In the following image (coming from SAT-Flare) you can see the orbit of the ISS (in red) and the day/night terminator (in blue). The terminator separates night from day along the surface of the Earth. How often will the ISS orbit align with the terminator? When it does so, how long will such an alignment last? And lastly, can it align any closer than it does in this image?
The short answer: 2-4 times per year the ISS enters a "full sun" period. Right before this happens, the ISS will line up near the terminator at the beginning and the end of this period.
Full answer:
I'll try and cover this without explaining too much orbital mechanics. The key thing is what is known as the beta angle, which is the angle between the sun and the orbital plane. A 0 degree beta will have the satellite in maximum eclipse, while a high enough beta angle will allow the satellite to actually be in the sun the entire time.
Okay, so how does this beta angle come about, and what value of it is required to see this effect? The first one is easier, so I'll answer it first. The angle has to be such that the beta angle is greater than the size of the Earth as appearing from that orbit. I have an offline table that has the value (Called Maximum NADIR angle), the Maximum NADIR angle for a satellite of the ISS's altitude is between 350 and 400 km, so the value is about 70 degrees. The higher up, the lower this angle becomes.
So, full sun happens when the beta angle is above 70, and it lines up with the terminator when the satellite is at 70 exactly. How often does this actually happen? There are two factors which determine the beta angle. The first is the Earth's contribution, which can go up to 23 degrees, depending on the season (Northern Summer +23, Winter -23). This corresponds to the Earth's axial tilt. The second corresponds to the orbital inclination, and this also varies. The math on the last part is somewhat complex, it rotates at the rate of nodal procession. The ISS's nodal precession period is about every 2 months, as I can tell from plotting it's beta angle in STK. So, the bottom line is that this can only happen when the inclination, of 51 degrees, adds to the relative Earth sun angle, at most 23 degrees, equals 70 degrees. Note that -70 is fine as well.
As you can see from this plot, it isn't very common. It seems to happen only once per peak season, although it might be possible to have it happen twice per half year, but no more than that. Thus, it happens between 2-4 times per year, always in near one of the solstices.
It's also worth noting that it's a pretty unique thing, this only tends to happen to satellites in LEO with inclinations between 40-70 degrees. There isn't a whole lot of satellites in that range.
