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Abstract
An unexplained centertolimb variation of solar wavelength has been
known
for 75 years. Many theories have been developed in order to
explain
its origin. Although recent studies reveal a large amount of new
information on the solar chromosphere, such as asymmetries of lines and
various mass motions in granules, which lead to wavelength shifts, no
theory
can consistently explain the observed centerto limb variation. It is
shown
that the theory considered in this paper not only explains this
variation
but also predicts its amplitude without the use of any adjustable
parameter.
1. Introduction.
The dedication of a Special Issue of this Transaction to Hannes
Alfvén
is a welldeserved acknowledgment for his extraordinary discoveries [1],
which have demonstrated the importance of plasma in the universe.
Alfvén's works led to the discovery of intergalactic plasma
(matter)
as the cause of the "cosmic static" observed by Reber [2].
Intergalactic matter is expected to produce inelastic transmission
(redshift)
when electrodynamical (bremsstrahlung) considerations are taken into
account
[3][5].
It has been shown
[3], [5]
that the amount of matter calculated by Reber [2]
is equal to the amount of matter required theoretically [3],
[5]
to produce a red shift equal to the Hubble constant. That same
redshift
theory [3],
[5]
also leads to other verifiable predictions when radiation crosses the
solar
chromosphere. In that case, a centertolimb variation (CLV) of
solar
lines is predicted, as seen below.
The centertolimb variation of solar lines was reported at the turn of
the Century by Halm [6]
and Adams
[7].
About 200 papers have appeared on the subject since then. The
impressive
number of independent observations and discussions by Adam [8][10],
FinleyFreundlich [11],
Schröter
[12],
Struve [13], Higgs [14],
[15],
SalmanZade [16],
Snider [17],
Schatzman and Magnan [18],
Pecker
[19],
Beckers and Cram [20],
Reboul
[21]
and many others, including extensive recent documentation of his
unexpected
phenomenon, plus the fact that there has been no contradicting
observation
of the red shift of the FeI lines, have firmly established that the
wavelengths
of the Fraunhofer lines in the solar spectrum are dependent upon
distance
from the solar limb. This CLV cannot be a consequence of
relativity,
which predicts that all solar line must be red shifted by a factor of 2.12
´10^{6}
and hence should be independent of the position on the solar
disk.
This CLV is detected superimposed on the known Doppler shift resulting
from the relative motion of the Sun and Earth. During those past
years, observers hoped in vain to discover new facts, but the basic
observations
of the CLV have not changed in 70 years, as is stated by Howard et al.
[22]
and Dravins [23]:
"The
early measurements of Evershed and Royds [24]
could adequately resolve differential line displacements and more
recent
photoelectric instruments yield practically identical curves".
The object of this paper is to describe and assess the results of exact
computations according to some models, and to compare them with
experimental
data. One of the models considered here follows the hypothesis
that
the CLV is caused by Doppler shifts due to radial mass motion in the
solar
photosphere and chromosphere [13],
[25]
or to granules, as calculated by Schatzman and Magnan [18],
Cavallini et al. [26],
Gurtovenko
et al. [27], Adam et
al. [28],
Brandt and Schröter [29],
Becker
and
Nelson [30],
Kaisig
and Durrant [31], and
others.
The model applied here and calculated by Marmet [3][5]
is based on the fact that the momentum transfer of the solar photons,
to
the electrons of the atoms of the solar atmosphere, produces secondary
radiation due to bremsstrahlung. The energy of this
bremsstrahlung
radiation is taken away from the energy of the initial photons and
leads
to a red shift. Such an energy loss is a natural phenomenon,
which
has been described recently by Marmet [3][5].
No adjustable parameters are used, and all constants are either
fundamental
physical constants or wellknown solar phenomena.
II. The Current Situation.
Measurements of solar lines (sodium D1 and potassium 7699Å) have
been made [32], [17],
and [25].
Initially, [17]
reported a disagreement with the gravitational red shift. Two
years
later, Shider [25]
claimed agreement
with the gravitational red shift following selection of his data (Earth
motion smaller than 12mA). The absence of any CLV as observed and
explained
[25]
is due to the fact that some lines "originate above that region
of
the photosphere."
Many other exotic explanations, such as photon decay
[20],
nonzero photon rest mass
[19],
and others, have been considered in order to explain all the observed
characteristics
of the redshifted Fe I lines.
Observations [14], [15],
[33]
(and by others) have definitely determined that the red shift of the
lines
near the limb is larger than the value predicted by relativity and is
accompanied
by an asymmetry of the lines profiles. Granules have been
observed
in the Sun for some years, but this phenomenon has been unsuccessful in
explaining the CLV. More recently, it has been confirmed [23]
that there is an upward mass motion in the brighter granules, and a
downward
motion in the darker intergranular lanes. The most recent data [34]
yield simultaneous resolution in both time and space in individual
granules
in several spectral lines, determining the exact mass motion in
granules.
Accurate new measurements [35]
led to the belief that there was meridional flow on the solar
surface.
Again, it was hoped that such meridional flow could contribute to the
explanation
of the CLV shift. More recently, in two different papers,
Cavallini
et al. [26], [33]
have shown that when several spectral lines are used, their data do not
fit with the sometimesobserved equatorward mass motion that gives "anomalous
CVL" and "contradictions".
Cavallini et al. [26],
[33]
state in their last article that "these contradictions somewhat
reflect
the contradictory results obtained in previous measurements by other
authors.
This shows that a linedependent unknown phenomenon introduces a
latitude
differential shift which is assumed as due to a mass motion".
There are different depths of comprehension of a physical
phenomenon.
After a reasonable number of observations, one normally tries to
explain
results (in this case CLV) in terms of previously known phenomena such
as relativity, mass flow, granules, erroneous measurements, etc.
When these attempts are unsuccessful, it is useful to make some data
analysis
and to compare the data with series of functions for which one can fit
arbitrary adjustable parameters. Due to the lack of a working
theory,
such a "parametrized model" has been used [23],
[36]
to fit the CLV experimental data. This mathematical model
provides
a good description of the data, but gives no real physical
understanding
of the phenomenon.
The polynomials fitted by Bruning [36]
and Cavallini et al. [26],
[33]
are
1 
III. Calculations
Calculations based on the theory which takes bremsstrahlung into
account
are considered first. In this case, Marmet [3], [5] has shown
that
when a photon encounters an atom or molecule, the energy of the
transmitted
photon is very slightly reduced. This follows from the fact that
a secondary photon of extremely low frequency is emitted [3],
[5]
which reduces very slightly the energy of the primary photon.
The fact that photon scattering is always inelastic has also been found
independently and was demonstrated several years ago by Bethe and
Salpeter
[41],
and has been recalled more recently by Jauch and Rohlich [42].
Because the energy loss is extremely small and the quantitative amount
of energy loss is difficult to calculate, this mechanism has been
generally
ignored. One can see that, due to bremsstrahlung, this inelastic
scattering leads always to an energy loss, and, consequently, the
transmitted
photon loses some energy. Therefore, the photon is slightly red
shifted.
It is seen that the energy loss is caused by the acceleration of an
electric
charge (and therefore follows Maxwell's equations) rather than by a
quantum
or a relativistic effect. Quantum and relativistic considerations
do not lead to energy losses in this case. Since calculations of
energy losses due to the acceleration of charges is treated by
electrodynamics,
we have used here classical electrodynamical considerations. The
results of those calculations have been reported [3],
[5].
It is seen that the red shift produced depends on the number of
collisions
with atoms and therefore on the amount of gas crossed in the trajectory
of the photon.
In order to calculate the energy loss (red shift)
produced
when photons cross the solar atmosphere, we must first determine the
amount
of gas above the base of the chromosphere. By definition, the
height
of the solar limb is the base of the chromosphere. The solar gas
density is known and given [43]
as a function of altitude.
The total amount of gas that the radiation has to pass through before
reaching
the Earth is integrated from different altitudes in the solar
chromosphere.
Thus, one obtains the column density, which is the number of atoms in a
column having a unit cross section and a length equal to the distance
between
the point where the absorption takes place and the observer on
Earth.
A second factor takes into account that, for different radii between
the
center of the disk and the solar limb, the column density increases
according
to the secant of the angle between the line of sight and the solar
radius
to the point where the line of sight cuts the surface. Finally,
the
solar curvature for the radiation tangent (or near tangent) to the
solar
disk at or near the limb is also taken into account. This last
consideration
makes the path length in the solar chromosphere become finite at the
limb.
In practice, we have found that this last correction adds a term that
has
a negligible contribution for our purpose and is limited by the
experimental
resolution near the solar limb.
The amount of red shift produced by the collision of one photon on an
atom
is given by b. This parameter is called R
in
previous papers [3], [5].
It is seen that each time a photon meets an individual hydrogen atom or
molecule on its path, the initial photon loses a very small fraction b»10^{13}
of its initial energy due to bremsstrahlung due to momentum transfer on
hydrogen in the solar chromosphere. This is consistent with the
prediction
of quantum electrodynamics as calculated by [42]
and [41].
This small energy loss [3],
[5]
is based on the following principles.
1) In the scattering process between electromagnetic radiation and
atoms
or molecules, one usually considers only the radiation emitted by the
transverse
dipole induced by the polarizing radiation.
2) It is seen that when electromagnetic radiation is transmitted
through
gases, the momentum transfer produces an axial dipole following
momentum
transfer.
3) This axial momentum transfer to the electrical charges (electrons)
of
the atoms of the medium also produces bremsstrahlung, which is
completely
perpendicular (orthogonal) to the conventional scattering process
(transverse
polarization).
4) Marmet [3], [5]
has shown that this bremsstrahlung removes a fraction b
of the energy of the initial photon.
We must realize that quite generally, when we deal with any waves, a
wavetrain
always necessarily possesses at least two frequency
components.
The frequency n_{o}
measured usually, is related to the rate of change of the electric and
magnetic field. However, since electromagnetic waves never last
during
an infinite time, the oscillating wave amplitude finally dies out after
the time of coherence. Therefore, since the amplitude of the wave
must decrease to zero after a time interval Dt
related
to
the
time
of
coherence,
(which is usually much longer than
the
time of one fundamental cycle), this gives the very low frequency n_{c}
(related to the time of coherence) component of the wave.
It
is that very low frequency component, which is responsible for the new
NonDoppler redshift [3],
[5]
and which has always been neglected in the past.
In the case of blackbody radiation, the relative energy b
is given by:
2 
3 
4 
5 
Since the red shift (Dl¤l) per collision is b, one finds that the total red shift is equal to:

6


7


8 

9



10


11


Figure 2
Fig. 2. Limb effect of solar lines as a function of
equivalent
radial velocity.
IV.
HighPressure Phenomenon
A. MultiAtom Interaction
We have seen that the red shift of spectral lines on
the sun starts as expected at an altitude just below the one of maximum
absorption of these lines. This altitude (277 km) above the base
of the chromosphere has also another important characteristic  it sets
a threshold to a highpressure phenomenon. As stated previously [5],
one can see that when the length of coherence of electromagnetic
radiation
is longer than the distance between atoms of the gas, one single wave
train
of radiation (one photon) has the possibility of interacting
simultaneously
on more than one atom. In other words, one single wave train can
polarize several atoms simultaneously, and the photon momentum is
transferred
to several electrons simultaneously.
For example, in the case of radiation interacting simultaneously on a
pair
of electrons (located on different atoms), the total energy emitted by
bremsstrahlung by the pair of electrons that has received a given
momentum
is smaller than the bremsstrahlung emitted in only one electron
(therefore
half the mass) has received the same total momentum. This is the
consequence of the Larmor law of radiation that says that the amount of
energy radiated by bremsstrahlung increases linearly with the charge
accelerated,
but quadratically with the acceleration given to the charge.
Consequently,
it is only in the case when the gas density reaches low value, such
that
the length of coherence is smaller that the distance between the atoms
of the gas, that one can be sure that only one atom interacts at a time
and that the red shift produced follows (10). Above that critical
pressure, the red shift (per collision) becomes smaller and must
practically
disappear at a much higher pressure. This means that a
photonatom
interaction leads to the emission of a smaller amount of bremsstrahlung
when other atoms are located within the coherence length of the
interacting
atom.
B. Critical Pressure
It is very difficult to determine precisely the critical pressure above
which bremsstrahlung is no longer emitted a full intensity. One
can
see [5, fig 1] that the time of coherence Tc of
blackbody
radiation at half intensity is given by the expression


C. Red Shift in the Earth's Atmosphere
We have explained that electromagnetic radiation having a length of
coherence
much longer that the interatomic distance should not produce
bremsstrahlung.
With the solar radiation the critical pressure gives about 10^{15}
at/cm^{3}. This same pressure
exists
in the Earth's atmosphere at 73 km above the Earth's surface [43],
and the column density above that altitude is 1.5×10^{19}
at/cm^{2}. Since this is 700 times
smaller
than the column density above the corresponding point on the Sun
(having
the same density), this leads to a red shift that is 700 times smaller
in the Earth's atmosphere than one observed on the Sun.
Consequently,
no observable red shift is expected to be detected in the Earth's
atmosphere.
V. Other Models
Let us now consider the assumption that the CLV is a consequence of a
Doppler
effect resulting from radial mass motion in the solar photosphere or
chromosphere,
as reported by FinlayFreundlich [11],
Struve [13], and Snider
[25].
It is clear that the velocity of such a mass motion must be adjusted in
order to compensate exactly for the redshift on the center of the disk,
and also be adjusted exactly to the relativistic value on the limb,
since
any radial motion has no effect on the limb as seen from Earth.
It
is easily seen in Fig 1 that between 0.5 and 0.9 of the solar disk
radius,
mass motion leads to a CLV in complete disagreement with experimental
data.
Again, in Fig 2, above 0.900 these other models are in total
disagreement
with the experimental data.
Finally, let us compare the experimental results with the ones
predicted
using the theory of microturbulence as calculated by Schatzman and
Magnan
[18].
From [18, eq. (29) fig. 1] one can see that the
red
shift varies almost linearly as a function of the position on the solar
radius. The same parameters used [18]
give a redshift on the limb of 0.35 km/s. Furthermore, the
redshift
at the center of the disk is adjusted with a proper choice of the
parameters
E/kT. The results are shown in Fig 1 and 2, and one can notice again a
serious discrepancy with the experimental data, as stated by Schatzman
and Magnan [18]
themselves: "In
fact, our theory predicts too much red shifts". Other
more
recent works [26],
[33],
[20]
show that both horizontal and vertical motion in the solar granulation
can explain the red shift resulting from mass motion inside granules,
but
does not lead to the observed CLV reported by so many observers.
Furthermore, Beckers and Nelson
[30] report:
"A quantitative interpretation of the limb shift in terms of our
convection model leads, however, to conclusions which are to some
extent
at variance with our present concept of solar granulation."
The difficulty of using the existing theories involving granules to
explain
CLV has been described in more detail at the beginning of this paper
and
can be summarized by the recent statement (1984) of Balthasar
[44]: "Up to now, no complete and
satisfactory
explanation for the line asymmetries known since 1956 and the limb
effect
known since 1907 exist."
Finally, another comment must be added since the absorption lines are
relatively
wide (several times wider than the redshift) and asymmetric.
Their
relative displacement leading to CLV has been accurately measured, but
their absolute position within the individual line profile is much more
difficult to establish. The difficulty of absolute measurements
in
the presence of pressure shifts and convection might account for a
small
constant shift across the solar disk. In these solar
measurements,
the wavelength reference is the minimum [43]
in the spectrum. It is well known to spectroscopists that, due to
line shapes and pressure broadening, precise location of atomic and
molecular
levels is not necessarily situated at the position of the maximum
absorption,
especially when the profile is asymmetric. The accurate
difference
in wavelength between the actual position of an atomic level and the
position
of maximum absorption is still an undetermined quantity in solar lines.
At the solar disk center, it is important to notice that Balthasar [44]
reports precise measurements showing that observed wavelengths "depends
strongly
on
the
formation
depths
of
the lines: Lines formed in higher
layers
show larger redshifts." Balthaser [44]
also reports that lines formed in the highest layers "have shifts
of about the gravitational red shift" To explain it more
clearly,
it is reported [45]
that the difference
between the wavelength of a given absorption line of iron observed at
the
disk center and formed in the highest layers of the solar chromosphere
and the laboratory wavelength measurement is exactly equal to the
gravitational
red shift. Furthermore, it is also suggested [44]
that the upward velocity of "the ascending matter is decelerated".
This
effect
explains
the
decreasing
blueshift
of the radiation issued
at
an increasing altitude, as observed experimentally by Crosswhite [45]
and reported by Balthasar [44].
It is extremely interesting to observe that relativistic corrections,
combined
with the upward velocity reported [44],
[45],
lead to zero vertical velocity in the "highest layers". At the
solar
disk center, it can also be calculated that the Doppler blueshift
resulting
from the upward velocity of the "lowers layers" overcompensates for the
redshift produced by the gas. We see now that the systematic
investigation
of the depth dependance of a lot of spectral lines done by Balthasar [44]
was necessary for the understanding of the problem. These
conclusions
are in exellent agreement with the observations of the new red shift
theory
presented above.
VI. Lines Profiles
As stated previously [14],
[15],
experimentally measured line profiles are modified and become more
asymmetric
near the limb. This asymmetry has not been clearly explained, as
indicated by Higgs [14],
[15]
and in more recent papers. In the case of red shifts due to
bremstrahlung
on atoms, we will see that the same method used for calculating the
amount
of gas in the solar chromosphere can be used to determine the line
profiles.
It is clear that the absorbing solar chromosphere has a nonnegligible
thickness, and the medium producing the absorption is the same as the
one
producing the red shift. Absorption lines produced in the gas in
the upper part of the solar chromosphere will not be red shifted
significantly
since the photons do not go through enough gas before reaching the
Earth.
On the other hand, gas at a lower altitude will produce specific
absorption
lines that will be red shifted due to the presence of gas at a higher
solar
altitude before reaching the Earth. The final result is that the
absorption features will appear at slightly different wavelengths as
seen
from Earth, depending on the altitude or the altitude range where the
absorption
takes place in the solar atmosphere. A larger altitude range will
produce a broadening of the absorption lines. Finally, if the
absorption
coefficient is not constant, as is the case when the FeI atoms have
different
temperatures and densities, the widened lines become asymmetric.
This is precisely what has been observed experimentally [15].
Let us assume that the line profile within a layer has a Gaussian
distribution.
The absorption profile coming from the lowest layer (#1) is red shifted
and added to the profile produced by the next layer (#2), and so
on.
One sees that the profile from layer (#1) is more redshifted when
reaching
the Earth than the profile of layer #2. A calculation using
40Gaussian
functions with linearly decreasing amplitudes has been used as an
example
in order to illustrate the contribution of uniformly spaced red shifts
(originating from layers giving different absorption). It is found that
this leads to a width three times as wide as the width of the Gaussian
function (at half height). The result of these calculations shows
that the calculated profile, shown as "B" in Fig. 3, is similar to a
typical
asymmetric profile :A" (Fig. 3) observed by Higgs [14,
fig.
4]
.
For
a
larger
column density, the calculated red shift
is
larger with respect to the width of the lines and has a larger
asymmetry,
as observed experimentally [14],
[15].
Finally, it can be shown that the asymmetry can be different and even
reversed
if the absorption takes place in different conditions. One must
conclude
that the red shift resulting from bremsstrahlung, as predicted [3],
[5],
leads naturally to asymmetries of the type observed in solar spectra.
Figure 3
Curve A is the line profile reported by Higgs
(ref.
[14])
(experimental)
Curve B is a typical line profile according to this
work
(theory)
VII. Further
Observations
Apart from the fact that the basic theory described by Marmet [3],
[5]explains
for the first time the absolute amplitude and the shape of the CLV of
spectral
lines across the solar disk without and ad hoc hypothesis or adjustable
parameters, several observations that have received no clear
explanation
previously support the theory applied here. Let us cite two of
them:
1) Very recently, Balthasar [44]
stated: "A missing limb effect for several lines formed in very
high
layers was reported by Schröter [46]*,
Appenzeller and Schröter [37],
RocaCortes el al. [47]*." Such a
result,
which has also been observed by Snider [25]
several years ago, is deduced naturally from the theory put forward
here,
since no red shift can be produced from the highest layers, because
there
is insufficient column density between the Sun and the Earth to produce
a measurable red shift.
2) Balthasar [44] also
reports
that "a stronger limb effect for the higher formed line at 5250
Å
than for the higher line at 5576 Å" has been found by
Bruning
[36].
This linedependent variation of red shift is easy to explain with our
present model. The larger limb effect of line 5250 Å is due
to the fact that the radiation has to go through a larger amount of
(red
shifting) gas before reaching the observer on Earth.
Thus, the theory which takes bremsstrahlung into account explains
precisely
the absolute amplitude of the CLV, its variation as a function of its
position
on the disk and near the limb, and even the change in lineshapes
especially
near the limb without having to introduce any ad hoc theory or
parameters.
Acknowledgment
The author wished to acknowledge fruitful discussions with Dr. G.
Herzberg
and Dr. P. Feldman of the Herzberg Institute of Astrophysics, Prof. P.
Marchand of Laval University, Dr. L. Higgs of the Division of Radio
Astronomical
Observatory in Pentincton, B.C. Can., Dr. J. W. Kierein of the
Aerospace
Systems Division, Dr. M. Proulx and Dr. P. Plessis of the National
Research
Council of Canada, Ottawa, and L. Marmet of the University of Toronto.
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