Analysis and Simulation of Four Frequent Halos

Analysis and Simulation of
Four Frequent Halos
Team Members
Wang Jun
Hu Jingjin
Hu Kun
Advisor
Deng Jingwu
School
Beijing National Day School, Beijing, China
July.2014
Analysis and Simulation of Four Frequent Halos
July. 2014
Abstract
Refracted or reflected by the ice crystals in the cirrostratus, the sunlight
enters the human eyes or other receivers, forming halos, light arcs or luminous spots, which are called solar halos. There are many kinds of solar halos,
such as 22◦ halos, circumscribed halos, tangent arcs and parhelia. The study
of them focuses on summarizing their characteristics, constructing models and
explaining how they form, subsequently simulating them using computer software. The whole process is representative as a general approach to the study
of a phenomenon.
This paper focuses on circumzenithal arcs (CZA), circumhorizon arcs
(CHA), tangent arcs and circumscribed halos. Firstly some preliminary results about crystals are reviewed. Secondly a geometrical optics model is constructed and, explanations for formation and the dispersion are made. Some
of the solar halos are described in the form of analytic expressions. Thirdly,
solar halos under given conditions are simulated. Based on these results, the
important variation of the shape with the change of the altitude of the sun are
discussed, with the critical points pointed out.
Although modern atmospheric optical theory has given a comprehensive
summary on the nature of the solar halos, and also there are some existing
conclusions about how they form, there is still room for further research, especially in quantitative aspects. The core focuses of this paper are the ray
tracing method and parametric equation, which is radically different from the
traditional method. The new idea delineates these halos in their entirety, instead of consider merely scattered points, providing a more holistic treatment
of halos. The approach arising from this new idea raises unprecedented results, especially in regard to circumzenithal arcs and circumhorizon arcs.
Key Words:
Circumzenithal arc
Circumhorizon arc
halo Quantitative analysis Simulation
1
Tangent arc
Circumscribed
1
Introduction
The solar halo is a kind of atmospheric optics phenomena caused by the refraction
and the reflection in ice crystals. There are four frequent halos: circumzenithal arcs
(CZA), circumhorizon arcs (CHA), tangent arcs and circumscribed halos. Tangent arcs
and circumscribed halo are irregular in shape, and the observations of them tend to be
imperfect due to some complicated atmospheric reasons.
This paper studies these four halos by providing a theoretical method and simulating the halos. The two aspects are inseparable, while their emphasis are distinct from
each other. The simulation is fully based on the theoretical analysis, because parametric
form is the main point of view in this paper. In this way the halos are more clear and
more holistic than a mass of scattered points. Some disadvantages of the observations
are remedied by the simulation results.These results are basic but crucial when it comes
to the trends and the critical points of the shape.
Section 2 gives out the physical picture of these phenomena, and sets the parameters. Section 3 puts forward the core method, which is used in theoretical derivation
and programing later. Section 4 gives an analytic expression of CZA and CHA, and
explains some properties of these two halos. Section 5 is a discussion of the simulation
of results. We summarise our findings in section 6.
2
Physical Picture and Parameters
Ignoring other atmospheric influences, this paper focuses on geometric optics problems. All the problems are discussed from the perspective of a observer facing the sun.
The sun is centered horizontally but vertically may not. The observer’s default state
is looking forward horizontally, the elevation angle is also considered when necessary.
The essential process is: set the vector representing the incoming ray, figure out the
emergent ray as refracted by one of the ice crystals , map the emergent light vector to a
dot on a plane perpendicular to the line of sight , and finally all these dots are related to
the parameters of the crystals, they form the halos finally. The parameters are retained
in the points, which helps the study significantly.
Some physical quantities are used throughout this paper:
1)ψ is the solar altitude. A is the elevation angle of the observer;
2)The angle of incidence and refraction of the refractions are respectively i, r; θ, φ
, while the unit normal vectors outward are n1 , n2 ;
3)Default coordinate system: z axis comes to the default observer head-on, and x
axis is the horizon, and y axis is straight up. Raising the observer’s head, the system
would be rotated 1 an angle of A on x axis.
2
Fig. 1: Plate Crystal
2.1
Circumzenithal Arc and Circumhorizon Arc
The hexagonal plate crystals are responsible for these halos. The rays of CZA
enter from top face and exit from side face, presenting the arc in a high altitude. In
contrast, CHA’s rays enter from the side face and exit from top face. These crystals are
approximately horizontally oriented [1], so the only one parameter needed is the angle
between the normal vector of the side face and z, called α. Due to the array of colors of
the two halos, the refractive index n is considered now or later.
2.2
Tangent Arcs and Circumscribed Halo
These halos arise in hexagonal column crystals. The planes of refraction are two
side faces neither adjacent nor opposite [2]. Angles α, β, γ describe the orientation of
crystal as follow. Set the axes parallel with x and rotate it as in Fig. 2, and then rotate
the whole crystal by β and γ in the y and z in sequence as in Fig. 3.
Fig. 2: Placing Column Crystal: Step 1
Fig. 3: Step 2 - Crystal Axes in Thick
Tangent arcs do not have z axis symmetry when ψ = 0, which directly points out
that columns are not uniformly distributed with respect to γ. Previous research shows
that the axes are nearly horizontal, or ‘singly oriented’[3]. Thus we will assume γ = 0
hereinafter.
1 Rotation
in this paper follows the right-hand rule.
3
3
Main Method
As above, the direction of the initial ray can be denoted as:
(
)
l0 = 0, − sin ψ, cos ψ
These four halos are all formed by light refracted twice. The incident angle and emergent
angle of the first refraction are 2 :
sin i
n
With the length guaranteed to be 1 by the sine theorem, we can get the line vector of the
first emergent ray:
sin r
sin(i − r)
l1 =
l0 −
n1
sin i
sin i
The second refraction is similar to the first:
i = π− < n1 , l0 > ;
r = arcsin
θ =< n2 , l1 > ;
φ = arcsin(n sin θ)
l2 =
sin φ
sin(θ − φ)
l1 +
n2
sin θ
sin θ
The emergent ray will emerge into a plane perpendicular to the observer’s line of
sight. Construct a horizontal x axis and a vertical y axis in this plane. When observing
at eye level, this plane is perpendicular to z axis. For instance, under default conditions,
the sun is exactly at (0, tan ψ) ; Therefore, the coordinates of the dot corresponding to
that emergent ray are:
( −l −l )
2x
2y
P =
,
l2z
l2z
As explained at the beginning of Section 2, dots are presented in a plane perpendicular to z axis, and they are in the opposite direction to that of the emergent ray, thus,
there is a coefficient −1/l2z .
In reality this image will be distorted, for example when viewing circumzenithal
arcs and circumscribed halos. The observer generally needs to look up to see them, so
we need an elevation angle correction:
(
)
l3 = l2x , l2y cos A + l2z sin A, −l2y sin A + l2z cos A
Then let
−l3y )
l3z
l3z
Based on the original rough range for parameters, we restrict our range in order to
guarantee the light rays reach both the faces of crystals and the observer.



l0 · n1 ≤ 0
l1 · n2 ≥ 0



l2z > 0 or l3z > 0
P =
2 In
this paper, < a, b >= arccos
a·b
|a||b|
( −l
3x
,
represents the included angle of two vectors.
4
This main method is essentially ray tracing, and we did not found it used in previous
literature. Taking it as our basis, we only need unit vectors for the light ray and range
of the parameters for the crystals.
4
Theoretical Analysis
4.1
Circumzenithal Arc and Circumhorizon Arc
Fig. 4: a and b are the ray paths of CZA and CHA respectively
Using the parameters as before, the outward unit normal vectors of the two refraction faces are:
(
)
(
)


n
=
0,
1,
0
n
=
sin
α,
0,
cos
α
 1
 1
(
) ; CZA
(
)
; CHA
n = sin α, 0, cos α
n = 0, −1, 0
2
2
A rough range is given −π < α < π
As for refractive index n(λ), this paper cites the empirical formula given by The
International Association for the Properties of Water and Steam (IAPWS) in 1997:
( 2
)
n − 1 ρ0
a1 ρ
a2 T
λ
T
λ
a5
a6
ρ
= a0 +
+
+ a3 ( )2
+ a4 ( )−2 + λ 2
+ λ 2
+ a7 ( )2
(n2 + 2) ρ
ρ0
T0
λ0 T0
λ0
ρ0
( λ0 ) − λ2U V
( λ0 ) − λ2IR
a1 = 0.00974634476;
a4 = 0.0015892057;
a2 = −0.00373234996; a3 = 0.000268678472;
a5 = 0.00245934259;
a7 = −0.0166626219; T0 = 273.15K;
λ0 = 589nm;
λIR = 5.432937;
a6 = 0.90070492;
ρ0 = 1000kg/m3 ;
λU V = 0.229202;
The temperature 3 and density of the crystal are taken as [5][6]
T = 248.2K; ρ = 921kg/m3 ;
From this the general range of the refractive index is 1.30536 < n < 1.32007. In fact,
the hexagonal column crystals are anisotropic to a certain extent. Although we treat the
3 Although
the applicable scope of the dispersion relation above is (261K, 773K), due to we not being able to find out a better
expression and 248.2K being not far from the applicable internal, we use it here as a extrapolation.
5
crystal as a isotropic medium, we will see later that the simulation result still accords
well with the observed result. The anisotropy of crystals does not have a large effect in
the phenomenon.
4.1.1
Analytic Expressions and Geometric Properties
First we study on the circumzenithal arc. Using n1 , n2 in the method given in
Section 3, we get
√
l2y = − n2 − cos2 ψ
2
Noticing 1 = |l2 |2 = (Px2 + Py2 + 1)l2z
, where Px = −l2x /l2z ; Py = −l2y /l2z , there
Py2
n2 − cos2 ψ
P is the point P (x, y) we observed, as α is eliminated, the analytic expression of monochromatic light arcs in CZA may be written as
Px2 + Py2 + 1 =
1 − n2 + cos2 ψ 2
y − x2 = 1,
n2 − cos2 ψ
(y > 0)
Substituting the corresponding n1 , n2 of the circumhorizon arc in, we focus on
√
l2y = − 1 − n2 + sin2 ψ
Similarly, we have
Px2 + Py2 + 1 =
Py2
Py2
1
=
=
2
2
l2z
l2y
1 − n2 + sin2 ψ
Namely, the analytic expression of monochromatic light arcs in CHA is
n2 − sin2 ψ 2
y − x2 = 1,
1 − n2 + sin2 ψ
(y > 0)
Nevertheless, circumzenithal arcs are difficult to observe when viewed at eyelevel, thus elevation angle correction is used. Corrected, we get Px = −l3x /l3z ; Py =
−l3y /l3z , writing l2 in terms of l3 , we have
(
)
l2 = l3x , l3y cos A − l3z sin A, l3y sin A + l3z cos A
Substituting this into the relations satisfied by the components of l2 , we may simplify
as
(Py cos A + sin A)2
l2x 2
l2y 2
(l2y /l2z )2
2
2
( ) +( ) +1= 2
⇒ Px + Py + 1 =
l2z
l2z
n − cos2 ψ
n2 − cos2 ψ
2
(Py cos A + sin A)2
l2x 2
l2y 2
(l2y /l2z )
2
2
⇒ Px + Py + 1 =
( ) +( ) +1=
l2z
l2z
1 − n2 + sin2 ψ
1 − n2 + sin2 ψ
Due to the restriction of incidence, the actual shape of monochromatic light arcs is only
part of the figure the analytic expression below:
(y cos A + sin A)2 = (n2 − cos2 ψ)(x2 + y 2 + 1);
CZA
(y cos A + sin A)2 = (1 − n2 + sin2 ψ)(x2 + y 2 + 1); CHA
6
This indicates that the monochromatic shapes of circumzenithal arcs and circumhorizon arcs are quadratic curves, which reflects the concision and beauty of the essence
of these two halos. Since there is no similar result in the literature, the validity of the
expression will be proved in Section 5.
Simplifying the expression above into a general formula, with eccentricity we get:
(
)2 n2 − cos2 A − cos2 ψ 2
(n2 − cos2 A − cos2 ψ)2
cos A sin A
y− 2
+
x ;
2
2
2
2
2
− cos ψ)(1 − n + cos ψ)
n − cos A − cos ψ
1 − n2 + cos2 ψ
(
)2 n2 − sin2 A − sin2 ψ 2
(n2 − sin2 A − sin2 ψ)2
cos A sin A
1=
y
+
−
x ;
2
2
2
2
(1 − n2 + sin ψ)(n2 − sin ψ)
n2 − sin A − sin ψ
n2 − sin2 ψ
1=
(n2
CZA
CHA
cos2 A
; CZA
n2 − cos2 ψ
cos2 A
e=
;CHA
1 − n2 + sin2 ψ
From this we know that when the elevation angle is not large, these two arcs are part
of a hyperbola whose real axis is vertical, when the elevation angle is large they will
become an elliptical arc, with the parabola in between as the critical point:
√
cos A = n2 − cos2 ψ;
CZA
√
cos A = 1 − n2 + sin2 ψ; CHA
e=
No matter whether the result is a hyperbola or ellipse, the coefficient of y 2 must
always be positive, otherwise there won’t be corresponding curve (taking n = 1.3094).
(n2 − cos2 A − cos2 ψ)2
>0
(n2 − cos2 ψ)(1 − n2 + cos2 ψ)
(n2 − sin2 A − sin2 ψ)2
>0
(1 − n2 + sin2 ψ)(n2 − sin2 ψ)
⇒
√
ψ < arccos n2 − 1 ≈ 32.3◦ ; CZA
⇒
√
ψ > arcsin n2 − 1 ≈ 57.7◦ ; CHA
The altitude of the sun deciding whether the circumzenithal arc and the circumhorizon arc can be produced is a important property of these two halos: the circumzenithal arc only forms when ψ < 32.3◦ , while the circumhorizon arc only forms when
ψ > 57.7◦ . This property is explained from the analytic expression above, it can be
illustrated through geometric deduction as well.
4.1.2 Conditions on solar altitude
It worth noting that 32.3◦ is complementary to 57.7◦ , while the formation of these
two halos are dual to a certain extent, so we conjectured that the conditions on solar
altitude are all because of total reflection. A 3-D geometrical approach comes much
closer to the essence of this phenomenon:
Lemma: The angle between diagonal b and a plane is the smallest among those
between b and every line in this plane, iff a is the projection of b in this plane.
From the lemma, for the second refraction face and the corresponding diagonal l1,
π
−θ ≤r
2
7
Fig. 5: Illustration of geometric deduction
To satisfy θ < arcsin 1/n
π
− arcsin 1/n < r
2
⇒
√
i > arcsin n2 − 1 = 57.7◦
For circumzenithal arcs, there ψ = π2 − i < 32.3◦ ;
For circumhorizon arcs, take the plane xz and the diagonal l0 , and we get ψ < i.
We need a lower bound for ψ, but using a geometric approach the only method seems
to be relate ψ to i . In this way, however, due to the above inequality it is not possible
to justify the condition ψ > 57.7◦ .
4.2
Tangent Arcs and Circumscribed Halo
Fig. 6: a and b are the ray paths of the upper and lower tangent arcs respectively
Taking our previous parameters, the outward unit normal vectors of the two refraction faces are:
(
(
)
(
(

π)
π)
π)
sin β, sin α +
, − cos α +
cos β
n1 = − cos α +
6
6
6
(
(π
)
(π
)
(π
)
)
n = cos
− α sin β, sin
− α , cos
− α cos β
2
6
6
6
8
A general range is
π
π
−π
π
<α<ψ+ ,
<β<
6
3
2
2
The figure has two parameters, each color corresponds to a plane area. Following are
further studies on these parameterized graphs through computer simulation.
ψ−
5
Simulation Based on Mathematica 9
Based on the idea of Section 3, we programmed a simulation of these halos in
Mathematica 9. However smooth the graphic is, computer cartography is at heart only
plotting large number of scattered points, so our simulation mainly focuses on gradual
variation and behaviour at critical points, which is different from theoretical analysis 4 .
In Section 5.1 and 5.2 we take n = 1.3094. The reason why dispersion is not taken
into account is that the graphic of tangent arc or circumscribed halo in different colors
has a large area of overlap, which makes the majority of the light arc look white and
dispersion can only be seen on the edge. Isolines of α, β form the grid in two dimensions.
Due to the crystals’ uniform distribution on these two parameters, it is brighter where the
isolines are denser. In addition, for convenience sake, when elevation angle correction is
needed, we take 5 A = ψ in simulations below. Some graphics turn out to have a jagged
edge, which the number of points and extraction method of data points are responsible
for. If the quantity of data point is large enough, or a more rational extraction method
is applied, the staircase on the edge can be eliminated. Fig.7 shows a comparison after
the quantity of data points is increased.
0.2
0.1
0.0
-0.1
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
(a) (nα , nβ ) = (50, 30)
0.2
0.1
0.0
-0.1
-0.6
-0.4
-0.2
0.0
0.2
(b) (nα , nβ ) = (200, 50)
0.4
0.6
Fig. 7: ψ = 0.6rad Lower tangent arc after quantity is increased
9
-0.10
-0.15
-0.20
-0.25
-0.30
-0.06 -0.04 -0.02
0.00
0.02
0.04
(b)
0.06
(a)
Fig. 8: ψ = 0.236rad = 13.5◦ Contrast of lower tangent arc
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
-0.6
-0.6
-0.7
-0.7
-0.8
-0.8
-0.9
-0.9
-1.0
0.0
-0.5
-1.0
0.5
-0.6
-0.4
0.0
-0.2
(a) ψ = 0
0.2
0.4
(b) ψ = 0.1rad
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
-0.6
-0.6
-0.4
0.0
-0.2
0.2
0.4
-0.2
(c) ψ = 0.188rad
0.0
-0.1
0.1
0.2
(d) ψ = 0.3rad
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.10 -0.05 0.00
0.05
0.10
-0.3
(e) ψ = 0.411rad
-0.2
-0.1
0.0
0.1
(f) ψ = 0.5rad
Fig. 9: Lower tangent arc as ψ increases
10
0.2
0.3
0.6
5.1
Important Properties of Lower Tangent Arc
In practice, noticing that the distribution and orientation of crystals are symmetric
about the plane xz,inverse the graphic corresponding to −ψ, the result is a lower tangent
arc. Regarding the density of the isolines, they are dense on one side where the edge
is more distinct, while isolines become spread out on the other side, this corresponds to
the luminous observed.
When the sun is rising from the horizon, the top of the lower tangent arc gradually
becomes sharper and when ψ = 0.188 ± 0.02rad the cusp curls and turns over. See
Fig. 9(c). At this critical point, the isolines become extremely dense, i.e. the luminosity
is high. The lower tangent arc is below the horizon at this point, so it can only be
observed at certain particular places, like the top of a hill. But when the sun is below
the horizonψ = −0.188 ± 0.02rad, the upper tangent arc has similar behaviour, which
means that if the atmospheric condition permit, theoretically a bright cusp can be seen
nearby the horizon under certain conditions before sun-rise. The roll then expands, when
ψ = 0.411 ± 0.01rad the roll is fully expanded. See Fig. 9(e). Nevertheless, because
the tails of solar halos are faint, it could hardly be distinguished in actual observation,
thus this property has little significance. When ψ is larger than this critical value, the
roll disappears and the lower tangent arc expands again.
Here, these critical values are for observers at eyelevel. Uncertainty exists because
the critical values are decided through simulation. Due to the number of data points, the
periphery of halos viewed in the simulation is not that clear whereas the inner edge is
sharp. Furthermore, the critical value is not the same for different colors, which makes
having an accurate value less relevant.
5.2
Conversion between Tangent Arcs and Circumscribed Halo
0.5
0.0
-0.5
0.0
-0.5
0.5
(a)
(b)
Fig. 10: ψ = 0.85rad Comparison of the circumscribed halo (after elevation angle
correction)
4 Technique
5 That
taken in programming see in appendix 1. Each photograph cited also has its copyright notice in appendix 2.
is, the sun is centered both horizontally and vertically.
11
As the sun climbs, the wings of the lower tangent arc go upwards after it expands
totally while the wings of upper tangent arc continually droop. When ψ = 0.70 ±
0.05rad, the edge of both arcs join and becomes a single circumscribed halo. Now that
the circumscribed halo appears only when the sun is high, and that it is a closed graph, an
elevation angle correction should be taken to better correspond with actual observation.
2.0
0.5
1.5
1.0
0.0
0.5
-0.5
0.0
0.0
-0.5
0.5
-1.0
(a) ψ = 0
-0.5
0.0
0.5
1.0
(b) ψ = 0.4rad
3.0
2.5
0.5
2.0
1.5
0.0
1.0
-0.5
0.5
0.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-0.5
(c) ψ = 0.7rad
0.5
0.5
0.0
0.0
-0.5
-0.5
-0.5
0.0
0.0
0.5
(d) ψ = 0.7rad
0.5
-0.5
(e) ψ = 0.8rad
0.0
0.5
(f) ψ = 1.2rad
Fig. 11: Conversion between tangent arcs and the circumscribed halo(d, e, f with elevation angle correction taken)
Fig. 11(d)(e)(f) are simulations of the circumscribed halo. Isolines turn out to be
denser on the inner edge, which accords with the observation that it is brighter. As
the sun climbs, it gradually approaches a circle from an oval shape. When ψ = π/2,
12
it coincides with the 22◦ halo. By the contrast with Fig. 11(c) and Fig. 11(d), the
elevation angle correction have little affect on the critical point where upper and lower
tangent arc join into the circumscribed halo. Since the periphery and wings of the upper
and lower tangent arcs are not so bright, they do not join into the circumscribed halos
whenψ = 0.70 ± 0.05rad, thus this critical value is only theoretical.
5.3
Circumzenithal Arc and Circumhorizon Arc
2.0
2.0
2.0
1.8
1.8
1.8
1.6
1.6
1.6
1.4
1.4
1.4
1.2
1.2
1.2
1.0
1.0
1.0
0.8
-1.0
0.0
-0.5
0.5
1.0
(a) CZA: ψ = 0.2rad
0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
(b) CZA: ψ = 0.4rad
0.6
0.8
-0.4
0.0
-0.2
0.2
0.4
(c) CZA: ψ = 0.5rad
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-2
0
-1
1
2
1
2
(d) CHA: ψ = 1.1rad
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-2
0
-1
(e) CHA: ψ = 1.3rad
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
(f) CHA: ψ = 1.5rad
Fig. 12: CZA and CHA (after elevation angle correction)
Seen from Fig. 12, the circumzenithal arc and circumhorizon arc change in width as
the sun rises, but the change is not observable. Furthermore, after some simulation, we
could not establish any values governing the width as there were no period of monotonic
behaviour. See Fig. 12 (d)(e)(f). The iridesence of the colors makes them easy to be
13
recognized by the naked eye. However, it is not easy to observe the circumhorizon arc
in reality. Its formation requires plate oriented crystals in the cirrus and the emergent ray
not being so inclined, which means only patches of iridesence can be seen on thin cirrus
most of the time6 . There may also be a little distortion in shape. Whereas at other times,
even if conditions are met, observers have to look up to receive even nearly horizontal7
light rays.
1.5
1.4
1.3
1.2
1.1
1.0
-0.4
-0.2
0.0
0.2
0.4
1.5
1.4
1.3
1.2
(b)
1.1
1.0
-0.4
-0.2
0.0
0.2
0.4
(a)
Fig. 13: ψ = 0.2rad Three ways comparison of CZA
-0.8
-0.9
-1.0
-1.1
-1.2
-1.0
-0.5
0.0
0.5
1.0
-0.8
-0.9
-1
-1.1
-1.2
-1
0
-0.5
0.5
1
(a)
(b)
Fig. 14: ψ = 1.3rad Three ways comparison of CHA
There are mainly two reasons why circumzenithal arcs and circumhorizon arcs are
colourful. Firstly, plate crystals have only one geometric parameters as they have only
6 Including
many existing photographs, see also in appendix.
light arcs in Fig. 13 and Fig. 14 correspond to λ = 600nm.
7 Monochromatic
14
one degree of freedom, also each colour corresponds to a curve not overlapping others;
however, for other halos formed by crystals with several degrees of freedom(such as
the 22◦ halos and the tangent arcs), different colours correspond to different areas which
have overlapped others; and for other halos also formed by crystals with a single degree
of freedom(like the parhelia), different colours correspond to different wirelike areas
overlapping others. Hence, the spectral colours of the CZA and CHA are clearer and
purer than those of other halos. Secondly, the angular dispersion power is quite large
when two refraction faces are inclined 90◦ to each other for a certain refractive index,
which separated different colours. This can be explained as follows:
Focus on a prism whose apical angle is A ≤ 90◦ , its dispersion power may be
expressed as:
sin A dn
Dφ =
cos φ cos r dλ
For a known incident angle i, we have
d
cos A cos2 φ + n2 sin θ cos θ dn
Dφ =
>0
dA
cos3 φ cos r
dλ
As a result, dispersion power of prisms whose apical angles are 90◦ is larger than those
of acute apical angle. Here, a prism represents the two refraction faces, that is, the
dispersion power of two refraction faces is quite large when they are perpendicular to
each other, at least larger than those inclined acutely to each other.
6
Conclusion
The result of our investigation may be summarized as:
1. Starting from our paramaterisal equation, we arrive at a method to calculate the
shape of tangent arcs, circumscribed halos, circumzenithal arcs and circumhorizon arcs
and construct a corresponding simulation.
2. The shape of monochromatic circumzenithal arcs as well as circumzenithal arcs
can be described as part of what the following analytical expressions.
(y cos A + sin A)2 = (n2 − cos2 ψ)(x2 + y 2 + 1);
2
2
2
2
CZA
2
(y cos A + sin A) = (1 − n + sin ψ)(x + y + 1); CHA
When A = 0, ie looking at eye level, we get quadratic curves symmetrical with repect
to the y axis, the eccentricities of which are respectively:
cos2 A
;
CZA
n2 − cos2 ψ
cos2 A
e=
; CHA
1 − n2 + sin2 ψ
e=
3. The eccentricities of both circumzenithal arcs and circumzenithal arcs decrease
with increasing altitude of the sun, and the shapes of which may change from hyperbolae
to parabolae or ellipses.
15
4. The main reason for the condition on the sun’s altitude which circumzenithal
arcs and circumhorizon arcs is to avoid total internal reflection. This paper makes explanations based on both analytical expressions and geometric deductions.
5. As solar altitude increases, lower tangent arcs experience changes from sharpening to the first critical point, whenψ = 0.188 ± 0.02rad, it exhibits ‘inverse stretching’
(See Fig. 9(d)(e)(f)), then to the second critical point, whenψ = 0.411±0.01rad, finally
to uptrending wings. The first criticality has its bright peak, and when ψ ≈ −0.188rad
ie the sun is lower than the horizon line, upper tangent arc may have a similar appearance
under proper conditions.
6. When ψ = 0.70 ± 0.05rad, lower and upper tangent arcs meet exactly. No
matter the elevation angle correction is taken or not, the circumscribed halo appears
when the sun is above this critical point, tangent arcs appear otherwise.
7. There are two major reasons to the iridescence of circumzenithal arcs and circumhorizon arcs. One is that ice crystals have a single degree of freedom, making
monochromatic arcs separated, the other is ice crystals have large angular dispersion
power during refraction.
7
References
[1][2]Tape, W. Atmospheric Halos. Washington: American Geophysical Union, 1994.
[3]Cowley, L. ”Tangent Arcs”. [Online] Avaliable:
http://www.atoptics.co.uk/halo/column.htm (June 28, 2014)
[4]The International Association for the Properties of Water and Steam. Release on the
Refractive Index of Ordinary Water Substance as a Function of Wavelength, Temperature and Pressure, Erlangen, Germany, September 1997.
[5]McGraw-Hill Editorial Staff. McGraw-Hill Yearbook of Science & Technology for
2005. McGraw-Hill Companies, Inc.
[6]Kohlrausch, F.W.G. and H. Ebert and F. Henning Praktische Physik, Band 3, Tabellen
und Diagramme. B.G.Teubner, 1955, p.370
16
Appendix 1: Simulating program in Mathematica 9
We are not skillful in computer programing, and we use Mathematica 9 only when
we need its function, so it may not work effectively and cleverly. As a result, the following program is probably unprofessional, but it plays an important role in our study.
Simulating program for circumzenithal arc and circumhorizon arc
Circumzenithal arc
Clear[ψ, n, A]
ψ=
π
180 Input[“ψ
A=
π
◦
180 Input[“A=? ”];
>58 ◦ ”];
n[λ_]:=
Module[{a0, a1, a2, a3, a4, a5, a6, a7, T0, ρ0, λ0, λIR, λUV, T1, T,
ρ1, ρ, λ1, B, Z, N },
a0 = 0.244257733;
a1 = 0.00974634476;
a2 = −0.00373234996;
a3 = 0.000268678472;
a4 = 0.0015892057;
a5 = 0.00245934259;
a6 = 0.90070492;
a7 = −0.0166626219;
T0 = 273.15;
ρ0 = 1000;
λ0 = 589;
λIR = 5.432937;
λUV = 0.229202;
T1 = T /T0;
ρ1 = ρ/ρ0;
λ1 = λ/λ0;
T = 248.5;
ρ = 921;
B = a0 + a1 ∗ ρ1 + a2 ∗ T1 + a7 ∗ ρ12 ;
Z = B + a3T1λ12 +
a4
λ12
+
a5
λ12 −λUV2
+
a6
;
λ12 −λIR2
1
√
]
N := 2ρ1Z+1
1−ρ1Z ; Return[N ]
l0:={0, −Sin[ψ], Cos[ψ]};
n2:={0, −1, 0};
n1:={Sin[α], 0, Cos[α]};
i:=π − VectorAngle[n1, l0];
[
]
r:=ArcSin Sin[i]
n[λ] ;
l1:= Sin[r]
Sin[i] l0 −
Sin[i−r]
Sin[i] n1;
θ:=VectorAngle[n2, l1];
φ:=ArcSin[n[λ]Sin[θ]];
l2:= Sin[φ]
Sin[θ] l1 +
Sin[θ−φ]
Sin[θ] n2;
l3:=RotationTransform[−A, {1, 0, 0}][l2];
[
]
−l3
P :=If l3[[3]] > 0&&n1.l0 < 0&&n2.l1 > 0, l3[[3]]
, Null ;
ParametricPlot[{P [[1]], P [[2]]}, {α, −π, π}, {λ, 380, 750},
ColorFunction → Function[{x, y, α, λ}, ColorData[“VisibleSpectrum”][λ]],
ColorFunctionScaling → {False, False}, AxesOrigin → {0, 0},
PlotRange → All, PlotPoints → {60, 10}, Mesh → {0, 0}, BoundaryStyle → None,
PlotStyle → Opacity[0.5]]
The program for circumhorizon arc is similar to the last one, and the differences
are:
ψ=
π
180 Input[ψ
< 32◦ ];
replaced with
ψ=
π
180 Input[ψ
> 58◦ ];
n2:={Sin[α], 0, Cos[α]};
n1:={Sin[α], 0, Cos[α]};
n1:={0, 1, 0};
n2:={0, −1, 0};
Simulating program for tangent arc and circumscribed halo
In this case, dispersion is not taken into account, and the dispersion relation is
needless. The program below is elevation-modified, and if Ψ ≤ 0.7rad, the output will
be tangent arcs, if Ψ > 0.7rad, the output will be circumscribed halo.
Clear[ψ, Ψ]
Ψ = Input[“Ψ/rad =?”];
A = Input[“A/rad =?”];
n:=1.3094;
l0:={0, −Sin[ψ], Cos[ψ]};
n1:={−Cos[α + π/6]Sin[β], Sin[α + π/6],
− Cos[α + π/6]Cos[β]};
2
n2:={Cos[−α + π/6]Sin[β], Sin[−α + π/6],
Cos[−α + π/6]Cos[β]};
i:=π − VectorAngle[n1, l0];
[
]
;
r:=ArcSin Sin[i]
n
l1:= Sin[r]
Sin[i] l0 −
Sin[i−r]
Sin[i] n1;
θ:=VectorAngle[n2, l1];
φ:=ArcSin[nSin[θ]];
l2:= Sin[φ]
Sin[θ] l1 +
Sin[θ−φ]
Sin[θ] n2;
l3:=RotationTransform[−ψ, {1, 0, 0}][l2];
]
[
−l3
, Null ;
P :=If l3[[3]] > 0&&n1.l0 < 0&&n2.l1 > 0, l3[[3]]
S = Graphics[{Orange, Disk[{0, Tan[Ψ − A]}, 0.08]}];
ψ = Ψ;
P1 = ParametricPlot[{P[[1]], P[[2]]}, {β, −π/2, π/2},
{
}
α, ψ − π6 , ψ + π3 , Axes → False,
PlotRange → All, PlotPoints → {30, 30},
BoundaryStyle → None];
ψ = −Ψ;
P2 = ParametricPlot[{−P[[1]], −P[[2]]}, {β, −π/2, π/2},
}
{
α, ψ − π6 , ψ + π3 , Axes → False,
PlotRange → All, PlotPoints → {30, 30},
BoundaryStyle → None];
Show[P1, P2, S]
Actually, P1 ,P2 in the above program are upper and lower tangent arc respectively.
It would run faster if plot them separately.
3
Appendix 2: Photography and results from other simulation software
Photograph
Circumzenithal arc, by Sylvain Rondi at Paranal, Chile on 21st March.
http://www.atoptics.co.uk/
Circumzenithal arc, at Fanshi County of Shanxi Province on 12st October, 2010.
http://club.city.travel.sohu.com/zz1112/thread/!818c67ee3a084089
3
Circumzenithal arc, by Taavi Babcock at Sooke on the southern tip of Vancouver Island,
British Columbia, Canada in May 16, 2001.
http://www.atoptics.co.uk/
4
Circumhorizon arc, at Yinchuan on 26st June,2011.
http://news.hf.fang.com/2011-06-27/5304680.htm
5
Circumhorizon arc, by Wang Ziai at Beijing National Day School on 11st July,2014.
Lower tangent arc, altitude of the sun: 13.5deg; by Wolfgang Hinz, D-Schwarzenber at
Wendelsteingebiet, Germany.
http://old.meteoros.de/bildarchiv/image.php?page=1&gallery_id=13&image_id=1742
6
Lower tangent arc, by Ryan Skorecki at Alaska. http://www.atoptics.co.uk/
Circumscribed halo, by Doug Short at Anchorage, Alaska, US.
http://cloudappreciationsociety.org/gallery/photo-n-793/comment-page-1/
7
Upper tangent arc, altitude of the sun: 29deg; by Geir T. Oye at Orsta, Norway.
http://epod.usra.edu
8
Results from HaloSim 3.6
Circumzenithal arc, altitude of the sun: 62deg
Circumhorizon arc, altitude of the sun: 10deg
9
Tangent arc, altitude of the sun: 10deg
Tangent arc, altitude of the sun: -10deg
10
Tangent arc, altitude of the sun: 30deg
Tangent arc, altitude of the sun: -30deg
11
Circumscribed halo, altitude of the sun: 50deg
12