P. Henrotay
Polar axis misalignment V1.0
16-Mar-2008
Polar misalignment
When the polar axis of an equatorial mount - the "instrumental pole" - is misaligned,
the following occurs:
 a drift in declination (declination error)
 a drift in right ascension (hour angle error), as if the drive rate were wrong
 a field rotation.
The present document focuses on deriving mathematical approximations for the hour
angle and declination errors, the misalignment being considered as small. The field
rotation is not addressed.
Observing the hour angle and declination errors (such as the difference between these
at different times or more) allows to determine the elevation and azimuth error. The
link between the maths and some polar alignment methods (Bigourdan, Challis, King,
Rambaut) is also made.
Note that this is only a partial picture for the generalized modeling of mount pointing
errors, ignoring all further aspects of geometric/flexure/refraction-induced errors.
Notations used







h
da
di
DX
DY
H
MA
ME
declination
latitude of observation site
polar coordinate, equivalent to declination in instrument referential
polar coordinate, equivalent to hour angle in instrument referential
amount of polar alignment error (the distance between the true and
instrumental pole)
instrumental pole coordinate, in relation with instrumental pole hour
angle
declination error due to polar misalignment
hour angle/right ascension error due to polar misalignment
azimuth error, equals MA sec 
elevation error, equals ME
drift in hour angle (eastward drift)
drift in declination (northward drift)
hour angle
misalignment of the polar axis to the left or right of the true pole; for
northern hemisphere, positive if right of due north
vertical misalignment of the polar axis; for northern hemisphere, positive
if below true pole
All angles in radians (180°=  radians).
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P. Henrotay
Polar axis misalignment V1.0
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Geometry
The misalignment can be characterized by:
 a rotation of the equatorial system clockwise around the polar axis with angle

 a rotation of this new coordinate system around the X axis by 
where
  is a measure of the hour angle of the instrumental pole (add 90° to obtain
hour angle for instrumental pole)
  is the amount of polar alignment error (the distance between the true and
instrumental pole)
(graphics from reference R1)
As a result, the transformation matrix is a product of rotations:
0
1

 0 cos 
 0 sin 

 cos 

 sin   sin 
cos   0
0
 sin 
cos 
0
0

0
1 
and finally we have:
 cos  cos    cos 

 
 cos  sin     cos  sin 
 sin    sin  sin 

 
 sin 
cos  cos 
sin  cos 
 cos  cos(  H ) 


 sin   cos  sin(  H ) 

cos  
sin 

0
(Eq. 1)
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P. Henrotay
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Declination drift
The last row of the above matrix equality (Eq. 1) is:
sin   sin  cos  cos(  H )  sin  cos  cos  sin(  H )  cos  sin 
which simplifies as:
sin   sin  cos  sin(   H )  cos  sin 
As we are dealing with small deviations, we have   1 , and we may thus use as
approximations:
sin   
cos   1
and introducing  (the declination error) as      with   1 being the
(small) declination error induced by the polar misalignment, we have:
sin   sin    cos 
We obtain that:
  northwarderror   sin(   H )
(Eq. 2)
Introducing:
u   sin   ME  di
v   cos    MA  da cos 
(Eq. 3)
we have, expressed in function of di/ME and da/MA:
northwarderror 
u cos H  v sin H  ME cos H  MA sin H  di cos H  da cos  sin H
(Eq. 4)
The northward error depends on H, the hour angle, which itself is dependent on time.
So if one measures the drift DY of a star in declination between two distinct times a
and b, this provides one relation between di and da:
DY   b   a     b  (   a )   b   a  di(cos H b  cos H a )  da cos  (sin H b  sin H a )
(the suffix is used to designate time a and time b).
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The (instantaneous) northward drift speed is obtained by taking the derivative of the
northward error expression (Eq. 4) and so:
northwarddrift _ speed 
 u sin H  v cos H  MA cos H  ME sin H  da cos  cos H  di sin H
(Eq. 5)
Right ascension drift
Dividing the first row of the matrix equality (Eq. 1) by the second row, we obtain:
tan  
cos  cos  sin(   H )  sin  sin 
sin 
 cos  tan(  H )  tan 
cos  cos(  H )
cos(  H )
As we are dealing with small deviations   1 , we may thus use as approximations:
sin   
cos   1
and introducing h as     H  h with h  1 being the (small) hour angle
error induced by the polar misalignment, we have:
tan   tan(  H ) 
h
cos (  H )
2
And we obtain that:
h   tan  cos(  H )
When measuring angular distances, we need to apply the cosine of the latitude, which
means that (dealing with small angular separations):
h cos   eastwarderror   sin  cos(  H )
(Eq. 6)
And so, expressed in function of da/MA and di/ME:
eastwarderror 
sin  (u sin H  v cos H ) 
sin  ( ME sin H  MA cos H )  sin  (di sin H  da cos  cos H )
(Eq. 7)
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The eastward error depends on H, the hour angle, which itself is dependent on time.
So if one monitors the drift DX of a star in right ascension between two distinct times
a and b, this provides one relation between di and da:
DX  sin  (di(sin H b  sin H a )  da cos  (cos H b  cos H a ))
(the suffix is used to designate time a and time b).
The (instantaneous) eastward drift speed is obtained by taking the derivative of the
eastward error expression (Eq. 7):
eastwarddrift _ speed 
sin  (u cos H  v sin H ) 
sin  ( ME cos H  MA sin H )  sin  (di cos H  da cos  sin H )
Application 1 - Bigourdan's method
Consider the northward (declination) drift speed above (Eq. 5).
At the meridian, we have H=0, the sine is 0 so the drift rate is solely related to MA,
the error in azimuth.
At H=-6 hours (East) or H=6 hours(West), that is H= 

or 
2
so the drift rate is solely related to ME, the error in elevation.

, the cosine is 0 and
2
This justifies the Bigourdan method, where the declination drift is observed and
corrections are in turn applied to azimuth then elevation:
 step1: watching a star drifting at meridian and correcting in azimuth till no
further declination drift is observed, then
 step2: watching a star at + or -6 Hours (so East or West) and adjusting
elevation till no further declination drift is observed.
Application 2 - Challis' method (1879)
In this method, one measures the northward (declination) drift only. Two
measurements are required to determine the polar misalignment: two stars over a
certain time interval or one star over two time intervals (illustrated below).
From the equation above for  (Eq. 2), one can write, for star at times a and b:
drift a to b   b   a  u (cos( H b )  cos( H a ))  v(sin( H b )  sin( H a ))
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and similarly for the same star at times c and d:
drift c to d   d   c  u (cos( H d )  cos( H c ))  v(sin( H d )  sin( H c ))
(subscript denotes the time).
Measuring the drifts over the time intervals allows thus to determine u and v (2 linear
equations with 2 unknowns) and hence MA/da/ME/di using (Eq. 3.
Note that the same principle may be used by monitoring two star over one time
interval.
Example (matching Reference R1)
star: alpha Boo
RA:14h15m49s
DEC:19°10'29"
time a: 2001-May-24, 21H00
time b: 21H50
drift a to b: -34.52 arcsec
time c: 21H50
time d: 22H23
drift c to d: -65.88 arcsec
latitude: 52°09'20.32"N
longitude: 0°0'38.36"E
One obtains:
u=1614 arcsec, hence di=1614 arcsec (the instrumental pole is below the pole;
it must be raised)
v=449 arcsec, hence da=-732 arcsec (the instrumental pole is west of the pole;
it must be rotated east)
Application 3 - King's method (1902)
In this method, one measures the northward (declination) and eastward (hour angle)
drifts of a star near the Pole. One measurement suffices to derive the polar
misalignment.
As we are close to the Pole, we have  

2
and so its sine is close to 1; the eastward
error (Eq. 6) simplifies to:
eastwarderror   cos(  H )
while the northward error (Eq. 2) remains:
northwarderror   sin(   H )
When deriving each expression to obtain the respective drift speeds, one notes that the
amplitude of the drift speed is constant, being  .
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P. Henrotay
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Let us designate by DX and DY the drift respectively in hour angle and declination,
over a time interval (from time a to time b): we have thus:
DY   sin(   H b )  sin(   H a )
DX   cos(  H b )  cos(  H a )
(subscript denotes the time, b after a)
From these, it is easily deduced that:

Ha  Hb
H  Ha
DX
DX
 arctg
 Ha  b
 arctg
2
DY
2
DY
and, as we have H b  H a <<1, implying sin

Hb  Ha Hb  Ha
:

2
2
DX 2  DY 2
Hb  Ha
From here, da/MA and di/ME remain easily be computed from (Eq. 3.
H  Hb
H  Hb
1
u  di 
( DX cos a
 DY sin a
)
Hb  Ha
2
2
v  da cos   
H  Hb
H  Hb
1
( DX sin a
 DY cos a
)
Hb  Ha
2
2
Application 4 - Rambaut's method (1893)
In this method, one measures the northward (declination) and eastward (hour angle)
drifts of a star. One measurement suffices to derive the polar misalignment. The
method will fail at the celestial equator, as the sine of the declination is zero. It can be
considered as a generalization of King's method to declination further away from the
pole.
The eastward error (Eq. 6) is:
eastwarderror   sin  cos(  H )
while the northward error (Eq. 2) is:
northwarderror   sin(   H )
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Let us designate by DX and DY the drift respectively in hour angle and declination,
over a time interval (from time a to time b): we have thus:
DY   sin(   H b )  sin(   H a )
DX   sin  cos(  H b )  cos(  H a )
(subscript denotes the time, b after a)
This can be re-expressed as :
Ha  Hb
H  Ha
) sin( b
)
2
2
H  Hb
H  Ha
DX  2 sin  sin(   a
) sin( b
)
2
2
DY  2 cos( 
From these we obtain, as we have H b  H a <<1, implying sin

Hb  Ha Hb  Ha
:

2
2
DX 2 / sin 2   DY 2
Hb  Ha
From here, da/MA and di/ME are easily computed via (Eq. 3:
u  di 
H  Hb
H  Hb
1
DX
(
cos a
 DY sin a
)
H b  H a sin 
2
2
v  da cos   
H  Hb
H  Hb
1
DX
(
sin a
 DY cos a
)
H b  H a sin 
2
2
This corresponds to the original article (Reference R10) where di corresponds to -k
and da to -A.
Example (matching Reference R4)
Latitude: +47°
Longitude: -7.601°
Date: 2001:09:30
Time: 22:04:21
Time zone: cest
Observing period: 1371s
Rightascension: 279.23458°
Declination: 38.783°
DX: -2.65 arcsec
DY: +14.10 arcsec
One obtains:
di=-126 arcsec (the instrumental pole is above the pole by 126 arscec - it must
be lowered)
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da=113 arcsec (the instrumental pole is east of the pole - it must be rotated
west)
Useful readings and links
R1. Matrix method for coordinate transformation - Toshimi Taki
http://www.asahi-net.or.jp/~zs3t-tk/matrix/matrix.htm
(the reference for this document)
R2. TPoint - Telescope Pointing - Patrick Wallace
http://www.tpsoft.demon.co.uk/pointing.htm
(table 1 in link above and equations above for  and h are fully in agreement)
R3. MESXI - la théorie, les équations - Jean-Claude Durand
http://pagesperso-orange.fr/mesxxi.durand/theorie.html
(drift rate equations are identical except for sign convention)
R4. The atmosphere as a prism - Markus Wildi
http://leq.one-arcsec.org/
R5. Publications of the Brayebrook observatory
http://www.brayebrookobservatory.org/BrayObsWebSite/HOMEPAGE/BRAYOBS%
20PUBLICATIONS.html#treediag
(in particular a description of the Challis method - unfortunately with some minor
typos)
R6. Polar axis alignment requirements for astronomical photography - R.N. Hook
http://adsabs.harvard.edu/abs/1989JBAA...99...19H
R7. Program to calculate effect of polar axis misalignment - Michael Covington
http://www.covingtoninnovations.com/astro/astrosoft.html
http://www.covingtoninnovations.com/astro/Polar.html (java applet)
R8. Stellar drift - Rob Johnson
http://www.whim.org/nebula/math/drift.html
R9. Astronomical algorithms - Jean Meeus
Willmann-Bell
ISBN 978-0943396613
R10. To adjust the Polar Axis of an Equatorial Telescope for Photographic Purposes.
Monthly Notices of the Royal Astronomical Society, 54:85-90, December 1893. Rambaut, A. A.
http://adsabs.harvard.edu/full/1893MNRAS..54...85R
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Polar axis misalignment V1.0
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Summary/Résumé
When the polar axis of an equatorial mount - the "instrumental pole" - is misaligned, a
drift in declination (declination error) and a drift in right ascension (hour angle error)
occur.
Quand l'axe polaire d'une monture équatoriale - le "pôle instrumental" - n'est pas
parfaitement aligné avec le Pôle, une dérive en déclinaison et en ascension droite
apparaissent.
Measuring these drifts over time allows to determine elevation and azimuth error.
Mesurer ces dérives permet de déterminer l'erreur d'alignement en élévation et en
azimut.
The declination drift can be expressed, for small misalignments, as:
Pour de petites erreurs d'alignement, la dérive en déclinaison peut s'exprimer par:
DY  di(cos H b  cos H a )  da cos  (sin H b  sin H a )
The hour angle/RA drift can be expressed as:
La dérive en ascension droite peut s'écrire:
DX  sin  (di(sin H b  sin H a )  da cos  (cos H b  cos H a ))
where:
où:
 the suffix is used to designate time a and time b
 H is the hour angle of the observed star
 di is the elevation error (positive if the polar axis must be raised)
 da is the azimuth error (positive if the polar axis must be moved west)
  is the latitude
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