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Axial tilt is an astronomical term regarding the inclination angle of a planet's rotational axis in relation to a perpendicular to its orbital plane. It is also called axial inclination or obliquity. The axial tilt is expressed as the angle made by the planet's axis and a line drawn through the planet's center perpendicular to the orbital plane.

## Obliquity

The axial tilt may equivalently be expressed in terms of the planet's orbital plane and a plane perpendicular to its axis. In our solar system, the Earth's orbital plane is known as the ecliptic, and so the Earth's axial tilt is officially called the obliquity of the ecliptic. In formulas it is abbreviated with the Greek letter ε.

The Earth has an axial tilt of about 23° 27’. The axis is tilted in the same direction throughout a year; however, as the Earth orbits the Sun, the hemisphere (half part of earth) tilted away from the Sun will gradually come to be tilted towards the Sun, and vice versa. This effect is the main cause of the seasons (see effect of sun angle on climate). Whichever hemisphere is currently tilted toward the Sun experiences more hours of sunlight each day, and the sunlight at midday also strikes the ground at an angle nearer the vertical and thus delivers more heat.

Through time, axial precession changes the position of the Earth in its orbit at which the seasons occur (precession of the equinoxes). This has little effect on the amount of solar influx (insolation) during times when the orbit is circular, but can have large effects on the strength of the seasons when the Earth's orbit is somewhat elliptical (see Milankovitch cycles).

The obliquity of the ecliptic is not a fixed quantity but changing over time. It is a very slow effect, but at the level of accuracy at which astronomers work, it does need to be taken into account on a daily basis. Note that the obliquity and the precession of the equinoxes are calculated from the same theory and are thus related to each other. A smaller ε means a larger p (precession in longitude) and vice versa. Yet the two movements act independent from each other, going in mutually perpendicular directions.

## Measurement

The obliquity of the ecliptic is such a pervasive element in positional astronomy that it must be used in the calculations and observations of all planetary positions, including Sun and Moon. However to quickly grasp an idea of its value one can look at the seasons. It suffices to consider that the extreme northern and southern declination of the Sun are per definition equal to the obliquity. Therefore the difference of the heights of the Sun above the horizon at noon on the longest and shortest day of the year is twice the obliquity. This was the way the Chinese astronomers determined it in 1000 BC.

Example: an observer on 40° latitude (either north or south) will see the Sun 73°5 above the horizon at noon on the longest day of the year, but only 26°5 the shortest day. The difference is 2ε = 47°, and so ε = 23.5°.

## Values

The Earth's axial tilt varies between 22.1° and 24.5° (but see below), with a 41,000-year periodicity, and at present, the tilt is decreasing. In addition to this steady decrease, there are also much smaller short term (18.6 years) variations, known as nutation.

Simon Newcomb's calculation at the end of the nineteenth century for the obliquity of the ecliptic gave a value of 23° 27’ 8.26” (epoch of 1900), and this was generally accepted until improved telescopes allowed more accurate observations, and electronic computers permitted more elaborate models to be calculated. Lieske came with an updated theory in 1976 with ε equal to 23° 26’ 21.44” (epoch of 2000), which became the officially approved theory by the International Astronomical Union in 2000:

ε = 84,381.448 − 46.84024T − (59 × 10−5)T2 + (1,813 × 10−6)T3, measured in seconds of arc, with T being the time in Julian centuries (that is, 36,525 days) since the ephemeris epoch of 2000 (which occurred on Julian day 2,451,545.0).

With the linear term in T being negative, at present the obliquity is slowly decreasing. It must be stressed that this formula is only valid over a limited time period. It is clear that if T gets large enough the T3-term will start to dominate and ε will go to positive values beyond 90° in the far future and dip below 0° in the distant past. Both are nonsense. In reality, more elaborate calculations on the numerical model of solar system shows that ε has a period of about 41,000 years, the same as the constants of the precession of the equinoxes (although not of the precession itself).

Other theoretical models may come with values for ε expressed with higher powers of T, but since no (finite) polynomial can ever represent a periodic function, they all go to either positive or negative infinity for large enough T. In that respect one can understand the decision of the International Astronomical Union to choose the simplest equation which agrees with most models. For up to 5,000 years in the past and the future all formulas agree, and up to 9,000 years in the past and the future, most agree to reasonable accuracy. For eras farther out discrepanies get too large.

## Long period variations

Nevertheless extrapolation of the average polynomials gives a fit to a sine curve with a period of 41,013 years, which, according to Wittmann, is equal to:

ε = A + B sin (C(T + D)), with A = 23.496932° ± 0.001200°, B = − 0.860° ± 0.005°, C = 0.01532 ± 0.0009 radians/Julian century, D = 4.40 ± 0.10 Julian centuries, and T, the time in centuries from the epoch of 2000 as above.

This means a range of the obliquity from 22° 38’ to 24° 21’, the last maximum was reached in 8700 BC, the mean value occurred around 1550 and the next minimum will be in 11800. This formula should give a reasonable approximation for the previous and next million years or so. Yet it remains an approximation in which the amplitude of the wave remains the same, while in reality, as seen from the results of the Milankovitch cycles, irregular variations occur. The quoted range for the obliquity is from 21° 30’ to 24° 30’, but the low value may have been a one-time overshot of the normal 22° 30’.

If we go back over the last 5 million years, the obliquity of the ecliptic (or more accurately, the obliquity of the equator on the moving ecliptic of date) has varied from 22.0425° to 24.5044°. But for the next one million years the range will be only from 22.2289° to 24.3472°

Other planets may have a variable obliquity too, for example on Mars the range is believed to be between 15° and 35°. The relatively small range for the Earth is due to the stabilizing influence of the Moon, but it will not remain so. According to Ward, when the distance of the Moon—which is continuously increasing due to tidal effects—will have gone from the current 60 to approximately 66.5 Earth radii in less than 2,000,000,000 years. Once this occurs, a resonance from planetary effects will follow, causing swings of up to 65° in the obliquity. More swings will occur again when the Moon reaches a distance of 68 Earth radii. This will have significant effects on climate.

## References

• Explanatory supplement to 'the Astronomical ephemeris' and 'the American ephemeris and nautical almanac'
• [1] for a comparison of values predicted by different theories
• A.L. Berger; Obliquity & precession for the last 5 million years; Astronomy & astrophysics 1976, 51, 127
• A. Wittmann; The obliquity of the ecliptic; Astronomy & astrophysics 73, 129-131 (1979)
• W.R. Ward; Comments on the long-term stability of the earth's obliquity; Icarus 1982, 50, 444