Basics of Celestial Navigation - stars Coordinate systems Observer based – azimuth and altitude Earth based – latitude and longitude Celestial – declination and right ascension (or sidereal hour angle) Relationship among three – star pillars Motions of the stars in the sky Major star groupings

Comments on coordinate systems All three are basically ways of describing locations on a sphere – inherently two dimensional Requires two parameters (e.g. latitude and longitude) Reality – three dimensionality Height of observer Oblateness of earth, mountains Stars at different distances (parallax) What you see in the sky depends on Date of year Time Latitude Longitude Which is how we can use the stars to navigate!!

Altitude-Azimuth coordinate system Based on what an observer sees in the sky. Zenith = point directly above the observer (90o) Nadir = point directly below the observer (-90o) – can’t be seen Horizon = plane (0o) Altitude = angle above the horizon to an object (star, sun, etc) (range = 0o to 90o) Azimuth = angle from true north (clockwise) to the perpendicular arc from star to horizon (range = 0o to 360o) Note: lines of azimuth converge at zenith

The arc in the sky from azimuth of 0o to 180o is called the local meridian

Point of view of the observer

Latitude Latitude – angle from the equator (0o) north (positive) or south (negative) to a point on the earth – (range = 90o = north pole to – 90o = south pole). 1 minute of latitude is always = 1 nautical mile (1.151 statute miles) Note: It’s more common to express Latitude as 26oS or 42oN

Longitude Longitude = angle from the prime meridian (=0o) parallel to the equator to a point on earth (range = -180o to 0 to +180o) East of PM = positive, West of PM is negative. Distance between lines of longitude depend on latitude!! Note: sometimes positive longitude is expressed as West, but this is inconsistent with math conventions. Avoid confusion: 40oW or 40o E

Comments on longitude Location of prime meridian is arbitrary = Greenwich observatory in UK 1 minute of longitude = 1 nautical mile * cosine(latitude) Lines of longitude converge at the north and south poles To find longitude typically requires a clock, although there is a technique, called the lunar method that relies on the fact that the moon moves ½ of a degree per hour.

Celestial coordinates - some definitions North celestial pole = point in sky directly above north pole on earth (i.e. zenith of north pole) South celestial pole = zenith of south pole on earth Celestial equator – circle surrounding equator on earth Ecliptic – path followed by the sun through the sky over the course of the year against a “fixed” background of stars

Declination – angle from celestial equator (=0o), positive going north (north celestial pole = + 90o), negative going south (south celestial pole = - 90o) Right ascension (RA) – angle from celestial “prime meridian” – equivalent of celestial longitude RA – typically expressed as a time going east – 0 to 24 hours is 360o “Prime meridian” – point where sun is located at the vernal equinox (spring) (called vernal equinoctial colure)

Declination and “star pillars” Declination “maps” onto latitude – At some point a star of a given declination will pass over the zenith at a point on the earth at its corresponding latitude. This happens once every 24 hours

Alternative to Right Ascension Sidereal Hour Angle (SHA) - same as RA, except measured in degrees, going from 0 to 360o – conversion is straightforward Note: RA is/was useful for navigation with clocks

As with longitude, the actual angular width between lines of SHA shrinks with higher declination as Cosine(declination)

John Huth’s alternative to SHA, RA Use same convention as for terrestrial longitude, with positive and negative angles. Prime meridian corresponds to 0o for SHA Same as SHA for 0o to 180o and (360o – SHA) for values of SHA from 180o to 360o Why? Easy to remember, and allows you to associate star coordinates with points on earth. Makes it easier to visualize and memorize. Also – declination and latitude go together.

Aldeberan (Taurus) = 69oE Rigel (Orion) = 78oE Example Aldeberan (Taurus) = 69oE Rigel (Orion) = 78oE Betelgeuse (Orion) = 89oE 69oE 78oE 89oE Aldeberan Betelgeuse Rigel Sirius Procyon Orion New Delhi Calcutta Dwarka Method – lie “on your back” look at the stars and visualize the locations on the globe (otherwise, it’s a mirror image)

Aldeberan (Taurus) = 69oE - Dwarka Rigel (Orion) = 78oE – New Delhi Example Aldeberan (Taurus) = 69oE - Dwarka Rigel (Orion) = 78oE – New Delhi Betelgeuse (Orion) = 89oE - Calcutta 89oE 78oE 69oE New Delhi Aldeberan Betelgeuse Calcutta Dwarka Orion Rigel

Can associate star coordinates with latitude and Longitude of locations on earth Note: don’t expect alignment with any star – this is just a way to memorize coordinates

Important Point Mariners had to/have to rely on tables for star coordinates You can memorize major navigational star coordinates and eliminate tables Helps identify stars, too On a desert island, with only a watch, can identify latitude and longitude – along with your memory! Tell that to the creators of “Lost”!!

Mapping of three coordinate systems onto each other

How stars move through the sky Stars move in arcs that parallel the celestial equator – angle perpendicular to celestial equator is the declination Star move across the sky at 15o per hour (4 minutes per degree) Each day star positions move 1o west Stars on the celestial equator rise and set with angles of (90o – Latitude) Some stars are “circumpolar” – never set

Star paths in the sky form arcs in the sky At the equator, stars rise and set at right angles to the Horizon.

At Boston (41oN), stars due east will rise and set at an angle (90o –Latitude) = 49o with respect to the horizon (i.e. on celestial equator) Stars always move in arcs parallel to the celestial equator

Paths of stars as seen from the N. Arctic Circle 66o N – few stars rise and set – most make complete circles

Rising/setting angle is (90o – Latitude) due east/west – along celestial equator Angles are smaller the further N/S one goes θ

Relation between Azimuth, Latitude and Declination of rising and setting stars Where Rz = rising azimuth d = declination L = Latitude So – at equator, L=0, cos(L) = 1, rising azimuth is the declination of the star – exploited by Polynesians in star compasses (near the equator cos(L) close to 1 Can use this to find latitude, if you’re willing to do the math, and find the azimuth of a rising star, knowing the star’s declination.

Notes on azimuth – when Then star is either circumpolar or below the horizon Example – at latitude 45oN, cos(L)=0.707, the star Capella (declination = 46o) just becomes circumpolar Then cos(Rz) is just slightly greater than 1. Largest rising/setting angles for Rz = 90/270 degrees (along celestial equator)

Circumpolar stars – never set

Knowing a star’s declination, can get latitude from horizon grazing stars. Latitude = (polar distance – minimum height) Polar distance = (90o – Declination) Min. star height Horizon (est)

Some star groupings If you can locate stars and know the declination you can find your latitude. With a watch, and SHA (or “stellar longitude”), you can find your longitude (must know date). Clustering into constellations and their stories help locate stars by name.

“Arc to Arcturus, spike to Spica” After sunset: Spring/summer Big dipper Arcturus (Decl = 19oN) and Spica (Decl = 11oS) “alone” in this part of the sky (“longitude” = 146oW and 159oW respectively) Arcturus Spica

Summer triangle and Antares Deneb Vega Altair Antares is only visible for a short period (hours) in mid summer. Declination = 26oS Good candidate for a horizon grazing star in the summer Antares Scorpio

Summer triangle, northern cross (Cygnus) Deneb Vega Summer Triangle Cygnus/ Northern Cross Altair Vega (Decl = 39oN) and Deneb (Decl = 45o) straddle zenith in Boston (Latitude = 42o), Altair is 9o N

Finding Polaris from the big dipper Schedar Schedar (Decl = 56o) and Dubhe (Decl = 62o) are circumpolar for Boston Cassiopeia Polaris Also can be used as the basis for a “clock” (project) Dubhe Big dipper/Ursa major

Constellation story about Orion Pleiades Aldeberan Betelgeuse Rigel Sirius Procyon Orion Mintaka – right star in belt is on the equator Winter constellations – Zeus’ daughters, Pleiades (24N, 57E) are guarded by Taurus (Aldeberan = orange eye – 17N, 69E), from Orion, the hunter (Betelgeuse = 7N, 89E, Rigel 8S,78E), followed by hunting dogs Canis Minor (Procyon = 5N, 115E) and Canis Major (Sirius = 17S and 101E)

Time lapse image of Orion Betelgeuse Arcturus Sirius Rigel

Late winter/early spring constellations Pollux/Procyon line (115E) forms good north-south arc Pollux (28N, 115E) is readily recognized with twin Castor Gemini Leo Pollux Regulus Procyon Regulus (12N, 152E) marks start of sparsely populated region of stars in N. hemisphere – closest is Arcturus (142W)