International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 4

Number 2

June 2010

217

1 INTRODUCTION

Assuming, that the movement of celestial bodies on

celestial sphere results only from rotary motion of

the earth, then these bodies are moving along circles,

which center is in the vicinity of closer pole and

their radius is equal to the complement of the decli-

nation to the right angle. This assumption is correct

during navigational or geodetic measurements due to

short time of their duration. When the low accuracy

of measurement is allowed (for example for the pur-

poses of celestial navigation accuracy of altitude of

0.1' and accuracy of time of 1 second is required),

then measuring series compound of several meas-

urements of the altitude or the azimuth and the time

can be equalized with straight line. The correction

for the curve of celestial latitude is taken into ac-

count in such series in methods of the astronomical

geodesy, and thanks to this it is possible to treat the-

se series as linear in relation to the center thread.

Both mentioned methods of the processing of meas-

uring data results from the tendency of reduction of

the amount of calculations connected with their pro-

cessing. In the case, when measuring data is pro-

cessed automatically, for the equalization one can

accept the path of celestial body along circle and de-

rive equation of the movement of the body in hori-

zontal coordinates system (approximating equation).

And then choose any location of the body on the cir-

cle, which data will be put to the reduction father.

2 APPROXIMATING EQUATION IN THE

FIELD OF VIEW

Celestial bodies in their daily movement should the-

oretically form the arcs of the small circles on the

celestial sphere with radiuses equal to the comple-

ment of the declination δ to the right angle and with

centers in the closer celestial pole. In the particular

case, when the body lies on the celestial equator it is

great circle and the path of the body form straight

line. The real path is influenced additionally by: the

change of refraction with the altitude of the body

and oscillations of its image, and at the measure-

ments random errors of the measurements. One uses

series n of the measurements of the position of celes-

tial body: zenith distance z

i

, the azimuth a

i

and the

time of registration t

i

appropriate for point P

i

, for

derivation of equation of the movement. Zenith dis-

tances z

i

have to be corrected for the refraction r(z

i

r

)

appropriate for z

i

r

( )

r

ii

r

ii

zrzz +=

. (1)

The variable z

i

r

is measured and burdened with

refraction, and z

i

already corrected for the value of

refraction.

The approximating equation is described by hori-

zontal coordinates z

P

(zenith distance) and a

P

(azi-

muth) of the center P of circle along which body

moves, with its radius r equal in first approximation

of its polar distance

δπ

−= 2/r

(2)

Equalization of the Measurements of the

Altitude, the Azimuth and the Time from

Observation of Passages of Celestial Bodies

P. Bobkiewicz

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: The article is describing the computational model serving equalization of the astronomical

measurements accomplished to navigational and geodetic purposes. Series of measuring data: the altitude, the

azimuth and the time from observation of passing of celestial bodies in the field of view of the observing de-

vice are input parameters to calculations. This data is burdened with random error of the measurement. The

equation of the movement of celestial body in the horizontal system is the result of the equalization. It is pos-

sible to calculate the azimuth and the altitude for the chosen moment or to fix the time of the given azimuth or

the altitude from this equation.

218

and by horizontal coordinates z

G

and a

G

and time t

G

of indicated point P

G

on this circle (Fig. 1). The

point P should theoretically agree with the pole

which is nearest to the celestial body. Additional pa-

rameters of which values are known these are a dec-

lination δ of celestial body and an angle speed of

change of right ascension v

max

equal 7,29212E-05

radian for a second result from rotation of the Earth.

Z

P (z

p

, a

p

)

W

E

S

r = π/2 - δ

O

P

G

(z

G

, a

G

, t

G

)

N

P

N

Figure 1 Parameters of the equation of the movement of celes-

tial body in the horizontal system: circle with radius r and with

centre in the point P as well as point P

G

on this circle and time

t

G

appropriate to this point.

2.1 Determination of the centre of the circle

P(z

P

, a

P

)

By the means of least square roots method, one seek

such point P on the spherical surface, so that the sum

Δ of square roots of the shortest great circle distanc-

es δ

i

between respective point P

i

(z

i

, a

i

) and circle

with the centre in the point P is minimal

min

1

2

==

∑

=

n

i

i

Δ

δ

. (3)

The function of distance δ

i

is difference of radius

r and distance r

i

of given point P

i

from centre of cir-

cle P (Fig. 2)

P

Z

P

i

z

P

Δa

O

a

P

z

i

r

δ

i

r

i

a

i

-a

P

P

N

a

i

Figure 2 The shortest great circle distances δ

i

between the point

P

i

and the circle with centre in the point P.

( )( )

PiPiPii

ii

aazzzzr

rr

−+−=

−=

cossinsincoscosarccos

δ

δ

(4)

The condition (3) is met if derivatives of varia-

bles z

P

and a

P

are equal 0

∑∑

==

=

∂

∂

=

∂

∂

n

i

P

i

n

i

P

i

az

1

2

1

2

0,0

δδ

. (5)

Differentiating (4) through variables z

P

and a

P

,

substituting to (5) and summing up for all points we

receive a pair of non-linear equations with two un-

known quantities z

P

and a

P

. It is possible to solve the

pair of equations with iteration method taking hori-

zontal coordinates of the pole nearest to celestial

body as first approximation of the centre. We re-

ceive values in demand z

P

and a

P

in the result of the

solution of the pair of equations.

2.2 Reducing measurements to time t

G

Each point P

i

is reduced to any chosen time t

G

.

This reduction is made by the rotation of the point

around determined centre of the circle for the angle

Δα

i

of the change of the right ascension for the dif-

ference of time Δt

i

between times t

G

and t

i

(Fig. 3)

ii

iGi

ΔtΔα

ttΔt

⋅=

−=

max

ν

. (6)

P

Z

P

i

z

P

ψ

i

a

P

z

i

Δα

i

γ

i

r

i

Δa

i

P

N

a

i

P

i

'

Figure 3 Reduction of points P

i

to time t

G

by the rotation of

points around P for the angle of the change of right ascension

Δα

i

.

It is necessary to calculate angle γ

i

and distance r

i

shown on figure 3 to determine coordinates of re-

duced point P

i

'(z

i

', a

i

'). Defining the angle Δa

i

as

Pii

aaΔa −=

(7)

and keeping its value in the range (0, 2π), then γ

i

and

r

i

are calculated from formulae

( )

iP

iPi

i

iPiPii

rz

rzz

Δazzzzr

sinsin

coscoscos

cos

cossinsincoscoscos

−

=

+=

γ

, (8)

in addition for Δa

i

< π

ii

γπγ

−= 2

. (9)

219

Defining the angle ψ as

iii

Δαb ⋅+=

γψ

(10)

where b is equal 1 for P lying near north pole and -1

for south pole, and then keeping its value in the

range (0, 2π), z

i

' and a

i

' of reduced point P

i

' are cal-

culated from formulae

( )

iP

iPi

i

iiPiPi

zz

zzr

aΔ

rzrzz

′

′

−

=

′

+=

′

sinsin

coscoscos

cos

cossinsincoscoscos

ψ

, (11)

in addition for ψ

i

< π

ii

aΔaΔ

′

−=

′

π

2

(12)

and then

iPi

aΔaa

′

+=

′

. (13)

2.3 Determination of the point P

G

(z

G

, a

G

) on the

circle for the time t

G

Point P

G

is made by averaging coordinates of re-

duced points P

i

'(z

i

', a

i

') (Fig. 4).

Figure 4 Point P

G

calculated as the average of reduced points

P

i

'(z

i

', a

i

').

Assuming that P

G

is in the considerable distance

from the Zenith and from the Nadir compared with

the error of position of P

i

', then the mean zenith dis-

tance and mean azimuth are calculated from formu-

lae

∑∑

==

′

=

′

=

n

i

i

n

i

i

aazz

11

. (14)

The standard deviation of position of the meas-

urement point along the vertical circle σ

z

, along al-

mucantar σ

l

and on the plane m

i

are calculated from

22

1

2

1

2

11

sin

lzi

n

i

i

l

n

i

i

z

ii

iiii

m

n

Δl

n

Δz

zΔal

aaΔazzΔz

σσ

σσ

+=

−

=

−

=

⋅=

′

−=

′

−=

∑∑

==

(15)

and the standard deviation of position of P

G

on the

plane

n

m

m

i

=

. (16)

Function (3) of determining of the centre P of the

circle is sensitive in the square roots of the distance

between the point P

i

and the arc of the circle but the

point P

G

is calculated as the average (14), it is pro-

portionally to the distance from mean point, so the

point P

G

doesn’t lie on the circle (Fig. 4). One can

move this point onto the circle, by projection along

radius r. Or one can determine new radius r' and

new angle γ' from (17), assuming that the arithmetic

mean is a better estimator for the measurement of

passage of celestial bodies then the square roots av-

erage.

( )

rz

rzz

Δazzzzr

aaΔa

P

PG

PGPG

PG

sinsin

coscoscos

cos

cossinsincoscoscos

−

=

′

+=

′

−=

γ

(17)

For Δa < π from (17) (value Δa kept in the range

(0, 2π))

γπγ

′

−=

′

2

. (18)

3 DETERMINATION OF TIME AND

COORDINATES FROM THE EQUATION OF

THE MOVEMENT

3.1 Calculation of coordinates on the circle for the

given time t

i

Having the point P

G

with coordinates z

G

and a

G

and

its time t

G

on the circle with the centre in the point

P(z

P

i a

P

) and with radius r', it is possible to deter-

mine coordinates z

i

and a

i

of the other point P

i

on

this circle for any given time t

i

, the same way as

measurement points were reduced to time t

G

– for-

mulae (6), (10)-(13). Appropriate formulae have the

form

ii

Gii

ΔtΔα

ttΔt

⋅=

−=

max

ν

, (19)

ii

Δαb ⋅+

′

=

γψ

, (20)

220

( )

iP

iP

i

iPPi

zz

zzr

Δa

rzr

zz

sinsin

coscoscos

cos

cossinsincoscos

cos

−

′

=

′

+

′

=

ψ

, (21)

in addition for ψ

i

< π

ii

ΔaΔa −=

π

2

, (22)

iPi

Δaaa +=

. (23)

3.2 Calculation of the time t

i

of reaching the zenith

distance z

i

Converting (21) with taking into consideration (20)

and (19), it is possible to calculate the appropriate

time t

i

for any given zenith distance z

i

. ψ value from

formula

rz

rzz

P

Pi

′

′

−

=

sinsin

coscoscos

cos

ψ

(24)

corresponds with two values of the angle ψ

i

from the

formula (20)

ψ

π

ψ

ψψ

−=

=

2

i2

i1

, (25)

so substituting each of them to

Gi

t

b

t +

⋅

′

−

=

max

ν

γψ

(26)

we receive two values t

i

. There is no solution of the

equation (24) for z

i

< |z

P

- r'| and z

i

> |z

P

+ r'|. Ex-

changing inequalities for equalities above formulae

describe conditions, by which (24) has only one so-

lution.

3.3 Calculation of the time t

i

of reaching the

azimuth a

i

In order to calculate t

i

appropriate to any given value

a

i

, it is necessary to determine from formula

iiPiP

Δazzzzr cossinsincoscoscos +=

′

(27)

involved value z

i

, where Δa

i

is calculated as (7).

There is one (for the equality) or two solutions (for

the inequality) of z

i

in the case, when

iP

Δazr sinsinsin ≥=

′

(28)

or there is no solution in remaining cases. The far-

ther proceedings comes down to the calculation of

the time of reaching the obtained zenith distance z

i

,

which was descried higher. To the formula (26) one

should substitute only one value calculated from the

equation (25), that is ψ

i2

for Δa

i

< π, and ψ

i1

in the

opposite case.

Time t

i

and coordinates z

i

, a

i

serve as data corrected

for refraction for the reduction in various methods of

making astronomical fix.

4 CONCLUSIONS

The described method of the equalization of meas-

urements assumes that celestial bodies rotate on cir-

cles with constant speed and only random errors of

measurement are found in measurement data. It is

possible to determine data for the reduction for the

any given point on the circle. If this point is in vicin-

ity of the arc containing measuring data, then the ac-

curacy of this point results directly from the accura-

cy of the measurement and the number of

measurements.

REFERENCES

Opalski, W. & Cichowicz, L. 1977. Astronomia geodezyjna.

Warszawa: PPWK.

Jurdziński, M. & Szczepanek, Z. 1975. Astronawigacja.

Gdańsk: Wydawnictwo Morskie.