Mathematics of Planet Earth

Mathematics allows us to explain some of Earth’s past climates. Indeed, they are linked in particular to variations of the orbit of the Earth. While the movement of the Earth is not quasi-periodic (i.e., a superposition of periodic movements), mainly due to the gravitational influence of Jupiter and Saturn, some periodic oscillations of reasonably short period are well known and called the Milankovitch cycles. These cycles change the insolation (the incident solar radiation) of the Earth, and hence its climate. The Earth’s axis has a precession movement (it rotates around an axis perpendicular to the ecliptic) with a period of 26,000 years, but the major axis of the elliptic orbit also rotates. This combined effect changes the time of the year where the seasons occur, with a cycle of 21,000 years. The obliquity (tilt) of the Earth’s axis oscillates between 22.1 and 24.5 degrees, with a period of 41,000 years. The present obliquity is 23.44 degrees, and is decreasing. A decrease in the obliquity favors warmer winters and cooler summers and, globally, a glaciation. The eccentricity of the orbit of the Earth around the Sun varies from 0.005 to 0.058 with a mean value of 0.028, this being a superposition of cycles of periods from 100,000 years to 413,000 years. The present eccentricity is 0.017, and it is decreasing. Other cycles are superimposed on these. Modeling these variations in the Earth’s movements is part of celestial mechanics. While relativistic effects cannot always be neglected, the main methods come from dynamical systems. To understand the influence of the Milankovitch cycles on the climate, other tools are required, since oceans, land and atmosphere react differently to variations of the insolation.