Obliquity of the ecliptic

Meaning

The obliquity of the ecliptic (written ε) is the angle between two great circles of the sky: the ecliptic, the Sun's apparent yearly path, and the celestial equator, the Earth's equator projected onto the sky. It is the same angle as the tilt of the Earth's axis to the plane of its orbit, which is why it is also called axial tilt.

The two circles cross at the equinox points, 0° Aries and 0° Libra, and are farthest apart at the solstice points, 0° Cancer and 0° Capricorn. There the gap between them is exactly ε. So the obliquity is the Sun's greatest declination: the Sun reaches about 23°26′ N at the June solstice and 23°26′ S at the December solstice. The tilt is also what gives the Earth its seasons and sets the tropics (at latitude ε) and the polar circles (at 90° − ε).

Current value. The mean obliquity by the IAU 2006 formula [5] is 23°26′21″ (23.4393°) at the start of 2000 and 23°26′09″ (23.4359°) in 2026. It is falling by about 47″ per century [5] [7], so the familiar round figure "23°27′", right for most of the 20th century, is now about a minute too large.

The long cycle. Over the last million years the tilt has swung between about 22.1° and 24.5°, in a cycle of about 41,000 years. It was last at its greatest about 10,000 years ago and will reach its least in about 10,000 years' time [6]. This slow swing is one of the Milankovitch cycles that pace the ice ages.

Mean and true obliquity. The Moon's pull on the Earth's bulge makes the axis nod slightly, an effect called nutation. Its largest term follows the Moon's Node round in 18.6 years and changes the obliquity by up to about ±9″ [7]. The smooth long-term value is the mean obliquity; with nutation added it is the true obliquity, the one that applies at a given moment.

Origin and history

Greek astronomers measured the obliquity from the Sun's noon height at the two solstices: half the difference is ε. Ptolemy reports in the Almagest that the arc between the two solstice points on the meridian is between 47°40′ and 47°45′, close to a ratio of 11 to 83 of the whole circle, which he says Eratosthenes found and Hipparchus also used [1]. That ratio gives an arc of about 47°42′40″ and an obliquity of 23°51′20″, the value behind Ptolemy's declination table [1]. The true value in his day was about 23°41′, so his figure was some 11′ too large. Historians disagree about how much of the 11/83 ratio goes back to Eratosthenes and how much to Hipparchus; Alexander Jones argues that it rests on rough round numbers rather than on a precise measurement [2].

Astronomers of the Islamic world measured it again with larger instruments. Al-Battānī, observing at Raqqa between 877 and 918, found 23°35′, within a few seconds of the true value for his time [3]. At Ulugh Beg's observatory in Samarkand, founded in 1420, the obliquity was measured as 23°30′17″, about half a minute below the true value [4]. Read in order today, these measurements show the slow fall that modern theory explains.

Today the value comes from dynamical theory fitted to modern observations. The IAU 2006 model gives the mean obliquity as a polynomial in time, starting from 84381.406″ (23°26′21.406″) at the year 2000 [5].

How it is used

The obliquity enters almost every step between the zodiac and the sky:

• Declination. Converting a zodiac position to declination needs ε: sin δ = sin β · cos ε + cos β · sin ε · sin λ (see declination).
• Out of bounds. The obliquity is the boundary itself: a planet beyond ε in declination is out of bounds.
• Lunar standstills. The Moon's farthest reach is roughly ε plus or minus the tilt of its orbit (see lunar standstill).
• Antiscia. A degree and its antiscion have the same declination whatever the value of ε, so the mirror points themselves do not depend on it; the declination they share does.
• Rising times and houses. How long each sign takes to rise, and so the Ascendant, the Midheaven and every house system built on them, depend on ε as well as on the latitude of the place.

On this site. Each chart carries its own obliquity for the moment of birth (the true obliquity, nutation included), and the Declination panel uses it both to work out declinations and as the out-of-bounds limit. A chart from 1900 is therefore measured against about 23°27′, a chart from 2026 against about 23°26′.

Examples

Mean obliquity by the IAU 2006 formula [5], rounded to the second:

• The Sun at 15° Gemini has declination 22°36′ N with ε = 23.44°. With Ptolemy's 23°51′20″ the same degree would come out at about 23°00′ N.