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.

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, 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, 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

Offset

0,1

Comments

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).

These coefficients are obtained by integrating over the sine of solar altitude over the course of one planetary year.

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.

Links

Table of n, a(n) for n=0..103.

David Ferreira, John Marshall, Paul A. O’Gorman, and Sara Seager, Climate at high-obliquity Icarus, vol. 243, pp. 236-248, 2014.

John Marshall, Climate at high-obliquity

D. M. Williams, J. F. Kasting, and L. A. Frakes, Low-latitude glaciation and rapid changes in the Earth's obliquity explained by obliquity-oblateness feedback, Nature, 1998 Dec 3;396(6710):453-5.

Worldbuilding Pasta, Climate Explorations: Obliquity

Formula

Equals A383141*Pi/180.

Example

0.9406667702399996632...

Mathematica

FindRoot[(EllipticE[Sin[x]^2] + Sqrt[Cos[x]^2] * EllipticE[-Tan[x]^2]) / Pi == Sin[x], {x, 0.94}, WorkingPrecision -> 100]

Prog

(PARI) \\ definition of ellM as in Mathematica's EllipticE[m]
ellM(k) = intnum(t=0, Pi/2, sqrt(1-k*sin(t)^2));
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

Crossrefs

Cf. A383141.

Sequence in context: A239349 A198675 A203131 * A259484 A307215 A393429

Adjacent sequences: A381251 A381252 A381253 * A381255 A381256 A381257

Keyword

nonn,cons

Author

Eliora Ben-Gurion, Apr 17 2025

Status

approved