asen5335- aerospace environments -- solar wind 1 the solar wind is the extension of the solar corona...

ASEN5335- Aerospace Environments -- Solar Wind 1

The solar wind is the extension of the solar corona to very large heliocentric distances.

The solar wind is ionized gas emitted from the Sun flowing radially outward through the solar system and into interstellar space.

The Solar Wind


Electron density 7.1 cm-3 Proton density 6.6 cm-3

He2+ density 0.25 cm-3 Flow speed 425 kms-1

Magnetic field 6.0 nT Proton temperature 1.2 x105 K

Electron temperature 1.4 x105 K

Observed Properties of the Solar Wind at 1 AU

The pressure in an ionized gas with equal proton and electron densities is

Pgas = nk (Tp + Te)

where k is the Boltzmann constant and Tp and Te are proton and electron temperatures. Thus,

Pgas = 2.5 x 10-10 dyn cm-2 = 25 pico pascals (pPa)

Similarly, a number of other solar wind properties can be derived (see following table)

Derived Properties of the Solar Wind


DERIVED SOLAR WIND PROPERTIES AT 1 AU

Gas Pressure 30 pPa

Dynamic Pressure Pd = u2 = 2 nPa (a.k.a. “momentum flux

density”, u•u)

Magnetic pressure PB = B2/8 = 14 pPa

Acoustic speed Cs = [p/]1/2 = [(cp/cv)p/]1/2 = 60 km s-1

cp/cv = 5/3 for ideal gas

Alfven speed VA =B/(4 )1/2 50 km s-1

Proton gyrating speed 45 km/s

Proton gyroradius 80 km

time for wind to travel to 1 AU 3.5 x 105 s (4 days)


The Solar Wind is Highly Variable - V

Historical Note:

The solar wind was first sporadically detected by the Soviet space probes Lunik 2 and 3.

Recent observations

fast streams

shock


The Solar Wind is Highly Variable - n

Recent observations

HistoricalNote:

The first continuous solar wind observations were made by Mariner 2 on its 1962 voyage to Venus. Nearly 3 months of data unequivocally confirmed the existence of the solar wind.


How Does the Solar Wind Escape the Sun’s Gravity ?

First, we will invoke several simplyfying assumptions:

The solar wind is an ideal isothermal gas;

The solar wind flows radially away from the sun;

Magnetic field effects are neglected;

Steady-state solution

Let us now outline the basic equations.

We will now quantify the basic ideas behind coronal heating and the solar wind through a simplified analytic model. More sophisticated treatments retain the fundamental ideas below.


Continuity Equation

€

∂∂t

+v

∇⋅ρ v u =0

€

1r2

∂∂r

ρr2ur ⎛

⎝ ⎜

⎞

⎠ ⎟+........=0

€

r2ur =const(1)

(conservation of mass for an outward expanding sphere)


The meaning of the above expression becomes clearer if it is

multiplied by 4to give

Because is the rate at which mass is carried

through a unit area on a sun-centered sphere, and is a surface area of such a sphere of radius r, then I is just the mass

flux (g s-1) through the entire sphere.

€

4πr2⋅ρur = I (a constant)

€

ur

€

4πr2

In other words, the total flux through all sun-centered

spheres must be the same.


€

∂ vu

∂t+ρ v u ⋅∇v u = −∇p−ρ

v F g +

v j ×

v B

€

urddr

ur =−1ρ

dpdr

−GMsr2

where Ms = mass of sun and G = gravitational constant.

Conservation of Momentum


One obvious solution to the above equations, and one that was accepted until the late 1950's, was that of static equilibrium, or that . The continuity equation is automatically satisfied, and the momentum equation becomes

which represents a balance between the pressure gradient and gravitational forces in a static atmosphere.

€

−dpdr

−ρ GMsr2

=0

€

ur ≡ 0

€

−dpdr

−ρg=0

In a planetary atmosphere, the variation in gravitational force is usually negligible over the depth of the atmosphere, and we can write


Writing

We obtain

and H = scale height. When the temperature is constant (isothermal atmosphere), the above can be integrated to give

where po is the surface pressure and z is the height above the surface ( = r - ro where ro is the planetary radius). Note that p ---> 0 as z ---> infinity.

€

p=nkT and ρ =nm

€

dpdr

=− 1H

where H = kTmg

€

p = p0e−z/H


Now, let us examine the situation with the sun. Starting with

€

−dpdr

−ρ GMsr2

=0

€

=n(me+mi)=nm

ρ=m p2kT

Solving for the mass density,

€

p=nk(Te+Ti)=2nkT

Assuming coronal electrons and protons to have the same temperature,


Substituting, we obtain (for an isothermal atmosphere)

€

1p

dpdr

= −GMsm2kT

1r2

If we let r ---> infinity, the pressure given by the above expression does not decay exponentially to zero (as in the planetary atmosphere case), but instead approaches the value

€

p(r)= p0exp{GMsm

2kT(1r− 1

R)}

The solution is


For a coronal temperature of 106 K, this is only about e-8 or 3 x 10-4 of the high pressure at the base of the corona.

This is many orders of magnitude higher than the pressure thought to exist in the interstellar medium (10-13 - 10-14 Pa), and thus could not represent an equilibrium state between the corona and that distant medium.

This problem arises from the variation in gravity over the great distance spanned by the corona (i.e., g is not constant, as in a planetary atmosphere)

The above inconsistency motivated E.N. Parker in the 1950's to consider solutions to the Equations on pages 7-9 that involved nonzero flow speeds.

We will now examine these alternative solutions.

€

p∞ = p0exp{−GMsm2kTR

}


Combining the radial continuity and momentum equations, and denoting v = ur , we obtain

€

dvv v2 −Cs

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥= dr

r 2Cs2−GMs

r ⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

rc =GMs2Cs

2

The "critical radius" is defined to be that value of r for which the numerator ----> 0,

€

dvdr

= vr

2Cs2 −GMs

r ⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

v2 −Cs2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

Cs2 = dp

dρ

Rewriting,


Class 1: Velocities approach zero near the sun and at great distances; however, the pressures at infinity corresponding to this solution are too large.

Class 2: Low velocity near the sun; high velocity at great distances; appears to be consistent with observations.

Class 3: High velocity near the sun, low velocity at great distances; however, the former are inconsistent with the low speeds at the coronal base inferred from absence of Doppler shifted spectral lines.

Class 4: High velocity near the sun and at great distances (same problem as Class 3).

A general analysis of the problem [Hundhausen, 1972], which we are not going to reproduce, admits four classes of solutions:

0 2 4 6

1

2

3

€

vvc

€

rrc

Class 2 solution represents the physically existing solar wind.


The Class 2 solution, corresponding to low velocity at the

sun, is one where everywhere; that is, the velocity

increases monotonically away from the sun.

Borrowing from this result, then

1.

(subsonic)

2.

(supersonic)

3. When , then for a mathematically valid solution.

€

dv

dr> 0

€

dvdr > 0

€

v <Cs

€

r < rc

€

dvdr > 0

€

v >Cs

€

r > rc

€

r = rc

€

v = Cs

€

dvdr

= vr

2Cs2−GMs

r ⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

v2 −Cs2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


The following condition is therefore required for transition to supersonic flow:

gravitational meanpotential thermal energy energy

( m = mass of a solar wind particle).

Therefore, for at the mean thermal energy of the

expanding solar wind must exceed the gravitational potential energy

of the gas.

€

GMsr =2Cs

2

€

⇒

€

GMsr m=2mCs

2

€

v>Cs

€

r>rc

€

(Cs2= kT

m)


The equation for dv/dr for the solar wind and the Class 1-4 velocity curves are reminiscent of the flow of a compressible fluid through a convergent-divergent nozzle:

€

du

u=

−dA A

1− M 2

u = fluid velocity

A = cross - sectional area

M = Mach number

Throat

M < 1 M = 1 M > 1

subsonic supersonic

At subsonic speeds (M < 1), a decrease in area (dA < 0) increases flow speed, and at supersonic speeds (M > 1) an increase in area (dA > 0) increases flow speed.

0 2 4

1

2

3

€

vvc

€

rrc This is like the Class 2 solution for the solar wind,

where the wind speed is subsonic until r = rC, whereupon the flow goes supersonic and continues to increase in speed for r > rC.

Note that dA = 0 at the throat implies M = 1 if du ≠ 0.

However M needn’t be = 1 at the throat (where dA = 0) if du = 0 there.

€

du

u=

−dA A

1− M 2

Throat

M < 1 M < 1

1.0

M

Throat

M > 1 M > 1

1.0

M

0 2 4

1

2

3

€

vvc

€

rrc

This is like the Class 1 solution for the solar wind, where the wind speed is initially subsonic, increases to a maximum subsonic velocity at r = rC, whereupon the flow decelerates for r > rC.

This is like the Class 4 solution for the solar wind, where the wind speed is initially supersonic, decreases to a minimum supersonic velocity at r = rC, whereupon the flow accelerates at supersonic speeds for > rC.


How does the corona acquire the necessary energy for the mean thermal energy of the coronal gas to increase

outward from the sun and overcome the sun's gravity ?

The currently favored mechanism is that reconnection of magnetic field line loops of small “magnetic patches”, covering the surface of the Sun and extending into the corona, and with lifetimes on the order of 40 hours, provide the energy necessary to raise the coronal temperatures to millions of degrees K. Microflares (nanoflares) are thought to accompany these reconnection events.

Four possibilities have been suggested:

• Acoustic wave dissipation

• Alfven wave dissipation

• MHD wave dissipation

• Microflares - “magnetic patches”

A source of coronal heating is required.


To understand the importance of a magnetic field to the behavior of a plasma, it is convenient to define a "magnetic pressure”

Pmag =

where B is the magnetic field strength and is the magnetic permeability.

€

B2

2μ0

€

μ0

€

β= PPmag

A relevant comparison then is between the plasma gas pressure P and Pmag . This ratio is defined as "beta”

WHAT ABOUT THE EFFECTS OF THE SOLAR MAGNETIC FIELD ?


A "high beta plasma" (β>> 1) is one which is controlled principally by the plasma gas dynamics .

€

β= PPmag

If the magnetic field is small, we would expect the expanding corona to drag the magnetic field with it --- this is called a "frozen-in" magnetic field, characteristic of a high-beta plasma.

A "low beta plasma" (β<< 1) is one which is dominated by the intrinsic magnetic field.

For the real corona, where the magnetic pressure is a few times the gas pressure, a mixture of these extreme behaviors is expected.

If the magnetic field is large, we expect the magnetic field to "contain" the plasma, or at least to inhibit its expansion.


MHD modeling shows that the inner magnetic field lines (R < 2) near the equator are closed, and that at higher latitudes the field lines are drawn outward and do not close.

These field lines that do not close nearly meet at low latitudes, but do not reconnect; this abrupt change in the magnetic field polarity is maintained by a thin region of high current density called the interplanetary current sheet.

Magnetic-field lines deduced from the isothermal MHD coronal expansion model of Pneuman and Kopp (1971) for a dipole field at the base of the corona. The dashed lines are field lines for the pure dipole field.

This current sheet separates the plasma flows and fields that originate from opposite ends of the dipole-like field.


Extension/generalization of the features indicated in the above model to more complicated solar fields at the lower boundary of the corona suggest the following:

• Closed magnetic structures should form over those locations where the vertical component of the field at the base of the corona changes sign (i.e., above so-called "neutral lines" in the solar magnetic field).

• These closed structures should be limited in extent to about 2 solar radii.

• Open magnetic structures should form over regions where the vertical component of the field at the base of the corona is of the same sense or sign over a large area (i.e., above so-called "unipolar regions" in the solar magnetic field.

• The open structures should spread laterally with increasing height to fill all space above closed regions with outward-flowing solar wind. Current sheets should form where these flows meet.


Closed magnetic structures should form over those locations where the vertical component of the field at the base of the corona changes sign (i.e., above so-called "neutral lines" in the solar

magnetic field).

Open magnetic structures should form over regions where the vertical component of the field at the base of the corona is of the same sense or sign over a large area (i.e., above so-called "unipolar regions" in the solar magnetic field).

asen5335- aerospace environments -- solar wind 1 the solar wind is the extension of the solar corona...

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