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 (dX2 + dY2)
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.