Total energy the star projects on a sphere is same at any radius. The formula for surface of a sphere is 4 * pi * r ^2. So if Earth was placed in Jupiter's orbit it would indeed receive 25 times less solar flux because the surface area of the sphere would be 25 times larger.
However, Jupiter's planetary radius is 11 times that of Earth (69,911 km vs 6,371). Area of the circle is pi * r^2, so Jupiter receives 121/25 ~= 5 times more total sunlight than Earth. Of course it also has 121 times larger surface area (see surface of the sphere equation above), so per square meter Jupiter does receive about 25 times less sunlight.
> Total energy the star projects on a sphere is same at any radius.
can you clarify? do you mean the total energy the sun projects onto a sphere centered at the center of the sun is the same for any radius greater than the radius of the sun?
Yes. The photons from the sun get uniformly emitted at the same speed in all directions (assuming spherical cows). Since there aren't any photons lost or gained the amount of energy at a certain distance from the origin must always be the same.
In other words, if you encapsulate the sun into a sphere, said sphere receives an equal amount of photons per hour regardless of its size, they are just spread over more surface area, hence the square term.
However, Jupiter's planetary radius is 11 times that of Earth (69,911 km vs 6,371). Area of the circle is pi * r^2, so Jupiter receives 121/25 ~= 5 times more total sunlight than Earth. Of course it also has 121 times larger surface area (see surface of the sphere equation above), so per square meter Jupiter does receive about 25 times less sunlight.