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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

10.3Introduction10.3.1 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

V 2=μ (2r−1a) ( 10-1)

1010

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

V 2

2− μr=−μ2a

( 10-2 )

Where it is evident that

Kinetic energy + potential energy = total energySatellite mass satellite mass satellite mass

Note that total energy/satellite mass is dependent only on a. as a increases, energy increases.

Figure 10-1. Conservation of energy relates r, V and a.

10.4Basic Orbital ManeuversOrbital 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.

Figure 10-2. Basic orbital maneuver.

10.4.1 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.

Figure 10-3. Delta-V Budget.

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From rocket theory,

Figure 10-4. Delta-V Budget.

F= ˙mνe=m I sp g0

M dVdt

=−dMdt

νe

dVνe

=−dMM

∆V=−ν e ln(M f

M i)=νe ln ¿)

( 10-3 )

∆V=−ν e ln(M f

M i)=νe ln ¿

Where I sp=specific impulse = thrust/rate of fuel consumption

M i= spacecraft initial mass M f= spacecraft final mass M p=propellant mass used g0=9.81m/s²

∆V=I sp g0 ln (1+M p

M f)

M p=M f [exp( ∆VI spg0 )−1] ( 10-4)

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¿M i[1−exp(−∆VI sp g0 )]

10.5Satellite LaunchHigh-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.

10.5.1 Single Coplanar Maneuver

A single burn can transfer a circular orbit into an elliptic orbit

DVVcircVp

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Vcirc=√ μR ,V p=√ 2 μR −μa ( 10-5)

Figure 10-6.Single coplanar maneuver.

10.5.2 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.

DV

DVa V

2V

1

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∆V=V 12+V 2

2−V 1V 2 cosα ( 10-6 )

10.5.3 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.

∆V 1=V p ,t−V 1=√ 2μr1 − μa−√ μr 1 (10-7)

∆V 2=V 2−V a , t=√ μr2−√ 2 μr2 −μa

(10-8)

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¿F=12Pt=π √ a3μ (10-9)

r p , t=r1 , ra , t=r2 (10-10)

Figure 10-8.Hohmann transfer.

10.5.4 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.

10.5.5 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.

10.5.6 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

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through the angle θ. The resulting velocity triangle is isosceles, as observed in Fig. from the right triangle,

∆V=2V sin θ2

(10-11)

|V i|=|V f|=V (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.

Figure 10-11.Transfer of orbits from 1 to 2; vector difference ΔV=V2-V1.

Figure 10-12.Velocity vector triangle for circular orbit plane change.

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10.5.7 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)

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.

∆V=∆V 1+∆V 2=V f (1+2sin θ2 )−V i (10-13)

(10-14)

Figure 10-13.Combined maneuver.

DV1

Vi

|Vf| DV2

|Vf|

Vf

q

DVc = DV1 + DV2

Vi

Vf

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∆V c=√V i2+V f2−2V iV f cosθ

Problems1. Given two circular orbits:

Initial Finalr1=6660km(h1=282km) rf=133200km i=30 deg 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:

b) Hohmann transfer and then plane change:

υ

c) Hohmann transfer with plane change at apogee in a vectorial combination (two impulses):υ

Descending node

Equator

Vc1

Vc1 ΔV1

ΔV2

30˚ Apoge

e

VATR ΔV3

Equator30˚

Vc1

ΔV1

ΔV3

ΔV2

VATR

Ω30˚

Vcf

30˚Vc1

ΔV1

Ω

VATR

Vcf

ΔV2

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d) Bi-elliptic transfer with vectorial plane change at rt=266400km (three impulses):

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).

30˚

ΔV1

Vc1 30˚

ΔV2

VATR2

VATR1Ω

VPTR2

υ

ΔV3

Vcf

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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 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).

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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.

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)?

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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?

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.

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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

ΔV1 and ΔV2 in meters per second. For the moon, Vc0 =√ μr 0=1679 m/s.

8.4 ReferencesChobotov, V. (2002). Orbital Mechanics. Reston, Virginia, American Institute of Aeronautics and Astronautics, Inc.