Estimate the illumination of the surface of Saturn’s moon Titan, facing Saturn, depending on the configuration of the Sun

Estimate the illumination of the surface of Saturn’s moon Titan, facing Saturn, depending on the configuration of the Sun, Saturn and Titan. What are the main periods of the day on Titan, what is their duration?

This hemisphere of Titan is illuminated by two bright luminaries – the Sun and Saturn, and Saturn is always located in the same region of the sky. This planet is 9.54 times farther from the Sun than Earth, and the magnitude of the Sun from the vicinity of Saturn is
m1 = –26.78 + 5 lg 9.54 = –21.88.
On average opposition with average ring opening, being at 8.54 AU. from Earth and having full phase, Saturn looks like a star of 0.4m. From Titan, located at a distance of 1221850 km or 0.0082 AU, the planet in the form of a full disk with a ring will have a luster
m2 = 0.4 + 5 lg (0.0082 / 8.54) = –14.7.
Saturn will have such a brilliance at night, when the Sun is below the horizon. The angular diameter of Saturn will be 5.7 °, but the night will still be 750 times darker than the day (night and day will last on Titan half the time of its revolution around Saturn, that is, almost 8 Earth days). However, the darkest time on Titan will be the middle of the day during the period when Saturn is on the line of intersection of the planes of the orbits of the satellites and its orbit – this happens twice during the period of Saturn’s revolution around the Sun (about 30 years). At this time, a total solar eclipse will occur on Titan, which can last up to 6 hours.

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