Mathematically defined, flux is actually thought of as a measure of fluid passing through some area. This can be the strength of a river's current moving it through a cross-section of streambed, but it can also be applied to electromagnetic radiation hitting some celestial body. The important thing to note, regardless of what you're using flux for, is that whatever you're measuring the flux of needs to be moving along a path that causes it to hit (or pass through) whatever area you're looking at.
The labels aren't really important in this image--what's important is to be able to visualize a flowing substance (represented by the vectors in the image) moving through an area. (Source)
Putting this into astronomical terms, this means that energy emitted from a star will only be received as flux on a surface that presents a nonzero cross-section to that star. If you were to imagine a 2-D plane in 3-D space oriented edge-on toward a star, it wouldn't actually be getting any flux from it. However, as you started to rotate that plane so that it presented more and more of a cross-section and was closer and closer to perpendicular to the direction light was coming from, you'd get more and more flux.
That's all very wordy, but hopefully its explaining a fairly intuitive idea--you don't need to tell me that you feel more warmth and see more light if you face toward the sun. When we've discussed blackbody flux and imagined spherical shells around stars this hasn't been an issue, because a spherical shell with a star at its center is perpendicular at all points to the light coming from that star (remember in geometry when you learned that a circle is always perpendicular to its radius?). However, if you're thinking about the way flux is distributed across, say, another sphere, it's a bit more complicated. Let's assume the sun is at the equator for simplicity's sake--we'll get to other declinations in a moment.
Apparently that's what I think a star looks like.
If you look at the image above, it should be fairly obvious that the most flux is going to be received at the equator, directly beneath the sun (remember to imagine that the Sun is infinitely far away; one drawback of this diagram is that it doesn't really show sunlight coming in parallel from the right at all latitudes). At the equator, the Earth' surface really is perpendicular to the incoming light, so a unit of surface area really is receiving the maximum possible flux. However, if you switch to examining the opposite extreme, at either of the poles, the Earth's surface is exactly parallel to incoming light, so the surface isn't receiving any flux at all. Thus, flux varies according to latitude by a factor of $\cos(\phi)$--at latitude $\phi = 0$, all of the incoming light is received as flux on the surface of the Earth, but this amount decreases steadily to zero as $\phi \rightarrow \pm \pi/2$.
b) Now, factoring in the tilt of the Earth (which is constant with respect to its orbital plane, at least on time scales less than 1000 years) qualitatively explain why we experience variations in temperature at a fixed latitude during the period the Earth's orbit (a.k.a. seasons during the year).
So yes, it does say to do this part of the problem qualitatively, but there's a very straightforward extension of the math we just worked out that explains this rather beautifully. The difference between the declination of the Sun and a given latitude--what we were assuming $\delta = 0$ for and calling $\phi$ in the previous problem--actually varies over the course of the year, as the Sun's declination varies sinusoidally from $0$ to $\pm 23.5$ degrees. Flux, then, isn't varying according to latitude by a factor of $cos(\phi)$. Rather, it's varying according to both latitude and declination.
It's actually a bit more complicated than $cos(\phi + \delta)$, but we'll get to that.
Since we define northern latitudes and declinations as positive, and their southern counterparts as negative, this means we actually need two separate equations for flux. In the north, the difference between your latitude and the declination of the Sun is $\phi - \delta$, while for southern latitudes the difference is $\phi + \delta$. Thus, in the northern hemisphere the amount of flux you receive varies by $cos(\phi - \delta)$, while in the southern hemisphere it varies by $cos(\phi + \delta)$ (if you don't believe me, try plugging in a few examples).
What does this have to do with seasons? Well, just about everything. The reason the sun's declination varies is because our rotational axis is 23.5 degrees away from being perpendicular to our orbital plane. Thus, at some time of year, a line from the center of the Earth to the Sun will be pointing through the equator, while at another it'll be pointing through +10 degrees, or +23.5 degrees, or -23.5 degrees (some of those times have names--do terms like equinox and solstice sound familiar?).
The variation of the Sun's declination over the course of a year. (Source)
The way that temperature varies from season to season is, then, a direct result of the flux being received at each latitude. From September to March, the Sun has a southern (negative) declination, and so much more energy is hitting a unit of area in the southern hemisphere than in the northern hemisphere. For the other six months of the year, the Earth has swung around so that the Sun's declination is northerly (positive), and the increased flux increases temperature as well. The result? Seasons.
Great way to explain this Tom! An interesting explanation from the point of view of the earth-based observer who sees the Sun's declination change, as opposed to the space-based observer who would see a change in which hemisphere points more directly towards the Sun. That's how I tend to think about it.) As you point out, it's important to remember that the Sun is very far away and so the light comes in as a plane wave. For future diagrams you could consider drawing incoming sunlight as parallel rays.
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