3 Positional measurement for the Jupiter and its Galilean satellites

  
3.1 Positional measurement for the Jupiter

For not over-saturated CCD images of Jupiter, the following steps are taken to measure the position of the geometric center of Jupiter.

Step 1. We choose a suitable rectangle that encloses a sub-image with each side approximately twice as size as the diameter of visual surface of Jupiter (see Fig. 3a). After searching all pixels in the rectangle, we can find the maximum gray level $\textit{M}$ and minimum one  $\textit{m}$.

Step 2. The sub-image is turned to a binary image after suitably thresholding. In order to obtain a precise position for the geometric center of Jupiter, an obvious prerequisite is to guarantee the measured position to be nearly constant when the same image is measured iteratively. That is to say, the measured position should hardly have relations to the size of the chosen rectangle and the tiny variation of the chosen threshold. Therefore, we have to search an optimal threshold by experiments. We assume a threshold in the form of

$$V=m+(M-m)(\alpha+\beta /300).$$

In this formula $\alpha$ is a small positive value (0.10 is adopted here) and $\beta$ is a factor varying in the range of (0, 1, ..., 200). The design of $\beta$ is to show a subtle graduation of Jupiter's geometric center measured in the following Step 4 and to search an optimal threshold. On the other hand, the design of a non-zero $\alpha$ is to guarantee a round figure on the whole for any binary image of Jupiter after thresholding. Otherwise (i.e. $\alpha=0$), the binary image of Jupiter would become a rectangle when $\beta = 0$. Obviously, the execution of the ellipse-fit procedure in the following Step 4 would fail! Therefore, a non-zero value for $\alpha$ is indispensable when $\beta$ begins at 0. Incidentally, the adopted 0.10 for $\alpha$is obtained by our experiments.

Step 3. An edge of Jupiter is detected. Since a binary image for the Jupiter is not convenient to fit with a regular curve, a four-neighborhood mask (see Appendix A) is used to detect precisely its edge. Figure 3b shows us an edge of Jupiter.

Step 4. Fit the edge with an ellipse by a least square and obtain its geometric center (x,y).

Figure 2: The relationships between the raw pixel geometric center (x,y) (unit: pixel) of Jupiter and different $\beta$ factors are given based on 3 CCD images obtained continuously with different exposure times. The left figures give the relationships between x and $\beta$ and the right ones between y and $\beta$. Abscissas are the range of $\beta$ factors.

After having obtained a series of centric positions of Jupiter corresponding to different $\beta$ factors, we have the following Fig. 2, which draws a figure with $\beta$ as its horizontal axis and with x (or y) as its vertical axis for a CCD image obtained in a given exposure time. Since we have 3 different types of exposure times, Fig. 2 gives us 6 sub-figures. They allow us to choose a suitable $\beta$ factor for each type of exposure time. We find that a very good stability for the measured center (x,y) in a large range of $\beta$ factors is shown for the image with a longest exposure time of 3 s (in the case of "3 s exp-time'' in Fig. 2), and also a quite good stability for the image with a median exposure time of 2 s (in the case of "2 s exp-time'' in Fig. 2), but poor one for the image with a shortest exposure time of 1 s (in the case of "1 s exp-time'' in Fig. 2). This phenomenon can be understood, since a shorter exposure time makes a less stable stellar image resulting from an abnormal atmosphere. In addition, irregular markings on the surface of Jupiter also make, sometimes greatly, this contribution (see the case of "1 s exp-time'' in Fig. 2, when $\beta$ is greater than 100). Even so, there still exists a small range for $\beta$ that keeps some good stability, such as $\beta \in ($0, 70) corresponding to both coordinates. For a conservative estimation, $\beta \in ($0, 30), (30, 100) and (50, 200) can be chosen for the images with 1, 2, and 3 second(s) of exposure time, respectively, in both coordinates. In our image-processing, the optimal $\beta$s of 15, 50 and 75 are adopted for the CCD images with 1, 2 and 3 second(s) of exposure time, respectively, in both coordinates.

  
3.2 Positional measurement for the Galilean satellites

As we mentioned in Sect. 1, a satellite, especially for the one near its primary planet (such as Jupiter, Saturn and Uranus), the removal of its halo is important. Here, the removal of halo of Jupiter is done in the same manner as the one Veiga & Martins (1995) used for their processing for Uranian satellites. Figure 4 displays an image before and after the halo-removal for the faint satellite, Europa (left to the Jupiter). Here, if (x₀,y₀) are the coordinates of the center of Jupiter, we subtract from every pixel (x',y') in the region of the satellite the gray level of the pixel ( x',2y₀-y') or ( 2x₀-x',y') if x or y, respectively, is the axis of symmetry. After that, a modified moment method is used to measure the raw-pixel position of a satellite. According to Stone (1989), the modified moment that suppresses the sky background below a prechosen threshold level is the best centering method when the sky-background level is significant. A further modified version of this method was made by Ji & Wang (1996). The new modified moment method (still called modified moment method later on) has successfully been used in our previous measurement for Saturnian satellites (Paper I). In order to show the halo-removal effect, the same measurement with the modified moment method before and after the halo-removal has been performed. Figure 5 shows systematic shifts for Io, Europa, Ganymede and Callisto based on the difference of raw pixel coordinates between halo-exitance and halo-removal for the same satellite. In Fig. 5, the independent variable along the horizontal axis is the distance between the geometric center of Jupiter and a satellite. The following phenomena are found from Fig. 5. First, greater shifts exist in x coordinate axis than that along y coordinate axis for the same exposure time. This is easy to understand since the four satellites almost locate in x coordinate on a CCD image (i.e. in the horizontal direction, also see Fig. 1). Second, an obvious greater shift is displayed for a shorter distance between a satellite and the Jupiter. For Io and Callisto, shifts along both axes are very small since their long distances from the Jupiter. Third, a longer exposure time results in a greater shift, Europa is the most obvious example. At last, a longer exposure time produces a greater dispersion for a mean systematic shift. Europa is also the most obvious example in this case. Incidently, the opposite sign of the shifts in x and y coordinates between Europa and the others results from their different positions. Europa is left to the Jupiter, but the others are right to the Jupiter. As we know, the center of any satellite is shifted towards the planetary center when the halo is not removed.

Figure 3: Image processing for measuring of the geometric center of Jupiter. a) A rectangle enclosing the Jupiter in an image. b) An edge after the edge-detection.

Figure 4: The halo removal for the satellite Europa. a) An image before image processing. b) The same image after processing.

Figure 5: Systematic shifts (unit: pixel) based on the difference of measured raw pixel coordinates before and after the halo-removal (i.e., in the sense of $(\Delta x,\Delta y)=(x,y)_{\rm halo}-(x,y)_{\rm no-halo}$). The left figures show the shifts along x-coordinate axis on an image and the right ones along y-coordinate axis. Abscissas give the distance (in pixel) between the geometric center of Jupiter and a satellite.