Celestial Sphere Jan 2011

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celestial sphere_mnk 1

Celestial sphere



2

•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



3



4



5

• 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.



6










γ

ZNcp

NS




γ

ZNcp

NS

δN




γ

ZNcp

NS

α

90-α




γ

ZNcp

NS

δS

90-(-δS )

α

90- α




γ

ZNcp

NS

δN

(90-δN )

α

R.A.

90- α




Z

Ncp

S

90-φN

90-δN90 - α




P

Z



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.



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



MNK 18

Spherical Triangle ABC

AB

C

c

ab



MNK 19

A

B Ca

c b



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.



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



MNK 22

A

BC

Spherical triangle ABC

b

c

a

O



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



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



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



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



MNK 27

Astronomical Triangle pzs

Z

P (NCP)

S (star)

ZS = 90o - α = ZENITH DISTANCESP = 90o - δ = POLAR DISTANCE

ZP = 90o - φ = COLATITUDE



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



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



MNK 30

Astronomical Triangle ABC

A

B

C

AC = 90o - α = Zenith distanceBC = 90o - δ = Polar Distance

AB = 90o - φ = COLATITUDE



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



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



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



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



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.



MNK 36

Right angle Astronomical triangle

A

B

C

AC

BC

AB

90 o



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)



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 )

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Celestial Sphere Jan 2011

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