Moon Halos.

On nights with diffuse high clouds and a bright moon, a halo can appear about the moon. This results from light refracting (through an angle of 22 degrees) through hexagonal ice crystals in the atmosphere. The mathematics is as follows:

Hexagonal ice crystals look something like this.

A side-on view of an ice crystal

When light strikes the boundary between two different substances (air and ice) it refracts.

Air-Ice boundary

Light ray

Snell’s Law is the physical relation governing refraction. It states that the angle of incidence and the angle of refraction are related by the formula

.sin

sin

2

1 k

The constant k depends on the nature of the two substances. For air-ice, it is about 1.31.

Light enters the crystal, and then leaves it, making two refractions.

θ1

θ2

θ3

θ4

Here, the angle α represents the total angle the light bends.

θ1

θ2

θ3

θ4

The necessary relations needed to determine α are on the next slide.

θ1

θ2

θ3

θ4

180180

,180120,sin

sin,

sin

sin

3421

323

4

2

1

kk

We solve the relations on the next slide.

180180

,180120,sin

sin,

sin

sin

3421

323

4

2

1

kk

11221

1

1111

1

1111

41

31

4

23

112

sinsin32

1sin60

sin2

1sinsincos

2

3sin60

sinsin60sinsin60

60

sinsin

60

sinsin

k

kk

kk

k

k

11221

1 sinsin32

1sin60 k

“Technically speaking”, there are about a zillion ice crystals in the atmosphere all oriented randomly. The angle

1

will take many different values.

11221

1 sinsin32

1sin60 k

This means that the light will be scattered through many different angles, though there is a catch.

11221

1 sinsin32

1sin60 k

If there are many different angles of incidence

1

that yield approximately the same α, light will seem to build up at that angle.

Light builds up most where the slope is smallest. This is 21.83929990 degrees.

This is the classic 22˚ moon halo.

22˚

Diffuse clouds

Apparent Halo

Moonlight

Light inside the halo is refracted out in a cone that misses the observer, making this region less bright to the observer.

Diffuse clouds

Apparent Halo

Moonlight

Light outside the halo is refracted out in a cone that misses the observer, making this region less bright to the observer.

Diffuse clouds

Apparent Halo

Moonlight

Light on the halo is refracted out in a cone that the observer sees, making this region brighter to the observer.

Diffuse clouds

Apparent Halo

Moonlight