Dumb physics question: "Jupiter's orbit is five times farther from the Sun than Earth's orbit, which means that the giant planet receives 25 times less sunlight than Earth."
Would the amount of energy from the sun be the square root or the cube root? I assumed the latter because space exists in three dimensions.
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.
Every instant the sun emits a certain amount of energy. After 1 second that energy is spread across the surface of a sphere 4pi(c)^2 big. After 2 seconds it's spread across a the surface of a sphere 4pi(2c)^2. And so forth.
Or another way to look at is that that at a certain distance from the sun all the energy leaving the sun is passing through a sphere that grows that same way.
EDIT: Thanks to grn for pointing out I was confusing the constant of a sphere's volume with its area. Fixed.
Here's one check: the area of a circle is pi r^2, and each hemisphere of a sphere has area greater than that (since it has the same circular boundary, but is inflated), so you know a sphere must have area greater than 2 pi r^2.
A bunch of comments have explained already, but I'd recommend this wikipedia article with some helpful illustrations for anyone who's having trouble visualizing it:
It's the inverse square (not a root), because the surface of the planet is two-dimensional. Think of moving a projector screen back and forth. For a fixed aperture angle, the area projected upon will grow quadratically with distance, so the intensity (inverse area) grows inverse-quadratically.
A given amount of light emitted by the Sun is spread over a sphere of radius r. As the light propagates the radius gets bigger and bigger. The light per area is proportional to 1 / 4πr². If the radius grows 5 times then the area of the sphere grows 25 times.
>"We learn that the force between two charges, two magnetic monopoles, or two masses all follow an inverse square law, however, most of the time, the scientific reader is not made aware of an important assumption, that of being able to model these entities as point objects. If the entities cannot be reduced to a point, then, the inverse square laws cannot be applied. As I shall mathematically show, the inverse square law changes into an inverse cube law approximation for the case of dipoles. In practice, a physicist finds that most of real life applications cannot be modelled by point entities, but only by dipoles."
I'll agree that this can be inaccurate/illogical/misleading/confusing/frustrating. Given that "50% more" is equivalent to the addition of 50% of the original value, and "50% less" is equivalent to its subtraction, "25 times less" is essentially the equivalent of "2500% less", which would mean Jupiter receives -2400% the light of Earth per square metre. Understandably nonsensical.
Granted there are more issues with that sentence than just the "25 times less". If I've done my math correctly, since Jupiter's radius is approximately 11 times that of Earth, Jupiter will have in the order of 380 times the surface area of Earth available to catch light. At 5 times the distance that actually works out at around 15 times more incident light than Earth in total, even if it is at about 1/25th the intensity.
This is a construction to be avoided. It's obvious that "25 times less" is impossible. So everyone knows that it means "a 25th". But "25 times more" is sometimes used for 25x, and not 26x.
