INDEX:

The Chopped and Nodded images
Mathematical model for Chopped and Nodded images

THE CHOPPED AND NODDED IMAGES

The figure below shows how the chopped and nodded technique works.

By pointing the telescope to the star target, we obtain an image containing both the signal from the target and the background due to atmosphere and telescope so that the former is hidden (image in a).
In order to extract the as In order to extract the astronomical signal we need to make the chopping of this image with the one obtained moving the secondary mirror only (given in figure b), i.e. we subtract the two images obtaining a chopped image.
The result obtained is an image with still part of the background signal (figure in e).
In order to remove the last "residuum" we need to acquire more information of the background moving, this time, the whole telescope (nodding) and repeat the subtraction (the image resulted is in figure f) between the two new loaded images (figure c and d).
At last we complete the procedure subtracting the frame of figure e and the one in figure f.

a) Image obtained by pointing the telescope on the source.
b) Image obtained by moving the secondary mirror; image b) is shifted with respect to a) by

arcsec in the direction of the columns.
c) Image obtained by moving the primary mirror of the telescope (nodded); image c) is shifted with respect to a) by -

arcsec in the direction oER> arcsec in the direction of the columns.
d) Image obtained moving the secondary mirror of the moved telescope (nodded).
e) Image obtained subtracting a) and b) (chopped).
f) Image obtained subtracting c) and d) (chopped).
g) Image obtained subtracting e) and f) (chopped and nodded).

MATHEMATICAL MODEL FOR CHOPPED AND NODDED IMAGES

If x, y are angular coordinates in the sky, the signal sp coming from the direction {x,y} at time t and detected on the corresponding pixel P of the detector can be expressed as:

(1) [figure a)]

where f is the unknown brightness distribution of the celestial source and a is the large and time-variable thermal background flux. The transfer function of the detection system Tp includesSIZE=-2>p includes the collecting area of the telescope, the field of view of each individual pixel and the overall optical transmission.
Under the conditions described by equation (1) it is clear that a small error in the estimate of a will dramatically affect the extraction of the signal f.

The background a can be obtained in principle by pointing the telescope to a sky area close to the region of interest at a time t' close to t. Assuming that this area corresponds to a shift in the y coordinate, then the new signal s'p detected at the pixel P is

(2) [figure b)]

The quantity is called chopping throw or chopping amplitude.
In order to remove the background signal, we use the chopping and nodding technique, obtaining the so-called chopping and nodding image:

(3) [figure g)]

i.e. an image which is independent of the atmospheric background and telescope thermal pattern.

is the subtraction of equation (2) from equation (1), i.e. the equation obtained by the chopping technique only [figure e)];
comes from the nodding technique: the telescope is pointed to a different point on the sky, in our notation the telescope is shifted by - arcseconds in the y coordinate. In this way, at the pixel P the signal s"p is obtained [figure c)] and subtracting the equation (1) from s"p we obtained [figure f)].

We take, for simplicity, Tp = 1 in equation (3). Then by computing the Fourier transform of both sides we get

(4)

where are the spatial frequencies associated with the variables x, y respectively.

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