2. The Celestial Sphere Goals Goals: 1. Gain familiarity with the basic co- ordinate systems used in astronomy. 2. Tackle simple problems in practical astronomy involving timekeeping and star positions. 3. Examine the timekeeping systems in current use.

Astronomical Co-ordinate Systems: All co-ordinate systems constructed on spheres are defined by a fundamental great circle (FGC) and a reference point (RP) on the FGC. All co-ordinates are angles measured: (i) between great circles perpendicular to the FGC, or (ii) between small circles parallel to the FGC. The FGC has two poles, and the RP is defined in a variety of ways, which accounts for slight differences from one system to another.

Terrestrial Co-ordinates. FGC = Earth’s equator: poles the North Pole and South Pole. RP = crossing point of equator by the Greenwich meridian. Co-ordinates: Longitude = angle measured east and west from the Greenwich meridian. Longitude meridians are great circles. Latitude = measured north and south (not plus or minus) from the Equator. Latitude parallels are small circles. Examples: Halifax. 63º 36'.0 W, 44º 36'.0 N Vancouver. 123º 04'.2 W, 49º 09'.0 N

Horizon System. FGC = horizon, with poles the zenith and nadir. RP = north point. Co-ordinates: azimuth = angle measured through east from 0º to 360 . Azimuth circles are great circles. altitude = measured from horizon towards zenith (positive) or nadir (negative) from +90º to –90º. Alternate: zenith distance, z = 90º – altitude.

Meridian = NS line running through zenith. Prime vertical = EW line running through zenith. Use. Airport runways are designated by azimuth  10°, i.e. runway 32 aligns along azimuth 320° magnetic. Air mass, for correction of photometry, is calculated from zenith distance z. Example Question. In what directions does runway 05/23 run? Solution: Given the name of the runway, 05/23, it must run along azimuth directions 50° (northeast) and 230° (southwest), which are 180° apart.

Equatorial System. FGC = celestial equator (CE, projection on the sky of Earth’s equator), with poles the north and south celestial poles, NCP and SCP. RP = intersection point of meridian with CE (observer-oriented), or vernal equinox γ (sky- oriented). Co-ordinates: declination = angle measured north or south of CE from 0º to +90  and  90° (δ). hour angle = angle measured west of meridian (HA), or right ascension = angle measured eastward from vernal equinox (RA).

From relations for angles associated with parallel lines, 90° – θ = 90° – , i.e. θ = .

Star Trails 1 h exposure

The orbit of Earth about the Sun and the 23½° obliquity of the ecliptic (its angle relative to the normal to Earth’s orbital plane) give rise to Earth’s seasons.

Solar insolation as a function of season.

A typical sky scene showing seasonal variations in the Sun’s diurnal motion.

RA(  ) and δ(  ) during the year are defined by the apparent motion of the Sun in the sky along the ecliptic = Sun’s apparent path, and can be calculated directly or from tables.

Solar day (24 h ) is rotation of Earth relative to Sun, sidereal day (23 h 56 m ) is rotation of Earth relative to stars.

HA and RA are measured in temporal units and are equivalent to angles. On the celestial equator: 1 h = 15°, 1 m = 15', and 1 s = 15″, with the equalities changing by cos δ with increasing declination. Because of their link to timekeeping, HA and RA are tied directly to sidereal (star) time and apparent solar time. Sidereal time (SidT)= HA(γ) Apparent solar time = HA(  ) + 12 h Now, HA(γ) = HA(*) + RA(*) = HA(  ) + RA(  ) Thus, SidT = HA(  ) + RA(  ) = Apparent solar time  12 h + RA(  )

By geometry and algebra, Sidereal Time = HA(γ) But HA(γ) = HA(*) + RA(*) So Sidereal Time = HA(*) + RA(*)

Useful values: Vernal Equinox, March 20: RA(  ) = 0 h, δ(  ) = 0° Summer Solstice, June 21: RA(  ) = 6 h, δ(  ) = +23½° Autumnal Equinox, September 23: RA(  ) = 12 h, δ(  ) = 0° Winter Solstice, December 22: RA(  ) = 18 h, δ(  ) =  23½° annual insolation The actual dates of the equinoxes and solstices slowly change with time. They were March 25, June 25, September 25, and December 25 when Julius Caesar modified the original Roman calendar system (Julian Calendar) in 46 BC.

An example. 1. Towards what directions on the co-ordinate axes must an equatorial telescope be set in order to point it towards Betelgeuse, RA = 5 h 56 m, δ = +7° 25' at 5 h sidereal time?

Solution: By definition, SidT = HA(γ) = HA(*) + RA(*)  5 h = HA(*) + 5 h 56 m And so HA(*) = 5 h – 5 h 56 m = –0 h 56 m The hour angle setting of the telescope should be set to –0 h 56 m, i.e. 56 m east of the meridian. And the declination setting of the telescope should be set to +7° 25'.

Another example. 2. Show that apparent solar time (AST) and sidereal time (SidT) are identical on the date of the autumnal equinox.

Solution: By definition, SidT = HA(γ) = HA(*) + RA(*) The Sun is also a star, so SidT = HA(  ) + RA(  ) But AST = HA(  ) + 12 h And RA(  ) = 12 h at the Autumnal Equinox. So at the Autumnal Equinox, SidT = HA(  ) + RA(  ) = HA(  ) + 12 h = AST i.e. Sidereal Time and Apparent Solar Time are identical on the date of the Autumnal Equinox.

3. When is the best time of year to observe the stars of Orion, RA = 5½ h ? Solution: The optimum time for observing any object is when it lies on the observer’s meridian at local midnight, which corresponds to 0 h local apparent solar time (LAST). i.e. LAST = HA(  ) + 12 h = 0 h (midnight) So HA(  ) = 0 h  12 h = 24 h  12 h = 12 h Orion is then on the meridian, so local sidereal time = RA(*) = 5½ h = HA(  ) + RA(  )  RA(  ) = 5½ h  HA(  ) = 5½ h  12 h = 29½ h  12 h = 17½ h The Sun is at RA = 17½ h approximately one week prior to the winter solstice, i.e. around Dec. 15.

Ordering the Planets Outwards ObjectMotion relative to the Stars Starsinfinite Saturn29.30 years Jupiter11.86 years Mars1.88 years Sun365¼ days Venus225 days Mercury88 days Moon27½ days

Saturn governs the 1 st hour of the 1 st day, Jupiter the 2 nd hour, Mars the 3 rd hour, etc., and Mars the 24 th hour. The Sun then governs the 1 st hour of the 2 nd day, the Moon the 1 st hour of the 3 rd day, Mars the 1 st hour of the 4 th day, Mercury the 1 st hour of the 5 th day, Jupiter the 1 st hour of the 6 th day, and Venus the 1 st hour of the 7 th day. The days of the week are therefore: Saturn-day Sun-day Moon-day Mars-day Mercury-day Jupiter-day Venus-day

North American time zones

World time zones ─ note the peculiarities

Calendars. Apparent solar time is defined by the passage of the Sun across the sky, but civil time is more closely related to the motion of the mean Sun, a fictitious object, across the sky. Mean solar time = HA(mean Sun) + 12 h The mean Sun differs from the true Sun in the following way. The true Sun travels along the ecliptic at a rate that varies according to the distance of Earth from the Sun. The mean Sun travels along the celestial equator at a uniform rate. Additional complications arise from the use of time zones and daylight saving time.

The analemma represents the equation of time = Apparent Solar Time – Mean Solar Time.

Length (days) JulianGregorian MonthCalendarCalendar January3131 February2928 March3131 April3030 May3131 June3030 Quintilus (July)3131 Sextilus (August)3031 September3130 October3031 November3130 December3031

Important Calendar Dates Calendar EventJulianModern Vernal EquinoxMarch 25March 20 Summer SolsticeJune 25June 21 Autumnal EquinoxSept. 25Sept. 23 Winter SolsticeDec. 25Dec. 22

The year length varies according to the calendar system, which has changed from lunar calendars, through luni-solar calendars, to solar calendars, such as the Julian Calendar, Gregorian Calendar, and current modified Gregorian Calendar. Variable star studies normally cite observations according to the Julian Date, JD, measured as the number of sequential days from noon, UT, on January 1, 4713 BC (named by Joseph Scaliger after his father Julius Scaliger), or, better yet, HJD = Heliocentric Julian Date (corrected to the barycentre of the solar system). Another term, modified Julian Date, MJD = JD  , is occasionally used.

Ecliptic System. FGC = ecliptic, with poles the north and south ecliptic poles, NEP and SEP. RP = vernal equinox γ. Co-ordinates: celestial (or ecliptic) longitude, λ = angle measured eastward from γ from 0º to 360 . celestial (or ecliptic) latitude, β = angle measured from ecliptic. The system is useful for studies of solar system objects.

Galactic System. FGC = Galactic equator (GE), defined by the Milky Way, with poles the north and south Galactic poles, NGP and SGP. RP = direction to the Galactic centre (GC), defined by Sgr A*. Co-ordinates: Galactic longitude, l = angle measured Eastward from GC from 0º to 360 . Galactic latitude, b = angle measured north or south of GE from 0º to +90  and  90°.

Precession of the Equinoxes. Earth’s axis of rotation precesses relative to the perpendicular to its orbit because of gravitational influences by the Sun and Moon, but not in the fashion implied by the Wikipedia figure below. The sense of precession is actually opposite the sense of Earth’s rotation. The period is ~25,725 years. A top’s precession.

Precession and the location of the NCP. Note that the NCP was near the bright star Thuban near 2700 BC, when the pyramids were built, and was once near Vega, a name that means “fallen.”

Astronomical Terminology Zenith. The point in the sky directly overhead. Nadir. The point directly beneath one’s feet. Azimuth. A measurement of angle increasing from north through east. Altitude (astronomical). A measurement of angular distance from the true horizon upwards. Ecliptic. The great circle in the sky along which the Sun appears to move because of Earth’s orbit about it. Right Ascension. A celestial co-ordinate like longitude on Earth, increasing eastwards. Declination. A celestial co-ordinate like latitude on Earth, measured from the celestial equator. Celestial Equator. The projection on the celestial sphere of Earth’s equator. Celestial Sphere. The imaginary sphere centred on the observer upon which the stars appear to be projected.

Astronomical Terminology (continued) Diurnal. = daily (once a day). Insolation. The amount of sunlight falling on Earth’s surface. Constellation. A group of conspicuous stars designated by ancient star gazers. Zodiacal Constellation. A constellation lying in the band of sky around the ecliptic, where the Moon and planets are always found. Solstice. Time of greatest or smallest declination for the Sun. Equinox. Time when the Sun crosses the celestial equator. (Vernal = spring) Stellar Aberration. The apparent displacement in a star’s location in the sky of at most 20½ seconds of arc resulting from Earth’s orbital motion about the Sun at a speed of 30 km/s.