Fig. 1. Example of the definition of the rectangular coordinate grid and solar radius (drawing by Galilei 1613). (a) Green rectangles define the frames where the rightmost, leftmost, topmost, and lowermost red points set the rectangular coordinate grid, with the center in the solar disk. (b) The rectangular coordinate system (Vokhmyanin and Zolotova 2018).
Rectangular coordinate system
In the first step, the analyzed historical image is resized to 1400 × 1400 pixels.
Next, the rectangular coordinate grid and solar radius R are set (Fig. 1a). In Green frames, the rightmost, leftmost, topmost, and lowermost intensity minima (red dots) are defined. These four points clearly denote the rectangular coordinate system, with the origin in the center of the solar disk and the fixed radius R of the Sun (Fig. 1b). The blue and pink lines are the x- and y-axes, and the red circle is the solar limb.
Fig. 2. Intensity of the whole image and a part of it. Red denotes the penumbra border, black is the umbra border.
The rectangular grid with the origin in the center of the Sun provides x-, y-, and z-coordinates of each object on the solar surface. Figure 2 shows the intensity I of an image in the rectangular coordinates. We calculated the x-coordinate of an object (group G, sunspot S, facula F, umbra, penumbra, etc.) as the intensity-weighted value:
$$x_{s}=\frac{\sum\limits_{i,j} x_{i,j} I_{i,j}}{\sum\limits_{i,j} I_{i,j}},$$where i and j define the position of each pixel with an intensity that is higher or lower than a certain threshold (see the color bar in Figure 2). For each drawing, and sometimes individually for an object, the threshold is set individually due to the difference in the texture and color of the images. The same is done for the y-coordinate. Since the Sun's surface is a sphere, the z-coordinate for each object S is defined as $$z_{s}= \sqrt{R^{2}-x_{s}^{2}-y_{s}^{2}}.$$
Area
For each object of a drawing, its area in pixels Apix is the sum of pixels with intensity I exceeding or falling below the threshold. Color gradient results in an uncertainty in determining the object boundaries (Fig. 3), which leads to an uncertainty in the sunspot area in dozens of percent. For some drawings, the image contrast has to be adapted.
Fig. 3. Example of the color gradient of the drawing from the Galilei's letters to Cardinal Maffeo Barberini (Pope Urban VIII) and to his nephew Cardinal Francesco Barberini (Manuscript Barb.lat.6479 of the Biblioteca Apostolica Vaticana). Different intensity thresholds result in different areas of the sunspot group.
Fig. 4. The angular distance α.
To calculate the area in microhemispheres Amsh, the correction for foreshortening toward the limb has to be taken into account (Meadows 2002). The angular distance α on the surface of the Sun from the center of the disk to a pixel (varies from 0 to 90o, Fig. 4) is calculated as follows: $$\alpha = \arcsin\left(\sqrt\frac{x^{2}+y^{2}}{R}\right),$$
where x and y are the coordinates of a pixel in the rectangular grid,
and R is the solar radius. In other words, the area in msh of an object is the sum of its pixels,
where each pixel is adjusted to its α. Commonly,
$$A_{\textrm{msh}} = \frac{A_{\textrm{pix}} \cdot 10^{6}}{2\pi R^{2} \cos(\alpha)}.$$
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