Drift alignment method (J. Scheiner)
During
the second part of the 19th century several methods to
align the hour axis were developed. One of the first practical was
the drift alignment method by J.
Scheiner published in the year 1889 in [1].
Probably this is the most frequently used method for the sake of
its simplicity. To carry it out only a eyepiece with a reticle is
needed. The corrections for the azimuth and elevation of the hour
axis are determined separately, the azimuth in the meridian and
the elevation at the local hour angle
= ±6h. Among many other similar descriptions
here the method is printed in the version of the Astronomical
League [2]:
Choose a bright, easily located star close to the equator (to maximize the drift rate) and near your meridian. Center it on the crosshair of your eyepiece.
Track the star until its drift in declination becomes noticeable. Ignore any drift in right ascension. If the star drifts northward, move the polar axis east. If the star drifts southward, move the polar axis west. Repeat, making finer adjustments, until the drift becomes negligible.
Choose another bright, easily located star near the eastern or western horizon and center it in your crosshair eyepiece.
Track the star until its drift in declination becomes noticeable. Ignore any drift in right ascension. If you are looking east and the star drifts northward, move the polar axis down. If you are looking east and the star drifts southward, move the polar axis up. (Reverse the corrective action if you are looking west.) Repeat, making finer adjustments, until the drift becomes negligible.
J.
Scheiner expressly pointed out that the refraction has to be
disarmed (“unschädlich”). Ignoring this hint then
the observations might be possible somewehre on the meridian and
in the East or the West. But this method relies on the fact that
the observation of the movement in the
direction takes place at the local hour angle of
=0 and ±6h.
If the above instructions are followed and one observes in the vicinity of the horizon the effect of the refraction contributes most. In the following chapter it will be shown that these instructions have to be taken with a pinch of salt and that the additional advice of J. Scheiner and A.A. Rambaut have to be considered.
Figure 1:
Star
trails of the drift alignment method for
= 0,10,20,30,40,50,60 and 80o (starting on the left) at
= 23h45m without atmosphere, exposure 30
minutes. The telescope's drive rate is sidereal. The length of the
scale is 25''.
Figure 2:
Star
trails of the drift alignment method for
= 0,10,20,30,40,50,60 and 80o (starting at the bottom)
at
= 18h00m. In order that the different trails
are distinguishable they have been moved manually, otherwise like
fig. 1.
In
fig. 1
and
2
the
intersection P' of the hour axis, this
is the instrumental pole, with the sphere is at the location
= 236'' and at the local hour angle h = 18h30m.
The polar distance
is counted from the true pole P. The actual star trails may
significantly differ from those presented here as a function of
the instrumental pole and the geographical latitude
.
Specially the movement in the
direction on the meridian(e.g. fig. 1)
is a consequence of the chosen position of the
instrumental pole P' which is at the location h = 18h30m
or at 277.5o. For other values the trails will
show all angles between 0...90o.
In fig. 1 the star moves towards South, that means that the instrumental pole lies East of the true pole. The direction of the movement can be recognized because the color of the trail changes for every full hour angle (1h, 2h...) from white to red.
In
fig. 1
the
observation begins at the local hour angle
= 23h45m (white) and at
= 0h (in the middle of the measurement)
it changes from white to red. The beginning in fig. 2
is
at
= 18h00m (red) and the color changes to
white twenty minutes later.
As
seen in both figures the star trails are similar among each other.
Specially the direction of the movement is independent
of the declination. The length of the trails in the
direction is in all cases identical as J. Scheiner described it.
In the case where
=
0o (see fig. 2)
the trails appear as straight lines and are parallel with the
direction in best agreement with the E.S.
King's prediction. The drift alignment method considers only
the movement in the
direction and so the variation of the length of the trail as a
function of the declination
is uncritical. That means that the observation can take place
anywhere between the horizon and the pole. In principle the method
seems to be suitable. What was not said so far is the fact, that
the star trails in
fig. 1
and
2
were calculated in the absence of the atmosphere. If the
atmosphere is taken into account the circumstances become more
complex.
Figure 3:
Star
trails of the drift alignment method for
= 0,10,20,30,40,50,60 and 80o (starting left) at
= 23h45m with atmosphere.
Figure 4:
Star
trails of the drift alignment method for
= 0,10,20,30,40,50,60 and 80o (starting
left) at
= 18h00m. The length of the trails for
= 0,10o are too long an can not be plotted to scale,
otherwise like fig. 3.
Fig.
3
and
4
show
the trails under the previous circumstances but in the presence
of the atmosphere. If the the consideration is confined to the
direction of the movement parallel to the
direction then independent of the
declination the trails show the same behavior.
The interpretation of the movement
provides in every case the same result despite the length and the
direction of them differ significantly. If one uses only this
qualitative result, that means that only the direction of the
movement parallel to the
direction is looked at, then the declination of the location where
the observation is carried out seems to be unimportant at first.
If one compares fig. 1,
and 3
,
that is the location where the azimuth is measured, then only a
minor difference in the tilt of the trails visible. This
difference does not influence the result of the interpretation.
If
one compares the two fig. 2
and
4
then
the movement in the
direction is fundamentally different. It is clearly visible that
the trails direct toward the South in the presence while they
where directing towards the North in the
absence of the atmosphere. If the given instructions are used to
interpret the star trails that means
that the instrumental pole lies now below the pole. What happened?
The
coordinates of the instrumental pole P' (
= 236'' , h = 18h30m) indicate, that the
hour axis intersects the sphere a little bit above and on the East
side of the true pole. This contradiction is quickly resolved
because the the location of the true
pole and the surroundig field is lifted by a small amount, in this
example 51.8'', by the refraction in the direction of the zenith.
And indeed the instrumental pole lies now below the apparent
(refracted) pole Pr and everything has its order.
This example shows that finally the apparent pole is reached if one carries out the observation in the vicinity of the pole.
Disturbing
is the fact that the movement in the
direction of the star depends on the declination itself which
makes the interpretation difficult. Normally one measure a certain
period and then the distance from the original location is
determined. It seems that the length of the trail in the case
= 80o is acceptable but in case
= 20o it seems not despite the polar distance remained
unchanged.
Figure 5:
Star
trails of the drift alignment method for
= 0,10,20,30,40,50,60 and 80o
(starting at the bottom) at
= 23h45m, with atmosphere. The hour axis
points on the apparent (refracted) pole. The length of the scale
is 6.25'' .
Figure 6:
Star
trails of the drift alignment method for
= 20,30,40,50,60 and 80o (starting at the bottom) at
= 18h00m, otherwise like fig. 5.
How
do the star trails look like if the hour axis points to the
apparent pole Pr ? Under this circumstance
the difference is zero in the direction of the azimuth and
following the given instructions the movement of the star is
allowed only parallel to the
direction.
Fig. 5 confirms the expectation with the measurement error of ±1'' for all declinations. This remark is only true in case where the instrumental and the apparent pole coincide. Is the hour axis a little bit off, but only in the azimuthal direction, then it is possible to observe the same behavior depending on the local hour angle of the observed location in the sky. As a consequence the practicle part of the observation is not that easy as the instructions may suggest.
The
mentioned instruction predict for the measurements at the local
hour angle
= ±6h star trails which are parallel to the
direction too. In fig. 6
it
is visible, that for declinations
> 80o the trails follow reasonably the prediction.
Form that figure one can conclude that the alignment of the hour
axis on the apparent pole Pr
does not succed in case the effect of the refraction are not
eliminated by calculation. If one likes to align the hour axis
with the help of the drift alignment method then only in the case
of azimuth one obtains satisfactory results but not in the by far
more important case of the elevation.
One
can ask oneself where the hour axis finally points to in case
where the observation is carried
out at e.g.
= 20o and the trails are parallel to the direction of
the rightascendion. In complete analogy
to the procedure at the telescope the position of the hour axis
was determined by trial and error with the help of the simulation
and the elevation was changed until the star trail remained
during 30 minutes within a band of 2'' (fig. 7).
The in this way obtained results for the polar distance
are compiled in tab. 1
on
the first row for the declinations between 20 and 80o.
Without restricting the general case the hour axis remained in the
plane of the meridian..
80
53.0
2
53.5
39.6
18
Table 1:
Polar
distance
as a function of the declination, obtained by applying the drift
alignment method, compared to the values at the local hour angle
= 18h and nominal values on the meridian (
= 0h) according to King's
equation (third and forth row) and the tolerable errors. The
apparent pole is at a distance of 51.8'' measured from the true
pole. The hour axis coincides with the plane of the meridian and
the geographical latitude is
= 47.5o.
From the second row of tab. 1, that means after subtracting the polar distance of the apparent pole Pr of 51.8'' , one can conclude, that only above the declination of 40o the difference between the instrumental and the apparent pole becomes smaller than 1 arc minute.
Figure 7:
The
more or less parallel to the
direction running star trails of the drift alignment method for
= 20,30,40,50,60 and 80o (starting at the bottom) at
= 18h00m with atmosphere. The white scale
bar on the right side indicate a length of 2'', otherwise like
fig. 2.
The hour axis points to the those in tab. 1
given
values.
It
should be mentioned that the final position of the hour axis
depends on the duration of the single measurement how it can be
easily seen in fig. 7
in
the case for
= 20o. This trail sags and one can conclude that there
is a certain freedom in the interpretation what is considered to
be in parallel to the
direction. This effect is less pronounced at higher declinations
but is still present. The closer the observatory is located to the
equator the more difficult the measurements becomes even in the
region of the pole itself because the great circle at ±6h
runs longer near the horizon.
E.S.
King derived in his article [3] an equation
how the position of the hour axis, that are the declination and
the local hour angle, has to be chosen in order that the movement
in the
direction vanishes. He obtained the following equation
Therein
is
the polar distance of instrumental pole from the true pole, n
the refractive index of the atmosphere,
the geographic latitude and
the zenith distance of the observed location in the sky.
The
drift alignment method by J. Scheiner is nothing else than the
rephrased equation of E.S. King.
This equation says that there exists to a given zenith distance
,
resp. a given pair local hour angle
and declination
of the observed location in the sky, a polar distance
,
in order that the movement in the
direction can be compensated for.
On
the third row of tab. 1
the
results are compiled of the calculation of E.S.
King's equation for
= 18h. In the polar region the values obtained from
E.S.
King's equation and the simulation are in good
agreement and become worse and worse for smaller declinations.
This behavior is a consequence of the
above mentioned difficulties with the interpretation of the star
trail among other things.
The
elevation of the hour axis is measured at the local hour angle
±6h. In case where the declination is held
constant the zenith distance is greater in the East or the West
compared to the position on the meridian. One may conclude that
the final polar distance
,
which is obtained by the uncorrected drift alignment method, is
normally greater. Normally a photograph is carried out in the
region of the meridian and for that purpose the setting of the
hour axis is not optimal. Compiled on the forth row are the polar
distances obtained from E.S.
King's equation which were calculated on the
meridian. The comparison of the third and the forth row in tab. 7
shows
that the values for small declinations differ significantly and
only for declinations
>
70o they fall below the tolerable error
(fifth row) of the
alignment of the hour axis. Considering
these arguments the conclusion is that the measurement
of the elevation of the hour axis has to be carried out in the
close vicinity of the pole at
= ±6h in case where the refraction is not
corrected by calculation.
How
already mentioned the measurement in the polar region is favored.
It could now happen that there are no appropriate stars visible at
the local hour angles
= 0h and ±6h. In this case one must
withstand the temptation to choose a star which is only a little
bit off the correct position because a shift of only 0.5o
in
direction may end up in a local hour angle which differs in the
maximum by 12h. The drift alignment
works only at the local hour angles
= 0,±6h, otherwise the results are wrong.
Because
of the zenith distance
varies constantly with the local hour
angle
,
one is runs behind a moving target. From the practical point of
view this is a drawback because the precision can not be increased
simply by making the measurement period longer. How shown above
the opposite is the case and therefore the observation of the star
should not last too long, e.g. a few minutes until the star leaves
the reticle. The observation is made best with a high power
eyepiece.
Figure 8:
The
deviation in arc second of the star from the reticle for
= 0 and at different declinations (left to right 0, 10, 20, 30,
40, 50 60 70 80o). This figure is valid for the
geographic latitude
= 47.5o.
Figure 9:
The
deviation in arc second of the star from the reticle for
calculated according to E.S.
King's equation, otherwise like fig. 8.
In
order to obtain a certain imagination how fast a star deviates
from the reticle results are plotted for two different positions
of the hour axis
in fig. 8
and 9.
The steep
slopes in fig. 8
are
due to the setting of the polar distance
= 0''. The hour axis points to the true pole P and is therefore
not as far away as with a rough polar alignment could be obtained
(
=
360'' ). That means that in the first observation after the
rough alignment a star deviates significantly faster in the
direction than shown here. The slope of the curves in fig.
9
are horizontal in the beginning because the polar distance of the
hour axis was set according to E.S.
King's equation. In the beginning the movement
in the
direction is completely compensated as predicted and therefore the
initial rate of deviation is zero but after a very short amount of
time it differs. One sees that in the best case (
= 80o) the deviation is larger than 0.5''
after one hour. The observation becomes more difficult in case it
is carried out at smaller declinations. Even if the position of
the hour axis is initially chosen optimally for the observed
location in the sky as it is shown in fig. 9
then after only 3 minutes a deviation is visible of the graph
representing
= 0o. From this practical point of view it becomes
gradually clear that the drift alignment method fails in the
vicinity of the horizon.
From
the fig. 8
and
9
follows
in addition, that each observation has to be carried out with
different star. The declination of the newly chosen star must have
within narrow limits the same declination and in the beginning the
same local hour angle
= ±6h. The reason for that is the following: it
is assumed that the hour axis is at the correct position for that
location. If then one checks the position
by a second observation then the same star does not remain on the
reticle. It inevitably deviates even faster from the reticle as it
can be seen in fig. 9.
Particularly well understandable is this behavior by means of the
graph for
= 50o. In the first half hour (
= 18h00m...18h30m) the
star deviates one arc second. To travel
the same distance it needs at the local hour angle
= 18h30m just about the half of the time. In
the mean time the zenith distance has changed and it follows from
E.S.
King's equation that the movement in the
direction can not be zero any more. By looking at the faster
movement of the star one does for certain conclude that the last
movement of the hour axis did not bring hoped success despite this
behavior is in absolute agreement with the underlying laws and the
hour axis pointed initially to the optimal location.
Who despite these practical drawbacks likes to use the unmodified drift alignment method should proceed with the following steps:
Chose an eyepiece that a deviation of 1...3'' can be measured easily.
Level the tripod. This step is not mandatory but it simplifies the whole alignment (see step 8).
The observation should only last as long as the direction of the deviation can be identified for sure. The passed amount of time is only a rough indicator for the accuracy of the alignment because no star remains for ever on the reticle.
Each observation has to be carried out with a different star which has within narrow limits a similar declination and the same local hour angle
= 0h (azimuth) and
= ±6h (elevation).
Each observation takes place at the local hour angle
= ±6h and not in the West or East. The
declination should be between 70o <
< 90o. The closer the observatory is located near
the equator the closer the observation has to take place near the
celestial pole.The correction of the position of the hour axis in azimuth is done on the meridian in the vicinity of the pole. In this case the star needs to be close to the pole but in the direction of the zenith.
The instructions mentioned in the introduction remain the same.
Because the two measurements are not independent the measurements have to be repeated several times until the correcting movements of the hour axis are small.
The
statements made in steps 3
and
8
let the drift alignment method appear in a bad light, because
there are no clear instructions when the aim is reached. The
reason for the statement in step 3
is,
that for each zenith distance
exists a polar distance
,
so that the movement in the
direction is compensated. In case of step 8
one
has to consider that the apparent movement of the star without
atmosphere relative to the true coordinates is described by a sum
of two terms. At the local hour angles
= 0h and ±6h one of these terms
becomes zero and that simplifies the interpretation. This limited
grip of the rules in step 3
and
8
follows
directly from the effects of the refraction and the simplified
manner how the observation is carried out. Even if one follows
these additional steps closely there is no guarantee
that the obtained position satisfies the requirements of the
astrophotography.
In the introduction the drift alignment method was described a a simple method. This holds true despite the above number of steps increased. It is still a little bit astonishing the the practical part of the observation comes along with all these difficulties. It is further a real drawback that this method says nothing about how much the hour axis have to be moved. As a conclusion this method remains in the realm of trial and error and at the end one does not know where to and with which precision the hour axis points to. The two methods described in the following chapters eliminate these deficiencies completely whereas the additional effort is confined within narrow limits perhaps it is the time to learn and to get used to them. In any case the result justifies these endeavors.
Literature
[1] SCHEINER, J.: Sur une méthode très simple permettant d’orienter un instrument a monture parallactique plus excactement qu’on ne peut le faire en général par des lecturs des cercles. In: Bulletin du Comité Permanent International pour l’Exécution Photographique de la Carte du Ciel, 6e fascicule, S. 385 - 388, 1889. Proceedings of the academy.
[2] THE ASTRONOMICAL LEAGUE: Astro Note 15: Accurate Polar Alignment. www.astroleague.org/al/astrnote/astnot15.html.
[3] KING, E. S.: Forms of images in stellar photography. Annals of Harvard College Observatory, 41:154-187, 1902.
Comments, questions, corrections: markus.wildi@one-arcsec.org