_{1}

^{*}

In the present paper, the theoretical frame work of magneto hydrodynamics (MHD) is used to give a solution of the problem about the origin, persistence and disappearance of the Sunspots; as well as their tendency to appear as bipolar magnetic couples. According to the results obtained, a possible explanation about the change of polarity in both solar hemispheres is given. Heuristic but logical arguments about the periodicity of the phenomenon of the observed magnetic polarity and the tendency of couples of Sunspots to appear solely in certain latitudes that can be called tropical regions of the Sun are presented. Finally, an indirect experimental test is proposed to show the possible process that produces the polarity of the Sunspots in a given cycle, as well as the invertion of that polarity in the next solar cycle.

The sun is a huge concentration of fluid at very high temperature, whose parts are kept together and in dynamic equilibrium with fluid’s gas pressure force by gravitational attraction. They are also under the influence of a self-generated magnetic field. Careful observation of the solar surface shows a granular texture given by short- lived objects known as granules, somewhat brighter than their neighbors. Also easily observed are the so called Sunspots which are colder and darker regions than the solar photosphere; discovered first by Chinese astronomers, then by Galileo and observed by himself and his contemporaries. These objects have a somewhat nebulous origin along with very peculiar properties and general behaviour, which leads to assume that under the surface of sun’s photosphere, certain local processes take place destabilizing it and creating the phenomenon.

The present work focuses on the study of the thermal energy conditions which create a thermal instability in some regions of the photosphere, and their relationship with the appearance, properties and life time of the Sunspots. A large amount of observational data has been accumulated of the subject [

The vast majority of the Sunspots are observed in certain regions of the sun. They appear predominantly in two stripes of equal latitude to the North and South of the Solar Equator in which can be called the tropical regions. They have the tendency to appear in big groups or in couples; in each of which its two members always have opposite magnetic polarity.

A typical Sunspot has the following observed structure and dimensions. It possesses a dark nucleus called Umbra of about 18,000 km in diameter, and a somewhat lighter halo called Penumbra, 20,000 km wide. The lower luminosity of the spot as compared to the photosphere is due to a decrease in its temperature.

A remarkable fact that should be noted is that the polarity of the Sunspot couples in the northern hemisphere is always opposite to the polarity of the Sunspot couples in the southern hemisphere. This disparity changes periodically with a period of about 11 years.

This problem is similar to that of the mechanical instability which occurs in some regions of the terrestrial atmosphere [

The non existence of mechanical equilibrium leads to the appearance a certain movement in the fluid; internal currents appear mixing the fluid until its temperature is uniform and reach a constant value in all its volume. This type of movement of a fluid in a gravitational field receives the name of Free or Thermal Convection [

A thermal instability which triggers convection currents can be produced in any real fluid with the above mentioned characteristics, as long as the fluid is exposed to non uniform heating from below [

For magnetomechanical equilibrium to be reached in any region of the solar photosphere, it is necessary that (see Appendix)

where ^{2}/8p the hydrostatic magnetic pressure [

If the Z-axis of the reference inertial frame used points vertically upwards, Equation (1) can be written as

where V = 1/r is the specific volume.

It is known that if temperature is not a constant throughout a fluid, the resulting mechanical equilibrium could be stable or unstable depending of certain conditions [

The theoretical treatment for the adiabatic displacement of a mass of solar fluid in the photosphere is identical to the one given for an air mass adiabatically displaced in the terrestrial atmosphere [

It can be shown that for both situations, the previous conditions can be written as follows [

where c_{P} is the specific heat of the fluid at a constant pressure and T its temperature. Then, according to Equation (2), from (4) we have that

In this case, the condition to be satisfied so that a thermal instability can be produced in some regions of the solar photosphere and strongly regulated by the magnetic field that generates the Sunspots can be given as follows:

In other words, the onset of convective movements in those regions can be substantially delayed if the magnitude of the Z-component of the gradient of the square of the existing magnetic field is larger than a certain value. The above condition must be fulfilled so that the thermal convection in the spots is magnetically regulated. It simply means that the persistence of the magnetic field in those regions, assures the permanency in time of the Sunspots in the solar disk. In fact as the magnetic field fades away, the mechanism of magnetic regulation weakens, allowing the thermal processes to take over the situation; as a result, the mechanism for the onset of convection begins generating and becomes more dominant so that, when

as is easily seen from (6) when

From Thermodynamics we have that [

Substituting this relation in (5), thereafter integrating the resultant expression and in the result obtained Equation (8) is used again, it is easy to see that

where z is some characteristic height in the photosphere where the Sunspots are produced. Now, consider that

Furthermore,

and

so that with (10) and the product of these two numbers the relation (9) is recovered. Thus, the persistence of each couple of sunspots on Sun’s surface as assured when the magnitude of the magnetic field which generates and maintains them is greater than a certain amount, that is

and

Clearly, both magnetic fields have the same magnitude. It can further be reasonably assumed that beyond sharing the previous quality, they also have the same direction. Thus, if for example

Consider a huge mass of viscous compressible and conducting fluid, which moves in the presence of a magnetic field and is under the influence of a gravitational field. Let _{o} the temperature of the Sunspots and T' the temperature at which the dynamic equilibrium between the mechanisms of magnetic regulation and the onset of thermal convection is broken, so that

where the proposed decomposition for density was used. Furthermore, by definition,

is the thermal expansion coefficient of the conductor fluid. With respect to hydrostatic pressure we have that_{o} is not a constant, since it represents the pressure corresponding to the dynamic equilibrium in the inner regions of the Sunspots whose temperatures and densities are constant and equal to T_{o} and r_{o}, respectively. Since the fluid is under to the gravitational attraction of the Sun, it will be assumed that p_{o} varies with height according to the hydrostatic equation [

Consider now the equation of movement of MHD, or generalized Cauchy equation [

where

Are the components of the generalized stress tensor [_{ij} the components of Kronecker’s delta and the third term of the right hand side of Equation (19) one has the components of Maxwell’s magnetic stress tensor [

are the components of the viscosity stress tensor [

where

is the specific viscous force and the following vectorial identity has been used

On the other hand,

where calculation up to the first order term for the small quantities was performed. Now, from (17) is clear that

and from (15)

On the other hand, consider that

Because is clear from the previous paragraph that

where,

where

so that, according to (15)

With (15) into (28) and with the approximations already made, it is easy to see that

where

is a constant. On the other hand, for the equilibrium situation between the regulatory and startup mechanisms we have that

only, because according the approximations already made,

is the explicit form of the required thermal equation of state. From this result, the following expression can be obtained

Then, from relation (26) the following result is obtained

where

On the other hand, the magnetic field of the Sunspots should satisfy the following field equation [

where

From the previous expression it is possible to determine the magnetohydrodynamic force which is responsible for the dynamic equilibrium between the onset of thermal convection and the mechanism of magnetic regulation, that is

Consider now that for the condition of dynamic equilibrium, the viscous force is not relevant so that in (39) the term _{o} to T, occur in the steady state. Then the temperature T' at which the dynamic equilibrium is broken and convection starts, depends only on the coordinates and not on the time. In consequence, throughout the whole life of the Sunspots the viscous force is not important and it can be ignored for the whole analysis of the problem.

Once the equilibrium is broken, convective movements are rapidly established mixing the fluid of the Sunspots with that of its surroundings until the average temperature T of the photosphere is reached. It is possible that during this process, the viscosity of the medium, and in consequence of the viscous force, becomes very important; but at this time, the Sunspots have disappeared because the magnetic field which generates them is zero and, clearly, the thermal gradient becomes null and, hence, the solar activity in those regions comes to an end. Since the remaining terms of the relation (39) are not explicitly depending on time, that equation can be integrated to obtain

However, _{o} = 0, one has that previous expression

If only the Z direction is considered, the following result is obtained

When the onset mechanism of the thermal convection begins to dominate the phenomenon, the magnetic part begins to decrease so that the term

and, its vertical component clearly is

where the subscript c indicates that the process of thermal convection is being considered.

So far the velocity field and the MHD force responsible for the dynamic equilibrium have been calculated. Also available is the scalar equation for the mass density, which is valid for any fluid, and of course, the thermal equation of state for the phenomenon of the solar spots. The general equation of heat transfer remains missing in order to have an analytical solution to the problem of origin, duration and disappearance of the spots. This last equation can be obtained from the law of energy conservation for MHD [

where

where

Then from Navier-Stokes equations for MHD in the absence of body forces

where

is the viscous force, with

because

and has been used the vectorial identity [

and the fact that [

By substituting (51) in (47) it’s easy to see that

From Equation (50) the following is obtained

where the previous approximation with relation to the mass density has been used and an integration by parts has been done. Substituting in relation (46) all the results previously obtained one gets

Now, let

where the following vectorial identity [

plus the fact that

because they are orthogonal vectors, and its scalar product is zero.

If in the second term of the right hand side member of (55) the previous vectorial identity is used again along with the following expression

we simply get,

just because

This is the general equation of heat transfer for MHD [

where j^{2}/s is the Joule heat per unit volume [

Then, in (59) we have that

In fact, because we can write that

is the corresponding viscosity stress tensor in terms of the perturbation in the velocity field. Equation (61) is the general equation of heat transfer for any fluid in terms of the perturbed quantities.

According to the previously given arguments, the velocity of the fluid inside the spots is damped, due to the stiffness in the fluid conferred by the magnetic viscosity which in turn is created by the magnetic field responsible for the generation of the spots; so that it is possible to assume that inside those regions the magnitude of the fluid velocity is smaller than the speed of sound in that continuous medium. Therefore, it could be thought that variations in hydrostatic pressure occurring as a consequence of the motion are so small, that any changes in density and in other thermodynamic quantities involved should be neglected. Nevertheless, it is clear that the solar fluid is subject to nonuniform heating so that its density undergoes changes which cannot be ignored, so that it is not possible to say that it is a constant. In consequence, it can be shown that [

where

is the specific heat of the fluid at constant pressure [

with

being the thermometric conductivity of the fluid [

As it is easy to see from (62),

This is the final form of the general equation of heat transfer for the full process of generation, duration and disappearance of the solar Spots. On the other hand, since T' has been assumed to depend on the coordinates only,

where an integration by parts has been made and the term

where the integration constant has been assumed to be zero without losing of generality. Now, and according to expression (41)

Let’s assume that for the condition of dynamic equilibrium, the thermometric conductivity can be neglected, so that only the other term is preserved. This is due to the fact that it could be very interesting to observe the phenomenon of the solar Spots as if it were frozen in time. Generally speaking, it is not possible to make such approximation as it will be seen in the next sections. Nevertheless, if for the moment that term is ignored, the following result is obtained in (70)

This is the analytical expression for the process of dynamic equilibrium in the solar spots. If only the z-component of the previous equation is considered, we get again that

Now, from the z-component of relation (71) the following is obtained

in which case

where z is any characteristic height of the photosphere in the region where the solar spots are produced. From the previous relationship, the following expressions can be obtained

and

where

and

In (75) and (76),

and

Clearly, both magnetic fields have the same magnitude. On the other hand, from (76) and (78) it is easy to see that

It can be seen from Equation (71) that as the intensity of the magnetic field generating the solar spots weakens its influence declines in favour of the thermal gradient basically because the other term is a constant for a particular couple of spots. Under such circumstances, the conditions for the startering mechanism of thermal convection are given; so that thermal processes become more important taking over the situation little by little. Again, in the limit when

As usual [

Finally, integrating Equation (81) the following result is obtained

This relation gives the temperature at which the thermal convection starts. Here, T_{o} is the characteristic temperature of the solar spots. According to some numerical data from specialized literature, it is possible to make an approximate calculation of that temperature. For example let T_{o} = 3700 K; a = 0.5 ´ 10^{-4} K^{-}^{1}; g = 2.74 ´ 10^{4} cm/seg^{2}; z = 8 ´ 10^{7} cm; g = 2.4 ´ 10^{8} cm^{2}/K×seg^{2}. In this case, from (83) we obtain T' = 2344 K; so that T = T_{o} + T' = 6.044 K which is the approximate average temperature of the solar photosphere. Additionally, and with the help of the previous data, it is easy to see that convective movements are generated when

Finally, considering that if r = 10^{−5} gr/cm^{3} [_{o} = 0.715 ´ 10^{−6} gr/cm^{3}, then,

as it is easy to see from (79) or (80). Certainly, the obtained value is within the order of magnitude of the magnetic fields of the solar spots measured with the aid of the Zeeman Effect [

From the Z-component of relation (70) the following result can be obtained

where

is the thermal gradient in the Z-direction.

Persistence of the Sunspots seems to depend mainly on the z-components of both the thermal gradient and the gradient of the square of the magnetic field which generates them. Nevertheless, the dominant mechanism at the beginning of the phenomenon is that of magnetic regulation of the convective movements; while at the end of it, disappearance of the Spots is determined by the predominance of the thermal gradient over the other quantities. In consequence it seems reasonable to assume that the average life time of the Sunspots can be determined from the dynamic equilibrium condition between the magnetic regulation and starting mechanisms of thermal convection; so that for the case in which

which is not other than the average life time of the spots. The previous relationship must at least provide an order of magnitude for the time duration of the Sunspots. Astronomical observations of the phenomenon made by other researchers show that the average life time of the phenomenon could be of days, weeks or even several months [^{18} cm^{2}/seg. Then, with the numerical data from the previous paragraph and with q = 2 ´ 10^{−5} K/cm [

Obviously, the permanency in time of a particular couple of spots in the solar disk, depends on the difference between its temperature T_{o} and that of the photosphere, T; this is, it depends on T'. Thus, the higher the spots temperature, the smaller the T', and consequently, the larger the spot average life time.

From the results obtained, it can be assumed that the Sunspots are phenomena very related with processes occuring on the surface of the sun, at a very shallow depth in the solar photosphere; that is, they are basically surface phenomena. Generation, evolution and disappearance of the Sunspots can be explained in terms of the competition established between two dynamic mechanisms: one of magnetic nature which controls and delays thermal convection, and the other one of thermal nature which propitiates convective movements. What happens there may be possibly as follows: the magnetic field characteristic of the solar spots starts to be created at any moment in certain regions of the photosphere. While its intensity increases, a growing magnetic viscosity is generated in the fluid which makes it stiff. Such stiffness damps thermal agitation in the fluid makes it difficult for the fluid to move in those regions. As a consequence, the temperature in those places decreases creating the required conditions for the appearance of a thermal gradient whose value is adjusted to the increasing value of the magnetic field intensity, being created. Thus, as the magnetic field intensity grows, the stiffness in the fluid increases and the magnitude of the thermal gradient grows bigger, firmly opposing the establishment of the said magnetic field. The fight between these two mechanisms leads to a gradual loss of thermal energy in those regions, resulting in areas which are darker and colder than their surroundings, which constitutes the Sunspots. On the other hand, the solar activity in those regions begins to grow and reaches its maximum value when the creation of the thermal and magnetic field gradients is completed.

Next an evolution period of the spots takes place, which is characterized by the existence of a state of dynamic equilibrium between both mechanisms, and by the fact that sun’s activity in those regions is at its maximum. In that intermediate stage, the dominant force is the huge solar gravity which acts over the fluid. Throughout of this period, the hot fluid flows more or less parallel to the surface of the sun due to its gravitational attraction, diverging from the center of the spot and overflowing through its boundaries [

In the final stage, the magnitude of the magnetic field begins to decrease; meanwhile the importance of the thermal processes increases until they become the dominant part when

Another interesting result explains the polarity of the Sunspots, as well as its tendency to appear as magnetic bipolar couples. Apparently, they can be considered as huge electromagnets produced by gigantic solenoids formed under the surface of the photosphere by the ionized fluid which rotates at high speed and creates the monstrous electric currents which generate and maintain the characteristic magnetic field in those regions. The following hypothesis is suggested by the polarity in the spots, their appearance, permanency in the solar disk and subsequent disappearance. The highly ionized solar fluid [

To explain the change in polarity at both hemispheres in each solar cycle, it’s enough to assume that in sun’s latitudes where the spots are usually observed, gigantic turbulent fluid flows take place, in a way similar to what occurs in the Earth at the cyclonic zones. The flows and the turbulent eddy in the northern hemisphere move in the opposite sense as they do in the southern hemisphere due to the Coriolis force [

The transport of ions from the convective zone up to the surface of the sun, is responsible of removing both the huge quantity of heat produced and products of the combustion process, while at the same time, feeding with new fuel the thermonuclear oven [^{+}), the difference between its mass and that of the free electron (e^{−}) produced during the process of ionization is around 2000. For other elements ionized by the loss of one or more orbital electrons, that difference could be even larger; such was the case of (He^{+}) whose mass could be four times bigger than that of (H^{+}). Then, it could be expected the migration of positive ions to be slower than that of negative ions; and such difference could make the former to reach the photosphere with a delay of 11.5 years with respect to the latter, in each Solar Cycle.

Thus, in one cycle there could be surges of ionized fluid mainly of negative charge and in the following, mainly of positive charge. This last argument is only one hypothesis derived from the results previously obtained, and is subjected to validation. It is put forward here as an heuristic attempt to explain the above mentioned phenomenon, because it is difficult to imagine some other processes occurring inside the sun that can have so regular effects. Nevertheless, maybe one could have an indirect proof of the periodicity and origin of this phenomenon by determining whether in the solar wind of the present solar cycle there is excess of a certain type of charge reaching the Earth and coming from both hemispheres of the sun, and if the sign of that charge coincides with the present polarity of the spots. The next step would be to measure whether in the next cycle also there is the said excess of charge and if it is of the opposite sign than the excess of charge of the previous cycle. If the answer is affirmative, the polarity of the spots in both hemispheres has to be as expected and, clearly, it must be opposite to that of the previous cycle. If the previous hypothesis is proven, the problem of finding a theoretical solution to the phenomenon of ionic migration in a star like the sun remains unsolved. Such task will be attempted in the near future. In the opposite case, another possible explanation would be searched for.

Angel FierrosPalacios, (2015) The Sunspots. Journal of High Energy Physics, Gravitation and Cosmology,01,72-87. doi: 10.4236/jhepgc.2015.12007

The influence of the external magnetic field on the system must be taken into account in order to determine its thermodynamic state [

where

is the square of the velocity of sound in the medium,

with

the coefficient of volumetric expansion and

the isothermic compressibility which is always positive [

On the other hand, it is affirmed in astrophysics that the solar plasma behaves as an ideal gas from the thermodynamic point of view; so that the equation

where

and R' = 2R. Let’s consider that the hydrostatic pressure in the interior region of the solar spots is such that it satisfies the hydrostatic Equation (17); this is,

where _{o} a constant and