A STUDY OF ATMOSPHERIC TURBIDITY
THROUGH OBSERVATIONS OF
ARAGO'S NEUTRAL POINT
By
Daniel H.
Lufkin
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Bachelor of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Signature
Department of Meteo6lofy, May 16, 1952
Certified by
Th(ess Supervisor
Chairman
Department Committee on Undergraduate
Students
ACKNOWLEDGMENT
The writer is indebted to Professor Hans Mueller of the
MIT Physics Department for his kind assistance in obtaining
from the Polaroid Corporation the retardation plate used in
the instrument, to Professor Keily of the Meteorology Department for making available the theodolite used in the
study, and to Professor J. C. Johnson for his advice and
encouragement through difficulties and adversities too frequent and harrowing to enumerate.
Thanks are due also to
Mr. Fred Sanders of the Meteorology Department for his uncannily accurate forecasts of cloudiness which were of
great value in preparing for observations.
4-
"Men judge by the complexion of the sky
The state and inclination of the day."
Richard II (III, 2,
194)
ERRATA:
P. 11, TABLE III:
PP. 3,
4,
5:
1914 sun depression should read -2.40
All subscript e's in Pe and Se should read
1 to be uniform with Stokes parameters.
P. 6:
TABLE I:
The S/P value for 300 should read .3143.
ii
-
1~
TABLE OF CONTENTS
I.
II.
III.
The Normal Polarization of the Sunlit Sky. . .
.
1
Distribution Criterion. . . . . . . . . . . .
7
An Hypothesis Toward a Large Particle
Instrument and Observation .
. . .
. .
. . .
. . 15
APPENDIX
I.
II.
The Stokes Parameters of Polarized Light
. . .
. 26
The Complete Form of the Rayleigh Scattering
Matrix
.
. . .
BIBLIOGRAPHY .
. .
. .
. . . . . . .
. .
. . .
. . .
. . .
. .
. . . . .
. 29
. . 31
PLATES
I,.
II.
A Typical Neutral-Point Curve
The Ratio of Secondary to Primary Scattering as
a Function of Neutral-Point Deflection
III.
The Height of the Lowest Layer of the Atmosphere
Illuminated by the Setting Sun
IV.
V.
VI.
The h vs. S/P Diagram of Turbidity Distribution
Fragmentary Neutral-Point Curves
The Neutral-Point Curve for January 30, 1952
iii
VII.
VIII.
The Neutral-Point Curve for November 27, 1951
The Neutral-Point Curve for January 11, 1952
iv
I.
THE NORMAL POLARIZATION OF THE SUNLIT-SKY
The principal features of the intensity and polarization
of the sky's radiation were set forth by Rayleigh in his classical theory of molecular scattering.
Rayleigh's conclusions
are exactly supported by observation within the limits of the
treatment of the classical theory.
The limits are these:
Orders of scattering higher than the first are not considered,
the size of the scattering particles must be small compared
to the wave-length of the radiation, and the particles must
be optically anisotropic, that is, the polarizability tensor
of the particle is considered to be constant in all orientations.
From classical theory, the light scattered from a particle illuminated by a beam of unpolarized light should exhibit
a polarization Po in the direction f from the axis of inci-
S
dence,
(1)
Thus, the polarization maximum of P-1 occurs at 90 degrees
from the sun, measured, as are all polarization measurements,
in the plane of the great circle through the sun, the sun's
meridian.
A maximum of Po=l, however, is valid for perfect
Rayleigh scattering only.
In the actual observation of the polarization of the
sky, the polarization maximum varies from about 80 per cent
to 50 per cent (i).
Chandrasekhar (ii) cites an average of
87 per cent and Bhagavantam (iii) 94 per cent.
Thus, a de-
polarization factor of anywhere from 6 per cent to 50 per cent
must be accounted for.
Raman and Bhagavantam (iii) assign a
value of roughly 4 per cent to the effect of optical anisotropy in nitrogen, oxygen and water vapor in the air.
The
depolarizing effects of multiple scattering and of ground
reflection have also been investigated, notably by Tichanowski
(iv).
Unfortunately, the mathematical treatment of scatter-
ing of partially polarized light by these investigators has
been shown to be incorrect by Chandrasekhar (ii, p. 264).
The nature of the error will be explained in the appendix,
and it is sufficient to note here that Chandrasekhar's revised
treatment of the problem of diffuse transmission of partially
polarized radiation brings the state of theory even closer to
observed distributions of polarization.
Even though the exact mathematical treatment of some,
indeed most of the features of anisotropy, multiple scattering, large particle scattering, and ground reflection is extremely complex and varies somewhat from authority to authority,
the physical effects of these phenomena are easily explained
in a qualitative manner.
- 2 -
Raman, after his experimental determination of the
anisotropic depolarization factor of the air (iii) made a
series of measurements at high altitudes in the Himalayas
to determine the maximum polarization at right angles from
the sun.
He concluded that, within the limits of observa-
tional error, all or nearly all, of the observed depolarization was attributable to anisotropy.
In other words,
secondary scattering by Rayleigh particles seemed to be
vanishingly small.
Now, with anisotropic scattering modify-
ing the original Rayleigh scattering, what sort of sky polarization have we?
Obviously, if we make the proviso that the anisotropic
scattering particles of the atmosphere are not ordered in
any way and are completely randomly chaotic in both orientation and position, anisotropic scattering may be regarded as
the superposition of isotropic scattering upon the Rayleigh
scattering.
If the incident light is partially plane-polarized,
each of the Stokes parameters Ir and Ie (See appendix) will be
scattered independently, thus reducing P, the polarization to
the value given by Ir-Ie.
Thus, as the factor of Rayleigh scattering introduces
partial polarization, the factor of isotropic scattering lessens the percentage polarization without changing either the
phase relationships (which would determine ellipticity) or
the plane of polarization.
Therefore, the total distribution
- 3 -
of polarization in the skylight remains unchanged except in the
local value of P.
The maximum P still falls at right angles
to the sun's direction and the Rayleigh neutral points, one in
the direction of the sun and one at the anti-solar point, remain undisturbed.
Thus, to explain the observed positions of the neutral
points, we must find another agency at work in the atmosphere
producing polarized light with the plane of polarization parallel to the horizon.
The addition of this negative polarization
to the field of positive polarization produced by primary
Rayleigh scattering will have the effect of moving the Rayleigh
neutral points to new equilibrium positions of stronger positive polarization background.
It can be shown (v) that the effect of multiple scattering in the air plus primary scattering by the air of light
which has been diffusely reflected by the ground will result
in a region of negative polarization.
(Again, positive polari-
zation is that having its plane in the principal meridian;
negative polarization, that having its plane parallel to the
horizon.)
Now, letting P=Pr+ Pe and S=Sr+ Se represent the total
intensities of the positively polarized light from primary
vi
scattering and the negatively polarized light from secondary
scattering, respectively, at an angle
0
from either the sun
of the anti-solar point, the ratio
CO~S2
P9
(2)
Since the effect of secondary scattering is proportional to
the second power of the optical path length of scattering
medium, the secondarily scattered radiation may be considered
to originate wholly in the illumination of the bright horizon
sky.
Now with the sun on the horizon and neglecting extinction
effects, the brightness and polarization of the primary scattered
radiation from a point on the horizon sky at an azimuth #
the sun will be given by the Rayleigh phase function:
Constant = K and I(hor) = KCos 2 0g .
from
I(vert)
The light reaching an ob-
server from this point after undergoing secondary scattering
in the plane of the horizon will have a polarization component
ratio of
TO.)
Ijcve'-t)
Cs4:
(3)
Evaluating the integral
yields a value of 8/3 for the average I(vert)/I(hor) for all
azimuths.
Thus, from the previous definition, Se/Sr = 8/3.
A neutral point will be found at that place in the sky where
the per cent negative polarization, Sr-Se equals the per cent
positive polarization, Pe-Pr.
From equation (2) and the defi-
netion of P:
P
(+ Co.so>)P
- 5 -
=
P=
C.o0s7
0)pr
(5)
and
(6)
Then, at a neutral point,
(Co-291)
and, substituting (5)
P =4 Sr
and (6)
I-Cosa
1+Cose
=
(7)
into (7)
$
I-P
A
_
:)
1 o4s?a
(8)
Then, since the neutral points fall within 35 degrees
of the sun or the anti-solar point, the following ratios of
secondary to primary scattering may be easily computed.
TABLE I (See also Plate II)
9
0
5
10
15
20
25
30
35
52014 '20"
s/P
0.0000
0.0084
0.0337
0.0763
0.1366
0.2158
0.3614
0.4322
1.0000
It must be remembered, however, that the formulae above
hold good only as an approximation for the special case of the
sun lying exactly on the horizon.
Since no secondary scatter-
ing flux downward was taken into consideration, the formula
- 6 -
7
probably gives a very poor indication of the state of the scattering ratio for positive solar elevations, at which time the
secondary scattering flux is predominantly in the downward
direction.
The effect of ground albedo and diffusivity is also not
considered.
While the action of a diffusely reflecting ground
surface, i.e., one closely obeying Lambert's law, consists entirely in the addition of completely unpolarized light to the
sky field, a ground surface which performs some specular reflection at low solar elevations, such as lake, sea, or smooth
snow-field, will introduce extra negatively polarized light
into the sky-field.
Since such light would have to undergo
depolarizing back-scattering to reach an observer of the Arago
point, while little depolarization would be introduced in the
apparent path from the Babinet point, it seems certain, although no experimental verification has been performed, that
the latter point would be very much more affected by variations in local ground-cover than would the former.
II.
AN HYPOTHESIS TOWARD A
LARGE PARTICLE
DISTRIBUTION CRITERION.
It will be observed from Plate I that the minimum Arago
point deviation occurs at small negative solar elevations.
Since this indicates a corresponding minimum in the secondaryto-primary scattering ratio, it may prove fruitful to examine
the mechanism of formation of such a minimum.
As the sug/ets, more and more of the atmosphere above an
observer is darkened by the shadow of the earth.
When the
lowest layers of haze and dust are no longer illuminated,
primary Rayleigh scattering becomes more predominant in the
distribution of polarized light.
This process of the increase
of positively polarized light continues until the predominant
light from the zenith is the result of back-scattering, at
which time the negative component of polarized illumination
increases rapidly, producing another maximum of neutral-point
displacement.
Thus, the solar depression at which the minimum neutralpoint deviation takes place is rather closely determined by
the distribution of dust and haze in the lower atmosphere.
Although the expression (8) given above is strictly valid
only for a solar depression of zero, its application to the
case of the deviation minimum cannot be wholly unjustified.
Thus, from neutral-point observations, two variables relating to the minimum deviation are obtainable:
the solar de-
pression at which the minimum value occurred and the value
of the minimum deviation itself, expressed either as an
angle
or converted into the scattering ratio S/P..
The lowest level of the atmosphere illuminated by a sun
below the horizon has been computed by Link (vii) on the
assumption that the layers below 3Km are opaque to grazing
(tangential) illumination.
Photometric observations of
zenith-brightness changes during the course of twilight have
upheld this assumption.
Link's values are presented in Plate III.
- 8 -
I
I1
So little is known about the multiple scattering effects
of large particles that it is difficult to attach any scheme
of actual physical distribution to values of the minimum.
The
most straightforward assumption, and one which does not seem
to be contradicted by any previous observational or theoretical
investigations is that the scattering ratio S/P given by the
value of the minimum deviation represents the fraction of nonRayleigh scattering particles above the height given by Link's
function from the solar depression.
Direct observational support for this interpretation of
the minimum is given by Neuberger (viii) in his analysis of a
long series of Arago-point measurements.
He found contiguous
the following variabilities in average values, variability defined as the mean absolute deviation from the average and
relative variability as these mean deviations divided by their
Diurnal variations are given by
respective average values.
average evening minus average morning values.
TABLE II
VARIABILITY
D. V.
SUN
DIURNAL
ELEVATION
VARIABILITY
(RELATIVE)
Degrees
Degrees
Per Cent
2.5
0.2
2.45
0.082
1.5
0.5
-0.5
-1.5
0.0
0.0
-0.2
-0.3
1.70
1.55
1.50
1.65
0.000
0.000
0.133
0.182
-2.5
-0.2
1.55
0.129
-3.5
-4.5
-0.1
-0.2
1.80
2.10
0.056
0.095
- 9 -
V.
Degrees
3
From TABLE II it is apparent that the greatest relative
diurnal variability is exhibited at a solar depression of 1.5
degrees --
the solar depression which corresponds to the
average minimum neutral-point deviation.
Although no correla-
tions are available for the variation of the sun depression at
the minimum neutral-point deviation
the atmospheric turbid-
ity, the absolute variability at -1.5 degrees is given by
Neuberger as 0.31 degrees.
Now it is unlikely that the gross turbidity should change
overnight or during the day by one-third its value.
It is
likely, however, that the day-to-night contrast in solar heating, reflected in the values of turbulent transport of dust
and smoke particles upward would produce a change of that great
magnitude in the concentration of dust particles at some particular level, in this case 4.2 Km.
It was observed by Jensen (ix) that the minimum Aragopoint deviation shifts toward larger solar depressions with
increasing turbidity, but the values of the shift which he observed were not so well correlated with measured values of
turbidity.
The uncertain correlation, together with the fact
that the changes observable were relatively small, led to the
abandonment of this criterion in favor of a measurement of the
Arago position at a relatively great sun's elevation, generally
in the neighborhood of 10.5 degrees.
- 10 -
Data taken by Jensen in Hamburg before and after the
Katmai volcanic explosion of June, 1912, show the following
minima data:
TABLE III
SUN
DEPRESSION
Degrees
YEAR
ARAGO-
DISTANCE
Degrees
S/P
Cm
1911
-1.3
19.5
3.8
.135
1912
-2.5
14.9
7.0
.076
1913
-2.6
19.0
7.3
.130
1914
-2.5
19.9
6.6
.137
(2d half)
If these data are arranged on a diagram of h vs. S/P
(See Plate IV), it is possible to connect them by a smooth curve
showing a rapid increase in h with a concommittant decrease in
S/P.
This may correspond to the physical case of a large amount
of dust introduced at low levels.
From 1912 to 1913, the curve
travels to a slightly higher value for h and a much higher one
for S/P.
This may correspond to further diffusion of the dust
particles upward, the general level of the concentration being
raised, and a greater proportion of non-Rayleigh scatters being
found above h.
h and in S/P.
The change from 1913 to 1914 is small, both in
This may indicate that the mean level of the
dust has begun to sink, while the ratio of non-Rayleigh particles found above h has slightly increased to a point above the
1911 normal.
The value of h=7.3 Km may be connected to the
- 11 -
fact that the average height of the tropopause over Hamburg is
fairly close to 7 Km.
Unfortunately, the advent of World War I
put an end to this series of observations, and the rest of the
curve must be postulated as a gradual return to the values of
1911, which may be regarded as normal.
The points plotted for
1934 and 1938 were computed from two separate series observed
by Neuberger at Sylt on the North Sea and at State College,
Pennsylvania, respectively.
Very little is known with any degree of certainty about
the diffusion of volcanic dust in the atmosphere.
The great
eruption of Krakatoa in August, 1883, was followed by worldwide optical effects which enabled observers to place the
upper limits of the dust layer at about 30 Km (x).
The mean
height of the glowing stratum of dust decreased about 15 Km
in the five months following the eruption, after which time
it presumably became too faintly defined to be followed.
Un-
fortunately no comparable studies of neutral point phenomena
seem to have been made during the period of intense turbidity
from 1883 to 1895.
No recent investigations have been made
under conditions of high dust concentrations due to a current
.ack of volcanoes.
The fact that S/P also decreases from
1911 to 1912 in a marked degree is more than offset by the
concommittant increase in h --
effectively placing our refer-
ence level for the determination of S/P much higher in the
atmosphere.
- 12 -
Also available are minima averages from a series of observations taken at Recklinghausen by V. Dahlkamp (xi) in 1941 and 1942.
The observations from each of these years are divided into three
categories of high, moderate and low turbidity, according to the
amplitude of the Arago curve.
Values in the vicinity of the minima
are given in TABLE IV.
TABLE IV
Sun Elevation, Degrees
1.5
0.5
-0.5
-1.5
-2.5
-3.5
Arago-distance: high turb.
"
"t
mod. turb.
"
"t
low turb.
17.6
18.4
19.7
17.2
17.8
18.8
16.6
17.2
17.9
16.6
16.8
17.0
16.7
17.0
16.8
17.1
17.9
17.5
Yearly Average
18.7
18.0
17.2
16.6
16.8
17.7
16.6
16.2
15.8
15.6
15.2
15.6*
18.2
19.0
17.5
18.5
16.9
17.9
16.3
17.4
16.5
15.9
17.4
17.0
18.7
18.1
17.6
17.0
16.6
17.4
1941
1942
Arago-distance: high turb.
if
mod.
low
it
Yearly Average
turb.
turb.
*1942 high turbidity values are from observations taken
through a blue filter which, because of the erratic and uncertain nature of the dispersion of polarized light in the
atmosphere,are impossible to compare exactly with the other
(white-light) observations.
The average minima, plotted on the S/P - h diagram, seem
to indicate a general decrease in upper-level large particles,
since S/P decreases while h remains nearly constant from 1938
to 1941-1942.
- 13 -
The question of whether or not neutral-point observations
made at different localities may be compared with one another
is still in some doubt; from theoretical considerations, only
the presence of large bodies of water or other surfaces approaching specular reflection should affect the neutral balance of
polarized light; however, many statistical comparisons (xi for
example) show that great variations in observed values may take
place even between two stations similarly situated.
Therefore,
curves have been drawn to connect 1934 with 1938 (Sylt and State
College, Pennsylvania), and 1938 with 1941-1942 (State College
and Recklinghausen) only after careful justification of observed similarity.
Although the hypothesis that the h and S/P values determined at the minimum neutral-point (specifically the Aragopoint) deviation are indicators of the distribution of large
particles with height rather than merely indicators of gross
turbidity may fail to be supported by subsequent observations
(assuming someone might be interested enough to perform the
task), the fact will remain that neutral-point observations
have a definite value in the study of atmospheric diffusion,
turbidity, and magneto-electric processes.
(viii)
Thus, it was with the hope of obtaining a series of neutralpoint measurements that I prepared the instrument described in
the following section and began my wait for a cloudless, smogless sunset.
- 14 -
INSTRUMENT AND OBSERVATION
III.
The instrument used in the observations of neutral-point
phenomena for this investigation was a modified Arago polarimeter attachment for a standard Keuffel and Esser pilot-balloon
theodolite.
The polarimeter was modified from the classical
model (xii, p. 673) in the following respects:
the pile of
glass plates serving as a depolarizer was omitted, since numerical values of the per cent polarization were not sought (therefore the instrument, strictly speaking,is a polariscope), the
quartz retardation plate was replaced by a Polaroid Corporation
quarter-wave plate made from laminated calcite, and the Nichol
prism was replaced by a Polaroid analyzing filter of the standard photographic, neutral absorption type.
The construction was childishly simple:
on the ocular of
the theodolite was fitted a standard series 5 filter-holder
with extension collars holding the quarter-wave plate and the
Polaroid filter.
A black cardboard diaphragm of 8 mm. aperture
to minimize internal reflections was fitted between the plate
and Polaroid.
The aperture stop also ensures that the light
passing through the plate will be divergent, an important
feature of the operation.
.Since the quarter-wave plate is a section of an uniaxial
crystal cut perpendicular to its optic axis, the optical action
of the system may be analyzed as follows:
- 15 -
S
P
FiG. I
Consider what happens when a cone of plane-polarized light
rays diverging from S passes through a crystal plate, the central ray SO of the cone coinciding with the optic axis of the
crystal.
The ray incident at 0 passes through the plate in
the direction of the optic axis, and its vibration plane remains unaltered.
Other rays in general, such as SQ, will undergo
double refraction and emerge with a phase difference between
the components of the vibration parallel and perpendicular to
the polarization plane.
This will not be true, however, for
certain rays, SP and SP', for example.
Consider the ray SP.
The direction of vibration is in the principal plane SOP, that
is, the plane containing the ray and the optic axis, and it
will be transmitted by the crystal without resolution into
ordinary and extraordinary rays.
The same holds true for the
ray SP', since in this case the vibration is perpendicular to
16
-
the principal plane.
Accordinaly, all rays striking the crystal
plate along OP or OP' will not be doubly refracted, and hence
will be either completely passed or completely absorbed by an
analyzer positioned behind the plate depending on whether the
plane of the analyzer is horizontal or vertical.
Consider now another ray incident upon some other point,
say Q.
The vibration a will be resolved into two components,
b and c, lying parallel and perpendicular respectively to the
principal plane SOQ.
The vibrations will traverse the crystal
with different velocities and will accordingly emerge with a
difference of phase, which will depend upon the thickness of
the crystal and upon the wave-length of the light.
Now the
thickness traversed by the ray will increase from 0 to Q,
owing to the increasing obliquity of the rays.
The phase
difference of the emerging components will therefore vary
along the line OQ, and the emergent light at some points on
this line will be polarized in the same plane as the incident
light; at other points, in a plane perpendicular to it.
The
analyzer will then absorb one or the other, according to its
orientation.
By symmetry the conditions of equal phase dif-
ference will occur along concentric circles with a common
center at 0.
The eye behind the analyzer thus sees a system
of colored fringes (for white incident light), interrupted
along OP and OP' by a cross which appears bright or dark according to the orientation of the analyzer.
-
17 -
(See Fig. II.)
Fi G. II
It should be noted that the actual thickness or retardation
of the plate is not a factor at all in the production of the
fringes.
A quarter-wave plate was used solely because it was
a standard commercial plate.
Although this polariscope is not the most sensitive, it
serves well enough to locate the position of a neutral point
in the following manner:
The theodolite is trained upon the
eastern horizon at an azimuth opposite that of the sun about
ninety minutes before sunset.
The initial settings and sub-
sequent checks are greatly simplified by the attachment of a
small bit of white card fastened upright just ahead of the
front sight.
The shadows of the front and rear sights cast
on the card by the sun may then be carefully aligned.
Note
is then taken of the azimuth and the depression of the antisolar point (toward which the theodolite will then, of course,
be pointing), and the local solar time, which may be computed
by subtracting 12 minutes 36 seconds from 75th Meridian time
(EST), and then adding the correction specified for that day
in the Nautical Almanac.
(The time correction to EST given
here is computed for 710 05' 32.145" west longitude, the
longitude of the MIT theodolite stand.)
About one and a half hours before sunset (the EST of
which should be found in the Almanac), the theodolite should
be trained on the eastern horizon, the azimuth being determined
by the shadow of the sights on the card.
- 18 -
The filter-holder containing the plate and the analyzer
is now mounted on the ocular of the instrument as shown in
The observer will now see a system of rings with
Fig. III.
a cross as in Fig. II superimposed on the image of the eastern
horizon.
The filter-holder should be rotated until the cross
is dark as shown when the theodolite is pointed at a region
of positive polarization; in other words, the plane of the
analyzer, which is indicated by a line on the filter rim,
should be horizontal.
It is not necessary to have the cross
centered exactly in the reticule, since only the direction of
the theodolite determines which part of the sky-light is
brought through the polariscope.
A careful survey of the sky is now begun, beginning at
the horizon and extending to about 30 degrees elevation.
If
the sky is cloudless and no industrial pollution is present,
a change from dark to light cross will be observed, with an
intermediate position in which the ring-figure disappears.
This is the position of Arago's neutral point.
Generally the disappearance of the rings takes place
over an area larger than that covered by the field of view of
the theodolite.
In such a case, it is necessary to determine
the upper and lower limits of the neutral area and to average
these for the position of the neutral point.
To estimate the
accuracy of the observations under such conditions is difficult,
since the disappearance of the rings is gradual and their
visibility is also a function of the brightness of the sky
background.
- 19 -
Fic.mII
Under very good conditions of visibility and cloudlessness,
the neutral point appears as a ring system with the cross poorly
defined, neither completely dark nor perfectly ringless, and
with the limbs crossed by the ring system.
This is because
the predominant polarization plane of the incident light is
neither vertical nor horizontal, but acts in a random manner
as elliptically polarized light with a major axis at 45-degrees.
An exposure made (on Ektachrome 3200 0 K film, f/4.5, 5 sec.) on
January 11, 1952, at sunset shows this phenomon well (Fig. IV).
The series of observations undertaken this past winter
(1951-1952) was, to say the least, disappointing.
Only eight
days from November through the middle of March were clear and
smoke-free enough to allow any observations at all to be made.
Of these, only three, November 27, January 11 and January 30
yielded full curves, and the remaining five, November 9 and
21, January 21, February 16 and March 3, yielded only spotty
values because of interference from smoke pollution on the
eastern horizon.
The observations were reduced for comparison by the following method:
The position of the neutral point was deter-
mined as often as possible -was about four minutes.
the average time between readings
The major observation was the eleva-
tion of the neutral point, but azimuth readings were taken
also, mainly as a check value, but also of occasional interest
in cases where the neutral point deviated from the principal
- 20 -
Fir.
H
meridian by as much as four degrees under the influence of
localized concentrations of smoke or of impending cloud formation.
Now, knowing the date and time, it is possible to com-
pute the sun's elevation for each observation.
Although
nomograms for performing this computation have been devised
(viii, p. 6),
it was found a simple matter to evaluate each
observation by means of the very fine navigation tables of
Cmdr. R. deAquino (xiii).
The sun's elevation is computed
only to the nearest tenth of a degree and the distance of the
Arago point from the anti-solar point to the nearest fifth of
a degree, this increment representing the best accuracy of
the neutral-point elevation measurements.
The final result, distance of the neutral point from the
anti-solar point as a function of the solar elevation, is presented in Plates V, VI, VII, and VIII.
Although too few observations were made to enable any
generalized conclusions to be drawn with much certainty, some
of the features of the curves deserve mention.
The minima of
the curves all fall at nearly the same solar elevation, even
including the fragmentary curve f~r February 16.
Applying
the height of illumination formula, h ranges from about 4 to
5 Km with S/P varying from 0.093 to 0.117.
Not enough data
for other days is available, however, to justify any attempt
to correlate these values with any observed meteorological
changes.
- 21 -
Perhaps it is significant to note that the observed curves
become smoother, the windier the day.
January 11, at 1630 EST
winds were reported as gusts to 32 MPH; November 27 had 1630EST
winds of 25 MPH; and January 30, the most irregular curve, had
winds of only 10 MPH at observation time.
It seems reasonable
to expect that the observed irregularities in the curves are
due to changing concentrations of scattering particles (in
Boston's case, mostly industrial and heating-fuel smoke).
A
further assumption that relatively higher winds would tend to
promote more uniform diffusion of the smoke and so to even out
any variations in concentration.
Thus, high wind speeds should
result in relatively smooth neutral-point curves.
Since strong
winds also result in greater turbulent upward diffusion of
dust and smoke particles, the high value of S/P (0.117) for
January 30 mat be of significance, although we should not expect to find the total local turbidity much altered since the
effect of wind in shifting the center of pollution is small
(xiv, p. 1146).
It is interesting to note that the two poorest observations -azimuth.
November 9 and March 3 --
both show deflections in
Since all theory outlined above was developed on the
assumption that all scattering particles in the atmosphere are
evenly distributed, it is logical to expect that departures
from even distribution in the horizontal plane will produce
deflections in the azimuth of a neutral point and cause it to
appear out of the principal meridian.
- 22 -
On both days that horizontal deflection appeared, heavy
concentrations of smoke in the south-east were visible.
The
observed deflections of the neutral points were all towards
the south, although we should expect the smoke to be an area
of strong secondary scattering and therefore of negative polarization, the addition of which would have the effect of moving
the neutral point to the other side of the principal meridian,
that is, towards more northerly azimuths.
Apparently visual
estimates of smoke concentration are not reliable in the prediction of areas of strong scattering.
Surveys of pollution in the atmosphere of large cities
show that a maximum of smoke concentration occurs in the early
morning (0630 to 0900) with a strong secondary maximum in the
late afternoon as turbulence associated with solar heating decreases before industrial emission of smoke has been reduced.
This is an annoying factor in neutral-point measurements,
since the observations must be made near sunset when the
secondary smoke maximum tends to make observation difficult,
as well as acting to obscure the more important long-term
changes in turbidity.
The fact that the annual maximum of
pollution occurs in winter when home and office heating plants
are in full operation does not make things easier for the investigator.
- 23 -
If any conclusion were to be drawn from this present
series of observations, it would be that neutral-point observation in the vicinity of a city, especially near one
located in a predominantly cloudy climate, is impracticable.
So few observations can actually be completed, and those
observations subject to so many unpredictable influences,
that any hopes for statistical validity must be discarded.
In a remote, relatively cloudless location, however,
neutral-point observation would certainly be worthwhile.
The present method is capable of great improvement -in the mechanics and the optics of the process.
both
The use of
photographic means for recording neutral-point positions
accurately and rapidly would permit the investigation of
polarization phenomena simultaneously at many wavelengths,
including those in the invisible ends of the spectrum.
Practically nothing is presently known about the dispersion
of atmospheric polarization in the infra-red --
a region
in which the proper choice of wavelength for observation
would enable the investigator to determine the role of water
droplets and water vapor,
of anisotropic
gase-s
such as
carbon dioxide, and of snow surfaces,which are almost comPhotographic methods could
pletely black in the infra-red.
also extend the observations into the later twilight -consequently into the higher atmosphere.
-
24
-
and
Certainly the study of neutral-point phonomena is no
magic key to the understanding of the mighty workings of the
atmosphere, as some of the earlier investigators believed.
The problems involved in the statement of a definitive theory
which could include all of the observed phenomena of atmospheric polarization make up a supreme test for the wisdom and
imagination of the investigator.
A unified, long-range plan
for integrated observation, while it could not guarantee success in the solution of these problems, would be a long, firm
step toward the ultimate achievement of that goal.
- 25 -
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APPENDIX
I.
The Stokes parameters of polarized light
It is evident that the complete characterization of a
radiation field must include the following parameters:
the in-
tensity, the degree of polarization, the plane of polarization, and
the ellipticity of the polarization of the radiation at any given
point in the field and for any given direction.
The difficulty
of formulating equations of transfer and of optical equilibrium
for a generalized radiation field will depend to a great extent
upon the system chosen to represent these parameters.
The most
convenient system yet devised is that of Sir George'Stokes, which
he introduced in 1852.
In an elliptically polarized beam of radiation the vibrations of the electric vector in the plane transverse to the
direction of propagation are such that the ratio of the amplitudes and the difference in phase of the components of vibration
in any two mutually perpendicular directions are absolute constants.
by
Thus the simple harmonic vibrations may be represented
54-.CL
- 26 -
where
,
and
6"
j,
,
are constants of the beam,
Er
1 and r represent the two perpendicular directions, and 4 and
4
are the components of the vibration, and w
the circular
,
frequency of the vibration.
If the principal axes of the ellipse described by (4, ,r)
are in directions making angles
'+
and
to the direction
1, the equations (1) take the simpler form
4
where
"0'o~5L~v
(2)
is the angle whose tangent equals the ratio of the
axes of the ellipse traced by the end-point of the E-vector.
Assume that the value of /3
and that
lies between 0 and
its sign is positive for right-handed elliptical polarization
is defined by
and negative for left-handed.
(3)
where I is the intensity of the beam.
*
From (2)
and the relations
l 1.
4:
~r 4(oJ(
then,
also,
t)L.
5*%
SX
uttx $Wt
os
=
C4 5s,A^4.
t
.
enIOt)
o4C~ct
+ 5.
-,
SK
-t
Crt ;K
(4)
(5)
and
4r
')-
(Cs'S
s,x +
- 27 -
(6
Then the intensities in the directions 1 and r, Il and Ir
are expressable as
IF,
and from (4)
A
L4 I = i
and (5)
Now, whenever the conditions postulated for (1) hold good,
the following parameters are defined:
Q=
R~v~~[~:'i~
-(9)
These are the Stokes parameters for that elliptically polarized
beam.
Notice that L=Q+y+V
The azimuth of the plane of
polarization from the direction 1,
/
and the ellipticity,
tion
is found from the relafumiig
may be found
.from Sin 2 (9 = v/I.
If two independent beams of light, that is, two beams
which have no permanent phase relation one with the other, are
mixed, the Stokes parameters of the mixture is the sum of the
respective Stokes parameters of the separate beams.
It follows,
then, that if independent beams of equal and opposite Q, U and
V are mixed, the resultant beam is characterized by its I only;
its Q = U = V = 0, and the beam is unpolarized natural light.
- 28 -
To satisfy the condition of equal and opposite Q, U, and
V, two elliptically polarized beams must be of equal ellipticity
and intensity, the major axes of the ellipses must be perpendicular, and the sense of rotation must be opposite.
In the
light of the sky, where retardation is negligible, only the
conditions of equal intensities and perpendicular planes of
polarization apply.
It is when these conditions are met that
neutral points appear.
II.
The complete form of the Rayleigh Scattering Matrix
In the atmosphere where retardation is negligible, the
skylight, as mentioned above, is not elliptically polarized,
the Stokes parameter V = 0.
Replacing I and Q by their defin-
ing components Il and Ir, we are left with three parameters:
Il, Ir and U.
In the original Rayleigh theory, the distribu-
tion of scattered intensities takes the analytic form:
Il
C Cos' a
Ir -
C
When C is a constant and 1 and r identify the components
parallel and perpendicular to the plane of scattering, respectively.
In a complete treatment of the atmospheric polariza-
tion, this simple distribution must be modified in two respects:
account must be taken of the depolarization by anisotropic
scattering and provision must be made for the treatment of
cases in which the incident light is not polarized.
- 29 -
The generalized Rayleigh scattering function then becomes:
where primed parameters refer to the scattered light and
where
eois
the coefficient of anisotropic depolarization.
Since partially polarized incident light in the atmosphere
could only reach a particle after having been scattered previously by another particle, the use of the complete Rayleigh
matrix demands consideration of the other effects of secondary
scattering, and the mathematical treatment of the problem becomes very complex indeed.
Chandrasekhar (ii) has developed a
solution for the case of a multiple Rayleigh scattering planeparallel atmos'\phere, but the effects of large particles, variation of density of the atmosphere with height, and curvature of
the boundary of the atmosphere are not included, although the
author has suggested means of taking these factors into account.
The practical and theoretical difficulties in such computations
will undoubtedly be overcome in time as methods for machine
solution of the rather cumbersome integral equations involved
are developed further.
- 30 -
BIBLIOGRAPHY
i.
Sekera, Z.; "Polarization of Skylight" in Compendium of
Meteorology, T. F. Malone, Ed., American Meteorological
Society, Boston, 1951.
ii.
iii.
Chandrasekhar, S.; Radiative Transfer, Oxford, 1950.
Bhagavantam, S.; Scattering of Light and, the Raman Effect,
Waltair, India, 1940.
iv.
Tichanowski, J.; "Die Bestimmung des optischen Anisotropiekoeffizienten der Luftmoleklen durch Messungen der
Himmelspolarisation." Phys. Z., 28:252-260 (1927).
v.
Soret, J.;
"Sur la polarisation atmospherique, " Arch. Sci.
phys. nat., 20:429-471 (1888).
vi.
Van de Hulst, H. C.;
"Scattering in the atmospheres of the
Earth and the Planets." The Atmospheres of the Earth and
the Planets, G. P. Kuiper, ed., Chicago, University of
Chicago Press, 1949.
vii.
Link, F.; Pub. Obs. nat. Prague, No. 14 (1940), No. 18
(1947).
viii.
Neuberger, H.; "Studies in Atmospheric Turbidity in Central
Pennsylvania", Penn. State College Studies, No. 9,
ix.
1940,.
Jensen, C.; 3 Beiheft z. Jahrb. Hamburg. Wiss. Anst. 33.
1915.
x.
Lettau, H.;
"Diffusion in the Upper Atmosphere". Compendium
of Meteorology, T. F. Malone, ed., American Meteorological
Society, Boston, 1951.
- 31 -
xi.
Dahlkamp, V.; "Beitrage zur Untersuchung des Atmospharischen
Reinheitsgrades." Z. f. Met., 3:11/12 (1949)
xii.
Jensen, C.;
"Die Apparate zur Untersuchung der Atmosph'ris-
chen Polarisationserscheinungen," in Kleinschmidt, E.;
Handbuch der meteorologischen Instrumente.
Berlin, Springer,
1935.
xiii.
Aquino, R. de; Sea and Air Navigation Tables. Annapolis, U.S.
Naval Institute, 1927.
xiv.
Hewson, E. W.; "Atmospheric Pollution" in Compendium of
Meteorology, Boston, American Meteorological Society, 1951.
- 32 -