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 systemBased 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 seenHorizon = plane (0o)Altitude = angle above the horizon to an object (star, sun, etc)(range = 0o to 90o)

Azimuth = angle fromtrue north (clockwise)to the perpendicular arc from star to horizon(range = 0o to 360o)

Note: lines of azimuthconverge at zenith

The arc in the sky from azimuth of 0o to 180o

is called the local meridian

Point of view of the observer

LatitudeLatitude – angle from the equator (0o) north (positive) orsouth (negative) to a point on the earth – (range = 90o = northpole to – 90o = south pole). 1 minute of latitude is always = 1 nautical mile (1.151 statute miles)

Note: It’s more common to expressLatitude as 26oS or42oN

Longitude

Longitude = angle from the prime meridian (=0o) parallelto 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 longitudeis expressed as West, but this isinconsistent withmath conventions.Avoid confusion:40oW or 40o E

Comments on longitude

Location of prime meridian is arbitrary = Greenwichobservatory 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 thereis a technique, called the lunar method that relies on the factthat the moon moves ½ of a degree per hour.

Celestial coordinates - some definitions

North celestial pole = point in sky directly above north poleon earth (i.e. zenith of north pole)South celestial pole = zenith of south pole on earth

Celestial equator – circlesurrounding equator on earth

Ecliptic – path followedby the sun through thesky over the course ofthe year against a “fixed” background ofstars

Declination – angle from celestial equator (=0o), positivegoing 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 expressedas a time going east – 0 to24 hours is 360o

“Prime meridian” – pointwhere sun is located at the vernal equinox (spring)(called vernal equinoctialcolure)

Declination and “star pillars”

Declination “maps” onto latitude – At some point a star of a given declination will pass over the zenithat a point on the earth at its corresponding latitude.

This happens once every24 hours

Alternative to Right Ascension

Sidereal Hour Angle (SHA) - same as RA, except measuredin degrees, going from 0 to 360o – conversion is straightforward

Note: RA is/was usefulfor 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 correspondsto 0o for SHA

Same as SHA for 0o to 180o and (360o – SHA) for valuesof SHA from 180o to 360o

Why? Easy to remember, and allows you to associatestar coordinates with pointson earth. Makes it easier tovisualize and memorize. Also – declination and latitudego together.

New Delhi

CalcuttaDwarka

69oE 78oE 89oE

Example

Aldeberan (Taurus) = 69oERigel (Orion) = 78oEBetelgeuse (Orion) = 89oE

Aldeberan

Betelgeuse

Rigel

Sirius

Procyon Orion

Method – lie “on your back”look at the stars and visualizethe locations on the globe (otherwise, it’s a mirror image)

Dwarka

New Delhi

Calcutta

69oE78oE89oE

Aldeberan

Betelgeuse

Rigel

Orion

Example

Aldeberan (Taurus) = 69oE - DwarkaRigel (Orion) = 78oE – New DelhiBetelgeuse (Orion) = 89oE - Calcutta

Can associate star coordinates with latitude andLongitude of locations on earth

Note: don’t expect alignment with any star – this is justa 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 atright angles to theHorizon.

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 arcsparallel to the celestialequator

Paths of stars as seenfrom the N. Arctic Circle66o N – few stars rise and set – most make completecircles

θ

Rising/setting angle is (90o – Latitude) dueeast/west – along celestial equator

Angles are smaller the further N/S one goes

Relation between Azimuth, Latitude and Declination ofrising and setting stars

)cos(

)sin()cos(

L

dRz

Where Rz = rising azimuthd = declinationL = Latitude

So – at equator, L=0, cos(L) = 1, rising azimuth is thedeclination of the star – exploited by Polynesians instar 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, knowingthe star’s declination.

Notes on azimuth – when )cos()sin( Ld Then star is either circumpolar or below the horizonExample – at latitude 45oN, cos(L)=0.707, the star Capella (declination = 46o) just becomes circumpolarThen 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 latitudefrom horizon grazing stars.

Horizon (est)

Min. star height

Polar distance =(90o – Declination)

Latitude = (polar distance – minimum height)

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.

Big dipper

Arcturus

Spica

“Arc to Arcturus, spike to Spica”

After sunset: Spring/summer

Arcturus (Decl = 19oN)and Spica (Decl = 11oS)“alone” in this part of the sky (“longitude” = 146oW and 159oW respectively)

Deneb Vega

Altair

Antares

Scorpio

Summer triangle and Antares

Antares is only visible for a shortperiod (hours) in mid summer.Declination = 26oS

Good candidate for ahorizon grazing star inthe summer

Altair

Vega

Deneb

SummerTriangle

Cygnus/NorthernCross

Summer triangle, northern cross (Cygnus)

Vega (Decl = 39oN) and Deneb (Decl = 45o) straddle zenithin Boston (Latitude = 42o), Altair is 9o N

Dubhe

Schedar

Cassiopeia

Big dipper/Ursa major

Polaris

Finding Polaris from the big dipper

Schedar (Decl = 56o) and Dubhe (Decl = 62o)are circumpolar for Boston

Also can be used asthe basis for a “clock”(project)

Aldeberan

Betelgeuse

Rigel

Sirius

Procyon Orion

Constellation story about Orion

Pleiades

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), followedby hunting dogs Canis Minor (Procyon = 5N, 115E) andCanis Major (Sirius = 17S and 101E)

Mintaka – right starin belt is on the equator

Time lapse image of Orion

Sirius

Betelgeuse

Rigel

Arcturus

Regulus

Leo Pollux

Gemini

Procyon

Late winter/early spring constellations

Pollux/Procyon line (115E) forms good north-south arcPollux (28N, 115E) is readily recognized with twin Castor

Regulus (12N, 152E) marks start of sparsely populatedregion of stars in N. hemisphere –closest is Arcturus (142W)