1. Latitude and longitude
You get a great circle when you cut through the centre
The equator is defined in terms of the poles, which is defined in terms of the rotation of the Earth
Same latitude: circles of latitude (or parallels of latitude) – small circles
Same longitude: meridians – great semi-circles
Prime meridian: meridian through Greenwich
Latitude defined by nature, longitude by politics (in 1884)

2. Angles
Distances on a sphere are measured in degrees and minutes and seconds of arc
1 degree = 60 min. of arc, 1° = 60’
1 min. of arc = 60 sec. of arc, 1’ = 60’’
1° 30’ = 1.5 °
Two great circles intersect at two antipodal points, and halfway between those two points the circles are maximally apart with distance equal to the angle between them

3. Different parts of the Earth
North Pole, lat = 90°N
Arctic region lat > 66.5°N, midnight sun
Arctic circle, lat = 66.5°N
North temperate zone, 23.5°N < lat < 66.5°N
Tropic of Cancer, lat = 23.5°N
Northern part of the tropics, 0°N< lat < 23.5°N, Sun can be in zenith (directly overhead)
Equator, lat = 0

4. Different parts of the Earth 2
Southern part of the tropics, 0°S< lat < 23.5°S, Sun can be in zenith (directly overhead)
Tropic of Capricorn, lat = 23.5°S
South temperate zone, 23.5°S < lat < 66.5°S
Antarctic circle, lat = 66.5°S
Antarctic region lat > 66.5°S, midnight sun
South Pole, lat = 90°S
The tropics is the region between the Tropics

5. The Celestial Sphere
Horizon: great circle that separates the visible part of the celestial sphere from the invisible (local)
Can also think of it as the tangent plane of the observer
The horizon circle is the intersection of the horizon plane with the celestial sphere

6. Celestial Sphere 2
North celestial pole (NCP): point directly above the north pole (global)
North point: point on the horizon straight north (local)
Zenith: point directly above (local)
Nadir: point directly below; invisible (local)

7. Celestial Sphere 3
Celestial equator: great circle above the earth’s equator (global)
Declination: distance from the celestial equator
(Celestial) meridian: great circle through the north point, zenith and the south point (local)

8. Celestial Sphere 4

9. Celestial Sphere 5
Infinitely many meridians on the Earth – only one in the Sky
a.m. = ante meridiem, p.m. = post meridiem

10. Celestial Sphere 6
Vertical circle: great circle perpendicular to the horizon
Prime vertical: vertical circle through the west point, zenith and the east point
Notice that the prime vertical and the meridian intersect perpendicularly at the zenith

11. The Celestial Sphere 7
The altitude (distance from the horizon) of the NCP = the latitude of the observer
If we use negative latitude for the southern hemisphere, this formula holds there, too

12. The Celestial Sphere 8
The angle between the celestial equator and the southern part of the horizon = 90 - the latitude of the observer, which is called the colatitude
This also holds for the southern hemisphere, where the colatitude is larger than 90

13. The Celestial Sphere 9
The altitude of the intersection between the celestial equator and the meridian = colatitude
If we measure the altitude from the south point and interpret an altitude of x > 90 as an altitude of 180 – x from the north, this will also holds for the southern hemisphere

14. Motion of the Stars
Stars move westward (across the visible part of the celestial sphere)
The daily (diurnal) path of the stars are parallel to the celestial equator
The stars cross the horizon at an angle equal to the colatitude of the observer

15. Circumpolar Stars
Circumpolar stars are stars that are so close to the pole that they never set
In Singapore there are no circumpolar stars
On the north pole, all stars are circumpolar
A star is circumpolar if the distance from the pole is less than the latitude

16. Circumpolar Stars 2
For an observer on the northern hemisphere, stars inside a disc around the NCP with radius equal to the observer’s latitude will never set
Stars inside a disc around the SCP with radius equal to the observer’s latitude will never rise
Stars in the middle will rise and set
Circumpolar does not mean visible all the time; a star is only visible if the Sky is dark and it is above the horizon