Transcript Three Coordinate Systems

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
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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.
Example
69oE
Aldeberan (Taurus) =
Rigel (Orion) = 78oE
Betelgeuse (Orion) = 89oE
69oE
78oE
89oE
Aldeberan
New Delhi
Betelgeuse
Orion
Procyon
Rigel
Sirius
Method – lie “on your back”
look at the stars and visualize
the locations on the globe
(otherwise, it’s a mirror image)
Calcutta
Dwarka
Example
69oE
Aldeberan (Taurus) =
- Dwarka
Rigel (Orion) = 78oE – New Delhi
Betelgeuse (Orion) = 89oE - Calcutta
89oE
78oE
69oE
New Delhi
Aldeberan
Betelgeuse
Calcutta
Orion
Rigel
Dwarka
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
sin(d )
cos(Rz ) 
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:
Spring/summer
Arcturus (Decl = 19oN)
and Spica (Decl = 11oS)
“alone” in this part of
the sky (“longitude” =
146oW and 159oW
respectively)
Big dipper
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
Also can be used as
the basis for a “clock”
(project)
Cassiopeia
Polaris
Dubhe
Big dipper/Ursa major
Constellation story about Orion
Pleiades
Aldeberan
Betelgeuse
Orion
Procyon
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)
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
Regulus (12N, 152E)
marks start of sparsely populated
region of stars in N. hemisphere –
closest is Arcturus (142W)
Procyon