# Parker Wind

Post-publication activity

Curator: Eugene N. Parker

Parker wind refers to Eugene N. Parker's classical model of the solar wind.

## Introduction

The million degree outer atmosphere of the Sun (the corona) continually expands to produce the supersonic solar wind. The solar wind sweeps out through the solar system, ruffling the magnetic fields of the planets and sweeping back the interstellar gas and magnetic field. The solar wind blows the gaseous tails of comets into the antisolar direction, regardless of the existence or absence of flares or the general level of magnetic activity of the Sun. This fact was one of the principal clues leading to recognition of the wind as a simple hydrodynamic phenomenon. Another indication of the solar wind is the ongoing rattling of the magnetic field of Earth, particularly at high geomagnetic latitudes, from where the lines of force extend far out into space. Were it not for the wind, that part of the field should quiet down between outbursts from solar flares. But even during a month of inactivity at the Sun, the geomagnetic activity never ceases, and there is always an aurora to be seen on any cloudless night at high latitude, implying a continuing wind.

The solar wind spreads out as it races off into space, eventually being halted at distances of the order of 102 AU by the feeble pressure of the interstellar wind and magnetic field. The region dominated by the solar wind is called the heliosphere.

Instruments carried on spacecraft give direct measures of the speed, density, temperature, and atomic composition of the solar wind. The first unambiguous verification of the existence of the solar wind was made from the Venus Mariner, or Mariner II, in 1962. Those measurements established the theoretical prediction from the hydrodynamics of the solar corona that the expansion of the corona produces an ongoing supersonic wind blowing outward through the solar system. The wind is mostly hydrogen, with a few percent helium by number, and smaller numbers of heavier elements. The helium abundance varies and is usually well below the normal solar abundance. The wind velocity and density vary widely between a slow dense phase, issuing from the vicinity of magnetic active regions on the Sun, and a fast tenuous phase issuing from the broad regions of weak magnetic field (<10 Gauss) that make up the so called coronal holes. Thus the slow dense phase is a low latitude phenomenon, shrinking at times of sunspot minimum, whereas the fast wind dominates throughout higher latitudes. Typically, at the orbit of Earth (1 AU) the number density $$N$$ of the wind is 5–15 ions/cm3 in the slow phase where the velocity $$v$$ is 250-500 km/sec and 1-5 ions/ cm3 in the fast phase where the velocity $$v$$ is 500 – 800 km/sec. Curiously, the particle flux $$Nv$$ varies much less than the density.

Coronal temperatures at the Sun are of the order of 1-2x106 K, while at 1 AU the ion temperatures are generally below 105 K and electrons above 105 K in the slow (300 km/sec) wind. This suggests adiabatic cooling of the ions and thermal conduction from the solar corona for the highly mobile electrons. However, with increasing wind velocity the ions are hotter, crossing over the electron temperatures of 105 K at about 500 km/sec and becoming as high as 3x105 K at 800 km/sec. It is interesting to note, then, that at the orbit of Earth the ion thermal velocities are anisotropic as a consequence of the transverse expansion of the outflowing wind. That is to say, the radial component of the thermal velocity (approximately parallel to the nearly radial magnetic field) generally exceeds the transverse thermal velocities (in the two dimensions perpendicular to the magnetic field). However, there are times when the transverse thermal velocities are heated to values in excess of the radial thermal velocities, indicating vigorous extended wave dissipation of some form in the outflowing wind. So the heat input to the expanding corona and solar wind is a complicated subject, and much remains yet to be learned about that heat input. Fortunately, observations of the vigorous small-scale magnetic activity at the Sun are becoming sharp enough to begin to show what happens there, with the expectation of better understanding of heating in the coming years. As we repeat later on, the heat input to the corona and the wind is the basic cause of the wind. Hence, understanding the heating is essential to understanding the physics of the solar wind.

## Magnetic fields

There are magnetic fields everywhere over the surface of the Sun. The magnetic fields are largely concentrated into intense (1–2x103 Gauss) filaments, or flux bundles, with diameters of about 102 km at the visible surface of the Sun, expanding above the visible surface to fill all the space. Some flux bundles extend out into space, while others arch over to connect with neighboring flux bundles, performing complicated reconnections with their neighbors. Some accumulate a large-scale coordinated twisting to such a degree that they participate in the spectacular coronal mass ejections. The mean field is determined by the spacing of the flux bundles, and lies below about 10 Gauss except for the magnetic active regions, where the mean rises to 102 Gauss and then to several thousand Gauss in the umbrae of sunspots. It is easy to show that the expanding solar corona, of perhaps 108 ions/cm3 and 1 – 2x106 K in the presence of fields of 5Gauss, would be confined by the stronger magnetic fields. Indeed, the magnetically confined (100 Gauss) corona builds up to 1010 ions/cm3 and temperatures of 1–5x106 K, emitting X-rays at substantial intensities. The emission is proportional to the square of the density $$N\ ,$$ so the expanding weak field coronal regions, at108 ions/cm3, emit few X-rays. Hence they are observed in X-rays as the dark coronal holes.

The weaker magnetic fields, allowing the expansion of the corona, are carried along in the expanding coronal gas. They are stretched out radially into space, providing a radial magnetic field component throughout the heliosphere, declining with radial distance $$r$$ as $$1/r^2\ .$$ Thus, for instance, a field of 5 Gauss in the low corona extrapolates to about 10-4 Gauss at the orbit of Earth at a distance of 220 solar radii. However, the Sun rotates, with a period of 25 days at the equator and something in excess of 30 days at the poles. The magnetic field in space is connected to the surface of the Sun, of course, so the base of the field at the visible surface rotates as the field is carried radially outward in the wind, giving the extended field a spiral form. In addition to the radial field component, then, there is an azimuthal component, whose strength is given by $$r \Omega \sin \theta / v$$ times the radial component, where the angular velocity of the Sun is denoted by $$\Omega$$ and the polar angle is $$\theta\ .$$ Thus, near the equatorial plane, where $$\Omega\simeq 3\times10^{-6}$$/sec, the azimuthal component is just $$r\Omega/v$$ times the radial component, declining outward only as $$1/r\ ,$$ and dominating the radial field beyond approximately the orbit of Mars, at 1.4 AU, depending upon the wind velocity. So the interplanetary magnetic field throughout the heliosphere lies principally perpendicular to the radial direction, except in the inner solar system. In the ideal case of a radial wind uniform around the Sun the field lines form Archimedean spirals, $$r = v(\phi - \phi_0)/\Omega$$ in the equatorial plane, where $$\phi$$ represents the azimuthal angle. The solar wind is generally not uniform around the Sun, so the spiral regions of faster wind ram into the trailing sides of the more tightly wound spirals of slower wind, forming shocks with fast particle acceleration, and generally referred to as corotating interaction regions.

## Outer termination of the wind

The total pressure $$P$$ of the interstellar gas and magnetic field outside the heliosphere is estimated to be of the order of 2x10-12 dynes/cm2. With the impact pressure $$NMv^2$$ of the solar wind proportional to $$N\ ,$$ which declines as $$1/r^2\ ,$$ it is evident that at some large distance from the Sun, where $$NMv^2$$ becomes comparable to $$P\ ,$$ the interstellar pressure blocks the wind. This causes the supersonic wind to form a standing terminal shock with the wind becoming subsonic beyond and making contact with the interstellar gas and field farther out. With $$N$$ = 5 ions/cm3 and $$v$$ = 500 km/sec at 1 AU the shock transition is estimated to lie somewhere in the vicinity of 100 AU. It takes about a year moving at 500 km/sec for the wind to reach 100 AU.

It is interesting to note, then, that Voyager I encountered the termination shock at 94 AU in the vicinity of the region of direct impact of the local interstellar wind against the heliosphere. The second encounter was made by Voyager II at 84 AU at a location off to the side, interpreted as an indication of the pressure of the galactic magnetic field lying obliquely across the interstellar wind. It is obvious that we need further exploration of the outer heliosphere if we are to develop an unambiguous and quantitative picture.

## Heat input and the origin of coronal expansion

The basic cause of the solar corona and solar wind is the heat input that raises the kinetic temperature to a million degrees or more. The convection beneath the surface of the Sun is a crude heat engine, converting some very small fraction of the outflowing energy into fluid motion. It is this turbulent convective overturning that drives the nonuniform rotation of the Sun and generates the magnetic fields of the Sun. The convection continually deforms the magnetic fields, causing ongoing magnetic reconnection on all scales from the giant flares at 1032 ergs to the myriads of nanoflares at 1023 ergs and less. There has been a continuing drive toward improving the angular resolution of ground based and space borne telescopes, for the purpose of studying the small-scale dissipative flaring activity of the magnetic fields extending through the surface of the Sun. The activity evidently produces enough agitation and outward propagating waves to provide the extended heating of the solar wind, inferred from the million degree temperature. As already noted, the heating process is expected to be much better understood in the next few years.

Now a million or two degrees K is impressive, but it is nowhere near enough to expel the ions and electrons from the Sun. For instance, the gravitational binding energy (negative gravitational potential energy) of a hydrogen atom at the base of the corona is 3x10-9 ergs, to be compared with the thermal energy $$3kT$$ of an ionized hydrogen atom, i.e. an electron -proton pair, of 0.8x10-9 ergs at 2x106 K. Thus the enthalpy, $$5kT\ ,$$ is 1. 3x10-9 ergs. This is substantially smaller than the gravitational binding energy, so how does the gas escape and then achieve a supersonic outflow velocity?

Close to the Sun the coronal gas is relatively dense, at 108 ions/cm3 , and more or less in hydrostatic equilibrium. At 2x106 K the pressure scale height is 1.2x1010 cm, to be compared with the solar radius of 7x1010 cm. Note, too, that for a given temperature the scale height increases in proportion to $$r^2\ .$$ So the density falls by perhaps ten scale heights- a factor of about 104-to the point at which the enthalpy of the gas becomes equal to the gravitational binding energy. Beyond that point the enthalpy exceeds the binding energy, so that the gas is free to expand away into space. The gas immediately below expands upward to replace the escaping gas, and a steady expansion is the result. The expansion velocity increases outward and becomes supersonic as the enthalpy is converted into kinetic energy and the temperature falls. Any additional heating of the gas, through thermal conduction, wave dissipation, etc., serves to increase the final velocity, of course. The essential necessary feature, then, is that the temperature of the corona falls off less rapidly than $$1/r$$ so that the enthalpy exceeds the gravitational binding beyond some value of $$r\ .$$

## Mathematical formulation

The mathematical formulation for purely radial flow $$v(r)$$ is easily accomplished for a steady flow. Conservation of particles requires that $$N(r) v(r) r^2 = N(a) v(a) a^2\ ,$$ where $$a$$ is some convenient reference level at the Sun. The pressure of ionized hydrogen at temperature $$T(r)$$ is $$2N(r)kT(r)\ ,$$ and the momentum equation becomes

$\tag{1} N M v \frac{dv}{dr} = - \frac{d}{dr} 2NkT - \frac{GM_0NM}{r^2},$

where $$M_0$$ represents the mass of the Sun and $$M$$ the mass of a hydrogen atom, and G is Newton’s gravitational constant. Use the equation for conservation of particles to eliminate $$N(r)$$ from the momentum equation, with the result

$\tag{2} \frac{dv}{dr} \left( v - \frac{U^2}{v} \right) = - r^2 \frac{d}{dr} \left( \frac{U^2}{r^2} \right) - \frac{GM_0}{r^2},$

where the characteristic ion thermal velocity is defined as $$U^2 = 2kT(r)/M\ .$$ We are interested in a solution to this differential equation that begins at small $$r$$ with $$v(r) \ll U(r)\ ,$$ from where it extends continuously and asymptotically to some finite value at $$r = \infty\ .$$ We expect such a solution provided that $$T(r)$$ does not decline with increasing $$r$$ as fast as the $$1/r$$ of the gravitational potential.

It is evident by inspection that for a strongly bound corona the first term in the parenthesis on the left hand side of the momentum equation is smaller in magnitude than the second term. For strong gravitational binding the right hand side is negative, providing the acceleration $$dv/dr > 0\ .$$ It is also evident that, if the temperature $$T(r)$$ varies asymptotically as $$1/r^\alpha\ ,$$ where $$\alpha < 1\ ,$$ the first term on the right hand side dominates the second term at large $$r\ ,$$ and the right hand side is positive. Hence the right hand side starts out negative at small $$r$$ and becomes positive at large $$r\ ,$$ passing through zero along the way. Note, then, that the factor in parenthesis on the left hand side crosses zero at the sonic point somewhere along the way from small to large $$r$$ and if $$dv/dr$$ is to remain finite where $$v$$ crosses over $$U\ ,$$ then that sonic point must occur at the same location as the zero of the right hand side.

To illustrate these principles with a simple solution, consider the isothermal corona, in which the temperature $$T(r)$$ is a constant. Then $$U$$ is a constant and it is convenient to write $$V(r) = v(r)/U\ .$$ Denote the critical radius $$R\ ,$$ where the right hand side vanishes, by $$R = GM_0/2U^2\ .$$ The momentum equation can then be written in the form

$\tag{3} \left( \frac{dV}{dr} \right) \left( V - \frac{1}{V} \right) = \frac{2}{r} \left( 1 - \frac{R}{r} \right).$

It is evident by inspection that the desired solution $$V(r)$$ must pass across the sonic point, or critical point, $$V = 1$$ at $$r = R\ .$$ Integrating the equation and choosing the integration constant to achieve this crossing yields the continuous solution

$\tag{4} V^2 - \ln V^2 = 4 \ln \frac{r}{R} + 4 \frac{R}{r} - 3.$

The asymptotic form at $$r/R \ll1$$ is

$\tag{5} V(r) \simeq \frac{R^2}{r^2} \exp \left( \frac{3}{2} - \frac{2R}{r} \right),$

from which it follows that the density varies in proportion to $$\exp(2R/r)\ .$$ This is, of course, just the solution to the hydrostatic barometric law. The exponential factor dominates and $$V(r)$$ declines rapidly with decreasing $$r/R\ .$$ The asymptotic form of for $$r/R\gg1$$ is

$\tag{6} V^2 \approx 4 \ln \left( \frac{r}{R} \right) - 3 + O(\ln V^2).$

The velocity grows slowly without bound for as far out as the temperature is maintained.

This simple exposition elucidates the basic principles of the expansion and escape at supersonic speeds of the tightly bound solar corona. The fundamental unknown at the present time is the heat input, beginning with the vigorous activity of the small-scale magnetic fields at the Sun and the outward propagation and dissipation of the waves created by that activity at the Sun. It is that heat input to the corona that causes the solar wind. The solar wind represents an energy loss of the general order of 4x1027 ergs/ sec, or about 10-6 of the total solar output of 4x1033 ergs/sec.

## Solar mass loss

Note that the solar wind represents a mass loss to the Sun of about 1012 gm/sec, or 106 tons/sec. Though an impressive figure this would amount to no more than 10-4 solar masses over the entire life expectancy of 1010 years for the Sun. There is reason to believe, however, that in the first 108 years on the main sequence, the solar wind was much more massive than at present. Observations of newly formed stars show rapid rotation, with rotation periods of a couple of days. Stars much older than 108 years rotate at a more leisurely pace, of the order of 15 – 30 days like the Sun. Evidently the angular momentum has been carried away by an early massive stellar wind flowing outward along extended strong rotating magnetic fields so that the angular momentum per unit mass carried away by the wind is very large.

The Sun is a typical type G dwarf main sequence star, so one infers that other stars of the same class have stellar winds similar to the solar wind, all too tenuous to be observed from a distance. In fact, one infers that any main sequence star with a significant convective zone has a corona that expands away to form a stellar wind of some sort. The fact that all stars, except for white dwarfs and red giants, emit X-rays is clear evidence of million degree coronas trapped in magnetic fields. This implies that there may be other parts of the coronas that are not effectively trapped, from which we would expect to find coronal gas expanding away to form a stellar wind of some sort.

## Cosmic ray modulation

The solar wind and its magnetic fields provide a number of curious effects. For instance, the outward transport of the irregular transverse magnetic fields sweeps back the galactic cosmic rays, substantially reducing their intensity throughout the inner solar system. This modulation produces the 11 – year cyclic variation of the cosmic ray intensity observed by neutron monitors at the surface of Earth. The effect is largest for the lower energy particles, of course, and amounts to about 15 percent at high geomagnetic latitudes, falling to a couple of percent at the geomagnetic equator where the field excludes most of the incoming cosmic rays below about 15 Gev/nucleon. The cosmic rays are at a minimum during the years of high magnetic activity and sunspot maximum, dominated by the slow, dense, and more irregular phase of the solar wind and its magnetic field. At sunspot minimum the magnetic fields are more regular and the cosmic rays more easily penetrate to the inner solar system. Indeed, at such times the anomalous cosmic rays make their subtle appearance, on which we have more to say later.

In addition to the general reduction of the galactic cosmic rays, the explosive coronal mass ejection carries its own magnetic fields and causes a transient localized reduction in the cosmic ray intensity, called a Forbush decrease, after their discoverer. Cosmic ray modulation and the interplanetary propagation of fast particles from solar flares and interplanetary shock fronts in the magnetic fields of the solar wind are an active and important ongoing field of research.

## Interaction with the interstellar wind

It must be appreciated that the interaction of the solar wind with the local interstellar wind blowing by the heliosphere is more than just a problem in hydrodynamics. In particular, the interstellar gas is only partially ionized, with densities of 0.1 hydrogen ions/cm3 and 0.24 neutral hydrogen atoms/cm3, all at about 6000 K, and collectively blowing past the heliosphere at the supersonic interstellar wind speed of 25 km/sec. The interstellar magnetic field carried in the wind may be presumed to be of the order of a few microgauss. The presence of the neutral atoms is interesting because they have no net charge and move freely across magnetic fields. They cannot feel the magnetic fields of the heliosphere, and the mean free path for collisions with ions and other neutral particles is comparable to the radius of the heliosphere. So they respond to the gravitational field of the Sun, falling freely in toward the Sun along hyperbolic trajectories. The interesting aspect of this is the ionization of those neutral atoms that come closer than approximately the orbit of Jupiter (5 AU). Ionization is provided by the intense solar UV and by charge exchange with the ions in the solar wind. Freely orbiting through interplanetary space, the neutral atom suddenly finds itself an ion. It is instantly grabbed by the magnetic field in the solar wind and transported outward again. Except that now the ion is very hot, with an energy of the order of 1 Kev. In the moving frame of reference of the solar wind, the neutral atom was moving inward with a speed comparable to the hundreds of km/sec of the solar wind. Upon becoming charged, this relative velocity becomes circular motion around the magnetic field in the frame of reference of the moving wind. The wind carries these hot pickup ions back out of the solar system, reaching the termination shock in approximately one year. The termination shock selectively accelerates these hot ions, some reaching energies well in excess of an Mev. They are cosmic rays of sorts, and with their new found vigor they are able to penetrate back into the heliosphere. At sunspot minimum, when magnetic activity is at a relatively low level and the wind is fast and the magnetic field is relatively smooth, enough of these anomalous cosmic rays reach the inner solar system to be detected with especially sensitive cosmic ray instruments. And these anomalous cosmic rays found here at the orbit of Earth are a direct product of the interaction of the solar wind and the neutral atoms in the interstellar wind at a distance of 100 AU.

In fact the interaction of the solar wind with the interstellar wind is a vast and complex phenomenon waiting to be explored. The heliosphere creates a wake in the interstellar wind that trails off downstream for an unknown distance. The so called hydrogen wall created by the interaction of the two winds is made known directly by the emitted Lyman alpha, which also shows the presence of the infalling neutral atoms. Needless to say, the vast dimensions of the heliosphere (100 AU = 13 light hours) make it technically difficult to carry out in situ measurements.

## References

M. Aschwanden 2004, Physics of the solar corona, 167 – 174, 703 – 737, Springer, Berlin.

P. C. Frisch (ed) 2006, Solar Journey: The significance of our galactic environment for the heliosphere and Earth, Springer, Dordrecht.

A. J. Hundhausen, 1972, Coronal expansion and the solar wind, Springer-Verlag, Berlin.

J. R. Jokipii, C. P. Sonett, and M. S. Giampapa (eds) 1997, Cosmic winds and the heliosphere, University of Arizona Press, Tucson.

N. F. Ness, C. S. Scearce, and J. B. Seek, 1964, Initial results of the Imp 1 magnetic experiment, Journal of Geophysical Research, 69, 3531 – 3569.

M. Neugebauer and C. W. Snyder, 1966, Mariner 2 observations of the solar wind: I Average Properties, Journal of Geophysical Research, 71, 4469 – 4484.

M. Neugebauer and C. W. Snyder, 1967, Mariner 2 observations of the solar wind: 2, Relation of the plasmas properties to the magnetic field, Journal of Geophysical Research, 72, 1823 – 1828.

E. N. Parker, 1958, Dynamics of the interplanetary gas and magnetic field, Astrophysical Journal 128, 644 - 676.

E. N. Parker, 1963, Interplanetary dynamical processes, Wiley Interscience, New York.

E. N. Parker, 1965, The dynamical theory of the solar wind, Space Science Reviews, 4, 666 – 708.

E. N. Parker, 2001, A history of the solar wind concept, The Century of Space Science, 225 – 255, Kluwer Academic Publishers.

C. W. Snyder and M. Neugebauer, 1964,Interplanetary solar wind measurements by Mariner II, Solar Research, 4, 89 – 113.

Internal references

• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.