Perturbation Theory Perturbation Theory (cont’d.): Approaches: at least 3 are commonly...

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  • 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)

    and instead of dε1 dt

    , 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