Solar Wind: Theory. Parker s solar wind theory

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1 Solar Wind: Theory The supersonic outflow of electrically charged particles, mainly electrons and protons from the solar CORONA, is called the SOLAR WIND. The solar wind was described theoretically by E N PARKER, in Parker s theory was verified experimentally by in situ observations by Soviet and American spaceprobes. On its way to Venus, in 1962, the MARINER II spacecraft observed the solar wind for 104 days. The average flow speed was more than 500 km s 1. This observation showed that the coronal plasma expands into a supersonic solar wind as Parker had predicted. Parker s solar wind theory Parker was familiar with the work on comet tails that had been carried out in Germany in the late 1940s and early 1950s under the leadership of LUDWIG BIERMANN (see COMETARY TAILS). The ionic comet tails were observed to be pointing radially out from the Sun. This required a radial force much larger than the force that the photons from the Sun can provide, and Biermann concluded that the corpuscular radiation from the Sun may play an important role in forming the radially pointing comet tails. Parker based his solar wind theory on the fact that the solar corona has a temperature of more than a million kelvin. He argued that the electron density and the pressure in such a hot atmosphere decrease rather slowly, and the pressure, far from the Sun, is orders of magnitude larger than the pressure of the interstellar gas surrounding the solar system. According to Parker, this imbalance between the pressure in the outer corona and the local interstellar medium would lead to an expansion of the coronal gas into a supersonic solar wind. The coronal gas, at a temperature of one million kelvin or more, is fully ionized; the number of neutral atoms is very small. However, the thermal energy of the plasma is not large enough to overcome the gravitation field and escape from the Sun. In the inner corona the pressure decrease is determined by gravity, so the gas is in static equilibrium. However, an ionized gas at a temperature of more than a million kelvin is a good conductor of heat. Hence, the coronal temperature does not decrease significantly with increasing distance from the Sun. The thermal energy of the plasma is almost constant whereas the energy needed to overcome the gravitational field decreases. In the outer corona the thermal energy is larger than the escape energy, and the gas can escape. In Parker s solar wind model, the energy transport (in the form of heat conduction) from the inner corona plays an important role for supplying the energy needed to bring the plasma out of the solar gravitational field. The electrons are much better at conducting this heat than are the ions, owing to their smaller mass, but most of the energy flux goes into increasing the energy of the ions. This transfer of energy from electrons to ions is achieved through an electric field. This electric field must be consistent with the force and energy balance in the flow. Parker described the coronal gas as a fluid. This allowed him to take into account the coupling of electrons and ions, without calculating the electric field explicitly. In his first study he assumed that the heat conductivity is so high that the coronal plasma has a constant temperature. On the basis of this assumption he could illustrate how the coronal plasma reaches supersonic speed around 5 solar radii from the Sun, and expands into interplanetary space with a steadily increasing flow speed. The equations also allow for subsonic flow, but the pressure far from the Sun in such solutions is almost the same as the pressure in a static, isothermal corona. Hence, the subsonic solutions do not describe an outflow that is in force balance with the local interstellar medium. In the supersonic flow the flow speed is larger than the thermal motion in the gas, and the dynamic pressure (associated with the directed motion) is larger than the thermal pressure (associated with the random motion). The dynamic pressure of the spherically expanding solar wind decreases with distance from the Sun. At a distance where the dynamic solar wind pressure is equal to the pressure of the interstellar gas a shock is formed: the flow speed is reduced, the density increases and most of the flow energy is transferred into thermal energy. The solar wind termination shock, at 100 AU or so (1 AU = Sun Earth distance), has not yet been observed, but the Voyager 1 and 2 spacecraft may cross this boundary in a not too distant future. Parker s solar wind theory has formed the basis for our understanding of the expanding solar corona as well as the outflow of ionized gas from galaxies and stars, and other celestial bodies, and the outflow of light ions from the polar regions of the Earth (the polar wind). History Studies of the Sun have a long history in many cultures, but it was the studies of aurora and geomagnetic storms that led to the studies of the solar wind. The AURORA was part of the mythology and the life of peoples living in the Arctic, but when Celsius and Hiorter in Uppsala, Sweden, started systematic observations of the the magnetic needle, in the 1740s, the correlation between aurora and fluctuations in the geomagnetic field was first found. Their collaboration with Graham in London showed that geomagnetic fluctuations occurred both in London and in Uppsala when aurora was seen in Sweden. These findings seem to be the first to establish the link between aurora and GEOMAGNETIC STORMS. It took some time to find the common cause of aurora and geomagnetic disturbances. The 11 year SUNSPOT CYCLE was noticed by the German amateur astronomer Schwabe in 1843, and several investigations showed a correlation between auroral and geomagnetic activity, and sunspot number. In the 1850s Broun found that geomagnetic storms had a tendency to recur after 27 days, a time close to the rotation period of the Sun, seen from the Earth. On 1 September 1859, a large solar flare was observed by Carrington and by Hodgson, and approximately 18 h later Dirac House, Temple Back, Bristol, BS1 6BE, UK 1

2 there was a large geomagnetic storm with aurora at very low latitudes. This event served as an indication that there is a connection between solar storms and geomagnetic storms (see MAGNETOSPHERE OF EARTH: GEOMAGNETIC STORMSAND SOLAR WIND ORIGINS). Throughout the last part of the 19th century there were several studies of the possible link between solar activity and geomagnetic storms. At the end of the century it was established that aurora and geomagnetic storms should be regarded as manifestations of an unknown cosmic agent of solar origin, Kristian Birkeland wrote in One hundred years ago Birkeland set up an impressive research program in Norway to investigate the effects of charged particles from the Sun on the near Earth environment. Together with his assistants he carried out laboratory experiments, observations from field stations around the world and theoretical studies. This was the first comprehensive research effort in solar terrestrial physics. (Birkeland financed the research program with money he earned from working with industry.) Birkeland is best known for his Terella experiments, where cathode rays interact with a magnetized sphere, placed in a low-density gas. The experiment showed that the cathode rays impacted on the sphere in regions around the two poles, much like the auroral zones at high latitudes. However, what is less known is that Birkeland also used his magnetized sphere as a cathode to study the emission of electric corpuscules from the Sun. On the basis of observations of continuous geomagnetic activity at stations in the Arctic, Birkeland concluded that there is a continuous flow of charged particles from the Sun. These particles interact with all bodies in interplanetary space, and the interaction with comets leads to the formation of comet tails. To study this process in the laboratory he let cathode rays impinge upon an anode of coal. On the basis of many years of studies of the Sun and of geomagnetic activity he was convinced that all stars, in the course of their evolution, emit electrons and ions into space, and he went on to speculate that most of the mass in the universe is not in stars and nebulae but in empty space. At the time when Birkeland carried out his work it was not known that the outer solar atmosphere, the corona, is hot. It was first in the 1930s that the picture of a corona with a temperature of more than a million kelvin began to emerge. This was made possible by the development of the CORONAGRAPH by BERNARD LYOT. By shielding the solar disk with a circular plate it was possible to carry out observations of the corona on a regular basis. Previously, such observations could be made only during eclipses. Lyot measured the width of the green line (5303 Å) to 0.9 Å (1 Å = m). He suggested that the broadening could be due to thermal motions, but the element emitting the line was not known, so a temperature could not be determined. Grotrian argued, from the early 1930s, that the corona is hot, but it was the identification of coronal lines, as emission lines from highly ionized elements, that established that the corona is hot. Edlén identified the green line (5303 Å) as an emission line from iron atoms that have lost 13 electrons, Fe XIV, and he showed that many of the other coronal lines were emitted by highly ionized elements. These elements cannot exist in the corona unless the temperature is a million kelvin or more. With such a high temperature one could describe the relatively slow electron density fall-off and the widths of the coronal spectral lines. However, it was not easy to understand how it is possible to maintain a hot corona overlaying a cold CHROMOSPHERE. The rapid increase in temperature from the chromosphere to the corona is consistent with a heat conductive energy loss from the corona. This energy loss must be balanced by coronal heating. Biermann and Schwarzschild claimed that acoustic waves could transport energy into the corona from the lower layers in the solar atmosphere, whereas ALFVÉN suggested that MAGNETOHYDRODYNAMIC WAVES (later called Alfvén waves) may carry the energy necessary to heat the corona. Already in 1941, Alfvén had published a model of a hot static corona, extending out to 10 solar radii, and in the early 1950s there were indications that the solar corona extends even further into interplanetary space. Hewish and Vitkevich observed fluctuations in radio signals that pass through interplanetary space relatively close to the Sun. They found electron density irregularities extending out to at least 20 solar radii and that the super corona changed over the sunspot cycle. Latitude variations were also observed. These observations, together with Biermann s observations of ionic comet tails and Forbush s observations of the variations of the low energy cosmic ray intensity over the sunspot cycle, were known when Parker formulated his solar wind theory. The solar wind In 1958 Parker published a paper on the dynamics of the interplanetary gas and magnetic field in the Astrophysical Journal. In this paper he presents a model of an expanding solar corona; the coronal gas is allowed to flow out from the Sun in the form of a solar wind. With this model Parker could describe the transition from a quasi-static inner corona to a supersonic solar wind with speeds of several hundred kilometers per second at the orbit of Earth. Parker used hydrodynamic equations, and he considered an isothermal, spherically symmetric, radial flow. This model suffers from several shortcomings, where the most severe may be that the energy balance in the flow is not addressed. However, with this very simple model, Parker could illustrate how the hot coronal gas expands and expels the interstellar gas from interplanetary space. Parker s solar wind model bears some similarities to Bondi s and McCrea s models of accretion of interstellar gas onto a central object. Formally, the two problems are identical, but the physics of Parker s solar wind model is more difficult to understand than the physics of the accretion flow. The simplicity of the model invited criticism. Chamberlain pointed out that the temperature in the flow is determined by degradation of the heat conductive flux Dirac House, Temple Back, Bristol, BS1 6BE, UK 2

3 and adiabatic cooling, whereas in the Parker model it was taken to be constant. This also implied that the energy per unit mass in Parker s model is infinite. Chamberlain argued that the energy per unit mass should be set to zero. This assumption led to a low-speed breeze solution of the fluid equations. Parker included the energy balance in the flow in his model and showed how heat is conducted outward from the inner corona and converted into flow energy by the pressure gradient force. This model could also describe the supersonic solar wind. Opponents argued that a fluid treatment of the coronal gas is not valid, and that a kinetic approach should be used. During the 1960s there were several attempts to construct exospheric solar wind models. Many of these studies gave results in better agreement with the breeze solution than with the supersonic wind solution. One reason some of the exospheric models gave low flow speeds is that the electric field in the models was too small. This electric field is set up by the plasma to balance the pressure gradient force in the electron gas. In the fluid model the electric field does not appear explicitly. In the kinetic models this is not the case. Many of the exospheric models used the electric field of a static corona. This is smaller than the field in a subsonic supersonic solar wind and gives rise to a smaller acceleration of the protons than is found in the hydrodynamic model. In Parker s model the solar magnetic field is dragged out by the solar wind. Because of the high electric conductivity the electric field in the expanding gas is small. As a consequence, the magnetic field is connected to the source region at the Sun. When the Sun rotates, the gas emitted from one region in the corona is situated on a spiralling field line, and the direction of the field depends on the polarity in the corona. In the ECLIPTIC, the average magnetic field is in the ecliptic plane, and the angle between the average field and the radial direction is around 45 at the orbit of Earth. It took only a few years before the solar wind controversy could be settled. In situ observations of the interplanetary plasma were carried out by the Soviet spacecraft Lunik 2 and the American spaceprobe Explorer 10. However, it was not until the fall of 1962, after the Mariner II observations of the solar wind, that Parker s solar wind model was accepted. On route to Venus, Mariner II obtained 104 days of observations. It measured an average solar wind flow speed of 504 km s 1. The proton density was around 5 cm 3, much lower than the density in a static corona. The average interplanetary magnetic field showed a spiral structure very similar to the model proposed by Parker. The Mariner II flight took place during the declining phase of the sunspot cycle. There were several high-speed solar wind streams, with a recurrence period of 27 days (seen from the Earth), during the mission, that gave rise to recurrent geomagnetic storms with the same period. Thus, the source of recurrent geomagnetic storms could be identified as high-speed solar wind streams, but the solar source regions of these streams, often called M-regions, were not identified. It was 10 years later that coronal holes, were first identified as the source regions of quasisteady high-speed solar wind; coronal holes are regions in the corona, with unipolar magnetic field and low electron density. During the declining part of the sunspot cycle the polar regions develop into large coronal holes, and the high-speed solar wind from these regions fills up a large fraction of interplanetary space. Solar wind from a given corona When Parker formulated his solar wind theory he took the inner solar corona to be a reservoir of particles and energy for the outflowing solar wind. Therefore it can be argued that he made the assumption that the energy balance in the corona is between coronal heating and inward heat conductive energy loss and that the energy loss in the solar wind is not large enough to significantly change the structure of the inner corona. This model can describe the basic dynamics of the corona solar wind system quite well, i.e. the expansion of the coronal gas into a supersonic solar wind. However, Parker realized that the model could not describe the quasi-steady high-speed solar wind streams that were observed, mainly during the declining phase of the sunspot cycle. He found that energy has to be added beyond the inner boundary to speed up the flow. Many years later Leer and Holzer showed that this energy has to be deposited in the supersonic region of the flow; energy deposition close to the Sun increases the solar wind particle flux whereas energy deposition in the supersonic flow increases the energy per unit mass in the flow and therefore the flow speed. There are several problems with a model where the electron density and temperature in the inner corona are taken to be independent of the solar wind outflow. If we assume that the electron (proton) density at the inner boundary is fixed, and let the temperature increase, the solar wind proton flux increases rapidly. By varying the temperature from one to two million kelvin the solar wind proton flux changes by a factor of 100 or so. However, the solar wind proton flux is observed to be fairly constant whereas the coronal temperature shows considerable variation. This inconsistency cannot be resolved in a reasonable manner within the framework of a traditional solar wind model. In order to obtain a more complete description of the solar wind we must include the coronal energy balance in the model. Formation of the corona and acceleration of the solar wind In Parker s theory the solar wind is a consequence of the coronal heating process, so the inclusion of the coronal energy balance would be a natural extension of his model. This can be done by moving the inner boundary from the corona into the upper chromosphere. A significant energy flux is needed to balance the radiative losses from the chromosphere, but the energy flux deposited in the outermost part of the solar atmosphere, where the electron Dirac House, Temple Back, Bristol, BS1 6BE, UK 3

4 density is very small, is not radiated away locally. The temperature increases until some other loss mechanism balances the heat input. In magnetically closed regions, where the coronal plasma is trapped by the magnetic field, inward heat conduction is the most significant energy loss, whereas in regions where the magnetic field extends into interplanetary space, and the coronal plasma is free to escape, energy may also be lost in the solar wind. How much of the energy that is lost as inward heat flux and as solar wind energy flux may depend on the amplitude of the energy flux and how and where this energy flux is deposited in the corona. Energy balance in a static corona Let us first consider the energy balance in magnetically closed regions, where the energy deposited as heat is lost as inward heat conduction. To make the problem as simple as possible we can consider a spherically symmetric corona, with a lid on it. This outer boundary does not allow transport of either plasma or energy. If this static corona is heated by an energy flux from the Sun, say 100 W m 2 at the solar surface, and this energy flux is deposited in the corona over a length scale comparable with a solar radius, we find a coronal temperature in the range million K; a corona at this temperature will lose 100 W m 2 into the TRANSITION REGION in the form of an electron heat flux. However, the heat conductive loss from the corona depends sensitively on the coronal temperature; a corona at 0.5 million K loses only 1 W m 2 as inward heat flux. Because of the very strong temperature dependence of the heat conduction in an ionized gas we find that a small heat input is sufficient to maintain a rather hot corona. In a corona where the electron (proton) density is so low that the electrons and protons are not thermally coupled, the proton heating must be balanced by heat conduction in the proton gas. As there is no reason to expect that the protons are heated less than the electrons, the lower heat conductivity in the protons leads to a proton temperature that is higher than the electron temperature. The electron density in the inner corona is determined by the pressure in the chromosphere corona transition region, and this pressure is determined by the heat conductive flux from the corona. If we assume that the radiative losses from the transition region are balanced by the heat flux from the corona, we find that an inward heat flux of 100 W m 2 corresponds to a transition region pressure of p TR = Nm 2. The pressure is proportional to the heat flux, so a heat flux of 10 W m 2 corresponds to Nm 2. A large inward heat flux leads to a high transition region pressure, a large electron density in the inner corona and strong collisional coupling between electrons and protons. A small inward heat flux is consistent with a low pressure in the transition region and a low electron density in the corona. Energy balance in coronal holes In magnetically open regions the energy that is deposited in the corona can be lost as inward heat flux and as solar wind energy flux. Which of these two loss mechanisms is the most important one may depend on how and where energy is transferred to the coronal plasma. If most of the energy is added to the electrons, as heat, a large fraction may be lost as inward heat flux. However, if most of the energy is added to the ions the inward heat flux may be reduced significantly. In order to investigate the energy balance of coronal hole regions we can construct mathematical models extending from the upper chromosphere, through the transition region and corona, and far into interplanetary space. The heating of the corona is specified through the amplitude of the energy flux and how and where this energy flux is deposited. These types of studies show that heating of the very inner corona leads to a large inward heat flux, whereas only a small fraction of the energy flux is lost in the solar wind. Heating further out from the Sun leads to a larger solar wind energy loss, but there is a significant difference between models with electron and proton heating. In models with extended electron heating the inward heat flux is a significant fraction of the energy flux, and because of the high heat conductivity in the electrons the temperature in the corona does not exceed 1.5 million K for a reasonable heat input. The large inward heat flux is consistent with a large transition region pressure and a quite large electron density in the inner corona. The solar wind emanating from such a corona does not reach a high flow speed far from the Sun. A typical value is 300 km s 1. In these models, where the electrons are heated, a large fraction of the outward energy flux from the corona is carried as electron heat conduction. This energy is transferred to the ions via the polarization electric field. This model has many similarities to the classical solar wind model, where electron heat conduction from the inner corona supplies the energy flux that is needed to drive the solar wind. If most of the energy is added to the protons, the inward heat flux is much smaller, the transition region pressure is low, the electron density in the inner corona is low, the thermal coupling may be weak and the coronal proton temperature may be quite high. In models with extended proton heating most of the energy deposited in the corona is lost in the solar wind. The density in the corona and the solar wind proton flux are small, so the energy per unit mass in the flow, and the flow speed far from the Sun, can be quite large. In order to obtain flow speeds of 800 km s 1, measured in the high-speed wind by the Ulysses spacecraft, a large fraction of the energy must be deposited in the outer corona where the proton density is so low that the heat cannot be conducted away. Then, the heating leads to a high proton temperature and rapid acceleration of the flow. This type of model study of the corona solar wind system is limited by the assumptions made, but some Dirac House, Temple Back, Bristol, BS1 6BE, UK 4

5 general results seem to emerge. High-speed solar wind streams cannot be found in models where most of the energy is deposited in the very inner corona. In models with significant electron heating the inward heat flux is too large, the coronal electron density is too high and the coronal temperature is too low to generate high-speed wind. The high-speed streams can be obtained in models where most of the energy is deposited in the protons. The energy may be deposited as heat or it may go into direct acceleration of the flow, but none of these processes will lead to a large flow speed unless the energy is deposited sufficiently far out in the corona. Most of the energy flux deposited in the corona, in these models, is lost in the solar wind. As the asymptotic flow speed is comparable with the escape speed of the Sun, the solar wind mass flux is proportional to the the energy flux deposited in the corona. Ulysses and SOHO In the 1990s there have been two successful space missions that have contributed significantly to our understanding of the solar wind. The Ulysses spacecraft has observed the solar wind at all solar latitudes (see SOLAR WIND: ULYSSES). The first orbital period was during the declining phase of the sunspot cycle. The observations showed a fast and steady solar wind at latitudes larger than 20. Close to the ecliptic the flow speed was lower and more variable. The solar wind mass flux showed little variation with latitude. The fast solar wind streams originate in large coronal holes. Observations of spectral lines from large polar coronal holes, with instruments on board the Solar and Heliospheric Observatory (SOHO), show that heavy ions are warmer than protons which themselves are warmer than the electrons. These observations support a model of high-speed wind, with significant energy deposition in the protons and not in electrons. The fact that we observe heavy ions in the corona may be taken as an indication that their temperature is high. If the temperature of heavy ions were equal to the proton temperature, the density of heavy ions would fall off so rapidly with heliocentric distance that it would be difficult to see them in the corona. However, does a high temperature tell us that a heavy ion is preferentially heated? Particle escape seems to be the most important energy loss process for protons and heavy ions in coronal holes. As the flux of escaping particles is an exponential function of the thermal energy over the escape energy a modest heating of heavy ions leads to a high temperature. If the energy input per unit mass is the same for protons and heavy ions the temperature should be proportional to mass. However, if the ion temperature increases with mass, more rapidly than proportional to mass, we can conclude that the heavy ions are preferentially heated. So far, only a few ion temperatures have been measured in large coronal holes with the Ultraviolet Coronagraph and Spectrograph (UVCS) on board SOHO, but observations of spectral lines from oxygen atoms that have lost 5 electrons, O VI, seem to indicate that there are heavy ions in the corona that are preferentially heated. Summary Parker s solar wind theory forms the basis for our understanding of the interplanetary plasma. The dynamics of the solar wind is described by models developed more than 40 years ago. The extension of these models, which includes the coronal energy balance, has led to a fairly good understanding of the acceleration of the solar wind. The solar wind is driven by the energy deposited in the corona. This process is not understood. In the future the emphasis will therefore be on trying to understand coronal heating. A better understanding of how energy is transported into the corona and transferred to the gas will also give us a deeper understanding of how the solar wind is accelerated. Bibliography Eather R H 1980 Majestic Lights (American Geophysical Union) Golub L and Pasachoff J M 1997 The Solar Corona (Cambridge: Cambridge University Press) Jokipii J R, Sonett C P and Giampapa M S (eds) 1997 Cosmic Winds and the Heliosphere (Tucson, AZ: University of Arizona Press) Parker E N 1958 Dynamics of the interplanetary gas and magnetic fields Astrophys. J Egil Leer Dirac House, Temple Back, Bristol, BS1 6BE, UK 5

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