This document summarizes a theoretical investigation of shadow bands that will be observed during the total solar eclipse of August 21, 2017. Standard scintillation theory provides quantitative and qualitative insight into the factors governing the visibility of shadow bands, including the geometry of the solar crescent near totality and the effects of atmospheric turbulence at different altitudes. The document also analyzes how parameters like the eclipse magnitude (ε) and time from totality (τ) will influence the appearance of shadow bands. Overall, the eclipse is expected to produce visible shadow bands in the minutes leading up to totality due to its high magnitude and the contribution of turbulence below 2 km in the atmosphere.

1. A Theoretical Investigation of Shadow Bands
for 2017 Eclipse
Katherine Stocker (kstocker@physics.montana.edu)
Physics Department, Montana State University
Eclipse Data: Total Solar Eclipse
Date August 21, 2017
Total Begins (UT) 17:16:03 (OR) 17:25:16 (ID)
17:34:18 (WY) 17:46:51 (NE)
18:05:23 (MO) 18:18:02 (IL)
18:22:12 (KT) 18:26:11 (TN)
18:36:24 (NC)
Longest Duration 2m40.25s Makanda, IL
Magnitude 1.03059
Ԑ 0.03059
Altitude 39 to 64 incline
Shadow Bands and Scintillation
Shadow bands are most likely a scintillation phenomenon caused by
turbulence in the Earth’s atmosphere [1].
Scintillation (most familiar to you as
twinkling stars) is described as
variations in apparent brightness or
position of a distant luminous object as
viewed through a medium (Earth’s
atmosphere). Shadow bands are the
visual effect from these intensity
variations backscattering off of the
ground or other surfaces.
Fig.(1): Illustration of intensity variations observed on the front surface of a
building.
For specific times, visit http://eclipse.gsfc.nasa.gov/SEgoogle/
SEgoogle2001/SE2017Aug21Tgoogle.html
Understanding shadow bands requires an understanding of “source
averaging.” Light from point sources (stars, lasers, etc.) passes through
turbulence collectively and is observed as twinkling. Light from extended
sources (Sun, planets, and crescents) passes through different patches of
turbulence and is visualized as a sum of variations “averaged out” and
twinkling is not observed.
How the Bands Work
However, as light from an extended
source reaches the lower levels of the
atmosphere, spacing between the rays
decreases. The converged rays will then
behave as rays from a point source;
collectively passing through turbulence
nearer the ground and producing the
twinkling seen as shadow bands.
Fig.(2): Converged light rays passing through turbulence nearer the ground creating
an intensity pattern on the ground.
Scintillation Theory
The salient features of shadow bands can be explained using standard,
weak scattering scintillation theory and standard models for turbulence.
The shadows on the ground are described as a random intensity pattern
I(x) written as the sum of the intensities contributed by the various parts
of the source [1]:
The intensity pattern shifts by -z when the direction of the incident
plane wave shifts by . Therefore, an extended source smears the
intensity pattern on the ground. Source averaging of an intensity
spectrum by an extended source can be applied to the weakly scattering
approximation to the intensity spectrum for a point source incident on a
thin phase screen to obtain the intensity spectrum of the extended
source as a spatial average [1]:
The spectrum is used in defining a measure of the pattern contrast given
by the “scintillation index, ” due to a thin screen, m [1]:
References
1. Codona, J.L. The scintillation theory of eclipse shadow bands. Astron. Astrophys. 164, 415-427, 1986.
Print.
2. Espenak, Fred. “Total Solar Eclipse of 2017 Aug 21.” EclipseWise.com: Predictions for solar and lunar
eclipses. Oct., 19 2014. Web.
3. Total Solar Eclipse of 2017 Aug 21. NASA. Web.
 =
𝑡𝑖𝑚𝑒 𝑎𝑤𝑎𝑦 𝑓𝑟𝑜𝑚 𝑡𝑜𝑡𝑎𝑙𝑖𝑡𝑦 𝑖𝑛 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
Standard scintillation theory provides both quantitative and
qualitative insight into the factors governing the visibility of shadow
bands. To analyze the different geometries of a solar crescent and
its rapid evolution near totality, two parameters are introduced [1]:
Turbulence extends to high altitudes, and the rapidly changing
solar crescent is affected by turbulence at certain altitudes more so
at certain times then others during its evolution. To see which
altitudes are most efficient for generating intensity fluctuations, a
“scintillation efficiency” parameter is defined [1]:
A few minutes from totality, the crescent is still considered a broad
source, and the turbulence important in forming shadow bands is
10s to 100s of feet above an observer. At 30s and less from totality,
the crescent as thinned to a point where the turbulence important
in forming bands extends up thousands of feet. Overall,
turbulence mainly responsible for the formation of shadow bands
is below 2 km [1].
𝜂 𝑚2� , , ,  =
𝑚2
𝑙𝐶 𝑛
2
What Can We Expect To See?
Shadow bands are usually seen for times less than two
minutes away from totality. The most striking
characteristic of total solar eclipses is the extent of the
crescent’s horns. Eclipses with larger values of  lose
their horns rapidly and form a better “slit.” A better
slit is expected to form crisper shadow bands due to
reduced source averaging [1]. Lucky, many have
reported shadow bands for cases of =.03.
Fig.(4): Evolution of solar crescents for  intervals of 0.01 (36s)
and  (a)0.01, (b)0.05, and (c)0.10 [1].

While Itotal is larger for eclipses with smaller , relative contrast is smaller.
Therefore, eclipses with larger values of  are better at producing the
narrow, wavelength-dependent patterns associated with shadow bands.
Eclipses with smaller  are governed by characteristic length scales of the
intensity spectrum affected only by turbulence nearer the ground.
Consequently, small  eclipses display random , smeared, smoky patterns.
The intensity pattern will snake across the ground if wind convects the
turbulence perpendicular to the line of sight. If the wind is parallel with the
bands, contrast and visibility will be reduced even if the intensity
fluctuations are strong. This eclipse will have less dependence on direction
due to larger crescent horns and therefore a higher probability of wind
blowing parallel to the solar crescent and presenting us with more of a
shimmering intensity pattern [1].
Hufnagel Model
To take into account the effects of
extended turbulence that dominates the
intensity fluctuations for times very near
totality, a generic turbulence profile is
constructed. Cn
2 is a structure constant
that models the strength of turbulence at
altitude h. Given this profile, we can see
which altitudes are contributing most to
the shadow bands [1].
𝑚2
 , , ,  = 𝐶 𝑛
2
𝑧𝑐𝑜𝑠  𝜂 𝑚2� , , ,  𝑑𝑧
∞
0
Fig.(3): Plot of the modified Hufnagel
turbulence profile [1].