978-0078112829 Chapter 12 Solution Manual Part 1

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.1 In Figure 12.1, the radius at perigee O is 7000 km and the speed is

v0=

m

r0

. The initial

mass of the spacecraft is

m0=2000 kg

and the thrust of the propulsion system is 0.5 kN. Using Cowell’s

method and ode45, find the perigee and the eccentricity of the osculating orbits at the following times af-

ter

t0

: (a) 1 h (b) 1.2 h (c) 1.4 h (d) 1.6 h.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.2 Repeat Problem 12.1 using ode45 and Encke’s method.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

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Problem 12.4 Find the zeros of each of the Legendre polynomials in Figure 12.7, namely

P

2x

 1

23x21

  P

3x

 1

25x33x

 

P

4x

 1

835x430x23

  P

5x

 1

863x570x315x

 

P

6x

 1

16 231x6315x4105x25

  P

7x

 1

16 429x7693x5315x335x

 

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

0.981781669965367

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Problem 12.5 Use Rodrigues’ formula to calculate Legendre polynomials

P

8x

( )

and

P

9x

( )

.

P

9x

 1

299!

d9

dx9x21

 

9

1

299!

d9

dx9x18 9x16 36x14 84x12 126x10 126x884x636x49x21

 

1

299!

18!

9! x9916!

7! x736 14!

5! x584 12!

3! x3126 10!x











1

299!

17,643, 225, 600x937, 362,124, 800x726,153, 487, 360x56,706,022, 400x3457, 228, 800x

 

1

9! 34, 459, 425x972,972,900x751,081,030x513,097,700x3893,025x

 

1

128 2835 34, 459, 425x972, 972,900x751,081,030x513,097,700x3893,025x

 

P

9x

 1

128 12,155x925,740x718,018x54620x3315x

 

Problem 12.6 Plot and find the zeros of the Legendre polynomials found in Problem 12.5.

Problem 12.7 Verify that the third zonal harmonic of the perturbing gravitational potential is

F

=J3

2

m

r

R

r

æ

è

çö

ø

÷

3

5cos3

f

–3cos

f

( )

3J3

2

r









Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.8 Show that the perturbing acceleration

p= – Ñ

F

due to the

J3

zonal harmonic is

p1

2

J3



R3

r55x

r7z

r











3

3z

r













ˆ

i5y

r7z

r











3

3z

r













ˆ

j335

3

z

r











4

10 z

r











2

1













ˆ

k





















where

z r =cos

f

.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12





x  1

2

J3



R3

r5

x

r35 z3

r315 z

r





























J3



R3

y









Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.9 For the orbit of Example 12.2, use Cowell’s method to determine the

J3

effect on the or-

bital parameters

W

,

w

,

h

,

e

and

i

for 48 hours after the initial epoch.