978-0078112829 Chapter 12 Solution Manual Part 3

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.21 Verify that for unperturbed two-body motion,

d



dt h r2

.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.22 Verify that for unperturbed two-body motion,

dM dt =n

.

dt n

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.23 Verify that for unperturbed two-body motion,

dE dt =na r

.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.24 Show that

da

dt =2a2

hesin

q

pr+h2

m

rps

æ

è

ç

ö

ø

÷

.

hesin



rps







Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.25 Show that there is no time-averaged

J2

perturbation of the semimajor axis (

&

a0

).

>>

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.26 Show that

¶v2¶v=2v

.

v2v

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.27 Show that

¶riv

( )

¶v=r

.

vr

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.28 Show that

¶h¶v=h´r

( )

h

.

vhr

h

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.29 Show that

¶h2¶v=2h´r

.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.30 Find the Gauss variational equation for

drpdt

.

prpgˆ

pspgˆ

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.31 Find the Gauss variational equation for

dradt

.

dt h



21e

 2h21e

1e







cos



r

1e



















ps

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

Problem 12.32 Find the Gauss variational equations for a radial acceleration perturbation,

p=prˆ

r

.

dt   1

eh













rsin

htan ipwd

dt   h2



eh cos

Problem 12.33 Numerically integrate the Gauss planetary equations for a given perturbation p and set

of initial conditions.

Let us choose an earth orbit in which a constant tangential thrust is the perturbing force. As shown in

Equation 6.26, the perturbing acceleration in that case is

pT

m

v

where T is the rocket thrust, m is the spacecraft mass, v is the velocity and v is the speed. For this problem

let the state vector in the geocentric equatorial frame at time t = 0 be

r

07000ˆ

I (km)

v01.1



r0

cos 45ˆ

Jcos 45ˆ

K

  km s

Plot the variation of the six orbital elements

h

,

e

,



,



,

i

,



over the next two hours by integrating

Equations 12.84. Use MATLAB’s RKF integrator ode45.

Select the following spacecraft parameters:

Thrust = 0.5 kN.

Specific impulse = 300 s.

Initial mass = 2000 kg.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 12

–––——––––———––—––———––––—––———–————–––