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Azimuth and Altitude – Earth Based – Latitude and Longitude – Celestial

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Azimuth and Altitude – Earth Based – Latitude and Longitude – Celestial 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 This slide not needed 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 This slide not needed 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. Example Aldeberan (Taurus) = 69oE 69oE 78oE 89oE Rigel (Orion) = 78oE Betelgeuse (Orion) = 89oE Aldeberan New Delhi Betelgeuse This slide not needed Procyon Orion Rigel Calcutta Dwarka Sirius Method – lie “on your back” look at the stars and visualize the locations on the globe (otherwise, it’s a mirror image) Example o o o Aldeberan (Taurus) = 69oE - Dwarka 89 E 78 E 69 E Rigel (Orion) = 78oE – New Delhi Betelgeuse (Orion) = 89oE - Calcutta New Delhi AldeberanThis slide not needed 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 This slide not needed 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 sin(d) cos(R ) z cos(L) 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 sin(d) cos(L) 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: Big dipper Spring/summer o Arcturus This slide Arcturus (Decl = 19 N) not needed and Spica (Decl = 11oS) “alone” in this part of the sky (“longitude” = 146oW and 159oW respectively) Spica Summer triangle and Antares Deneb Vega Altair This slide not needed Antares is only visible for a short Antares period (hours) in mid summer. Declination = 26oS Good candidate for a Scorpio horizon grazing star in the summer Summer triangle, northern cross (Cygnus) Deneb Vega Summer This slide Triangle not needed 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) Cassiopeia are circumpolar for Boston Polaris This slide Also can be used as not needed the basis for a “clock” (project) Dubhe Big dipper/Ursa major Constellation story about Orion Pleiades This slide not needed Aldeberan Betelgeuse Mintaka – right star Procyon Orion in belt is on the equator Rigel Sirius 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 This slide not needed Leo Pollux Regulus Procyon Regulus (12N, 152E) marks start of sparsely populated region of stars in N. hemisphere – closest is Arcturus (142W).
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