Reduction ? What' that ?
It is the outcome of all efforts made until there ! Reduction
consists in transforming the rectangular coordinates of the couple
in relative polar coordinates to the main component.
The principles
By analogy with the visual methods, the average of 10 measurements in the
same evening provides the results in angle and distance for what we call
an observation. The averages of three observations give the measurement
of the couple.
Proceeding in a same way as for visual is a deliberated choice of
my own. There are other ways and this choice raises more from the
habit and the confidence acquired with numerous filar micrometer
sessions.
What facilitates the task
The calibration frames give immediately the sampling and the
camera orientation. The sampling can also be determined
'definitively' by numerous measurements of calibration stars with
the same optical configuration.
Where it is necessary to be prudent
It will be necessary to verify with care that the results on the two extreme
calibration stars are equal or at the less in good agreement.
If the gaps are too important, we can think that it occurred
something wrong during the session and that the reduction will
probably give bad results.
The most frequent problem concerns the angles of position. A bad polar alignment
or an inopportune rotation of the camera (ie. when refocusing) can reduce
all the imaging session to nothing.
However we can take care to record a trail on the visited stars.
This is an alternative way to determine the orientation of the
camera by measuring the position of several points on this trail. A
linear regression permits to deduct the leading coefficient that is
the tangent of the shift angle. The rest of the reduction is
identical to what is said otherwise, the only difference is that
the shift angle is provided by the trail and not by the calibration
stars.
Determination of the centroïds
It is the most delicate phase of the reduction. It is necessary to
find the center of the components in the most precise way. Several
methods exist and, there also, if we want a certain constancy in
the results, it is necessary to use the same method systematically.
Beware of the false precision of the measures, a big number of
decimals doesn't present any interest. We can say that the best
measurements with a CCD on an observatory instrument can reach a
precision about 1/20 of pixel. With the webcam on amateur's
instrument a precision of 1/5 to 1/10 is something usual.
How to do that ?
By using statistical functions and modelling of the stars. These
functions are present on all astronomical software. Let's mention
the Centroid functions of WinMips and PSF of iris for example.
Practical aspect of the reduction
The narrow field of webcam's sensor doesn't permit a rigorous astrometric
reduction (determination of the position of the components in relation to
reference stars).
The process described here is a process by default that need
probably to be deepened on some points.
It goes from the assumption that the projection of the celestial sphere
on the narrow field is assimilated to a plan. Nothing is less sure, but
while waiting better, it is the one that is usually used. The coordinates
of the centroids kept preciously will always undergo the test of new algorithms
in the future.
The following table show all steps of a reduction :
| |
What
|
How
|
| 0 |
Capture the images |
|
| 1 |
Select the best ones |
|
| 2 |
Measure of the centroïds |
IRIS (PSF function) |
| 3 |
Eventual correction for rectangular
pixels |
X = X * 8.2/7.6 (case of the
Quickcam VC) |
| 4 |
Calculation of the differences of
rectangular coordinates (only the differences is of interest) |
dX = Xb - Xa
dY = Yb - Ya |
| 5 |
Determination of the position angle
on the matrix |
a = arctg(dY/dX) |
| 6 |
Determination of the position angle
on the sky (d=shift in relation to the true North) |
t = a + d |
| 7 |
Determination of the distance (e =
sampling) |
r = e * square root
(dX2 + dY2) |
| 8 |
Determination of the the angle of
position for the observation |
thetai = Average
(t) |
| 9 |
Determination of the distance for the
observation |
rhoi = Average (r) |
| 10 |
The averages of three close evenings
give the definitive values of the measure |
theta = Average
(thetai)
rho = Average (rhoi) |
When processing the images of one evening, we
reduce first the calibration stars. They give the values of
sampling and the shift of the matrix.
The principle remains the same that in the table, but step 6 is
replaced by the calculation of the shift as follow:
d = angle of position of the
calibration star - a
and the step 7 determines the sampling as follows
e = separation of the
calibration star / square root (dX
2 + dY
2)
the d and e values are reintroduced in the
formulas for the reduction of the stars of the program.
Very convenient aspect of the
reduction
All these steps require many attentions and manipulations, they are a
good way to make numerous mistakes and to disguss us to measure double
stars ! So, I choose to develop a software that process directly the bitmap/fits
files delivered by the cameras.
It integrates in a single environment the stages 1 to 9 the
previous table.
Its main functionalities are therefore:
- Sorting of the pictures
- Calibration on the calibration stars and determination of the
quadrants
- Reductions either manual plots or automatic
- Measure of the internal dispersions
- Generation of importable log for other software
The program and its documentation are distributed
freely on simple demand.