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### Transcript of Perturbation Theory Perturbation Theory (contâ€™d.): Approaches: at least 3 are commonly...

• Astro 6570

Physics of the Planets

Planetary Perturbation Theory

• Keplerian Orbital Elements

i = inclination Ω = longitude of the (ascending) node ω = argument of pericenter (ν = true anomaly) a = semi-major axis e = eccentricity T = time of pericenter passage

Also, !ω ≡ Ω +ω , the longitude of pericenter L= !ω + ν , the true longitude

For the 2-body problem, all of these elements (except v and L) are constants of the motion.

• Perturbation Theory Purpose: calculate deviations from Keplerian 2-body

motion due to external influences. Examples: • Gravitational effects of a 3rd body (e.g., another planet or

satellite) • Non-spherical primary body (J2) • Atmospheric drag or thruster firings • Radiation forces (e.g., light pressure on small particles) • EM forces (e.g., charged particles in a planet s

magnetosphere) • Relativistic corrections (e.g., Mercury s orbit)

• From a numerical integration of the Solar System.

• Note: In this plot and the next, the reference plane is the Invariable Plane of the solar system, i.e, the plane perpendicular to its total orbital angular momentum, H.

This differs by ~1.5o from the Ecliptic plane, which is not fixed over long time intervals.

The invariable plane is largely determined by the orbits & masses of Jupiter & Saturn.

• Note: The variations for Jupiter and Saturn are almost perfectly anti-correlated, as they exchange angular momentum periodically. The dominant period for this exchange is ~50,000 yrs.

• Perturbation Theory (cont’d.): Approaches: at least 3 are commonly employed: 1. Calculate perturbing forces directly, resolve in radial,

azimuthal & normal directions (R, B, N) and derive the effect of each on the orbital elements (a, e, etc.) • Simple, but inelegant & generally messy…

2. Write perturbation in terms of a potential (the “Disturbing Function”, R) and derive the effect on the orbital elements.

• Conservative forces only, but generally simpler and more direct.

3. Rewrite Kepler problem in terms of Hamiltonian (canonical coordinates & momenta) and apply canonical perturbation theory.

• Elegant & simple, but non-intuitive.

• Perturbation Theory: Lagrange’s Planetary Equations

(Danby pp 238-252)

u

• Method 1: the perturbing force approach:

p = a(1-e2) = h2/GM

u = w + v

• Notes on perturbation equations:

• The equations for di/dt and dW/dt involve only the normal force, N and the argument of latitude, u = w + v.

• The equation for da/dt involves only the radial and transverse forces, R and B, but mostly depends on B for small e, as this directly affects the orbital energy.

• The equations for de/dt and dw/dt involve both the radial and transverse forces, R and B, as well as (indirectly) the normal force, N via dW/dt .

• R strongly affects de/dt at v = 90 and 270o, while B strongly affects de/dt at v = 0 and 180o.

• The effects of R and B on dw/dt are opposite those on de/dt. • The dependence on N is stronger for dw/dt, as it is measured from

the node.

• Note on longitude perturbations The most troublesome orbital element is the 6th, the time of pericenter

passage (T) or equivalently the longitude at t = 0. This is most often written as the mean longitude at epoch , ε:

λ t( ) = Ω +ω + M ≡ !ω + n t − T( ) = ε + nt From Kepler's equation, E − esin E = M , we get ε = !ω + E − esin E − nt and when the orbit is perturbed "ε = !"ω + 1− ecos E( ) "E − "esin E − n − "nt Even if !"ω , "E, "e, and "n remain small, the last term gets large as t →∞. This is circumvented by introducing a different quantity, which depends on the history of the perturbation:

λ t( ) ≡ ε1 + n t( )0 t

∫ dt so that "ε1 satisfies the same equation, minus the "nt term. See Danby's text on Celestial Mechanics for more details.

Note that, since n2a3 = GM exactly for the unperturbed Kepler problem, we have "nn ≡ −

3 2 "a a( ).

• Method 2: the Disturbing Function approach:

Danby, p. 251

Note: See J. Burns, Amer. J. Phys. 44, 944 for a simplified derivation of the perturbation equations

• Perturbations for small e or i

Note: In many papers h & k are reversed, and (h2, k2) à (p, q)

Note that the 2nd and 3rd terms can often be neglected if h1 and k1 are small.

Ditto for 2nd term here.

• Satellite Orbits Around an Oblate Planet As an example of perturbation techniques, we calculate the effects of a planet s oblateness on a close

satellite (natural or artificial). The planet s gravitational potential can be expanded in a series of multipole terms; we keep only the first 2 terms in this expansion:

Where R is the planet s equatorial radius, J2 is a dimensionless constant of order 10-3 to 10-2, and P2 is the 2nd Legendre polynomial:

( is the usual spherical polar co-ordinate.)

The 2nd term in V is our perturbing potential, , which we must express in terms of a, e, i, … !

V r,θ( ) = − µr 1− J2 Rr( )

2 P2 cosθ( ){ } (17)

P2 cosθ( ) = 12 3cos2θ −1( ).

θ

• i.e.,! r,θ( ) = − µJ2 R 2

2r3 3cos2θ −1( )

From geometry, we have cosθ = sin isin ω +υ( ), so

!= − µJ2 R

2

4r3 3sin2 i 1− cos2 ω +υ( )⎡⎣ ⎤⎦ − 2{ } and

r −3 = a 1− e2( )⎡⎣ ⎤⎦ −3

1+ ecosυ( )3

Since the perturbation equations involve ε (i.e., mean longitude) rather than the true anomaly, υ, it is necessary to expand the cosine

terms in terms of the mean anomaly, M = n t − T( ). We can further simplify the problem, however, by considering only the cumulative perturbations over many orbits and neglecting the various short-period perturbations.

Accordingly, we average ! over one orbital period as follows:

! = − µJ2 R

2

4i2π

3sin2 i 1− cos2 ω +υ( )⎡⎣ ⎤⎦ − 2 r30

∫ dM

the integrals involved are

cos2υ r3

dM 0

∫ , sin2υ

r3 dM

0

∫ , 1 r3

dM 0

∫ ,

of which the 1st and 2nd are equal to zero (verification left as an exercise to the student). To evaluate the 3rd integral, we employ Kepler's 2nd law again:

1 r3

dM 0

∫ = 1 r3

dυ dM

⎝⎜ ⎞

⎠⎟

−1

dυ = n 1 r3 "υ( )−1

0

∫0 2π

∫ dυ = n h

dυ r0

∫ , since r 2 "υ = h

(Note that R is conventionally defined as the negative of the perturbing potential.)

• ∴ 1 r3

dM 0

∫ = n

ha 1− e2( ) 1+ ecosυ( )dυ = 2π

a3 1− e2( ) 3 2

0

∴! = − µJ2 R

2

2a3 1− e2( ) 3 2

3 2 sin2 i −1⎡⎣ ⎤⎦ (18)

Note the ! is a function of a,e, and i only, so Lagrange's Equations ⇒

"a = "e = di dt = 0

i.e., there are no long-term perturbations in a, e or i. (There are short - term perturbations due to the parts of ! that averaged to zero over one orbit, however.) Applying Lagrange's Equation, we obtain

dΩ dt

= − 3 2

n J2 R a( )

2 cos i 1− e2( )

⎧ ⎨ ⎪

⎩⎪

⎫ ⎬ ⎪

⎭⎪ (19)

d #ω dt

= 3 2

n J2 R a( )

2 1− 32 sin

2 i − 1 2

sin2i tan i2( ) 1− e2( )2

⎨ ⎪⎪

⎩ ⎪ ⎪

⎬ ⎪⎪

⎭ ⎪ ⎪

(20)

, we calculate

dM dt

= d dt

ε1 + n dt − #ω∫( ) =

dε1 dt

− d #ω dt

+ n

= n 1+ 3 2

J2 R a( )

2 1− 32 sin 2 i

1− e2( ) 3 2

⎨ ⎪

⎩ ⎪

⎬ ⎪

⎭ ⎪

⎨ ⎪

⎩ ⎪ ⎪

⎬ ⎪

⎭ ⎪ ⎪

(21)

Note that all factors enclosed in { } are ! 1 for small e and small i.

• These results have considerable practical significance. Since J2 > 0 for any planet

flattened by rotation, we see that

(i) the line of nodes (i.e., Ω) regresses for prograde orbits, except for i = 90o , so the plane of the obit rotates backwards in inertial space, relative to a fixed direction.

(ii) for all i