Planetary Theory

Classical set of elements: (a,e,i,ε,π,Ω)

• a = semimajor axis
• e = eccentricity
• i = inclination
• ε =
• π = argument of perihelion (usually symbolized as ω with strike-over [pi variant or "varpi"], that renders well in Firefox [ϖ] but not in Internet Explorer; so instead of strike-over I will use strike-through ω.)
• Ω = argument of ascending node

Alternatively, λ (= ...) is used instead of ε, with

• dλ/dt = n + dε/dt
• n = ...

New set of (real) elements: (a,λ,h,k,p,q)
h = e sin(ω)
k = e cos(ω)
p = sin(i/2) sin(Ω)
q = sin(i/2) cos(Ω)

New set of (complex) elements: (a,ε,z,ζ)
z = k + jh = e exp(jω)
ζ = q + jp = sin(i/2) exp(jΩ)

(? indices not per-planet)
z = Σ(M exp(gt+β))
ζ = Σ(N exp(st+δ))

• [Chapront_1975|disk*] Un formulaire pour le calcul des perturbations d'ordres élevés dans les problèmes planétaires
• Simon 1974 ?
• [Bretagnon_1974|disk*] Termes à longues périodes dans le système solaire
• [Berger_1977a|disk*] Long-term variations of the Earth's orbital elements
• [Berger_1976|disk*] Obliquity and Precession for the Last 5 000 000 Years

Notes on papers

Berger_1978a

• Prime source for insolation formulas. Introduces daily insolation at "mid-month" (longitude) & at "calendar" days.
• Quasiperiodic series expansions for e, ε etc.

Berger_2000

• Good intro to Milankovitch theory, in particular to forcing elements e, ε, e sin(π)
• Figures for ice sheet volume vs eustatic sea level (δicevolume of 10e6 km³ -> δlevel of 25 m), with typ. ice volumes for Greenland and Antarctica. Also see Bentley M.J.
• Orbital curves 0-4 MaBP according to Loutre & Berger 1993 model, to be compared with Laskar
• Summer insolation @ 65N as driver for ice sheets

Issues

No secular variation of semi-major axis

(to second order in masses)

• Poisson's theorem
• See Duriez_1978
• Berger_1978a refers to Brouwer & Clemence (1961): Methods of Celestial Mechanics (book)

Seminal papers

• Hays, Imbrie & Shackleton. Variations in the Earth's Orbit: Pacemaker of the Ice Ages. Science, 1978 [ads]