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What do the twelve Zodiac constellations represent in Astronomy
You may know the Zodiac from its usage in Astrology (the Sun Signs). The Zodiac is an imaginary belt in the sky. It is a set of 12 (sometimes referred as 13) constellations that serve as the backdrop to the Sun through the year.
During the year, the Sun appears to move from one Zodiac Constellation to the next. In reality, it is the Earth's movement around the Sun that generates the effect. The path of the Sun through the sky is called the Ecliptic. Most of the planets of the Solar System orbit the Sun in a similar plane. Hence even the planets appear to be moving from one Zodiac constellation to the next.
Observing a Zodiac Sign
Familiarity with the night-sky helps.For beginners, get hold of a star-chart that depicts the position of the sky in your location or latitude for a given month - SkyMaps publishes monthly updates and includes position of planets and the moon cycles. The SFA Star Charts are quite comprehensive and popular.
The planets tend to move in the zodiac constellations. The planets are also quite bright and stand out because they don't twinkle like the stars. So they help draw attention to a section of the sky.
Linking this to an Astrology Duration (say - Leo being 23 July-22 Aug) means that the Sun would be in the constellation of Leo during this period - so Leo would "not be visible" in the night sky (It would be up in the sky during the day). A rough rule-of-thumb would be to watch out for the constellation in a 3-month period prior to the sun sign. During April-June, you would be able to see Leo in the evening skies.
The 360 Degree view is divided into 12 sectors and the primary constellation that occupies the sector in the ecliptic is marked as the Zodiac Constellation. From an Astrological perspective, each zodiac sign is given a 30-degree space (and hence the Sun spends equal time in each zodiac). In reality, the constellations have varying spaces and have varying lengths at the ecliptic. Astronomers don't work with a fixed 30-degree segregation of zodiac constellations. The constellation boundaries are as drawn out and updated by the International Astronomical Union (IAU).
The Earth's movement around its axis and around the Sun has a set of variations. Typically they are observed over hundreds of years. These variations tend to add up over larger time frame. These are referred to as precession of the ecliptic and precession of the equator.
Precession brings in a change in the Pole Star over cycles of 26,000 years. It also brings differences in the constellation positions as referenced on a calendar date, sometimes called a shift in the equinoxes. Over a 2,000 year period, the sun-sign corresponding to the month shifts by one sign. This is depicted beautifully in a set of computed images at dyonisia.org.