total velocity to 0. Determine
𝜇
ΔV1 and ΔV2 in meters per second. For the moon, Vc0 =√𝑟0=1679 m/s.
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8.4 References
Chobotov, V. (2002). Orbital Mechanics. Reston, Virginia, American Institute
of Aeronautics and Astronautics, Inc.
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