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1
Dynamics of the Solar Wind: Eugene Parker's Treatment
and the Laws of Thermodynamics
Pierre-Marie Robitaille
a
Department of Radiology and Chemical Physics Program, The Ohio State University, Columbus, Ohio, 43210,
USA,
Stephen J. Crothers
b
PO Box 1546, Sunshine Plaza 4558, QLD, Australia,
ABSTRACT
In 1958, Eugene Parker advanced that the solar wind must be produced through the
thermal expansion of coronal gas. At the time, he introduced a dimensionless
parameter, aTkMGM
BHS 0
2/=
λ
, where G corresponds to the universal constant of
gravitation, M
S
to the solar mass, M
H
to the mass of the hydrogen atom, k
B
to
Boltzmann’s constant, T
0
to the temperature at the location of interest, and a is the
distance to the effective surface, or the radial distance, to the outer solar corona, the
location of interest, relative to the centre of the Sun. It is straightforward to
demonstrate that this equation stands in violation of the 0
th
and 2
nd
laws of
thermodynamics by simply rearranging the expression in terms of
temperature: akMGMT
BHS
λ
2/
0
=. In that case, then temperature, an intensive
property, is now being defined in terms of an extensive property, M
S
, and the radial
position, a, which is neither intensive nor extensive. All other terms in this
expression are constants and unable to affect the character of a thermodynamic
property. As a result, temperature in this expression is not intensive. Consequently,
the expression advanced by Parker is not compatible with the laws of
arobitaille.1@osu.edu
b
thenarmis@yahoo.com
2
thermodynamics. This analysis demonstrates that solar winds cannot originate from
the thermal expansion of coronal gas, as is currently accepted.
RÉSUMÉ
En 1958, Eugene Parker a avancé que le vent solaire devait être produit par la
dilatation thermique du gaz coronal. A l'époque, il introduisit un paramètre sans
dimension, aTkMGM
BHS 0
2/=
λ
, où G correspond à la constante universelle de
gravitation, M
S
à la masse solaire, M
H
à la masse de l'atome d'hydrogène, k
B
à la
constante de Boltzmann, T
0
à la température à l'emplacement d'intérêt, et a est la
distance à la surface effective, ou la distance radiale à la couronne solaire extérieure,
la position d'intérêt, par rapport au centre du soleil. Il est simple de démontrer que
cette équation viole les 1
ière
et 2
ième
lois de la thermodynamique en réarrangeant
simplement l'expression en termes de température: akMGMT
BHS
λ
2/
0
=. Dans ce
cas, la température, une propriété intensive, est maintenant définie en termes d’une
propriété extensive, M
S
, et de la position radiale, a, qui n'est ni intensive ni éxtensive.
Tous les autres termes de cette expression sont constants et incapables d'affecter le
caractère d'une function d’état thermodynamique. En conséquence, la température
dans cette expression n'est pas intensive. De même, l'expression avancée par Parker
n'est pas compatible avec les lois de la thermodynamique. Cette analyse démontre que
les vents solaires ne peuvent provenir de la dilatation thermique du gaz coronal,
comme cela est actuellement accepté
Key words: Solar Wind, Thermodynamics, Parker Solar Probe
3
DEDICATION
This work is dedicated to, Robert George Murdoch Crothers
(16
th
October 1924 - 20
th
June 2018) and Marion Crothers.
I. INTRODUCTION
In thermodynamics, systems are described in terms of properties which are classified
as either intensive or extensive
1-5
. Intensive properties can be determined at every
spatial location and are independent of any changes in the mass of a system by
definition. Temperature, pressure, velocity, thermal conductivity, and density are
examples. It is well recognized that temperature maintains the same value at all spatial
locations within a system in thermodynamic equilibrium. However, it might take on
varying values in a system out of equilibrium. In either case, temperature always
remains intensive, as it can be defined at every spatial location. Conversely, extensive
properties are defined over a certain spatial extent. Typical examples are mass,
volume, internal energy, and heat capacity. Extensive properties are additive. The
thermodynamic coordinates necessary and sufficient to describe any thermodynamic
system are determined by experiment.
Consider a homogeneous system in thermodynamic equilibrium. Divide it into two
equal parts, each having equal mass. Those properties of the original system that
remain unchanged in each half of the original system are intensive. Those properties
that are halved are extensive. Those which change but not by half are neither intensive
nor extensive
4
. It is also important to note that the quotient of two extensive properties
4
is intensive. Density,
VM=
ρ
, is the best known example of such a quotient. Mass
and volume are both extensive, but their quotient results in density, which is intensive.
The quotient of two non-extensive properties, which behave identically with changes
in spatial extent, is also intensive. However, the quotient (or the product) of an
intensive property and an extensive property is always extensive.
Constants such as Boltzmann's constant, k
B
, the Stefan-Boltzmann constant, σ, and
Planck's constant, h, do not alter the intensive or extensive nature of a thermodynamic
expression. For instance, the total energy of a simple monoatomic gas, E, can be
given by the following simple expression,
23 TNkE
B
=
. In this expression, the
extensive nature of E on the left hand side is imparted by the extensive nature of the
number of particles, N, on the right, given that temperature must remain intensive.
Boltzmann's constant does not contribute towards establishing the thermodynamic
nature of energy as extensive in this expression. At the same time, variables that are
neither intensive nor extensive, but have units, affect the thermodynamic balance of
expressions. Variables without units have no influence on thermodynamic character.
In the end, it is important to remember the following rules: 1) extensive and intensive
properties exist, 2) some properties are neither extensive nor intensive, and 3)
physical constants (e.g. G, k
B
, h) play no role in establishing whether a property is
intensive, extensive, or neither. Landsberg
2
has argued that the nature of properties as
intensive or extensive is so important to the study of thermodynamics that the concept
should be adopted as the 4
th
law.
It is also true that any proposed thermodynamic equation must be thermodynamically
balanced, as just demonstrated. If one side of the equation is intensive, or extensive,
5
then the other side must also be intensive or extensive, respectively
5
. When a
thermodynamic property in an expression is being defined in terms of other
thermodynamic properties, the correct nature of the sought property must be obtained.
Temperature cannot become non-intensive simply as a result of a mathematical
expression. Temperature must always be intensive, in keeping with its role relative to
defining the laws of thermodynamics.
The 0
th
law of thermodynamics requires thermal equilibrium between objects in
defining temperature. Consider two isolated systems
c
, each in thermodynamic
equilibrium. Remove a section of the thermal insulating material from the surface of
each system and place them in contact via the uncovered sections. When there are no
observable changes in any thermodynamic properties of either system, they are each
at the same temperature. The 0
th
law of thermodynamics not only makes a
statement about thermal equilibrium of systems, it also includes the intensive
character of temperature: “when two systems are at the same temperature as a third,
they are at the same temperature as each other”
6
, “Two systems in thermal
equilibrium with a third are in thermal equilibrium with each other”
7
. Take two
systems, A and B, at the same temperature in accordance with the foregoing method.
The temperature of A is the same as that at every spatial location in B: as every part of
B. Divide the system B into two parts, B
1
and B
2
. Since A has the temperature of B, it
has the temperature of B
1
and B
2
: parts of B. Therefore, B
1
and B
2
are at the same
temperature. Thus, the intensive nature of temperature is contained within the very
definition of the 0
th
law of thermodynamics. Such equilibrium cannot exist if
c
An isolated system does not exchange any energy, either by mechanical work or flow of heat, with its
surroundings.
6
temperature is no longer intensive. Similarly, entropy must always remain extensive,
in order to preserve the 2
nd
law.
If a system has spherical symmetry, its area can be expressed as
2
4rA
π
=. Clearly, r
is neither intensive nor extensive, as it is not additive. This can also be established
relative to a volumetric system with spherical symmetry. The radius is not extensive
since volume, V, is given by 34
3
rV
π
=. In this expression, it is the volume of a
sphere which is an extensive property, along with r
3
. It is clear that radius r is not
additive. Hence, the radius of a sphere can never be considered as an extensive
property.
Length is generally not extensive, as radius attests. However, length can become
extensive in certain limited circumstances, as for example, in stretched wires
7
, having
the thermodynamic coordinates of tension (intensive), length (extensive), and
temperature (intensive). The spatial extent of this system is length. It is extensive, in
this case, as it is directly related to the mass of the system. Any change in length of
the wire is directly associated with a change in its mass.
It is also true that extensive properties in one system might not be extensive in
another. A prime example is surface area. For a planar system composed of a single
monolayer, area is extensive. Such systems arise when considering surface tension
which, in turn, is an intensive property. However, the area of a sphere is never
extensive. That is because such area is not additive. If one takes a sphere and divides
it into two spheres of equal volume, the area of each sphere is not half of the initial.
7
As an additional example, consider the Stefan-Boltzmann law
8
describing a system in
which area is a thermodynamic coordinate,
.
4
ATL
εσ
= (1)
In this expression, L is the luminosity of the object, ε is emissivity of the material (a
unitless property), σ is the Stefan-Boltzmann constant, A is the area, and T is
temperature. In this case, note that neither luminosity nor area are extensive, because
these properties are not additive. However, both luminosity and area change in
identical fashion relative to spatial extent. Temperature is defined at every spatial
location in the system and remains intensive. For any given temperature, the
luminosity is directly proportional to the area. Hence, the luminosity per unit area
(L/A), also known as the emissive power, is intensive and so is the temperature, as
required by the laws of thermodynamics. Equation (1) is therefore thermodynamically
balanced.
The intensive nature of temperature is also necessary to the understanding of entropy
as defined in the 2
nd
law of thermodynamics, stated mathematically as,
,
T
Q
dS
δ
= (2)
where S is entropy, Q is heat, and T is temperature. Entropy and heat are extensive.
By the 2
nd
law, temperature is always intensive, for, if it is not, then entropy would
not be extensive, contrary to the definition of entropy and the character of heat.
8
II. PARKER’S EQUATION FOR THE SOLAR WIND
In 1958, Eugene N. Parker attempted to account for the production of the solar wind
by invoking thermal expansion of coronal gas
9
. It is simple to demonstrate that this
work constitutes a violation of the laws of thermodynamics. Parker has proposed the
following equation, with dimensionless variables, for the solar wind
9
,
,
2
ln4
2
ln43ln
ξ
λ
ξ
λ
ψψ
++−−=− (3)
where the dimensionless variables are defined as,
0
2
2Tk
vM
B
H
=
ψ
, (4)
,
a
r
=
ξ
(5)
,
2
0
Tak
MGM
B
HΘ
=
λ
(6)
wherein
a
r
≥
is the radial distance from the centre of the Sun,
a
is the coronal radial
position from which the solar wind emanates, or the effective surface of the Sun,
T0
is
the temperature at
r
=
a
,
MH
is the mass of a hydrogen atom,
MΘ
is the mass of the
Sun,
kB
is Boltzmann's constant,
v
is the speed of the solar wind, and
G
is the
universal constant of gravitation. Rearranging Eq. (4) for temperature gives,
.
2
2
0
ψ
B
H
k
vM
T=
(7)
9
The mass
MH
of the hydrogen atom is a constant,
ψ
is a dimensionless variable, and as
the velocity
v
and the temperature
T0
are both intensive coordinates, Eq.(7) is
thermodynamically balanced. Thus Eq.(4) is admissible.
Rearranging Eq. (5) for the radius
r
yields,
.ar
ξ
=
(8)
Since ξ is a dimensionless variable, it cannot influence the thermodynamic character
of r or a, the latter the lower bound on radius r relative to the solar wind. For ξ = 1, r
= a, giving a spherical surface of area
2
4aA
π
= from which the solar wind emanates,
enclosing the Sun of volume 34
3
Θ
=RV
π
, aRr <=
Θ
the radius of the Sun. Hence,
both r and a are neither intensive nor extensive, but have the same thermodynamic
character. Thus, Eq.(5), relative to Eq.(3), does not violate the laws of
thermodynamics and is admissible.
That Eq.(3) violates the laws of thermodynamics is made clear by solving Eq.(6) for
T
0
,
,
2
0
λ
B
H
ak
MGM
T
Θ
=
(9)
which reveals that temperature T
0
,
an intensive property, is being defined in terms of
the extensive property M
Θ
and the radius a which is neither intensive nor extensive.
This is essentially the same as the problem previously highlighted relative to the
equation defining the temperature within a gaseous star
10-13
. This violation of
10
thermodynamics is amplified by substituting into Eq.(3) the explicit values of the
dimensionless variables, by which one obtains,
.
3
128
ln
2
434
3
0
432
2
0
−
−
=
Θ
Θ
MMG
Trkv
k
r
MGM
vM
T
H
B
B
H
H
(10)
Prima facie, this expression seems correct. Dimensionally, both sides are expressed
in terms of Kelvin. However, on closer examination, it becomes evident that this
expression is thermodynamically unbalanced. Note that temperature on the left must
be intensive, as required by the laws of thermodynamics. However, while the first
term in the numerator on the right is intensive
d
, the second term is not intensive, since
M
Θ
is extensive but r is neither intensive nor extensive. The term in the natural
logarithm of the denominator, although variable, has no units (is a pure number).
Hence, the right side of Eq.(10) is not intensive, even though the laws of
thermodynamics require that temperature always remains intensive. Consequently,
Eq.(3) is inadmissible.
III. CONCLUSIONS
From this simple analysis, it has been demonstrated that Eugene Parker's expression
for the production of the solar wind, through the thermal expansion of coronal gas,
violates the laws of thermodynamics. Temperature must always be an intensive
property. When astronomers first advanced the theory of a gaseous star
14-16
, they
d
Because M
H
is a constant and velocity is intensive.
11
proposed an equation similar to Eq.(9). Robitaille
10-13
has demonstrated that, in
analogous fashion, that expression is also thermodynamically invalid. It is not
appropriate to utilise the virial theorem and introduce temperature through kinetic
theory, when balancing kinetic energy with potential energy. Such an approach results
in direct violations of the laws of thermodynamics. Gravitational collapse (i.e. self-
compression) of a gas cannot occur: gravitational collapse of an ideal gas produces a
perpetual motion machine of the first kind
10-13
. Eugene Parker's treatment has fallen
victim to the same type of error in failing to respect the laws of thermodynamics.
NOTE IN ADDED PROOF
After reading this work, one might be left with a sense of “Why did such an error
relative to intensive and extensive properties arise in astrophysics?” The answer lies
in the inappropriate treatment of the gravitational fields and their effect on the
temperature of a gas. External forces, including gravity, must never be permitted to
alter this property. From a historical perspective, the introduction of this type of error
is evident by contrasting the analysis provided by Boltzmann
17
versus Jeans
18
for a
column of air within an adiabatic cylinder under the influence of a gravitational field.
Boltzmann correctly argues
17
that the temperature distribution within this column
remains uniform and indeed, that the entire column remains in thermal equilibrium.
This remains the case even though the density and pressure of the gas assume
gradients with respect to height
17
. However, the ratio of pressure and density remains
constant throughout the column, and therefore, by the ideal gas law, so does the
temperature: “…the temperature is also the same everywhere in spite of the action of
external forces”
17
. Boltzmann’s temperature remains intensive
17
, as it never becomes
12
dependent of the force of gravity. Conversely, Jeans argues
18
that the temperature
varies with elevation by assuming that the gas never reaches equilibrium. This
directly leads to a violation of the 0
th
law of thermodynamics, as manifested by the
analysis of his expressions involving both the force of gravity and temperature
18
. The
temperatures which Jeans obtains are not intensive as a direct result. This problem
has drawn the attention of educational works
19,20
demonstrating that the correct
answer does indeed rest with Boltzmann
17
.
Parker has allowed temperature to become affected by an external force, namely
gravity, and has committed the same error as Jeans
18
. The fact that his temperature is
not intensive in Eq. 9 is a direct reflection of this misstep. Parker makes temperature
dependent on gravity which is not allowed by Boltzmann
17
. In fact, it is interesting to
note that while Parker’s temperature in Eq. 9 is not intensive, his temperature in Eq. 7
is, in fact, intensive. As a result, Parker is simultaneously advancing that temperature
can be both intensive and non-intensive simultaneously at the same location. This
emphasizes, once again, that Parker’s treatment of this problem cannot be correct.
REFERENCES
1
P. Atkins, The Law of Thermodynamics: A Very Short Introduction, (Oxford
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2
P. T. Landsberg, Thermodynamics with Quantum Statistical Illustrations,
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13
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4
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5
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6
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7
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8
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9
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10
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11
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14
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13
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14
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15
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16
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