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8/4/2019 Celestial Sphere Jan 2011
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celestial sphere_mnk 1
Celestial sphere
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•It is an imaginary sphere covering the whole universe.
•The radius is infinity and the earth as well as the solar
system is at the center of this sphere.•This is used widely in astronomy where all the heavenly
bodies are projected on to the surface of the sphere.
•All the points and terms used to identify this sphere are
similar to the points on the earth.
•Example: North pole – North celestial pole
Earth equator - Celestial equator
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• Assuming that the earth is at the center of this sphere,
then a point exactly above the observer is called zenith.
• Another point exactly below and opposite to the zenith iscalled nadir.
• All the stars or heavenly bodies are located on the
surface of this imaginary sphere called the celestial
sphere.
• The plane which passes the observer and perpendicular
to the axis (zenith and nadir) is the horizon.
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γ
ZNcp
NS
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γ
ZNcp
NS
δN
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celestial sphere_mnk 11
γ
ZNcp
NS
α
90-α
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celestial sphere_mnk 12
γ
ZNcp
NS
δS
90-(-δS )
α
90- α
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celestial sphere_mnk 13
γ
ZNcp
NS
δN
(90-δN )
α
R.A.
90- α
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celestial sphere_mnk 14
Z
Ncp
S
90-φN
90-δN90 - α
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celestial sphere_mnk 15
P
Z
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MNK 16
Spherical Triangle
• Spherical triangle is a triangle that is formed from three
great circle on the spherical surface.
• The concept of the formation of spherical triangle :-
from the diagram, the three great circle
EABDE, GACFG and HBCJH.
• Great circle EABDE intersect with the great circle GACFG
at point A
• Great circle EABDE intersect with the great circle HBCJHat point B
• Great circle HBCJH intersect with great circle GACFG at
point C.
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MNK 17
• The spherical triangle ABC is a triangle that is formed by
these point A,B,C.
E
A
B
C
F
G
H
J
Oa
b
c
G
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MNK 18
Spherical Triangle ABC
AB
C
c
ab
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MNK 19
A
B Ca
c b
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MNK 20
A
B
C
A
B Ca
c b
b
c
a
Many shaper of spherical triangle but all sides must be
parts of great circle.
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MNK 21
Characteristic of spherical Triangle :
• All the sides of a spherical triangle are in angulardegree. Sides of a plane triangle are in linear
measurement.
• There are six parts involved:-
3 angular sides : a, b and c ;
3 included angles : A, B, and C.
Spherical triangle
Plane triangle
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MNK 22
A
BC
Spherical triangle ABC
b
c
a
O
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MNK 23
NCP
SCPE
W
Nadir
Z
Observer
Meridian
Let the observer be at latitude φ (about 55 deg)
Altitude α NCP = latitude observer φ
φ
Horizon
li i Ci l δ i i l h h C S SC
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MNK 24
Declination Circle δN is a great circle that passes the NCP, Star, SCP
and is perpendicular with the celestial equator
NCP
SCPE
W
δ
Z
Declination circle
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MNK 25
α = altitude of the star is the vertical angle or
elevation angle from the horizon to the star alongthe arc of the altitude circle.
δ= declination of the star is the angle measured from
celestial equator to the star along the arc of thedeclination circle.
ϕ= observer latitude = altitude NCP
Altit d i l i t i l th t th t
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MNK 26
Altitude circle is a great circle that passes the star,zenith and nadir. This circle is perpendicular withthe horizontal plane.
NCP
SCPE
W
Z
αHorizon
Nadir
Altitude circle
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MNK 27
Astronomical Triangle pzs
Z
P (NCP)
S (star)
ZS = 90o - α = ZENITH DISTANCESP = 90o - δ = POLAR DISTANCE
ZP = 90o - φ = COLATITUDE
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MNK 28
• Referring to the Astronomical triangle (PZS), the
related angles :
sides angles
• ZS = 90o - α = ZENITH DISTANCE
SP = 90o - δ = POLAR DISTANCE
• ZP = 90o - φ = COLATITUDE
Included angles
PZS = AZIMUTH
ZPS = t , ( angular time )
ZSP = PARALATIC ANGLE
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MNK 29
Spherical Trigonometry Formula:
• Several important formula that can be used
depending the suitation.
(a) Sine Formula
( sine A / sine a ) = ( sine B / sine b ) = ( sine C / sine c )
i.e: sine of the included angle is propotional to the sine of the
opposite side
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MNK 30
Astronomical Triangle ABC
A
B
C
AC = 90o - α = Zenith distanceBC = 90o - δ = Polar Distance
AB = 90o - φ = COLATITUDE
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MNK 31
(b) Cosine Formula
• Cos a = Cos b Cos c + Sin b Sin c Cos A
Cos b = Cos a Cos c + Sin a Sin c Cos BCos c = Cos a Cos b + Sin a Sin b Cos C
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MNK 32
(c) Four Parts Formula
• This formula is a manupulation of SINE and COSINE.General formula :
• Cos (IS) Cos (IA) = Sin(IS) Cot (OS) - Sin (IA) Cot (OA)
where-
IS = Inner Side
IA = Inner Angle
OS = Outer Side
OA = Outer Angle
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MNK 33
A
B
C
ba
c
Cos (IS) Cos (IA) = Sin(IS) Cot (OS) - Sin (IA) Cot (OA)
IS = Inner Side
IA = Inner AngleOS = Outer Side
OA = Outer Angle
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MNK 34
A
B
C
ba
c
Cos (IS) Cos (IA) = Sin(IS) Cot (OS) - Sin (IA) Cot (OA)
IS = Inner Side
IA = Inner AngleOS = Outer Side
OA = Outer Angle
Example: Given b, c, A in a spherical triangle.
Compute the angle C
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MNK 35
Formula for right angle spherical triangle:
Defination:
A right angle spherical triangle is a triangle where
one of the included angle is 90o and fullfill
Napier’s Anologies.
• According to Napier anolagies, the spherical
triangle must be divided into five parts that is two
parts of adjacent sides angle with the right angle
between them. Three parts will be opposite to theangle.
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MNK 36
Right angle Astronomical triangle
A
B
C
AC
BC
AB
90 o
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MNK 37
• Right angle is C then 2 adjacent sides is CB ( i.e. a )
and CA( i.e. b).
• Three parts opposite to C is given as suplement angles toA, c and B is ( 90o - A ) ,( 90o - c ) and ( 90o - B ).
• There will be two formula in Napier’s Anologies :
1) SINE ( middle part) = product TAN (adjacent parts)
2) SINE( middle part) = prtoduct COS(opposite part)
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MNK 38
Example :
Given the value a , B dan C = 90o. Compute the value b.
Solution :
• Using Napier’s anologies :-
a = middle part
b and ( 90o - B ) = adjacent sides
Then:-
sin a = tan b tan ( 90o - B )
• ortan b = sin a
tan ( 90o - B )