# Why do sundogs move to larger angles at high sun elevations?

As stated (and visible) here: http://www.atoptics.co.uk/halo/dogalt.htm sundogs move to larger angles if the sun is at a higher elevation angle. I don't understand the explanation though. Why can't a sun ray enter at the angle of minimum deviation?

I mean, the 22° halo is always at 22° and the only real difference between sundogs and 22° the halo is that sundog-forming crystals are plates so the light has to be reflected internally in contrast to 22° halo-forming columns where the light can propagate diagonally through the crystal. So ultimately it must have to do with the reflection inside the crystal, but what makes the difference?

• The crystals which give rise to sun dogs have a fixed (or at least strongly preferred) orientation with respect to the surface of the Earth, or sun dogs would be rings around the Sun. So it's reasonably obvious that things will change as the lline from the sun to the observer changes orientation with respect to the Earth's surface (ie how high the sun is above the horizon). – tfb Jul 8 '17 at 13:30
• But what changes? I aware of the fact that the crystals are oriented, but I don't see why light can't enter in the angle of minimum deviation. Why does the change of angle in one plane influence the angle in the other plane? – lou_liu Jul 9 '17 at 10:08
• I haven't looked at it in detail, but obviously the whole geometry of the system is different (consider the two extremes, when the Sun is on the horizon and when it is directly overhead) and it would be entirely extraordinary if the effect didn't change as a result. – tfb Jul 9 '17 at 10:47

The website referred to in the question (http://www.atoptics.co.uk) is an excellent resource for sun dogs, solar halos of all sorts and various other atmospheric optics. I will try and fill in some of the details it doesn't include.

## 22° Halo

The 22° halo is formed by sunlight refracting in hexagonal ice crystals entering through one of the side faces and exiting a side face that's oriented 60° to the first face. See the diagram below. The 22° halo is formed when the orientation of the crystals is randon. For now we'll just consider the case of rays of light entering the crystal in a plane parallel to the hexagonal faces and call the initial angle of incidence $\alpha$. Taking the index of refraction of air as 1 and of ice as n=1.31 Snell's law gives us the first angle of refraction $\beta = \arcsin(\frac{\sin\alpha}{n})$. The second angle of incidence, $\gamma$, is found by the simple relationship $\gamma=60°-\beta$. Using Snell's Law one more time for the second refraction gives $\delta = \arcsin(n\sin\gamma)$.

The angle of deviation is difference in direction of the emerging ray compared to the initial incident ray. It's labelled $\theta$ and is related to the other angles by the simple relation $\theta=\alpha+\delta-60°$. Putting it all together we get the the following relationship between the initial angle of incidence and the angle of deviation: $$\theta = \alpha - \arcsin(n\sin(60°-\arcsin(\frac{\sin\alpha}{n})))-60°$$

Graphing the angle of deviation vs. the incident angle reveals that the angle of deviation is never less than about 22° and that, in fact, it's pretty close to 22° for a fairly broad range of angles of incidence.

For an interactive ray diagram check out the Atmospheric Optics website. So, when we look towards the sun through bunch of these crystals light gets refracted away from us (at least light that passes through the crystals in this particular way). On the other hand, it we look 22° away from the sun light gets refracted towards us. If the crystals are randomly oriented the results is a dark circle around the sun with a bright halo 22° from the sun.

## Sun Dogs

The formation of sun dogs is closely related to the formation of the 22° halo. They are formed when the atmospheric ice crystals are oriented with the hexagonal faces on the top and bottom and the rectangular faces vertical. In the case where the sun is not elevated (i.e. near the horizon) the analysis given above applies and the the sun dogs appear as part a bright patches of the 22° halo. However, when the sun is elevated light enters the side faces inclined at some angle which changes the geometry in a couple of ways. Firstly, and especially when the ice crystals are more platelike than columnlike, the light must reflect internally off of the top and bottom hexagonal faces before emerging from the the crystal.

These internal reflections however are not the reason the sun dogs are further away from the sun at higher elevations. This effect would happen even without reflection.

When incident rays enter the crystal inclined they refract vertically as well as bending in the horizontal plane. The final emerging ray is not coplanar with the initial incident ray. Below are are a couple images showing the refractions of an inclined ray. To get a much better feel for how the refractions work go to this GeoGebra interactive worksheet. In all cases (as projected onto the horizontal plane) the final angle of refraction as well as the angle deviation increase as the angle of inclination increase. Since the minimum angle of deviation is increased the sun dogs appear farther from the sun.

• Aaaaahhhhh, thank you! The last paragraph was the missing piece… now it's kind of obvious (as it often is as soon as somebody explains it to you). The GeoGebra worksheet/figures are really great! – lou_liu Aug 6 '17 at 18:10