Theory of Orbit Perturbations From Madrid Polytechnic

8/22/2019 Theory of Orbit Perturbations From Madrid Polytechnic

1/44

Basics of Orbital Mechanics II

Modeling the Space Environment

Manuel Ruiz Delgado

European Masters in Aeronautics and Space

E.T.S.I. Aeronauticos

Universidad Politecnica de Madrid

April 2008

Basics of Orbital Mechanics II p. 1/24


2/44

Basics of Orbital Mechanics II

Keplerian and Perturbed Motion

Magnitude of the Perturbations

Special Perturbations all, numerical

Enckes Method

Cowells Method

General Perturbations some, analytical, approximate

Osculating OrbitVariation of Parameters

Lagrange Equations potential

Gauss Equations potential & not potentialGeneral Perturbations: Analytical approx/Semianalytical

Numerical Integration



3/44

Keplerian and Perturbed Motion

r = G (M + m) r|r|3

Kepler Problem+

P1

m1 P2

m2

Perturbation

r

k=

ak

G (M + m)rk

|rk|3rp = G (M + m) rp|rp|3

+ ap

rp

rk

m

M

Usually, |ap| |ak| rp rk How small?



4/44

Perturbations (LEO)

1e008

1e006

0.0001

0.01

1

100

10000

1e+006

0 100 200 300 400 500 600 700 800 900

Acceleratio

n(m/s2)

Height (km)

Accelerations of the Satellite (BC=50)

Shuttle

ISS

KeplerJ2

C22Sun

MoonDrag (low)

Drag (high)Prad



5/44

Perturbations (GEO)

1e008

1e006

0.0001

0.01

1

100

10000

1e+006

0 5000 10000 15000 20000 25000 30000 35000 40000

Acceleratio

n(m/s2)

Height (km)

Accelerations of the Satellite (BC=50)

GEOGPS

KeplerJ2

C22Sun

MoonDrag (low)

Drag (high)Prad



6/44

Enckes Method

keple

ria

n

perturbe

d

rk

r

r0

v0

Epoch

rp

M

Compute only the difference r

rk = rk|rk|3rp = rp|rp|3

+ ap

r = rp rk |r| |rp|



7/44

Enckes Method

keple

ria

n

perturbe

d

rk

r

r0

v0

Epoch

rp

M



+ ap

r = rp rk |r| |rp|r = rp rk =

rp

|rp|3+

rk

|rk|3+ ap =



8/44

Enckes Method

keple

ria

n

perturbe

d

rk

r

r0

v0

Epoch

rp

M



+ ap


rp

|rp|3+

rk

|rk|3+ ap =

r = |rk|3r + |rk|3

1 |rk|3|rp|3

rp + ap



9/44

Enckes Method

About f(q), cf. Battin, p. 389 and 449

keple

ria

n

perturbe

d

rk

r

r0

v0

Epoch

rp

M



+ ap


rp

|rp|3+

rk

|rk|3+ ap =

r = |rk|3r + |rk|3

1 |rk|3|rp|3

rp + ap

1 |rk

|3

|rp|3 = f(q) = q3 + 3q+ q2

1 + (1 + q)3

2q =

r

(r

2rp)

rp rp


k h d


10/44

Enckes Method


keple

ria

n

perturbe

d

rk

r

r0

v0

Epoch

rp

M



+ ap


rp

|rp|3+

rk

|rk|3+ ap =

r = |rk|3r + |rk|3

1 |rk|3|rp|3

rp + ap

1 |rk

|3

|rp|3 = f(q) = q3 + 3q+ q2

1 + (1 + q)3

2q =

r

(r

2rp)

rp rp

r =

|rk|3 r

|rk|3 f(q) rp + ap


E k M h d


11/44

Enckes Method


keple

ri

an

Epoch2

perturbe

d

rp

M



+ ap

r

=rp rk |r| |rp|

r = rp rk = rp

|rp|3+

rk

|rk|3+ ap =

r = |rk|3r + |rk|3

1 |rk|3|rp|3

rp + ap

1 |rk

|3

|rp|3 = f(q) = q3 + 3q+ q2

1 + (1 + q)3

2q =

r

(r

2rp)

rp rp

r =

|rk|3 r

|rk|3 f(q) rp + ap

if r , rectify: r = 0rk1 rk2


L f P i i


12/44

Loss of Precision

REAL*4 = Single-Precision = 6-7 DIGITS

REAL*8 = Double-Precision = 15-16 DIGITS

0.100000000000000 E+00

+ 0.123456789012345 E-10

= 0.100000000000000 E+00

+ 0.000000000012345 E+00= 0.100000000012345 E+00

0.123456789012345 E+00

- 0.123456789000000 E+00

= 0.000000000012345 E+00

=0.123450000000000 E-10


L f P i i


13/44

Loss of Precision

1 |rk|3|rp|3

REAL*4 = Single-Precision = 6-7 DIGITS

REAL*8 = Double-Precision = 15-16 DIGITS

0.100000000000000 E+00

+ 0.123456789012345 E-10

= 0.100000000000000 E+00

+ 0.000000000012345 E+00= 0.100000000012345 E+00

0.123456789012345 E+00

- 0.123456789000000 E+00

= 0.000000000012345 E+00

=0.123450000000000 E-10


C ll F l ti


14/44

Cowells Formulation

Direct numerical integration of the equations

ODE: r = r

|r

|3

+ ap (r, r, t)

IC: t0, r0, r0 r = r (t, t0, r0, r0)

x =

x

yz

vxvyvz

x =

vx

vyvzx

y

z

=

vx

vyvz

r3 x + ax

r3 y + a

y r3 z+ az

x = f (x, t)


Osculating Orbit Variation of Parameters


15/44

Osculating Orbit - Variation of Parameters

perturbe

d

M

r0

v0

Epoch

Satellite in r0, v0 at Epoch t0

Follows perturbed trajectoryrp(t)




16/44


keple

rian

perturbe

d

M

r0

v0

Epoch



Osculating Orbit at r0, v0:

The Keplerian orbit followed by the satellite

if all perturbations become zero from thispoint on.




17/44


keple

rian

oscul

atin

g

perturbe

d

rp(t)

M

r0

v0

Epoch



Osculating Orbit at r0, v0:

The Keplerian orbit followed by the satellite

if all perturbations become zero from thispoint on.

Osculating orbit elements can be used as coordinates

r0,v0 , t0 i, , , a, e, , , t0rp(t),vp(t) , t

i(t), (t), (t), a(t), e(t), (t) , (t), t


Variation of Parameters: Fast/Slow variables


18/44

Variation of Parameters: Fast/Slow variables

M,

Fast Variables:

, M, , t

r(t) ,v(t)

Slow Variables:

i, , , a, e, (M0)


Variation of Parameters: Secular/Periodic


19/44

Variation of Parameters: Secular/Periodic

Secular

Secular + Long periodic

Secular + Long periodic + Short periodic

Short Orbital period

OrbitalParam

eter

t


Variation of Parameters - Lagrange


20/44

Variation of Parameters Lagrange

Variation of Parameters:

r = r|r|3

+ ap

r = r (i(t), (t), (t), a(t), e(t), t)

x = i, , , a, e, Tx = f (x, t)

Lagrange Planetary Equations: Conservative perturbations

ap = R R(i, , , a, e, M0) M0 = n x = i, , , a, e, M0Tx = f (x,

R)


Lagrange Planetary Equations


21/44


Singularities forlow eccentricity

or inclination

di

dt=

1

na2

1 e2 sin i

cos i

R

R

ddt = 1na2

1 e2 sin i Rid

dt

=

1 e2

na2 e

R

e cos i

na21 e2 sin iR

ida

dt=

2

na

R

M0

dedt

= 1 e2na2 e

RM0

1 e2na2 e

R

dM0

dt = 1

e2

na2 e

R

e 2

na

R

a




22/44


Singularities forlow eccentricity

or inclination

di

dt=

1

na2

1 e2 sin i

cos i

R

R

ddt = 1na2

1 e2 sin i Rid

dt

=

1 e2

na2

e

R

e cos i

na21 e2 sin iR

ida

dt=

2

na

R

M0

dedt

= 1 e2na2 e

RM0

1 e2na2 e

R

dM0

dt = 1

e2

na2 e

R

e 2

na

R

a M = n 1e2

na2 e

R

e 2

na

R

a aBasics of Orbital Mechanics II p. 13/24

Lagrange VOP: Kozais Method


23/44

Lagrange VOP: Kozai s Method

Separate disturbing potential R into constant/periodic, and orders ofmagnitude: R = R1 + R2 + R3 + R4

R1 =

3

2

J2 R2E

a3 13 12 sin2 i1 e21/2 R2 = 0R3 =

3

2

J3 R3E

a4sin i1

5

4sin2 i e 1 e

2

5/2

sin

R4 =3

2

J2 R2E

a3

ar

3 13

12

sin2 i

1

ra

3 1 e23/2+

+ 12

sin2 i cos2(+ )Only gravitational perturbations J2 (flattening) and J3 (pear-shape)

are included.


Lagrange VOP: Kozais Method (secular)


24/44

g g ( )

didt

= 38n J3 RE

p3 cos i 4 5sin2 i sin2 i cos da

dt= 0

d

dt= 3

2n J2

REp

2

cos i 38n J3

REp

3

15 sin2 i 4 e cot i sin

d

dt=

3

4n J2

REp

2 4 5sin2 i + 3

8n J3

REp

3 4 5sin2 i

sin2 i e2 cos2 ie sin i + 2 sin i 13 15 sin2 i e sinde

dt= 3

8n J3

REp

3

sin i 4 5sin2 i 1 e

2

cosdM

dt= n

1 +

3

2J2

REp

2 1 3

2sin2 i

1 e21/2

38n J3 RE

p3 sin i 4 5sin2 i 1 4e2 1 e21/2

esin


Gauss Planetary Equations


25/44

y q

Conservative and not conservative perturbationsUse the Orbital Frame for ap

ap = ar ur + a u + az uz

Peric

.

Sat.

h

ur

u

e

uN

i

i

x1y1

z1


Gauss Planetary Equations


26/44

y q

Singularities for loweccentricity or inclination

didt

= r cos na2

1 e2 az

d

dt=

r sin

na21 e2 sin iaz

d

dt=

1 e2na e

cos ar + sin 1 +

r

pa

r cos i sin h sin i

az

da

dt=

2

n

1 e2

e sin ar +p

ra

de

dt =

1

e2

nasin ar + cos + e + cos 1 + e cos a

dM0dt

=1

na2 e[(p cos 2er) ar (p + r) sin a]M =n+ b

ah e[(p cos

2re) ar(p+r) sin a]


Numerical Methods: Euler


27/44

t

y

y0

y(t1)

y1

t0 t1

h

y = f(y, t)

y0 = y(t0)

y1 = y0 + f[y(t0), t0] h. . .

yn = yn1 + f[yn1, t0 + (n

1)h]

h

. . .

Error = O(h2)


Numerical Methods: Midpoint


28/44

t

y

y0

y(t1)

t0 t1

h

y = f(y, t)

y0 = y(t0)




29/44

y1

t

y

y0

y(t1)

t0 t1

h

y = f(y, t)

y0 = y(t0)

y1 = y0 + f[y(t0), t0]

h/2




30/44

y1

t

y

y0

y(t1)

t0 t1

h

y = f(y, t)

y0 = y(t0)

y1 = y0 + f[y(t0), t0]

h/2

y1 = f[y1, t0 + h/2]




31/44

y1

y2

t

y

y0

y(t1)

t0 t1

h

y = f(y, t)

y0 = y(t0)

y1 = y0 + f[y(t0), t0]

h/2

y1 = f[y1, t0 + h/2]

y2 = y0 + y1 h. . .

Error = O(h3)


Numerical Methods: Runge-Kutta 4


32/44

tn tn+1h

yn

y(tn+1)

y = f(y, t)




33/44

y1

tn tn+1h

yn

y(tn+1)

y = f(y, t)

k1 = h f(yn, tn) y1 = yn + k1/2




34/44

y1

y2

tn tn+1h

yn

y(tn+1)

y = f(y, t)

k1 = h f(yn, tn) y1 = yn + k1/2

k2 = h f(y1, tn + h/2) y2 = yn + k2/2




35/44

y1

y2

y3

tn tn+1h

yn

y(tn+1)

y = f(y, t)

k1 = h f(yn, tn) y1 = yn + k1/2

k2 = h f(y1, tn + h/2) y2 = yn + k2/2

k3 = h f(y2, tn + h/2) y3 = yn + k3




36/44

y1

y2

y3

y4

tn tn+1h

yn

y(tn+1)

y = f(y, t)

k1 = h f(yn, tn) y1 = yn + k1/2

k2 = h f(y1, tn + h/2) y2 = yn + k2/2

k3 = h f(y2, tn + h/2) y3 = yn + k3

k4 = h f(y3, tn + h) y4 = yn + k4




37/44

y1

y2

y3

y4

yn+1

tn tn+1h

yn

y(tn+1)

y = f(y, t)

k1 = h f(yn, tn) y1 = yn + k1/2

k2 = h f(y1, tn + h/2) y2 = yn + k2/2

k3 = h f(y2, tn + h/2) y3 = yn + k3

k4 = h f(y3, tn + h) y4 = yn + k4

yn+1 = yn +k1

6+ k2

3+ k3

3+ k4

6

Error = O(h5)


Numerical Methods: Burlish-Stoer


38/44

tn tn+1h

yn

y = f(y, t), yn, tn




39/44

n = 2

n = 4n = 6

tn tn+1h

yn

y = f(y, t), yn, tnCompute the interval h with n steps hn , n =

k 2, 4, 6 . . .




40/44

n = 2

n = 4n = 6

tn tn+1h

ynh2h6 h40

y


k 2, 4, 6 . . .




41/44

n = 2

n = 4n = 6

tn tn+1h

yn

y(tn+1)

h2h6 h40

y

Error = O

h2k+1


k 2, 4, 6 . . .

Polynomial extrapolation to n , h 0


Adaptive Stepsize Control


42/44

Set a truncation error and stepsize hGive a step with a method of order n

Repeat the step with order n + 1

If the difference is > , decrease hIf the difference is < , increase h

Each section of the curve is integrated with the maximum h

compatible with This reduces the number of steps, but may require more derivativeevaluations


COWELL Program


43/44

Begin y = f(y, t)

Initializations

Input dataKB/File

ODE Integrator Call Int step Call Derivs

Compute elements

Compute Kepler

Save Data

INTTRAJ.DAT

OSCELEM.DAT

KEPTRAJ.DAT

Plot

End

aKep

agrava3Body

aDrag

aPrad...


ODE Integrator


44/44

Fixed Step

ti = ti1 + t

Dumb Integr Step Derivs

t = tf ?

Yes

No

Adaptive Stepsize

ti = ti1 + t

Adjust t

QS Integr Step Derivs

Error

t = tf ?

OK

Yes

>

Theory of Orbit Perturbations From Madrid Polytechnic

Documents

Transcript of Theory of Orbit Perturbations From Madrid Polytechnic