978-0078112829 Chapter 6 Solution Manual Part 2

Problem 6.14 The Space Shuttle was launched on a fifteen-day mission. There were four orbits after

injection, all of them at 39° inclination.

Orbit 1: 302 by 296 km.

Orbit 2 (day 11): 291 by 259 km.

Orbit 3 (day 12): 259 km circular.

Orbit 4 (day 13): 255 by 194 km.

Calculate the total delta-v, which should be as small as possible, assuming Hohmann transfers.

h22



r

apogee2rperigee2

vperigee2h2

฀

rperigee2

6637 7.759 km s

v3



r3

398 600

6637 7.750 km s

h42



r

apogee4rperigee4

r

apogee4rperigee4

2398 600 6633 6572

6633 6572 51300 km 2s

vapogee4h4

r

apogee4

51300

6633 7.734 km s

vperigee4h4

rperigee4

51300

6572 7.806 km s

vtotal  v12  v23  v34 0.01393 0.009313 0.02026 0.04351km s

Problem 6.15 Calculate the total delta-v required for a Hohmann transfer from a circular orbit of radi-

us r to a circular orbit of radius 12r.

A

B

3

1

2

12r

r

1vB3vB20.3587

r0.1754

r0.5342

r

Problem 6.16 A spacecraft in circular orbit 1 of radius r leaves for infinity on parabolic trajectory 2 and

returns from infinity on a parabolic trajectory 3 to a circular orbit 4 of radius 12r. Find the total delta–v

required for this non-Hohmann orbit change maneuver.

1vB4vB30.4142

r0.1196

r0.5338

r

Problem 6.17 A spacecraft is in a 300 km circular earth orbit. Calculate

(a) the total delta-v required for the bielliptical transfer to a 3000 km altitude coplanar circular or-

bit shown, and

(b) the total transfer time.

฀

r

B

12 402 4.743 km s

Orbit 3:

h32



r

Br

C

r

Br

C

2398 600 12 402 9378

12 402 9378 65 250 km 2s

vB3h3

r

B

65 250

12 402 5.261km s

300 km

e = 0.3

1

4

3000 km

2

3

AB C



vC



vB



vA

2T

1T2

Problem 6.18 Verify Equations 6.4.

1h1

rA

A

BC

1

2

3

4

rA

rC

rB

F

D

5

r

C

r

C

rAr

C

rA

r

C

rAr

C

1

 



1



 1







v0

Problem 6.19 The space station and spacecraft A and B are all in the same circular earth orbit of 350 km

altitude. Spacecraft A is 600 km behind the space station and Spacecraft B is 600 km ahead of the space

station. At the same instant, both spacecraft apply a



v

so as to arrive at the space station in one revolu-

tion of their phasing orbits.

(a) Calculate the times required for each spacecraft to reach the space station.

(b) Calculate the total delta-v requirement for each spacecraft.

฀



฀

Problem 6.20 Satellites A and B are in the same circular orbit of radius r. B is 180° ahead of A. Calculate

the semimajor axis of a phasing orbit in which A will rendezvous with B after just one revolution in the

phasing orbit.

a3/ 2 1

2r3/ 2

a1

2r3/ 2











2/ 3

 a0.63r

A

B

r

F

Problem 6.21 Two spacecraft are in the same elliptical earth orbit with perigee radius 8000 km and ap-

ogee radius 13 000 km. Spacecraft 1 is at perigee and spacecraft 2 is 30° ahead. Calculate the total delta–v

required for spacecraft 1 to intercept and rendezvous with spacecraft 2 when spacecraft 2 has traveled

60°.

฀

P

C

D

30°

60°

Spacecraft 1

Spacecraft 2

intercept

8000 km

13 000 km

A

1

2

EDtan 11e1

1e1

tan



D

2









tan 110.2381

10.2381 tan 90

2









1.330 rad

% Problem 6_21

% ~~~~~~~~~~~~

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 6

Problem 6.22 At the instant shown, spacecraft

S1

is at point A of circular orbit 1 and spacecraft

S2

is at

point B of circular orbit 2. At that instant,

S1

executes a Hohmann transfer so as to arrive at point C of

orbit 2. After arriving at C,

S1

immediately executes a phasing maneuver in order to rendezvous with

S2

after one revolution of its phasing orbit. What is the total delta-v requirement?

S1

2







2







vtotal 2.1594 km s

Problem 6.23 Spacecraft B and C, which are in the same elliptical earth orbit 1, are located at the true

anomalies shown. At this instant, spacecraft B executes a phasing maneuver so as to rendezvous with

spacecraft C after one revolution of its phasing orbit 2. Calculate the total delta-v required. Note that the

apse line of orbit 2 is at 45° to that of orbit 1.

8100 km

18 900 km

B

C

Earth

1

2

Phasing

orbit

Ap se line

150°

45°

Ap se line

of orbit 2

of orbit 1

1vB

1

B

1