Basics of Orbital Mechanics II - OCW...

Basics of Orbital Mechanics II

Modeling the Space Environment

Manuel Ruiz Delgado

European Masters in Aeronautics and SpaceE.T.S.I. Aeronauticos

Universidad Politecnica de Madrid

April 2008

Basics of Orbital Mechanics II – p. 1/24

Basics of Orbital Mechanics II

Keplerian and Perturbed Motion

Magnitude of the Perturbations

Special Perturbations all, numericalEncke’s MethodCowell’s Method

General Perturbations some, analytical, approximateOsculating OrbitVariation of ParametersLagrange Equations potentialGauss Equations potential & not potential

General Perturbations: Analytical approx/Semianalytical

Numerical Integration


Keplerian and Perturbed Motion

r = −G (M + m)r

|r|3︸︷︷︸

Kepler Problem

+P1

m1− P2

m2︸︷︷︸

Perturbation

rk =

ak︷︸︸︷

−G (M + m)rk

|rk|3

rp = −G (M + m)rp

|rp|3+ ap

rp

rk

m

M

Usually, |ap| ≪ |ak| ⇒ rp ≃ rk How small?


Perturbations (LEO)

1e−008

1e−006

0.0001

0.01

1

100

10000

1e+006

0 100 200 300 400 500 600 700 800 900

Acc

eler

atio

n (m

/s2 )

Height (km)

Accelerations of the Satellite (BC=50)

Shuttle

ISS

KeplerJ2

C22Sun

MoonDrag (low)

Drag (high)Prad


Perturbations (GEO)

1e−008

1e−006

0.0001

0.01

1

100

10000

1e+006

0 5000 10000 15000 20000 25000 30000 35000 40000

Acc

eler

atio

n (m

/s2 )

Height (km)

Accelerations of the Satellite (BC=50)

GEOGPS

KeplerJ2

C22Sun

MoonDrag (low)

Drag (high)Prad


Encke’s Method

kepl

eria

n

perturbed

rk

δr

r0

v0

Epoch

rp

M

Compute only the differenceδr

rk = −µrk

|rk|3rp = −µ

rp

|rp|3+ ap

δr = rp − rk |δr| ≪ |rp|


Encke’s Method

kepl

eria

n

perturbed

rk

δr

r0

v0

Epoch

rp

M


rk = −µrk

|rk|3rp = −µ

rp

|rp|3+ ap


δr = rp − rk = −µrp

|rp|3+ µ

rk

|rk|3+ ap =


Encke’s Method

kepl

eria

n

perturbed

rk

δr

r0

v0

Epoch

rp

M


rk = −µrk

|rk|3rp = −µ

rp

|rp|3+ ap



|rp|3+ µ

rk

|rk|3+ ap =

δr = − µ

|rk|3δr +

µ

|rk|3

(

1 − |rk|3

|rp|3

)

rp + ap


Encke’s Method

Aboutf(q), cf. Battin, p. 389 and 449

kepl

eria

n

perturbed

rk

δr

r0

v0

Epoch

rp

M


rk = −µrk

|rk|3rp = −µ

rp

|rp|3+ ap



|rp|3+ µ

rk

|rk|3+ ap =

δr = − µ

|rk|3δr +

µ

|rk|3

(

1 − |rk|3

|rp|3

)

rp + ap

1 − |rk|3

|rp|3= −f(q) = −q

3 + 3q + q2

1 + (1 + q)3

2

q =δr · (δr− 2rp)

rp · rp


Encke’s Method


kepl

eria

n

perturbed

rk

δr

r0

v0

Epoch

rp

M


rk = −µrk

|rk|3rp = −µ

rp

|rp|3+ ap



|rp|3+ µ

rk

|rk|3+ ap =

δr = − µ

|rk|3δr +

µ

|rk|3

(

1 − |rk|3

|rp|3

)

rp + ap

1 − |rk|3

|rp|3= −f(q) = −q

3 + 3q + q2

1 + (1 + q)3

2


rp · rp

δr = − µ

|rk|3δr− µ

|rk|3f(q) rp + ap


Encke’s Method


kepl

eria

n

Epoch∣∣2

perturbed

rp

M


rk = −µrk

|rk|3rp = −µ

rp

|rp|3+ ap



|rp|3+ µ

rk

|rk|3+ ap =

δr = − µ

|rk|3δr +

µ

|rk|3

(

1 − |rk|3

|rp|3

)

rp + ap

1 − |rk|3

|rp|3= −f(q) = −q

3 + 3q + q2

1 + (1 + q)3

2


rp · rp

δr = − µ

|rk|3δr− µ

|rk|3f(q) rp + ap

if δr ↑, rectify: δr = 0

rk

∣∣1→ rk

∣∣2


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


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


Cowell’s Formulation

Direct numerical integration of the equations

ODE: r = −µr

|r|3+ ap (r, r, t)

IC: t0, r0, r0

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

x =

x

y

z

vx

vy

vz

x =

vx

vy

vz

x

y

z

=

vx

vy

vz

− µr3 x + ax

− µr3 y + ay

− µr3 z + az

x = f (x, t)


Osculating Orbit - Variation of Parameters

perturbed

M

r0

v0

Epoch

Satellite inr0, v0 atEpocht0

Followsperturbedtrajectoryrp(t)



ke

pler

ian

perturbed

M

r0

v0

Epoch



Osculating Orbitat r0, v0:

TheKeplerianorbit followed by the satelliteif all perturbationsbecome zero from thispoint on.



ke

pler

ian

oscu

lati

ng

perturbed

rp(t)

M

r0

v0

Epoch



Osculating Orbitat r0, v0:

TheKeplerianorbit followed by the satelliteif all perturbationsbecome zero from thispoint on.

Osculating orbit elements can be used ascoordinates

r0,v0 , t0 ⇒ i,Ω, ω, a, e, τ , θ, t0

rp(t),vp(t) , t ⇒ i(t),Ω(t), ω(t), a(t), e(t), τ(t) , θ(t), t


Variation of Parameters:Fast/Slow variables

θM , Æ

ω

FastVariables:

θ, M , Æ, t

r(t) ,v(t)

SlowVariables:

i, Ω, ω, a, e, τ (M0)


Variation of Parameters:Secular/Periodic

Secular

Secular+ Long periodic

Secular+ Long periodic+ Short periodic

“Short” ∼ Orbital period

Orb

italP

aram

eter

t


Variation of Parameters - Lagrange

Variation of Parameters:

r = −µr

|r|3+ ap

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

x = ⌊i,Ω, ω, a, e, τ⌋T

x = f (x, t)

Lagrange Planetary Equations: Conservative perturbations

ap = ∇R R(i,Ω, ω, a, e,M0) M0 = n τ

x = ⌊i,Ω, ω, a, e,M0⌋T

x = f (x,∇R)


Lagrange Planetary Equations

Singularities forlow eccentricity

or inclination

di

dt=

1

na2√

1 − e2 sin i

(

cos i∂R

∂ω− ∂R

∂Ω

)

dΩ

dt=

1

na2√

1 − e2 sin i

∂R

∂i

dω

dt=

√1 − e2

na2 e

∂R

∂e− cos i

na2√

1 − e2 sin i

∂R

∂i

da

dt=

2

na

∂R

∂M0

de

dt=

1 − e2

na2 e

∂R

∂M0−

√1 − e2

na2 e

∂R

∂ω

dM0

dt= −1 − e2

na2 e

∂R

∂e− 2

na

∂R

∂a


Lagrange Planetary Equations

Singularities forlow eccentricity

or inclination

di

dt=

1

na2√

1 − e2 sin i

(

cos i∂R

∂ω− ∂R

∂Ω

)

dΩ

dt=

1

na2√

1 − e2 sin i

∂R

∂i

dω

dt=

√1 − e2

na2 e

∂R

∂e− cos i

na2√

1 − e2 sin i

∂R

∂i

da

dt=

2

na

∂R

∂M0

de

dt=

1 − e2

na2 e

∂R

∂M0−

√1 − e2

na2 e

∂R

∂ω

dM0

dt= −1 − e2

na2 e

∂R

∂e− 2

na

∂R

∂a

(

M = n − 1−e2

na2 e∂R∂e − 2

na∂R∂a

∣∣a

)


Lagrange VOP: Kozai’s Method

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

R1 =3

2

µ J2 R2E

a3

(1

3− 1

2sin2 i

)(1 − e2

)1/2R2 = 0

R3 =3

2

µ J3 R3E

a4sin i

(

1 − 5

4sin2 i

)

e(1 − e2

)−5/2

sin ω

R4 =3

2

µ J2 R2E

a3

(a

r

)3(

1

3− 1

2sin2 i

)[

1 −(r

a

)3 (1 − e2

)−3/2

]

+

+1

2sin2 i cos 2 (ν + ω)

Only gravitational perturbationsJ2 (flattening) andJ3 (pear-shape)are included.


Lagrange VOP: Kozai’s Method (secular)

di

dt=

3

8n J3

(RE

p

)3

cos i(4 − 5 sin2 i

)sin2 i cos ω

da

dt= 0

dΩ

dt= −3

2n J2

(RE

p

)2

cos i − 3

8n J3

(RE

p

)3(15 sin2 i − 4

)e cot i sin ω

dω

dt=

3

4n J2

(RE

p

)2(4 − 5 sin2 i

)+

3

8n J3

(RE

p

)3 [(4 − 5 sin2 i

)·

·(sin2 i − e2 cos2 i

)

e sin i+ 2 sin i

(13 − 15 sin2 i

)e

]

sin ω

de

dt= −3

8n J3

(RE

p

)3

sin i(4 − 5 sin2 i

) (1 − e2

)cos ω

dM

dt= n

[

1 +3

2J2

(RE

p

)2(

1 − 3

2sin2 i

)(1 − e2

)1/2]

−

− 3

8n J3

(RE

p

)3

sin i(4 − 5 sin2 i

) (1 − 4e2

)(1 − e2

)1/2

esin ω


Gauss Planetary Equations

Conservative and not conservative perturbations

Use the Orbital Frame forap

ap = ar ur + aθ uθ + az uz

Peric.

Sat.

hur

uθ

eω

Ω

θ

uN

i

i

x1

y1

z1


Gauss Planetary Equations

Singularities for loweccentricity or inclination

di

dt=

r cos θ

na2√

1 − e2az

dΩ

dt=

r sin θ

na2√

1 − e2 sin iaz

dω

dt=

√1 − e2

na e

[

− cos θ ar + sin θ

(

1 +r

p

)

aθ

]

− r cos i sin θ

h sin iaz

da

dt=

2

n√

1 − e2

(

e sin θ ar +p

raθ

)

de

dt=

√1 − e2

na

[

sin θ ar +

(

cos θ +e + cos θ

1 + e cos θ

)

aθ

]

dM0

dt=

1

na2 e[(p cos θ − 2er) ar − (p + r) sin θ aθ]

M =n+ b

ah e[(p cos θ−2re) ar−(p+r) sin θ aθ]


Numerical Methods: Euler

t

y

y0

y(t1)

y1

t0 t1

h

y = f(y, t)

y0 = y(t0)

y1 = y0 + f [y(t0), t0] · h. . .

yn = yn−1 + f [yn−1, t0 + (n − 1)h] · h. . .

Error = O(h2)


Numerical Methods: Midpoint

t

y

y0

y(t1)

t0 t1

h

y = f(y, t)

y0 = y(t0)



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

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]



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

tn tn+1h

yn

y(tn+1)

y = f(y, t)



y1

tn tn+1h

yn

y(tn+1)

y = f(y, t)

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



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



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



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



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

tn tn+1h

yn

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



b

b

b

b

b

b

b

b

b

b

b

b

b

bb

n = 2

n = 4n = 6

tn tn+1h

yn

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

Compute the intervalh with n stepshn , n =

k︷︸︸︷

2, 4, 6 . . .



b

b

b

b

b

b

b

b

b

b

b

b

b

bb

n = 2

n = 4n = 6

tn tn+1h

yn

h2

h6

h4

0

y

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


k︷︸︸︷

2, 4, 6 . . .



b

b

b

b

b

b

b

b

b

b

b

b

b

bb

n = 2

n = 4n = 6

tn tn+1h

yn

y(tn+1)

h2

h6

h4

0

y

Error =O(h2k+1

)

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


k︷︸︸︷

2, 4, 6 . . .

Polynomial extrapolation ton → ∞, h → 0


Adaptive Stepsize Control

Set a truncation errorǫ and stepsizeh

Give a step with a method of ordern

Repeat the step with ordern + 1

If the difference is> ǫ, decreaseh

If the difference is< ǫ, increaseh

Each section of the curve is integrated with the maximumhcompatible withǫ

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


COWELL Program

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

agrav

a3Body

aDrag

aPrad...


ODE Integrator

Fixed Step

ti = ti−1 + ∆t

Dumb Integr Step Derivs

t = tf ?

Yes

No

Adaptive Stepsize

ti = ti−1 + ∆t

Adjust∆t

QS Integr Step Derivs

Error⋚ ǫ

t = tf ?

OK

Yes

> <No


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