SARI
Sebuah sumber-cahaya-titik buatan telah dibentuk pada lensa-koreksi guna tes-pelempangan kaset pelat potret Schmidt, di Observatorium Bosscha. Cara Dewhirst dan Yates diadaptasi-kan untuk keperluan tes. Dalam tes ini, diselidiki jalan berkas cahaya yang terpantul sekali \((I_1)\) dan 3 kali \((I_3)\) oleh cermin utama. Kalau kedudukan kaset pelat baik, maka \(I_1\) akan berimpit dengan \(I_3\).
Tetapi hasil tes kita memperlihatkan bahwa, akibat kedudukan sumber cahaya di luar sumbu, terjadi aberasi pada \(\mathbf{I}_1\) dan \(\mathbf{I}_3\).
** Bosscha Observatory, Dept. Of Astronomi, Institute Technology Bandung, Lembang, Indonesia
To detect tilt of plateholder in a Schmidt telescope, Dewhirst and Yates (1954) proposed to use a point source on the correcting lens. An auxiliary mirror replaces the photographic plate in the plateholder, to participate in the forming of image \(I_3\). This point is the image formed by three reflections, due to the concave primary mirror, convex mirror in the plateholder and again due to the concave mirror. \(I_1\) is the image formed by only one reflection in the concave mirror. Fig. 1 shows the light path producing \(I_1\) and \(I_3\). According to Dewhirst and Yates, the mirror plateholder adjusment is correct when \(I_1\) coincides with \(I_3\).
Haffner (1954) shows that \(I_1\) is astigmatic, and \(I_3\) is free of aberration. Furthermore, Andersen and Clausen (1974) showed that \(I_3\) occured as two lines, because of the astigmatism and vignetting, while \(I_1\) is still a point and free of aberration. So, according to them, the adjusment is correct when \(I_3\) is in the centre of \(I_1\).
The aberration function, originally known as the Scidel aberration, can be expressed as follows.
\[W(Q') = -\frac{1}{4} C \left( -\frac{D_{12}}{L} \right)^4 \left\{ r'^4 + 4bhr'^3 \cos \phi + 4b^3 h^3 r' \cos \phi + 2b^2 h^2 r'^2 \left( 2 \cos^2 \phi + 1 \right) \right\}\] (1)
where
\[C = \frac{1}{2} \left\{ \left( \frac{1}{D_{01}} - \frac{1}{R} \right)^2 \cdot \frac{1}{D_{01}} + \left( \frac{1}{D_{12}} - \frac{1}{R} \right)^2 \frac{1}{D_{12}} \right\}\] (2)
\[b = \frac{(R + L' - D_{12})}{(D_{12} - R)}\] (3)
Q' is the position at exit pupil
\(D_{01}\) is the distance of object to mirror
\(D_{12}\) is the distance of image to mirror
R is the radius of curvature of the mirror
L' is the distance of exit pupil to image
\(\mathbf{r}'\) and \(\phi\) are polar coordinates of point Q' at the exit pupil
h is the distance object to the axis of system
For the object on the correcting lens, it can be shown that
\[R - D_{01} = \lim_{n \to \infty} d \tag{4a}\]
\[R - D_{12} = \lim_{d \to 0} d \tag{4b}\]
\[L' \approx R\] (4c)
Thus we will obtain
\[W(Q') = -\frac{1}{4} \lim_{d \to 0} \left(\frac{d^2}{R^5}\right) \left(\frac{D_{12}}{R}\right)^4 \left\{r'^4 - 4\left(\frac{R}{d}\right)hr'^3 \cos\phi + 4\left(\frac{R}{d}\right)^3 H^3 r' \cos\phi + 2\left(\frac{R}{d}\right)^2 h^2 r'^2 (2\cos^2\phi + 1)\right\}\](5)
Furthermore, the first and the second term become zero, while the third becomes infinitely large, which does not have physical meaning. Therefore, the aberrations formed are astigmatism and field curvature, explained by
\[W(Q') = -\frac{1}{2} \frac{h^2}{R^3} r^{-2} (2 \cos^2 \phi + 1)\] (6a)
In the Cartesian coordinate, it becomes
\[W(Q') \equiv (x' \cdot y') = -\frac{1}{2} \frac{h^2}{R^3} (3x'^2 + y'^2)\] (6b)
The deviations in the x and y directions are
\[\Delta(\mathbf{x}) = -\frac{D_{12}}{\cos\alpha} \left( \frac{\partial W(\mathbf{x}', \mathbf{y}')}{\partial \mathbf{x}'} \right)\] (7a)
\[\Delta(y) = -\frac{D_{12}}{\cos \alpha} \left( \frac{\partial W(x', y')}{\partial y'} \right)\] (7b)
where
\[\sin \alpha = \frac{h}{R} \tag{8}\]
The Bosscha Schmidt Telescope used for this experiment has been described by The (1961). The technical data of this instrument are as follows: diameter of the mirror 71.12 cm, radius of curvature 253.33 cm and h is 17.40 cm. Inserting these values into aquations (6) and (7), we obtain \(I_1\) as an ellipse with dimension of the axes: \(\Delta x_{max} \approx 0.504\) cm, and \(\Delta y_{max} \approx 0.168\) cm as round out by Andersen and Clausen (1974) the image \(Y_1\) would become a ellipse which is cut in its centre, and resemble a capital letter M, due to ustig \(x_1\) on and vignetting.
According to the additive theorem (Klein, 1970), the image formed by three reflections is simply the total aberration function and is the sum of the respective aberration function caused by each reflection.
From Lig. 2, we get
\[W_{1}(x', y') = W^{(1)}(x' + 2h, y') + W^{(2)}(x', z') + W^{(3)}(x', y')\] (9)
where
\[W^{(+)}(x',y') \equiv W^{(3)}(x',y') = -\frac{1}{2} \frac{h^2}{R^3} r^2 (3x'^2 + y')\] (10)
Similarly \(W^{(2)}(x', y')\) can be obtained here \(R^{(2)} = -R/2\).
\[D_{12}^{(2)} \approx -D_{12/2}' - D_{01}^{(2)} \approx -D_{01/2}' \text{ and } h^{(2)} = h \ 30 \text{ that}\] \[W^{(+2)}(Q') = -\frac{h^2}{R^3} r'^2 (3x'^2 + y'^2)\] (11)
Therefore
\[W_{1}(x', y') = \frac{1}{2} \frac{h^{2}}{R^{3}} \left\{ 3(x' + 2h)^{2} \right\} + \frac{h^{2}}{R^{3}} \left( 3x'^{2} + y'^{2} \right) + \frac{1}{2} \frac{h_{2}}{R^{3}} \left( 3x'^{2} + y'^{2} \right)\] (12a)
\[W_{t}(x', y') = -\frac{6h^{3}}{R^{3}}(x' + h)\] (12b)
The aberration that occurs in the image formed by three reflections is a distortion. A deviation appears with the presence of aberration function, in accordance with (7). Therefore
\[\triangle x \approx 0.494 \text{ cm}\]
\(\triangle y = 0\)
From the above derivation, we know that the image formed by three reflections has no astigmatism and field curvature. This image still retains the form of a point, but it moves radially at distance \(\frac{6 \text{ h}^3}{\cos \alpha}\) away from the image of the centre of the correcting lens.
In our experiment we found the image \(I_1\) looks like "H", while \(I_3\) is distorted. The distortion will displace \(I_3\) a certain distance away. The present result supports the finding by Andersen and Clausen (1974) in which they asserted that the method is sensitive for tangential shifts. Here we found again that the adjustment is correct, due to the off-axis position of the source, if \(I_3\) is not in the centre of \(I_1\).
Figure 1
Figure 2
We would like to aknowledge our thanks to Mr. Djoko Pitono, of the National Institute for Instrumentation (LIN), for providing us with test mirrors and to Mr. Sutia for his technical help. Part of this project is supported by ITB Project no. 1401.