## Summary of the calculations of the positions of the celestial objects in the sky

#### Introduction

This is intended as a summary of the first 5 weeks or so of the class.

#### Definitions

• Local coordinates
• Altitude - the angle of an object above the horizon
• altitude of any point on the horizon = 0o
• altitude of the point above you = 90o
• Azimuth - the angle of an object around the horizon, measured eastward from the north.
• Celestial coordinates
• Declination - the angle of an object north or south of the Celestial Equator
• declination of any point on the equator = 0o
• declination of the North Celestial Pole = 90o
• declination of the South Celestial Pole = -90o
• Right Ascension - the angle of an object round the Celestial Equator, measured from the vernal equinox
• usually expressed as a time, at a rate of 15o per hour.

#### Altitude of the Sun

• Only for latitudes inside the tropics, defined by the Tropic of Cancer (23½o N) and Tropic of Capricorn (23½o S), can the Sun be directly overhead.
• At the Summer Solstice, the altitude of the Sun = 90 - latitude + 23½
• At the Winter Solstice, the altitude of the Sun = 90 - latitude - 23½
• At either Equinox days, the altitude of the Sun = 90 - latitude

#### Declination and Right Ascension of the Sun

• Declination
• At the Summer Solstice, the declination of the Sun =  23½o
• At either of the Equinox days, the declination of the Sun = 0o
• At the Winter Solstice, the declination of the Sun = -23½o
• Right Ascension
• At the Vernal Equinox, the RA of the Sun = 0 h (equivalent to 0o)
• At the Summer Solstice, the RA of the Sun =  6 h (equivalent to 90o)
• At the Autumnal Equinox, the RA of the Sun =  12 h (equivalent to 180o)
• At the Winter Solstice, the RA of the Sun = 18 h (equivalent to 270o)

#### Circumpolar Stars

• Altitude of the North Celestial Pole = your latitude
• at the North Pole the North Celestial Pole is directly above you
• on the Equator, the North Celestial Pole is on the northern horizon
• Declination of the North Celestial Pole = 90o
• Within the course of a day, circumpolar stars make a circle around the North Celestial Pole, never rising or setting. They can be seen every night of the year.
• Declination of the star which just touches the north point on the horizon = 90 - latitude
• Stars with declination between (90 - latitude) and -(90 - latitude) will rise and set. They can be seen on at least one night per year
• Stars with declination less than -(90 - latitude) will never rise. They cannot be seen at this latitude.

#### Rotation of the Celestial Sphere

• All objects on the Celestial Sphere move across the sky at a rate of 15o per hour
• This includes the Sun and the planets, whose motion relative to the Celestial Sphere is too slow to be noticed during the course of a single day
• The exception is the Moon, because it is so close to us.
• More detailed description

#### Motion of the Moon

• Because of its orbit around the Earth the Moon moves relative tot he Celestial Sphere at a rate of about 12o per day, or about ½o per hour.
• When combined with the rotation of the Celestial Sphere, the Moon moves across the sky at a rate of 14½o per hour.
• More detailed description

#### Phases of the Moon

• The phase of the Moon is defined by the Moon's elongation, the angle between a line from the Earth to the Moon and one from the Earth to the Sun
• if the elongation = 0o  then it is a new Moon
• if the elongation is between 0o and 90o the it is a crescent Moon
• if the elongation is 90o the it is a quarter Moon
• if the elongation is between 90o and 180o the it is a gibbous Moon
• if the elongation is 180o the it is a full Moon
• if the Moon follows the Sun across the sky then the phase of the Moon is waxing

#### Time of Day

• If the Sun is on the eastern horizon, then it is sunrise, nominally 6 am on an equinox day
• If the Sun is at its highest point, directly to the south of you, then it is noon
• If the Sun is on the western horizon, then it is sunset, nominally 6 pm on an equinox day
• If the Sun is at its lowest point, directly below you on the other side of the Earth, then it is midnight