978-0078112829 Chapter 6 Solution Manual Part 3

B

1

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 6

vtotal 2vB3.3054 km s

Problem 6.24 (a) With a single delta-v maneuver, the earth orbit of a satellite is to be changed from a

circle of radius 15 000 km to a collinear ellipse with perigee altitude of 500 km and apogee radius of 22000

km. Calculate the magnitude of the required delta-v and the change in the flight path angle



.

(b) What is the minimum total delta-v if the orbit change is accomplished instead by a Hohmann transfer?

฀

15 000 km

22 000 km 6878 km

2

A

B C D E



2

vA1

v

A2



v

Common apse

line

Earth

3

1

4

rAh22



1

1e2cos



A

15000 64 6302

398 600

1

10.5237 cos



A





A125.1

vA2

h2

rA

64 630

15000 4.309 km s

vA2

r



h2

e2sin



A398 600

64 630 0.5237 sin 125.1  2.641km s

vA2vA2



2vA2

r

24.30922.64125.054 km s



A2tan 1vA2

r

vA2



tan 12.641

4.309 0.5499 



A231.51





A



A2



A131.51  031.51

vAvA1

2vA2

22vA1vA2cos 



A5.15525.054225.1555.054 cos 



A2.773 km s

T

(b)

Try Hohmann transfer (orbit 3) from point E on orbit 1 to point B on orbit 2.

h32



r

Er

B

r

Er

B

2398 600 15000 22000

15000 22000 84 320 km 2s

vE1vA15.155 km s

vE3h3

r

E

84 320

15 000 5.621 km s

vB3h3

r

B

84 320

22000 3.833 km s

vB2h2

r

B

64 630

22000 2.938 km s

vtotal vE3vE1vB2vB30.4665 0.985 1.362 km s

Try Hohmann transfer (orbit 4) from point C on orbit 1 to point D on orbit 2.

฀

h42



rCrD

rCr

D

2398600 15000 6878

15000 6878 61310 km 2s

vC1vA15.155 km s

vC4h4

r

C

61 310

15000 4.088 km s

v1.362 km s

Problem 6.25 An earth satellite has a perigee altitude of 1270 km and a perigee speed of 9 km/s. It is

required to change its orbital eccentricity to 0.4, without rotating the apse line, by a delta-v maneuver at



100

. Calculate the magnitude of the required



v

and the change in flight path angle



.

฀

v2v22v2r

25.31122.24825.767 km s



2tan 1v2r

v2

tan 12.248

5.767 22.94









2



122.94  31.13  8.181

vv12v222v1v2cos 



6.11425.767226.114 5.767 cos 8.181

  0.9155km s

Problem 6.26 The velocities at points A and B on orbits 1, 2 and 3, respectively, are (relative to the

perifocal frame)

vA

1 3.7730 ˆ

p6.5351ˆ

q (km/ s)

vA2 3.2675 ˆ

p8.1749 ˆ

q (km/ s)

vB2 3.2675 ˆ

p3.1442 ˆ

q (km/ s)

vB3 2.6679 ˆ

p4.6210 ˆ

q (km/ s)

Calculate the total

v

for a transfer from orbit 1 to orbit 3 by means of orbit 2.

vtotal 3.3098 km/ s

Problem 6.27 Trajectories 1 and 2 are ellipses with eccentricity 0.4 and the same angular momentum h.

Their speed at B is v. Calculate, in terms of v, the

v

required at B to transfer from orbit 1 to orbit 2.



B

B2

B1 2 tan 1e

cos 



Bcos 2 tan 1e

 cos2tan 1e

 sin 2tan 1e

 

10.42v0.7428v

Problem 6.28 A satellite is in a circular earth orbit of altitude 400 km. Determine the new perigee and

apogee altitudes if the satellite’s onboard rocket

(a) Provides a delta-v in the tangential direction of 240 m/s.

(b) Provides a delta-v in the radial (outward) direction of 240 m/s.

vr



hesin







r

0

e1cos2







r

0

e

za6997.0 6378 618.98km

Problem 6.29 At point A on its earth orbit, the radius, speed and flight path angle of a satellite are

rA12 756 km

,

vA6.5992 km/ s

and



A20

. At point B, at which the true anomaly is 150°, an im-

pulsive maneuver causes



v 0.75820 km/ s

and



vr0

.

(a) What is the time of flight from A to B?

(b) What is the rotation of the apse line as a result of this maneuver?

B

150°

1

2





P1

P2

A

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 6

1h1

r

B

27 848.9 2.84043km s

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 6



Problem 6.30 A satellite is in elliptical orbit 1. Calculate the true anomaly



(relative to the apse line of

orbit 1) of an impulsive maneuver that rotates the apse line an angle



counterclockwise but leaves the

eccentricity and the angular momentum unchanged.













 2















2

Original apse

line

2

1





Problem 6.31 A satellite in orbit 1 undergoes a delta-v maneuver at perigee

P

1

such that the new orbit

2 has the same eccentricity

e

, but its apse line is rotated 90° clockwise from the original one. Calculate the

specific angular momentum of orbit 2 in terms of that of orbit 1 and the eccentricity

e

.

฀

1e

1

2

P

1

P

2

F

Problem 6.32 Calculate the delta-v required at A in orbit 1 for a single impulsive maneuver to rotate

the apse line 180° counterclockwise (to become orbit 2), but keep the eccentricity e and the angular mo-

mentum

h

the same.

Problem 6.33 Spacecraft A and B are in concentric, coplanar circular orbits 1 and 2, respectively. At the

instant shown, spacecraft A executes an impulsive delta-v maneuver to embark on orbit 3 in order to in-

tercept and rendezvous with spacecraft B in a time equal to the period of orbit 1. Calculate the total delta–

v required.

 12 470 ˆ

p5804.51ˆ

q km

 

% Problem 6_33

% ––——–––—–

clear

global mu

v2(1), v2(2), v2(3))



vA



vA2.16866 km s