Axial tilt

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 also called the obliquity of the ecliptic. In formulas it is abbreviated with the Greek letter ε.

The Earth has an axial tilt of about 23.4°. 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 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 already did 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° on the shortest day. The difference is 2ε = 47°.

Values

The Earth's axial tilt varies between 22.1° and 24.5° (but see below), with a 41000 year periodicity, currently 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 19th century for the obliquity of the ecliptic: ε = 23.452294 degrees or 23°27'08.26" (epoche 1900) was the general accepted value until improved telescopes delivered more accurate observations and electronic computers allowed more elaborate models to be calculated. Lieske came with an updated theory in 1976: ε = 23.43929 degrees (epoch 2000), which became the officially approved theory by the IAU in 2000:

ε = 84381.448 -46.84024 * T -59×10-5 * T^2 +1813×10-6 * T^3 in arcseconds, with T the time in Julian centuries (36525 days) since the ephemeris epoch 2000 (JED = 2451545.0).

Indeed with the linear term in T being negative, the obliquity is slowly decreasing nowadays. 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 T^3 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 41000 years, the same as the constants of the precession of the equinoxes (not the precession itself!). Other theoretical models may come with values for ε expressed with higher powers of T, but since no polynomial can ever represent a periodic function, they all go to either plus or minus infinite for T large enough. In that respect one can understand the decision of the IAU to choose the simplest equation which agrees with most models. Up to 5000 years in the past and the future all formulas agree. Up to 9000 years in past and future most agree to reasonable accuracy. For eras farther out discrepanies are getting too large.

Long period variations

Nevertheless extrapolation of the average polynomials gives a fit to a sine curve with a period of 41013 years, according to Wittmann 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 rad/julian century, D = 4.40 ± 0.10 julian centuries, T the time in centuries from epoch 2000 as above.

This means a range of the obliquity from 22.64° to 24.36°, the last maximum was reached in -8700, 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 21.5° to 24.5°, but the low value of 21.5° may have been a one time shootout of the normal 22.5°.

If we go back over the last 5 million years, the obliquity of the ecliptic (more accurate: 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 our Moon. But that is not going to 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 Earth radii to approximately 66.5 in less than 2 000 000 000 years from now, a resonance from planetary effects will occur. Huge swings in the obliquity, up to 65° will then occur. And again when the Moon reaches 68 Earth radii distance. This will have significant effects on the climates.

References

Explanatory supplement to 'the Astronomical ephemeris' and 'the American ephemeris and nautical almanac'
^* 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

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Obliquity

Measurement

Values

Long period variations

References