Let's start by understanding what solar flux is. It's the amount of solar electromagnetic radiation per unit area incident upon Saturn (in this case). It's given by the inverse of the square of the distance from the source (the Sun) because it's dissipating on a spherical surface as it moves outwards.
The distances given here in astronomical units (AU) are Saturn’s closest and farthest points of its orbit around the Sun, 9.05 AU and 10.12 AU, respectively.
First, we must calculate each solar flux. The solar flux at the closest distance (9.05 AU) is calculated as 1 / (9.05^2). The result of this operation gives us the final flux value of approximately 0.012209639510393453.
Similarly, the solar flux at the farthest distance (10.12 AU) is calculated as 1 / (10.12^2). The result of this calculation gives us a flux of approximately 0.009764251902076271.
Having calculated the solar flux able at both the closest and farthest distances, we can now determine the difference between them. This is reasonably straightforward: we just subtract the smaller flux from the larger one. The difference in flux, therefore, is approximately 0.0024453876083171816.
Finally, we end by calculating the percentage difference between the two flux values. The first step is to calculate the average solar flux, which we get by adding the two flux calculations together and dividing by 2. We then take the absolute difference between the two fluxes, and divide that by the average. To express it as a percentage, we multiply the whole thing by 100. The percentage difference in solar flux between the closest and farthest distances from Saturn to the Sun, therefore, is approximately 22.257210272091136%.
In conclusion, we find that the solar flux on Saturn varies significantly over the course of its orbit, with approximately a 22% difference between the closest and farthest distances to the Sun.