cp's OEIS Frontend

A381254 Decimal expansion of the obliquity (in radians) of a planet at which the annual instellations received by the poles and the equator are identical.

%I A381254 #30 May 03 2025 22:25:50
%S A381254 9,4,0,6,6,6,7,7,0,2,3,9,9,9,9,6,6,3,2,1,5,8,8,1,8,6,7,9,9,3,8,5,7,9,
%T A381254 0,5,3,2,8,8,2,0,5,4,7,1,7,1,6,9,0,5,6,4,6,8,5,0,5,4,7,1,2,0,1,2,7,4,
%U A381254 6,7,1,4,1,3,7,7,7,8,8,7,0,7,3,4,3,7,6,7,0,3,2,1,6,3,0,8,0,7,2,4,3,4,4,7
%N A381254 Decimal expansion of the obliquity (in radians) of a planet at which the annual instellations received by the poles and the equator are identical.
%C A381254 For an obliquity x, the normalized annual instellation coefficient at the equator is e(x) = (EllipticE(sin(x)^2) + sqrt(cos(x)^2) * EllipticE(-tan(x)^2)) / Pi, and at the poles is p(x) = sin(x), and the present constant is x where e(x) = p(x).
%C A381254 These coefficients are obtained by integrating over the sine of solar altitude over the course of one planetary year.
%C A381254 If the obliquity of a planet is greater than this value (for example, Uranus), then the poles would receive more instellation per year than the equator, which would result in a climate that inverts typical perceptions of those latitudes and the polar regions would be hotter than equatorial ones, in some cases resulting in an "ice belt" planet. However, these seasonal means would be accompanied by intense seasonal variations, as opposed to purely "tropical" polar regions.
%H A381254 David Ferreira, John Marshall, Paul A. O’Gorman, and Sara Seager, <a href="https://doi.org/10.1016/j.icarus.2014.09.015">Climate at high-obliquity</a> Icarus, vol. 243, pp. 236-248, 2014.
%H A381254 John Marshall, <a href="https://oceans.mit.edu/JohnMarshall/eapsdb/a045d7637b7c22298f0374bdf9d3c899">Climate at high-obliquity</a>
%H A381254 D. M. Williams, J. F. Kasting, and L. A. Frakes, <a href="https://doi.org/10.1038/24845">Low-latitude glaciation and rapid changes in the Earth's obliquity explained by obliquity-oblateness feedback</a>, Nature, 1998 Dec 3;396(6710):453-5.
%H A381254 Worldbuilding Pasta, <a href="https://worldbuildingpasta.blogspot.com/2022/08/climate-explorations-obliquity.html">Climate Explorations: Obliquity</a>
%F A381254 Equals A383141*Pi/180.
%e A381254 0.9406667702399996632...
%t A381254 FindRoot[(EllipticE[Sin[x]^2] + Sqrt[Cos[x]^2] * EllipticE[-Tan[x]^2]) / Pi == Sin[x], {x, 0.94}, WorkingPrecision -> 100]
%o A381254 (PARI) \\ definition of ellM as in Mathematica's EllipticE[m]
%o A381254 ellM(k) = intnum(t=0, Pi/2, sqrt(1-k*sin(t)^2));
%o A381254 solve (x=0.9, 0.95, (ellM(sin(x)^2) + sqrt(cos(x)^2)*ellM(-tan(x)^2))/Pi - sin(x)) \\ _Hugo Pfoertner_, Apr 26 2025
%Y A381254 Cf. A383141.
%K A381254 nonn,cons
%O A381254 0,1
%A A381254 _Eliora Ben-Gurion_, Apr 17 2025