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

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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.

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New Delhi

CalcuttaDwarka

69oE 78oE 89oE

Example

Aldeberan (Taurus) = 69oE

Rigel (Orion) = 78oE

Betelgeuse (Orion) = 89oE

Aldeberan

Betelgeuse

Rigel

Sirius

Procyon Orion

Method – lie “on your back”

look at the stars and visualize

the locations on the globe

(otherwise, it’s a mirror image)

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Dwarka

New Delhi

Calcutta

69oE78oE89oE

Aldeberan

Betelgeuse

Rigel

Orion

Example

Aldeberan (Taurus) = 69oE - Dwarka

Rigel (Orion) = 78oE – New Delhi

Betelgeuse (Orion) = 89oE - Calcutta

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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

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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

θ

Rising/setting angle is (90o – Latitude) due

east/west – along celestial equator

Angles are smaller the further N/S one goes

Relation between Azimuth, Latitude and Declination of

rising and setting stars

)cos(

)sin()cos(

L

dRz

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 )cos()sin( Ld

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.

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)

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Deneb Vega

Altair

Antares

Scorpio

Summer triangle and Antares

Antares is only

visible for a short

period (hours) in

mid summer.

Declination = 26oS

Good candidate for a

horizon grazing star in

the summer

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Altair

Vega

Deneb

Summer

Triangle

Cygnus/

Northern

Cross

Summer triangle, northern cross (Cygnus)

Vega (Decl = 39oN) and Deneb (Decl = 45o) straddle zenith

in Boston (Latitude = 42o), Altair is 9o N

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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 as

the basis for a “clock”

(project)

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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), followed

by hunting dogs Canis Minor (Procyon = 5N, 115E) and

Canis Major (Sirius = 17S and 101E)

Mintaka – right star

in belt is on the equator

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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 arc

Pollux (28N, 115E) is readily recognized with twin Castor

Regulus (12N, 152E)

marks start of sparsely populated

region of stars in N. hemisphere –

closest is Arcturus (142W)

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