Basics of Celestial Navigation - stars

Transcription

1 Basics of Celestial Navigation - Coordinate systems stars 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

2 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!!

3 Altitude-Azimuth coordinate system Based on what an observer sees in the sky. Zenith = point directly above the observer (90 o ) Nadir = point directly below the observer (-90 o ) can t be seen Horizon = plane (0 o ) Altitude = angle above the horizon to an object (star, sun, etc) (range = 0 o to 90 o ) Azimuth = angle from true north (clockwise) to the perpendicular arc from star to horizon (range = 0 o to 360 o ) Note: lines of azimuth converge at zenith

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

5 Point of view of the observer

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

7 Longitude Longitude = angle from the prime meridian (=0 o ) parallel to the equator to a point on earth (range = -180 o to 0 to +180 o ) 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: 40 o W or 40 o E

8 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.

9 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

10 Declination angle from celestial equator (=0 o ), positive going north (north celestial pole = + 90 o ), negative going south (south celestial pole = - 90 o ) 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 360 o Prime meridian point where sun is located at the vernal equinox (spring) (called vernal equinoctial colure)

11 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

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

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

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15 John Huth s alternative to SHA, RA Use same convention as for terrestrial longitude, with positive and negative angles. Prime meridian corresponds to 0 o for SHA Same as SHA for 0 o to 180 o and (360 o SHA) for values of SHA from 180 o to 360 o 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. This slide not needed

16 Example Aldeberan (Taurus) = 69 o E Rigel (Orion) = 78 o E Betelgeuse (Orion) = 89 o E Aldeberan 69 o E 78 o E 89 o E Procyon Betelgeuse Orion This slide not needed New Delhi Rigel Dwarka Calcutta Sirius Method lie on your back look at the stars and visualize the locations on the globe (otherwise, it s a mirror image)

17 Example Aldeberan (Taurus) = 69 o E - Dwarka Rigel (Orion) = 78 o E New Delhi Betelgeuse (Orion) = 89 o E - Calcutta 89 o E 78 o E 69 o E Betelgeuse AldeberanThis slide not needed New Delhi Orion Calcutta Dwarka Rigel

18 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

19 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!!

20 Mapping of three coordinate systems onto each other

21 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 15 o per hour (4 minutes per degree) Each day star positions move 1 o west Stars on the celestial equator rise and set with angles of (90 o Latitude) Some stars are circumpolar never set

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

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

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

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

26 Relation between Azimuth, Latitude and Declination of rising and setting stars cos( R z ) sin( d) cos( L) Where R z = 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.

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

28 Circumpolar stars never set

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

30 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.

31 Arc to Arcturus, spike to Spica After sunset: Spring/summer Big dipper Arcturus (Decl = 19 o N) and Spica (Decl = 11 o S) alone in this part of the sky ( longitude = 146 o W and 159 o W respectively) Arcturus Spica This slide not needed

32 Summer triangle and Antares Deneb Vega Antares is only visible for a short period (hours) in mid summer. Declination = 26 o S Altair Antares This slide not needed Good candidate for a horizon grazing star in the summer Scorpio

33 Summer triangle, northern cross (Cygnus) Deneb Vega Summer Triangle This slide not needed Cygnus/ Northern Cross Altair Vega (Decl = 39 o N) and Deneb (Decl = 45 o ) straddle zenith in Boston (Latitude = 42 o ), Altair is 9 o N

34 Finding Polaris from the big dipper Schedar (Decl = 56 o ) and Dubhe (Decl = 62 o ) are circumpolar for Boston Schedar Cassiopeia Also can be used as the basis for a clock (project) Polaris This slide not needed Dubhe Big dipper/ursa major

35 Constellation story about Orion This slide not needed Aldeberan Pleiades Procyon Betelgeuse Orion Mintaka right star 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)

36 Time lapse image of Orion Betelgeuse Arcturus Sirius Rigel

37 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 Regulus (12N, 152E) marks start of sparsely populated region of stars in N. hemisphere closest is Arcturus (142W) Procyon